Sylvain Nathan Michel

Giorgio Bertrand

Sorin Valmorbida

Didier Dumur, Design Olaru

Nathan Michel

Sorin Olaru

Sylvain Bertrand

Giorgio Valmorbida

Mes remerciements vont tout d'abord à celles et ceux qui ont accepté de faire partie de mon jury de thèse : Saïd Mammar pour avoir présidé ce jury ; George Bitsoris et Konstantinos Ampountolas pour avoir été rapporteurs ; et enfin Ionela Prodan et Benoit Clément. Merci à tous pour vos retours et remarques pertinentes qui ont donné lieu à de très riches échanges lors de la soutenance.

En deuxième lieu, je souhaite remercier très chaleureusement mes (nombreux) encadrants de thèse. Merci Sorin, je n'oublierai jamais nos discussions interminables sur les ensembles invariants. Ta pédagogie, ton sens de l'écoute, et ton expertise sans limite font que je chérie ces moments (qui me manquent déjà !). Merci Giorgio de m'avoir toujours poussé à chercher plus loin dans la théorie et à m'améliorer chaque jour un peu plus (notamment dans la rédaction !). Tu as été d'une aide et d'un soutien moral précieux. Merci Didier pour avoir participé à l'encadrement, pour la sagesse de tes remarques sur les directions à suivre. Et surtout, merci à toi Sylvain ! Tu as toujours été là pour prêter main forte pour discuter théorie, mais également (et surtout !) lorsque ce satané drone me menait la vie dure. Tu resteras toute ma vie un exemple, que ça soit professionnellement ou humainement. Mille mercis.

Vient maintenant le moment de « remercier » mes collègues doctorants de l'ONERA. Le couple le plus mignon, Vincent et Romain. Puisse le sac poubelle de Nantes vous unir à jamais. Élinirina et Irina. Ton sens du tact, ton sens froid et ta poésie Ricardo. Camille, toi la grande. Camille, toi la petite. Léon le meilleur partenaire de bureau que j'aurais pu espérer. Julien, l'homme qui meme plus vite que son ombre. Émilien pour ne pas m'avoir aidé à faire ma thèse. Denis et tes 20 plans de mariage. Baptiste, et ton jeûne qui restera dans les annales. Esteban, bien plus grand et bien meilleur au babyfoot qu'Enzo. Enzo, tes fake news et ton absence totale de talent au babyfoot. Profite bien de tes 18 mois à Coventry (c'est bien ça non ?). Ali, je suis heureux d'avoir pu te voir éclabousser les terrains de ta grâce et de ton talent. N'oublie jamais les saintes paroles « AOE2>AOE3 ». Mon fils Jean Lynce. Que je suis fier de toi. Tu as grandi si vite. Mon père Alain, et nos rencontres à la cantine. Sergioooooo et ton ouverture d'esprit concernant la paella. Ioannis, et nos discussions dans la navette de l'ONERA. Merci pour le support moral et les encouragements ! Étienne . . . quelle barbe ! Antonello, qui a enfin pu

Introduction

Les quadrotors sont des plateformes versatiles, capable de planer et de réaliser des manoeuvres acrobatiques. Leur utilisation pour des applications civiles et industrielles a considérablement augmenté ces dernières années, et cette tendance ne devrait pas s'arrêter [Commercial 2016, Hanlon 2018]. La conception de quadrotors qui soient autonomes nécessite de prendre en compte de nombreuses contraintes, notamment en termes de sûreté et d'embarquabilité (évitement d'obstacles, présence de saturation au niveau des actuateurs, capacité de calcul limité). Nous nous intéressons à la partie commande de l'automatisation d'un quadrotor. Les lois de commande présentées dans ce manuscrit reposent sur la connaissance d'un modèle des dynamiques du quadrotor. Les équations sont obtenues à partir des principes de la physique, et les paramètres sont identifiés à partir de données expérimentales. Les dynamiques d'un quadrotor sont composées des dynamiques en translation et des dynamiques en rotation. Dans le cadre d'application civile sans manoeuvre acrobatiques, les équations décrivant les dynamiques du quadrotor peuvent être linéarisées, sous l'hypothèse de petits angles. Dans la suite, nous nous focalisons sur les dynamiques en translation, et nous utilisons un modèle linéaire et discret.

La présence de perturbations (vent, effet sol,...) et d'erreurs de modèle peuvent nuire à la réalisation de la mission, et leur impact sur les performances de vol doivent être mesuré. Différentes stratégies ont été adoptées dans la littérature pour élaborer des lois de commande robustes aux perturbations pour des quadrotors, par exemple le switching Model Predictive Control [START_REF][END_REF]] ou encore le PID robuste [START_REF] Kada | Robust PID Control Design for an UAV Flight Control System[END_REF]]. Ces stratégies permettent de limiter l'impact des perturbations mais ne garantissent pas le respect des contraintes.

Problématiques

Nos travaux de thèse s'inscrivent dans le contexte de la construction de loi de commande garantissant le respect de contraintes en présence de perturbations pour un quadrotor, et se placent plus généralement dans le cadre de la commande d'un système linéaire discret soumis à des perturbations additives bornées et à des contraintes sur l'état et la commande.

Il y a deux principales problématiques dans la synthèse de lois de commande pour un système dynamique soumis à des perturbations. La première est la caractérisation de la région de convergence, non réduite au point d'équilibre. La seconde est la caractérisation de l'ensemble des conditions initiales convergeant vers ce voisinage, appelé région contrôlable. Dit autrement, la synthèse de la loi de commande nécessite

Ensembles invariants

Diverses méthodes ont été envisagées pour résoudre ces problèmes : Model Predictive Control avec un filtre de Kalman [START_REF] Huang | [END_REF]], MPC stocastique [START_REF] Van Hessem | [END_REF], des méthodes ensemblistes [Blanchini 1999], ou encore de la MPC robuste [Mayne 2005]. Les trois dernières méthodes utilisent des ensembles invariants. Les ensembles invariants sont un outil théorique fort pour étudier l'impact des perturbations sur un système, en particulier en présence de contraintes.

Pour un système en boucle fermé, un ensemble Positively Invariant (PI) est une région de l'espace d'état telle que si elle contient l'état à un instant alors elle le contiendra dans le futur. Il est Robustly Positively Invariant (RPI) s'il a cette propriété quelles que soient les perturbations qui affectent le système. Cette propriété prend en compte les contraintes (état et commande) qui affectent le système. Ainsi, toute trajectoire initialisée dans un RPI reste dans cet ensemble et respecte les contraintes. La figure 1 illustre le concept d'ensemble invariant : la trajectoire reste dans l'ensemble invariant et respecte les contraintes.

Étude locale

Un ensemble RPI d'intérêt est le minimal RPI (mRPI), défini comme le plus petit RPI (en termes d'inclusion) qui contienne le point d'équilibre. Il borne les variations autour du point d'équilibre dues aux perturbations. Cet ensemble est à la fois un outils d'analyse et un objectif dans de nombreuses méthodes de contrôle robuste [START_REF] Bertsekas | [END_REF], Blanchini 1992, Mayne 2006]. Cet ensemble a été le sujet de nombreuses recherches lorsque les dynamiques et la loi de commande sont linéaires [Kolmanovsky 1998a, Rakovic 2005b[START_REF] Olaru | [END_REF]].

Résumé étendu de la thèse xiii

Lorsque la loi de commande est pré-imposée pour notre quadrotor, les ensembles RPI représentent des zones de vols sûres (safe flight zone), et le mRPI borne les déviations maximales du quadrotor autour du point d'équilibre. Notre objectif est de synthétiser la loi de commande ayant les meilleures propriétés possible en termes de rejet de perturbation, notamment autour des points d'équilibre. C'est-à-dire, synthétiser la loi de commande menant à l'ensemble mRPI le plus petit possible.

Le concept d'ensemble invariant peut être étendu aux systèmes en boucle ouverte, c'est à dire lorsque la loi de commande n'est pas fixée. Un ensemble est dit Robust Control Invariant (RCI) si toute trajectoire initialisée dans cet ensemble peut y être maintenue avec le bon choix de commande. Un ensemble RPI pour un système en boucle fermé est un ensemble RCI pour le système en boucle ouverte associé, et réciproquement. Ainsi, la synthèse d'une loi de commande menant à un mRPI qui soit le plus petit possible consiste à construire un ensemble RCI dit « minimal ».

La notion de minimalité des ensembles RCI est complexe puisque l'intersection d'ensembles RCI n'est pas RCI en général. Le critère d'inclusion ne permet pas de définir le concept de minimalité de manière satisfaisante. La notion d'ensemble RCI minimal n'est pas très étudiée dans la littérature à quelques exceptions [Rakovic 2010[START_REF] Cannon | [END_REF], Blanco 2010, Chen 2018]. Ces recherches utilisent le volume pour mesurer la minimalité. Le principal inconvénient est qu'il est complexe de calculer le volume de polytopes, classe d'ensemble très présente dans la littérature des ensembles invariants.

Dans ces travaux, la phase d'identification des paramètres du modèle précède la synthèse de la loi de commande. Cet ordre est classique dans la littérature, puisque la loi de commande utilise la connaissance du modèle. Cependant, les ensembles RCI dépendent à la fois du choix des paramètres du modèle (et des bornes des perturbations), et du choix de la loi de commande. Il n'existe pas, à notre connaissance, de méthodes d'identification qui utilise les paramètres du modèle pour construire des ensembles invariants optimisés.

Le calcul d'un ensemble invariant minimal correspond à l'étude du comportement local du système, puisqu'il donne la loi de commande à appliquer dans le voisinage des points d'équilibre. Cette étude locale doit être complétée par une étude plus globale qui étudie le passage d'un point d'équilibre au suivant en garantissant la satisfaction des contraintes.

Étude globale

La commande prédictive (MPC) est une méthode de contrôle basée sur la résolution de problèmes d'optimisation adaptée à la présence de contraintes. Pour un système discret, la commande est calculée en résolvant un problème d'optimisation à chaque pas de temps, comme décrit dans l'Algorithme 0.1.

Les trajectoires prédites par la MPC respectent les contraintes tant que les problèmes d'optimisation restent faisables. En présence de perturbation, la trajectoire réelle diffère de la trajectoire prédite et peut violer les contraintes comme illustré sur la Figure 2.

Les deux méthodes les plus répandues pour robustifier la MPC, c'est-à-dire garantir que la trajectoire réelle satisfasse les contraintes, sont respectivement le min-max MPC [Campo 1987] et le Tube Based MPC (TBMPC) [Mayne 2006]. La première méthode est très couteuse en termes de temps de calcul, et pour cette raison n'a pas été retenue pour notre application. Nous avons choisi d'appliquer le TBMPC, dont les principes sont brièvement décrits ci-après.

L'entrée de contrôle est la somme de deux termes. Le premier est l'entrée de contrôle obtenue en résolvant un problème d'optimisation pour un modèle sans perturbation, appelé nominal. Le second est un contrôleur local de l'erreur entre le système réel et le système nominal. Cette erreur est bornée par le mRPI associé au contrôleur local. Avec cette stratégie de commande, la trajectoire réelle est contenue dans un tube de trajectoire, définie par la somme de la trajectoire prédite et du mRPI. Des contraintes plus serrées, qui prennent en compte la taille du mRPI, sont introduites dans le problème d'optimisation pour garantir que le tube de trajectoire satisfasse les contraintes réelles. Le contrôleur local, et l'ensemble mRPI associé, sont calculés en amont de la phase de vol. La Figure 3 Le choix du contrôleur local et le calcul du mRPI associé sont des paramètres clés du TBMPC. En effet, le mRPI est responsable de la largeur du tube de trajectoires, que l'on veut le plus fin possible pour deux raisons. La première est que la largeur du tube de trajectoires représente la précision de la loi de commande puisque le tube borne la déviation de la trajectoire réelle autour de la trajectoire prédite. La seconde est que plus le tube est large, plus les contraintes du problème d'optimisation doivent être serrées. Cela renforce notre intérêt pour la synthèse d'une loi de commande conduisant à un ensemble mRPI le plus petit possible. C'est à dire la construction d'un ensemble RCI minimal.

Objectifs de la thèse

L'objectif de la thèse est double. Le premier est l'implémentation et la validation expérimentale d'une loi de commande robuste pour la stabilisation des dynamiques en translation d'un quadrotor. Pour cela, nous avons choisi pour une loi de commande TBMPC. Le second est la construction d'ensemble RCI minimal pour des systèmes linéaires discrets soumis à des perturbations bornées et à des contraintes sur l'état et la commande. Ces deux objectifs sont étroitement liés, puisque l'ensemble RCI est un paramètre de la loi TBMPC.

Structure du manuscrit

Ce manuscrit présente des résultats théoriques et des méthodes pratiques pour la construction d'ensembles invariants optimisés pour des systèmes linéaires discrets soumis à des perturbations additives bornées et des contraintes sur l'état et la commande. La construction d'ensembles invariants est intéressant pour notre ap-xvi Résumé étendu de la thèse plication, puisqu'ils représentent des zones de vol où l'on garantit le respect des contraintes (obstacles, saturation des actuateurs,...) en présence de perturbations (vent, effet sol, erreurs de modèle...). De plus, les ensembles invariants jouent un rôle central dans l'implémentation de loi Tube Based MPC, technique adaptée à notre application.

La première difficulté de la construction d'ensembles invariants minimaux est la définition même de minimalité pour des systèmes en boucle ouverte. En effet, l'inclusion n'est pas adaptée, et un autre critère doit être considéré. Nous proposons un nouveau critère dans le Chapitre 1. Ce critère est adapté aux polytopes et aux ellipses, deux classes d'ensemble omniprésentes dans la littérature des ensembles invariants.

Les ensembles invariants dépendent des éléments suivants

• La valeur des paramètres du modèle retenu.

• Les bornes des perturbations affectant le système.

• Le choix d'un contrôleur local.

Dans la plupart des études sur le sujet, et plus généralement dans la littérature de l'automatique, les phases de modélisation et d'identification précèdent la phase de synthèse de la loi de commande. Le contrôleur local est le seul paramètre d'optimisation des ensembles invariants. Nous présentons des résultats dans ce cadre théorique dans les chapitres 5 et 6. Dans le Chapitre 3 nous utilisons la phase d'identification pour optimiser les ensembles invariants.

L'organisation de ce manuscrit est détaillée ci-après : Le Chapitre 1 présente les outils mathématiques, le formalisme, et les définitions relatifs aux ensembles invariants et au TBMPC. Ces éléments sont nécessaires pour établir les résultats présentés dans le reste du manuscrit. Nous proposons un nouveau critère pour mesurer la taille des ensembles invariants permettant de définir mathématiquement l'ensemble RCI minimal pour un système en boucle ouverte. Ce critère est utilisé tout au long du manuscrit. Le chapitre est conclu par une analyse en simulation de l'impact du contrôleur local sur les performances du TBMPC pour un modèle linéarisé des dynamiques en translation d'un quadrotor [Michel 2017].

Par la suite, le manuscrit est divisé en deux parties ayant chacune son propre cadre d'étude.

Partie 1 : Implémentation d'une loi de commande prédictive robuste pour la stabilisation d'un quadrotor Le Chaptire 2 présente un état de l'art sur l'application de loi MPC robuste pour le contrôle de quadrotors. Ces lois dépendant du choix d'un modèle, nous énumérons les différents choix de modélisation des dynamiques d'un quadrotor présents dans la littérature. Ensuite, nous étudions les différentes stratégies de robustification des Résumé étendu de la thèse xvii lois MPC, et en particulier leur capacité à garantir la satisfaction de contraintes en présence de perturbations.

Le Chapitre 3 aborde les détails de notre implémentation d'une loi TBMPC pour la stabilisation des dynamiques en translation du Parrot AR.Drone 2.0. Le contrôleur local est un gain choisi à partir de données expérimentales générées en boucle fermée avec une large gamme de gain linéaires. Nous proposons ensuite une méthode d'identification adaptée à l'implémentation d'une loi TBMPC, visant à minimiser la taille de l'ensemble mRPI. L'identification obtenue est moins précise que des techniques classique d'identification (bornes de perturbation plus larges), mais permettent d'obtenir des ensembles invariants plus petits. Les résultats expérimentaux des vols effectués avec la loi TBMPC sont présentés. Les résultats théoriques et expérimentaux de ce chapitre sont partiellement retranscrits dans [START_REF] Michel | [END_REF]].

Construction d'ensemble RCI minimaux pour un système linéaire discret soumis à des perturbations additives bornées

Dans la seconde partie du manuscrit nous rentrons dans un cadre théorique plus conventionel pour l'étude des ensembles invariants. Nous faisons l'hypothèse classique que les paramètres du modèle ont déjà été identifiés, et le contrôleur local est l'unique paramètre d'optimisation des ensembles invariants.

Le Chapitre 3 présente un état de l'art des recherches sur le calcul d'ensemble RCI optimal (minimal et/ou maximal) pour des systèmes linéaires discrets soumis à des perturbations et des contraintes sur l'état et la commande. Les approches présentées utilisent une paramétrisation linéaire de la loi de commande et/ou une paramétrisation polytopique des ensembles RCI pour obtenir une solution sousoptimale.

Le Chaptitre 4 est dédié à l'étude des ensembles invariants obtenus avec des lois de commande par mode glissant (SMC). Ces lois de commande sont reconnues pour les systèmes continus pour leur capacité à totalement annuler l'impact des perturbations. Nous étudions les propriétés géométriques des ensembles invariants pour des systèmes discrets. Cette étude nous permet d'établir de premiers résultats sur l'existence et la charactérisation d'ensembles RCI minimaux. En particulier, nous utilisons le choix de la surface de glissement pour orienter le mRPI associé comme illustré Figure 4. Ces résultats sont illustrés en simulation. Une partie des résultats théoriques sont publiés dans [Michel 2018b] et [Michel 2018a].

Le Chapitre 5 aborde le raffinement itératif des ensembles RCI en vue de leur minimisation via des méthodes d'optimisation. Les résultats présentés dans ce chapitre se basent sur la création de fonction associant à chaque ensemble RCI un unique ensemble RCI plus petit. Ces fonctions peuvent être itérées pour construire des suites décroissantes d'ensembles RCI, comme illustré sur la Figure 5. Nous détaillons la construction de telles fonctions, et étudions les propriétés de ces suites décroissantes. Les résultats sont présentés en simulation, et ont été partiellement publiés dans [Michel 2018a]. La validation expérimentale du TBMPC a été réalisée sur un scénario de vol basique consistant en la stabilisation du Parrot AR.Drone 2.0 sur une succession de points d'équilibre. Afin de réaliser des missions plus complexes, il serait intéressant d'élargir à des scénarios plus complexes, par exemple

• Augmenter la distance entre les points d'équilibre. Cette distance ne peut dépasser une certaine valeur liée à la taille de la région de faisabilité des problèmes d'optimisation du TBMPC. Une manière d'élargir cette région de faisabilité est d'augmenter l'horizon de prédiction. Cette augmentation ne serait pas sans conséquence, puisque cela ajoute du temps de calcul. Une autre manière est d'implémenter un reference governor.

• Prendre en compte une gamme plus large de perturbations en volant près d'obstacles (mur, sol, autres quadrotors,...) ou en générant des rafales de vent.

• Étendre à des scénarios de suivi de trajectoire. Une première approche serait de garder des stabilisations successives selon une direction (par exemple x) avec le TBMPC et de faire du suivi de trajectoire avec une autre loi de commande selon l'autre direction (y).

Il est important à ce stade de préciser que n'importe quelle modification du scénario de vol nécessite de refaire une campagne de captation de données pour identifier les nouvelles bornes de perturbation et les paramètres du modèle. Nous avons proposé une technique d'identification dans le Chapitre 2 qui consiste à minimiser la norme quadratique du signal de perturbation tout en bornant les valeurs propres de la matrice d'évolution. Ce choix bénéficit de la structure LP/QP du problème d'optimisation, mais requiert de faire du grid search sur la borne. Il serait intéressant de chercher des techniques d'identification plus sophistiquées, qui utiliseraient notamment la connaissance du contrôleur local.

Partie 2

Les premiers résultats concernant la construction d'ensembles invariants minimaux reposent sur une étude approfondie des ensembles mRPI obtenus avec des loi de commande par mode glissant. Il s'agit d'un cas particulier d'eigenstructure assignement. Il serait intéressant d'étudier plus en profondeur la relation entre les vecteurs et valeurs propres de la matrice d'évolution et les propriétés géométriques des ensembles invariants associés. Cette étude serait bénéfique à la fois d'un point de vue théorique et d'un point de vue algorithmique pour simplifier les algorithmes itératifs présentés au Chapitre 6.

Un deuxième axe d'étude prometteur est l'étude des propriétés des fonctions introduites dans le Chapitre 6. Par exemple xx Résumé étendu de la thèse

• Charactériser de nouvelles selection function adaptées à la nature itérative de l'approche. Les fonctions proposées dans ce manuscrit ne cherchent qu'à minimiser la taille de l'image du RCI, et non la N -ème image.

• Étudier la relation entre l'ensemble RCI initial et la limite de la suite décroissante d'ensemble RCI.

Perspective générale

Enfin, il serait prometteur de développer une approche qui unifie les apports théoriques et expérimentaux de cette thèse, où les paramètres du modèle et la loi de commande seraient utilisés conjointement dans le but unique de construire l'ensemble invariant minimal à partir de données expérimentales. Cela n'a pas été possible avec notre quadrotor, à cause de la présence de délais dans les dynamiques en translation : il n'est pas possible d'utiliser les résultats théoriques des chapitres 5 et 6 pour la construction d'ensembles invariants minimaux. Pour mener à bien une telle approche, il est nécessaire d'avoir un quadrotor capable d'embarquer la loi de commande pour réduire le délai de la commande, et dont les performances en termes de stabilisation des dynamiques en attitude soient nettement supérieures pour réduire les erreurs de modélisation.

Introduction

Motivation

Unmanned Aerial Vehicles (UAVs) quadrotors are versatile platforms capable of agile motion and stable hovering. They have been applied to surveillance [START_REF] Gohari | [END_REF], Leahy 2016], reconnaissance [Bhandari 2015[START_REF] Korpela | A framework for autonomous and continuous aerial intelligence, surveillance, and reconnaissance operations[END_REF], delivery [START_REF] Mo | Study on control method of a rotor UAV transportation with slung-load[END_REF]], and visual inspection (gas leakage [Rossi 2014], railway network verification [Bertrand 2017], building thermal profiling [Mauriello 2014], water stress in agriculture fields [START_REF] Gago | [END_REF]), border patrolling or infrastructure monitoring, media cover of sport events (ski, cycling) and forest fire control and monitoring [Zhang 2019]. Initially developed for military purposes, the use of drones in civil application and industry has considerably increased in the last years, and is foreseen to continue growing [Commercial 2016, Hanlon 2018]. In most cases, the quadrotor is controlled remotely by an operator.

The design of autonomous UAVs should take into account safety and technological constraints, such as distance to obstacles, actuator limitations or real-time computational constraints for embedded implementation. The automation of a quadrotor raises several issues, from the structural to the mechanical [Çetinsoy 2012] and electronical ones [Jang 2007], and regarding the guidance [Goerzen 2010], navigation [Nikolos 2003] and control.

Here we will focus on quadrotor control for an indoor application in a cluttered environment, where we want to account for the presence of external disturbances (wind, ground effect) and physical constraints (actuator limitations, presence of obstacles). The goal is to compute control laws that generate collision-free trajectories by bounding them within safe flight regions, where all constraints satisfaction is guaranteed.

The control laws for UAVs presented in this manuscript are model-based. A model for the quadrotor dynamics is obtained from physical principles, and its parameters are obtained from experimental data. The quadrotor dynamics are composed by its translational dynamics, that describe the evolution of its three dimensional position, and its orientation dynamics, that describe the evolution of its attitude. The orientation dynamics are non-linear, and the resulting thrust vector, which depends on the orientation, is used as input to the translational dynamics. Under the assumption that the closed-loop orientation dynamics is faster than the closed-loop translational dynamics [START_REF] Bertrand | [END_REF]], it is possible to design separately the position and the attitude controllers with a hierarchical control structure [Arleo 2013[START_REF] Liang | [END_REF][START_REF] Bertrand | [END_REF]]. Further, for most civil applications, the quasi-stationarity nature of the manoeuvres can rely on linearized models of the translational dynamics under the small angles assumption [START_REF] Dydek | [END_REF][START_REF] Hernandez | [END_REF][START_REF] Bertrand | [END_REF].

In this manuscript we focus on the stabilization of the translational dynamics of the quadrotor, and we use a linearized and discretized model of these dynamics for the control design.

External disturbances and modelling mismatches can affect the execution of a mission and its impact on the closed-loop trajectories must be assessed. This is particularly important whenever the system evolves in an environment with obstacles. In this context, work has been carried out on the design of robust control for UAVs, from a switching model predictive strategy [START_REF][END_REF]] to a robust PID controller [START_REF] Kada | Robust PID Control Design for an UAV Flight Control System[END_REF]]. These strategies are designed to guarantee robustness with respect to bounded disturbances but suffer from limitations in constraint handling. The control of a quadrotor in presence of both external disturbances and constraints can thus be positioned in the larger topic of robust constrained control.

Robust constrained control

Constrained control of dynamical systems in presence of bounded disturbance faces two main challenges, namely the convergence to an attractive set instead of a single equilibrium point and the characterization of the controllable region to this set. These challenges have been addressed using several methods: Model Predictive Control with Kalman filter [START_REF] Huang | [END_REF]], stochastic MPC [START_REF] Van Hessem | [END_REF], set theoretic methods [Blanchini 1999], interval based approaches [START_REF] Jaulin | Interval constraint propagation with application to bounded-error estimation[END_REF]], or robust MPC [Mayne 2005]. Invariant set theory is a solid theoretical tool for the analysis of the robustness of control laws, in particular in the presence of constraints, and underpins the last three cited approaches.

Quantification of the disturbances influence using invariant sets

Linear robust control relies on transfer function gains to evaluate the impact of disturbances and uncertainties. When accounting for input or state constraints, these gains are not representative of the behavior, and a better description of the impact of the disturbances is obtained with invariant sets. A systematic way to assess the influence of disturbances is to compute invariant sets [Blanchini 1999]. For closed-loop systems, a positively invariant (PI) set is a region of the state space with the property that if it contains the system state at a given time, then it will contain it in the future. This set is said Robustly Positively Invariant (RPI) if it has the same property regardless of the bounded disturbances affecting the system. These strong properties allow to account implicitly for the presence of state and/or input constraints. Hence, an initial condition that belongs to an RPI set avoids constraints violation in the future. The control law ensures robust recursive constraints satisfaction to the closed-loop system once it reaches an RPI set. Figure 6 shows the principles of set invariance, where the closed-loop trajectory remains in an RPI set contained in the state constraints. An RPI set of particular interest is the minimal RPI (mRPI) set. The mRPI set, also called the 0 reachable set, is defined as the smallest RPI set in terms of set inclusion that contains the equilibrium [Kolmanovsky 1998a]. This definition of mRPI set is relevant since the intersection of two RPI sets retains the RPI property. It represents the set of states that can be reached from the equilibrium under unknown-but-bounded additive disturbances [Blanchini 1999, Rakovic 2005b]. This set is of particular interest for performance analysis [Blanchini 1999]. Moreover, it is a suitable target set in robust time-optimal control [START_REF] Bertsekas | [END_REF], Blanchini 1992[START_REF] Mayne | [END_REF]], and it is a key element in the synthesis of robust predictive control methods [START_REF] Langson | [END_REF], Mayne 2005]. This set has been extensively studied in the literature, in particular for linear discrete-time systems, linear control laws and additive disturbances contained in a polytopic set [Kolmanovsky 1998a, Rakovic 2005a[START_REF] Olaru | [END_REF]]. The computation of RPI sets for a given system depends of the choice of the control law.

For a quadrotor application, whenever the feedback loop is endowed with a pre-imposed control law, the RPI sets will define safe flight regions. The mRPI set measures the maximal deviations in stationary flight, and thus measure the disturbance rejection properties of the control law. Our goal, once the analysis is carried out, will be to design a control law that provides good disturbance rejection properties, characterized set-wised by an mRPI set that is as small as possible. In addition, the control law should ensure convergence of the closed-loop system toward this mRPI set while respecting the state and input constraints despite the presence of unknown-but-bounded disturbances.

Introduction

as the initial condition of the predicted trajectories {ξ[kT ], ..., ξ[(k + N )T ]}. The predicted trajectories that satisfy the state and input constraints are called feasible, and among them we use the input that minimizes a cost function. The solution of the optimal control problem yields an optimal control sequence, and only the first element is applied to the input of the real system. In general, its implementation follows Algorithm 0.2. A requirement for an MPC scheme is the underlying optimization problem to be solved between two samples. This is particularly important because quadrotors have limited embedded computational capacity and fast dynamics. A key advantage of MPC is that it allows efficient closed loop plant operation in the presence of constraints. The design of the MPC must ensure strong theoretical properties, among which the recursive feasibility of the successive optimization problems and the stability of the closed-loop system [Mayne 2000, Mayne 2014].

The ideas to ensure the recursive feasibility and closed-loop stability are recalled as follows, in a historical perspective. It first consisted of an equality constraint [Thomas 1975[START_REF] Keerthi | [END_REF], where the last element of the predicted trajectory had to be equal to the equilibrium point. The strict equality was later relaxed, and the last element had to be contained in a terminal constraint set that contains the equilibrium [START_REF] Sznaier | [END_REF]]. The terminal constraint set is designed to ensure recursive feasibility of the optimal control problem. In case of linear dynamics and polytopic constraints, the terminal set is usually a Maximal Output Admissible Set [Gilbert 1991], which is a polytope and can be described based on a finite construction procedure. In the more general setting [Kolmanovsky 1998a], its computation relies on the use of basic set operations, among which the Minkowski addition and the Pontryagin difference. For polytopic constraints, linear dynamics and polytopic terminal set, the feasible region of the optimization problem can be computed using similar basic set operations. A terminal cost was introduced in the cost function to ensure the stability of the closed-loop system, where the cost function plays the role of a Lyapunov function [Chen 1982[START_REF] Mayne | [END_REF].

In presence of disturbance, a trajectory of the actual system differs from the nominal predicted trajectory of the optimization problem. This deviation can lead to constraints violation, as illustrated in Figure 7. The design of MPC schemes that ensure recursive feasibility and constraints satisfaction, and closed-loop stability despite the presence of disturbances is the topic of robust MPC [START_REF] Bemporad | [END_REF]]. There exist two main approaches for the robustification of MPC schemes for systems subject to bounded additive disturbances.

The first, called min-max MPC, was originally proposed in [Campo 1987] and consists in solving a min-max optimization problem drawing inspiration from game theory, where the disturbances is the adversary. The optimal control problem seeks to minimize the value function in the worst case scenario (min-max open loop MPC). The complexity of these robust MPC schemes increases exponentially with the lenght of the prediction horizon. Feedback MPC was introduced to reduce the conservatism and complexity of min-max MPC, where decision variable is not the control input sequence but a control policy (a sequence of control laws) [Mayne 1995[START_REF] Kothare | [END_REF]]. However, this class of approaches remains prohibitively complex to these days for real-time implementation on fast dynamical systems.

The second approach is Tube Based MPC and uses the concept of tube of trajectories which originates from [START_REF] Bertsekas | [END_REF]]. The first designs of MPC using tubes were presented in [START_REF] Chisci | [END_REF], Mayne 2001[START_REF] Langson | [END_REF]. In contrast to the previous approach that enforces the open-loop trajectories to satisfy the constraints explicitly, the tube based approaches ensure that the closed-loop trajectory remains inside a tube that subsequently satisfies the constraints. Here, the control action is the sum of two terms, each computed with a distinct purpose. The first term is computed from a nominal MPC scheme for a disturbance-free version of the system called nominal system. The second term is a feedback involving the error between the current nominal system state and the current actual system state. In this manuscript, the second feedback is called the local controller.

This robustification requires the computation of an RPI set Z for the local controller to bound the maximal discrepancy between the two systems' states. With this control strategy, the state of the actual system remains in a tube around the nominal trajectory, characterized set-wise by the RPI set Z. The possible future trajectories of the real system satisfy the constraints provided the tube respects the constraints. This is accounted for in the MPC scheme of the nominal system, where the state and input constraints are tighten accordingly. Figure 8 provides an illustration of the principle of Tube Based MPC, where the tube of trajectories, and thus the actual trajectory, satisfy the constraints. The tube of trajectories define safe flight corridors in our quadrotor application.

Tube Based MPC has been extensively studied in the literature for linear dynamics [Mayne 2005, Mayne 2006[START_REF] Raković | Invention of prediction structures and categorization of robust MPC syntheses[END_REF], and has been extended to certain classes of non-linear systems [Mayne 2011[START_REF][END_REF], Köhler 2019]. This robustification method does not extensively increase the online computational complexity of the original non-robust MPC scheme, as the RPI set Z is computed offline. For this reason, it is more suitable for real-time computation.

Invariant sets as a design knob of Tube Based MPC

The RPI set has a double role in the Tube Based MPC scheme. First, it bounds the error between the nominal system state and the real system state to define the tube of trajectories. Hence, this set represents the disturbance rejection properties of the Tube Based MPC. Second, its (set) difference to the original constraints defines the tightened constraints of the MPC scheme for the control of the nominal system. For both reasons, it is preferable to design an RPI set that is as small as possible. As discussed above, for a given control law, the smallest RPI set is the mRPI set. This further motivates our goal of computing a control law whose mRPI set is as small as possible.

We have the following definition for invariant sets where the control law is a design parameter. A set is said Robust Control Invariant (RCI) for an open-loop system if it is possible to keep the trajectory inside the set with suitable control action. An RPI set for a closed-loop system is an RCI set for the associated openloop system, and conversely. Hence, the computation of a control law whose mRPI set is as small as possible relates to the construction of an RCI set that is minimal.

As opposed to RPI sets, the definition of minimal RCI (mRCI) set raises theoretical issues. Given two RPI sets, it is always possible to construct a smaller RPI set by considering their intersection. However, the intersection of two RCI sets does not preserve the RCI property. Given that set inclusion is a partial order, it cannot be used to compare every pair of RCI sets. Hence, other criteria are to be sought to evaluate the minimality of RCI sets. The notion of minimality for RCI sets, and thus their computation, is less addressed in the literature with some exceptions [Rakovic 2010[START_REF] Cannon | [END_REF], Blanco 2010, Chen 2018]. In these researches, the measure to compare the size of RCI sets is either chosen as the radius of the minimal p norm ball containing the RCI set, as in [Rakovic 2010], or the volume of the RCI set, as in [START_REF] Cannon | [END_REF], Blanco 2010].

The main difficulty of the computation of optimized RCI sets is that we do not have an explicit characterization of the collection of RCI sets in general. The above approaches to compute optimized RCI sets use a parametrization of the control law (linear or piece-wise affine) and/or of the RCI set (low/high complexity polytopes, ellipsoids) to propose candidate solutions.

Computation of RCI sets from experimental data

In the above researches, and more generally in control theory, it is assumed that the system identification step precedes the control synthesis. As a result, the control law is the only design parameter of the RCI set. However, the computation of invariant sets (RPI and/or RCI) also depends the model of the system, which includes the choice for a model itself, its structure, and the choice on the hypothesis (models/parameters/bounds) for the disturbance.

We could say there is a gap in the literature for identification methods that use the model parameters to compute minimal RCI sets. A notable exception is [Chen 2018] who proposes an iterative system identification method. The goal of such system identification approaches is to minimize the RCI set at the expense of an inaccurate model (larger disturbance bounds). This is of particular interest for the experimental application of Tube Based MPC schemes, where the RCI set is a key design parameter of the control law.

To the author's knowledge, there are two studies that provide experimental application of Tube Based MPC, for autonomous ground vehicle [Gao 2014] and mobile robots [González 2011]. These studies use classic LQR techniques for the design of the local controller, and classic methods for the identification of the model parameters. Both papers acknowledge the key role of the RCI set, and note that more sophisticated techniques for the identification of the model parameters and the design of the control law could lead to a smaller RCI set.

Objectives of the thesis

The general goal of this thesis is the design of minimal RCI sets for constrained discrete-time linear systems subject to bounded additive disturbances, in particular in quadrotor applications. This goal takes motivation both from the importance of computing small RCI set in the implementation of Tube Based MPC, and in the more general context of designing control laws that mitigate the influence of disturbances on the closed-loop performances of a system in the same field of application. The criterion of minimality used in this thesis is presented in Chapter 1. The manuscript is divided into two parts, according to the design parameters used in the minimization of the RCI sets.

The first part considers the broader topics of the implementation and experimental validation of a control law for the stabilization of the translational dynamics of a quadrotor that ensures recursive constraint satisfaction and closed-loop stability in presence of disturbances. To this task, we chose Tube Based MPC as control scheme for its strong theoretical properties. The main steps of the practical implementation of Tube Based MPC are

• The identification of the model parameters and the associated disturbance bounds.

• The choice of a local controller.

• The computation of the mRPI set for the identified model and local controller.

• The design of the terminal elements to ensure robust recursive feasibility of the optimization problems.

• The online computation of the solution to an optimal control problem for the control of the nominal system.

We derive a model of the translational dynamics from physical principles and we identify the model parameters using data from experimental flights. We propose a novel data-driven system identification whose goal is to obtain the smallest possible RCI set, in view of its use in the Tube Based MPC. Here, the local controller is fixed and the model parameters are the optimization variable in the computation of minimal RCI set. The experiments are performed on the Parrot AR.Drone 2.0 quadrotor.

The second part considers the class of constrained discrete-time linear systems subject to bounded additive disturbances from a theoretic point of view. We assume that the system identification step precedes the control synthesis. That is, the model and disturbance bounds are known and the control law is the only design parameter of the RCI set. The main contributions are

• Proposing conditions for the characterization of RCI sets that are minimal.

• The computation of candidate solutions when a formal characterization is not available.

• The exploration of set-iterative procedures for the refinement of any initial RCI set towards its minimization.

The results presented in the second part of the manuscript are illustrated in various simulations.

Outline and summary of contributions

Chapter 1: Theoretical background on invariant sets and Tube Based MPC.

The necessary mathematical background regarding invariant set theory and Tube Based MPC for constrained discrete-time linear systems subject to bounded additive disturbances is presented. A simulation analysis of the impact of the Tube Based MPC parameters on the closed-loop performances is provided to showcase the key role of the RCI set. Details on this simulation analysis have been presented in [Michel 2017].

The structure of the manuscript is then divided according to the two main areas of research that have been studied.

Part I: Implementation of a robust MPC scheme for the stabilization of a quadrotor UAV Chapter 2: State of the art on the design of robust MPC in quadrotor applications

We give a state of the art on the application of robust MPC scheme in quadrotor applications. Given that MPC schemes are model-based control strategies, we first present an overview of the quadrotor models used in the literature. Then, we provide an overview of the design of MPC schemes for quadrotor applications that have robustness properties with regards to disturbances. We focus in particular on results that present experimental applications, and on studies that use the Parrot AR.Drone 2.0.

Chapter 3: Implementation and experimental validation of Tube Based MPC for the AR.Drone 2.0 quadrotor In this chapter, we detail the implementation of a Tube Based MPC scheme for the stabilization of the translational dynamics of the AR.Drone 2.0 quadrotor. The tuning of the local controller is based on closed-loop performances from experimental data. Then, we propose a data-driven identification method adapted to the design Introduction and implementation of a Tube Based MPC, that aims at minimizing the RCI set. We provide an algorithm for the real-time computation of the terminal constraints. The experimental results of the Tube Based MPC is presented to showcase the robust recursive constraints satisfaction. Some of the experimental and theoretical developments in this chapter have been presented in [START_REF] Michel | [END_REF]].

Part II: Computation of minimal RCI sets for constrained discretetime linear systems subject to bounded additive disturbances Chapter 4: State of the art on the computation of optimal RCI sets This chapter provides a state of the art on the methods in the literature for the computation of optimimal (minimal or maximal) RCI sets for constrained discretetime linear systems subject to bounded additive disturbances, where the model and disturbance bounds are known. The main difficulty comes from the fact the direct computation of optimized RCI sets is not tractable. The approaches available in the literature use a parametrization of the control law (linear, piece-wise affine) and/or of the RCI set (polytope), to reduce the computational complexity at the expense of conservatism.

Chapter 5: Invariant sets for linear discrete-time systems subject to bounded additive disturbance using Sliding Mode Control This chapter presents the theoretical framework of discrete-time sliding mode control. This type of control laws is well-known in continuous-time for its ability to mitigate matched disturbances [Edwards 1998]. We study the geometrical properties of the mRPI sets obtained with such control laws. These geometrical properties are inherited from the choice of the stable sliding surface. We provide condition for the characterization of minimal RCI sets using the geometrical properties of the mRPI sets for discrete-time sliding mode control laws. In the absence of a characterization, we propose methods for the computation of candidate solutions, where we restrain the control law to the class of discrete-time sliding mode control laws. A simulation example is provided to illustrate the results presented in this chapter. Some of the developments in this chapter have been presented in [Michel 2018b] and [Michel 2018a].

Chapter 6: Set mapping within the class of invariant sets for constrained discrete-time linear systems subject to bounded additive disturbance of the decreasing sequences are a refinement in terms of disturbance rejection. We provide a first characterization of the limit set of these set mappings, which consists of iteratively refined RCI sets. Some first elements among these results have been included in [Michel 2018a]. The submission of a journal publication is ongoing as of the writing of this manuscript.

This manuscript ends with a conclusion and perspectives for future works, in relation with both the practical and theoretical developments.

Introduction

This chapter presents the notations used in the manuscript, as well as a number of definitions. We define polytopes as a particular class of convex sets in Section 1.3 ; Section 1.4 provides the definitions of invariant sets for constrained discrete-time linear systems subject to bounded additive disturbances. In particular, we propose a novel criterion to measure the minimality of RCI sets. Formulations of Tube Based MPC applied to this class of systems are given in Section 1.5, as originally presented in [Mayne 2005]. A simulation analysis of the impact of the Tube Based MPC parameters on the closed-loop performances is provided in Section 1.6 to showcase the key role of the RCI set. This chapter ends with an overview of the challenges for the computation of minimal RCI sets in view of its use for the implementation of a Tube Based MPC scheme.

Notation and basic definitions

• The set of natural integers is denoted N. We write N + the set of strictly positive natural integers, N k = {0, 1, ..., k}, N + k = {1, 2, ..., k}, and N+ = N + ∪ {∞}.

• The set of real numbers is denoted R. We write R + the set of non-negative real numbers, and R ++ the set of strictly positive real numbers.

• In general, we use latin and greek letters for vectors and scalars (e.g. ξ, u, w, α), capital letters for matrices (e.g. A, B, R, J, P ), and calligraphic letters for sets (e.g. X , W, U).

• The i th component of a vector ξ ∈ R n is denoted ξ i .

• The i th row of a matrix M ∈ R n×m is denoted M i (thus M i ∈ R 1×m ).
• For a matrix M ∈ R n×m , (M ) i,j denotes the component on the i th row and j th column.

• The transpose of a matrix M ∈ R n×m is denoted M ∈ R m×n , and verifies

(M ) i,j = M j,i , ∀i ∈ N + n , ∀j ∈ N + m .
• Given a vector ξ ∈ R n , we denote diag(ξ) the diagonal matrix whose entries starting in the upper left corner are ξ 1 , ..., ξ n .

• A unit vector of length k is denoted 1 k .

Notation and basic definitions

• The identity square matrix of dimension k is denoted I k .

• A zero matrix of m rows and n columns, respectively a vector of length m, is denoted 0 m,n , respectively 0 m .

• Given a set A ⊆ R n , {a[i]} i∈N + k ∈ A k reads a[i] ∈ A, ∀i ∈ N + k . In a similar fashion, {a[i]} i∈N ∈ A ∞ reads a[i] ∈ A, ∀i ∈ N.
• The set of eigenvalues of a square matrix M ∈ R n×n is denoted λ(M ). The spectral radius of M is the largest absolute value of its eigenvalues, and is denoted

ρ(M ) = max {|λ| | λ ∈ λ(M )}. • The set of invertible matrices of R n×n is GL n = {M ∈ R n×n | 0 ∈ λ(M )}. • The set of Schur matrices of R n×n is S n = {M ∈ R n×n | ρ(M ) < 1}.
• Given a matrix M ∈ R n×m and a set A ⊆ R m , we define the linear set mapping

M A = {M a ∈ R n | a ∈ A} .
• Given a set A, 2 A denotes the set containing all the subsets of A, called the power set. The subset of 2 A that comprises of bounded sets is denoted 2A .

• Given two sets A ⊆ R n and B ⊆ R n , the Minkowski sum is defined as

A ⊕ B = {a + b | a ∈ A, b ∈ B}.
• Given two sets A ⊆ R n and B ⊆ R n , the Pontryagin difference is defined as

A B = {a | {a} ⊕ B ⊆ A}.
• The closure of a set A ⊆ R n consists of all points in A together with all limit points of A, and is denoted Cl(A).

• The interior of a set A ⊆ R n is the union of all the open subsets of A, and is denoted Int(A).

• The boundary of a set A ⊆ R n is the the sets of points in the closure that do not belong in the interior, and is denoted δA.

• Given a set A ⊂ R n , we denote Co (A) its convex hull.

• Given a collection of points {ξ

[i]} i∈N + p ∈ (R n ) p , we denote Co {ξ[i]} i∈N + p ⊂ R n its convex hull. • The p ∈ N+ norm in R n is denoted |ξ| p = i∈N + n |ξ i | p 1/p . The induced p matrix norm is given by ∀M ∈ R n×n , |M | p = max {|M ξ| p | |ξ| p = 1} .
• The p-norm ball of radius > 0 and centered on the origin in R n is denoted

B n p ( ) = {ξ ∈ R n | |ξ| p ≤ } .

Theoretical background on invariant set and Tube Based MPC

• For a positive definite, or positive semi-definite, matrix P ∈ R n×n and a vector ξ ∈ R n , we denote ||ξ|| P = ξ P ξ ∈ R + .

• If A is a finite set that contains n elements, then card(A) = n.

• Given two sets P and Q, we denote P Q the set of functions from P into Q.

Definition and properties of polyhedra

This manuscript considers a particular class of sets generally encountered in Tube Based MPC and invariant set literature related to linear dynamical systems, namely the polyhedra. We recall next definitions and refer to the literature for their properties and in-depths structural analysis.

Definition 1.1 (Hyperplane) An affine hyperplane of R n is an affine subspace of dimension n -1.

An hyperplane can be described as the collection of solutions for a linear equation of the form hξ = g where h ∈ R 1×n and g ∈ R are given. This affine hyperplane separates the space into two half-spaces, respectively described by the linear inequalities hξ ≤ g and hξ ≥ g. As a direct consequence of their definition, the hyperplanes and the half-spaces are convex sets.

Definition 1.2 (Polyhedron) A polyhedron in R n is the intersection of a finite number of half-spaces.

From this definition it follows an essential property: as an intersection of convex sets, a polyhedron in R n is a convex set.

Definition 1.3 (H-representation) Any polyhedron P ⊆ R n can be written P = {ξ ∈ R n | G p ξ ≤ g p } for some matrices G p and g p of suitable dimensions.

There exists an infinite number of H-representations of a polyhedron. However, the H-representation is said minimal if the dimension of the vector g p is minimal. A minimal H-representation avoids redundant linear inequalities in the H-representation.

Definition 1.4 (H-complexity) The H-complexity of a polyhedron is defined as the dimension of its minimal H-representation.

In this thesis, for a polyhedron P ⊂ R n , we denote l p its H-complexity, and G p ∈ R lp×n and g p ∈ R lp a pair of matrices defining a minimal H-representation. Note that this choice is not unique.

We have the following property for the H-representation of a polytope containing the origin in its interior.

Proposition 1 (Polyhedron containing the origin) The polytope P ⊂ R n contains the origin in its interior if and only if

(g p ) i > 0, ∀i ∈ N + lp . Moreover, P = ξ ∈ R n | Gp ξ ≤ 1 m , where Gp ∈ R lp×n , Gp i,j = (G p ) i,j (g p ) i , ∀i ∈ N + lp , ∀j ∈ N + n .
The following definition relates to the class of bounded and closed polyhedra.

Definition 1.5 (Polytope) A polytope in R n is a closed and bounded polyhedron.

Theorem 1.1 [START_REF] Motzkin | [END_REF]] (V-representation) A polytope P ⊂ R n can be defined as the convex hull of a finite set of points.

There exists an infinite number of V-representations of a polytope. The minimal V-representation, in terms of number of extreme points (which cannot be expressed as a convex combination of othe points in P), is unique and is given by the set of vertices of the polytope, which we denote V (P).

Definition 1.6 (V-complexity) The V-complexity of a polytope is defined as the number of its vertices, in short card(V (P)).

This manuscript uses mathematical operations that benefit from the computation of the H and V representations of polytopes. For instance, the set inclusion ξ ∈ P consists of the l p linear inequalities G p ξ ≤ g p . The Minkowski Sum of two polytopes

P and Q uses the V-representation, P ⊕ Q = Co({v p + v q | v p ∈ V (P), v q ∈ V (Q)}).
The Pontryagin difference of a polytope P and a set W uses the H-representation of the polytope P, yielding [Kolmanovsky 1998b]

P W = {ξ ∈ X | G p ξ ≤ g p -g(W)} ,
where the i th row of g(W) is given by g i (W) = sup w∈W (G p ) i w. In particular, the set P W is a polytope whose H-complexity is at most l p . If the set W is a polytope, we use its V-representation to obtain g i (W) = max

w∈V (W) (G p ) i w.
We use the Multi-Parametric Toolbox 3.0 [Herceg 2013] to compute the changes of representation, and to illustrate graphically the polyhedra and polytopes.

Invariant set theory

We will present the definitions and results regarding set invariance in the context of linear discrete-time constrained systems. Let us consider a linear discrete-time system subject to additive disturbances,

ξ + = Aξ + Bu + w, (1.1)
where ξ ∈ R n is the known current state, u ∈ R m is the control input, and w ∈ R n is an exogenous disturbance. We assume that the system is subject to constraints, characterized set-wise by

(ξ, u, w) ∈ (X × U × W) , (1.2)
Theoretical background on invariant set and Tube Based MPC

with X ⊆ R n , U ⊆ R m , W ⊆ R n .
In this chapter, we assume that the model parameters A ∈ R n×n , B ∈ R n×m and the disturbance bounds, represented set-wise by the set W, are known. We are interested in the local behavior of the system around an equilibrium {ξ eq , u eq } ∈ (X × U) satisfying ξ eq = Aξ eq + Bu eq . By considering ξ = ξ -ξ eq , u = u -u eq , X = X ⊕ {-ξ eq } and U = U ⊕ {-u eq }, we have

ξ + = Aξ + Bu + w, (1.3) (ξ , u , w) ∈ (X , U , W). (1.4)
Hence, the local behavior of the system (1.1) subject to the constraints (1.2) around (ξ eq , u eq ) is similar to the local behavior of the system (1.3) subject to the translated constraints (1.4) around the origin {0 n , 0 m }. For this reason, we consider in the following that {ξ eq , u eq } = {0 n , 0 m }.

The results presented in this section are obtained under the following assumption.

Assumption 1.1 We assume that the sets U and W are compact, the set X is closed, 0 n ∈ Int(X ), 0 m ∈ Int(U) and 0 n ∈ W.

The local behavior depends on the choice for a control law, and will be characterized in terms of invariant sets. As a first step, we provide the definition regarding set invariance for closed-loop systems.

Closed-loop systems

In this section we consider the system (1.1) with a state feedback control law ν :

R n → R m , yielding ξ + = Aξ + Bν(ξ) + w.
(1.5)

With the control law, the input constraints become additional state constraints. We thus consider the set

X ν = {ξ ∈ X | ν(ξ) ∈ U} ,
and the constraints (1.2) become

(ξ, w) ∈ (X ν × W) .
(1.6)

General definitions

As a first step, let us define the following operator related to (1.5) and (1.6) below.

Definition 1.7 (One-step image set) For a non-empty set Z and a control law ν : R n → R m such that Z ⊆ X ν , the one-step image set for the system (1.5) and constraints

(1.6) is Image(Z, ν) = {Aξ + Bν(ξ) + w | ξ ∈ Z, w ∈ W} = {Aξ + Bν(ξ) | ξ ∈ Z} ⊕ W.
The one-step image set of Z and control law ν consists of the image through (1.5) of all the elements of Z. This construction accounts for the presence of disturbances w ∈ W. The notion of one-step image set can be iterated to define the i-step image set for all i ∈ N as follows

Image 0 (Z, ν) = Z, Image i+1 (Z, ν) = Image(Image i (Z, ν), ν), ∀i ∈ N.
For the closed-loop system (1.5) an invariant set is defined as follows.

Definition 1.8 (Robust Positively Invariant set) A set Z is a Robust Positively Invariant (RPI) set for the system (1.5) and constraints (1.6) if Z ⊆ X ν and for all ξ ∈ Z, Aξ + Bν(ξ) + w ∈ Z, ∀w ∈ W.

We can use the concept of one-step image to define RPI sets. Indeed, the set Z is an RPI set for the system (1.5) and constraints (1.6) if and only if

Z ⊆ X , ν(Z) ⊆ U, Image(Z, ν) ⊆ Z.
An RPI set is a region of the state space with the property that if it contains the state of the closed-loop system at a given time, then it will contain it in the future for any w ∈ W. We have the following result regarding the intersection of RPI sets containing the equilibrium (here chosen as the origin).

Proposition 2 (Intersection of RPI sets) Let Z 1 and Z 2 be two RPI sets for the closed-loop system (1.5) and constraints (1.6) that contain the origin. Then, the set Z 1 ∩ Z 2 is an RPI set for the closed-loop system (1.5) and constraints (1.6).

This result allows us to define the concept of minimal RPI sets using set inclusion as a criterion for minimality.

Definition 1.9 (minimal Robust Positively Invariant set) The minimal Robust Positively Invariant (mRPI) set for the system (1.5) and constraints (1.6) is defined as the intersection of all the RPI sets for the system (1.5) and constraints (1.6) that contain the origin.

This set is the smallest neighborhood of the origin that has the property to recursively contain the state of the closed-loop system regardless of the disturbances. Any trajectory initialized in the mRPI set remains in the mRPI set. Note that, due to the nonlinearity in the feedback, this set may not be the limit set of all trajectories initialized in any RPI set, as illustrated in the following example.

Theoretical background on invariant set and Tube Based MPC

Example -mRPI set and limit set Consider the closed-loop system ξ

[k + 1] = 0.5ξ[k] + ν(ξ[k]) + w[k], ∀k ∈ N, where ξ[k] ∈ R, w[k] ∈ [-1, 1] and ν(ξ[k]) = 0, if ξ[k] ∈ [-2, 2], ν(ξ[k]) = 1, if ξ[k] ∈ [-2, 2].
The mRPI set is [-2, 2], and [-2, 4] is an RPI set. With the initial condition

ξ[0] = 4 and the persistent disturbance realizations w[k] = 1, ∀k ∈ N, we have ξ[k] = 4, ∀k ∈ N.
This shows that the mRPI set is not always the limit set of all trajectories initialized in an RPI set.

The mRPI set is a measure of the disturbance rejection properties of a control law around the origin (or around an equilibrium). Comparing the disturbance rejection properties of control laws consists in comparing the size of the associated mRPI sets, and thus requires their caracterization. Unfortunatly, the computation of the mRPI set for general non-linear control laws can be complex.

In the following section we focus on mRPI sets for the particular case of linear systems with linear control laws.

Linear control laws

We now consider a linear control law ν(ξ) = Kξ. In the following, we assume that the feedback gain is stabilizing, that is A + BK is Schur. The closed-loop system (1.5) and the constraints (1.6) read

ξ + = (A + BK)ξ + w, (1.7a) (ξ, w) ∈ (X K , W), (1.7b)
where X K = {ξ ∈ X | Kξ ∈ U}. With this choice for the control law, the one-step image set of a set Z is given by Image(Z, K) = (A + BK)Z ⊕ W.

Remark 1.1 If the set Z and W are polytopes, we can use their V-representation to compute the one-step image set as follows

Image(Z, K) = Co ({ξ + w | ξ ∈ (A + BK)V (Z), w ∈ V (W)}) ,
and the image set is also a polytope.

The set Z is an RPI set for the closed-loop system (1.7a) and constraints (1.7b) if and only if

(A + BK)Z ⊕ W ⊆ Z, (1.8a) Z ⊆ X K . (1.8b)
For a given feedback gain K, the smallest set (in terms of set inclusion) that satisfies (1.8a) is given by

Z ∞ (K) = ∞ i=0 (A + BK) i W.
The mRPI set for the system (1.7a) and constraints (1.7b) exists, is unique, and is given by

Z ∞ (K) if and only if Z ∞ (K) ⊆ X K . Note that 0 ∈ Z ∞ (K) as long as 0 ∈ W.
This expression of the mRPI set uses the Minkowski sum of an infinite number of terms. Hence, it is not possible to compute explicitly this set in general. Nevertheless, if there exist r ∈ N + and 0 ≤ α < 1 such that (A + BK) r = αI n , then the following representation with a finite number of terms [Lasserre 1993]. For more information on finite definiteness of the mRPI set, the reader is refered to [Seron 2019]. In the general case, we would need an infinite number of operations to compute the mRPI set. For this reason, we instead rely on outer approximations to estimate the mRPI set.

Z ∞ (K) = (1 -α) -1 r-1 i=0 (A + BK) i W is obtained

Approximations of the mRPI set

In this section we provide a non-exhaustive list of results and algorithms for the computation of approximations of the mRPI set for the linear system (1.7a). In order to quantify how accurate is an approximation, we introduce the following definition.

Definition 1.10 [Rakovic 2005a] (( , p)-outer approximation) For a given > 0 and p ∈ N+ , a set Z is said to be an ( , p)-outer approximation of the mRPI

Z ∞ (K) if Z ∞ (K) ⊆ Z ⊆ Z ∞ (K) ⊕ B n p (
). This definition uses the concept of p-norm ball of radius , where the pair ( , p) define the quality of the approximation.

In this section we give two methods for the computation of ( , p)-outer approximations of mRPI sets from the literature. These methods compute ( , p)-outer approximations that are also RPI sets. Indeed, the RPI property of this approximation is a key element in the synthesis of robust predictive control methods [START_REF] Langson | [END_REF], Mayne 2005].

Scaling of the partial sums

The first method presented here, as described in [Rakovic 2005a], consists of first computing partial sums Z r (K) = r-1 i=0 (A + BK) i W for a sufficiently large integer r ∈ N, and then scaling the set by a suitable value.

Theoretical background on invariant set and Tube Based MPC

For a given r ∈ N, if there exists 0 ≤ α < 1 such that (A + BK) r W ⊆ αW, then the set (1 -α) -1 Z r (K) satisfies (1.8a) (i.e. it is an RPI set for the system (1.7a) and constraints (R n , W)). We define the smallest scalar α > 0 that satisfies the above condition as

α 0 (r) inf {0 ≤ α < 1 | (A + BK) r W ⊆ αW} .
The following result allows for a priori determination of the integer r and the scalar α to have a sufficiently accurate approximation of the mRPI.

Theorem 1.2 (RPI ( , p)-outer approximation of the mRPI) [Rakovic 2005a] For all > 0 there exists 0 ≤ α < 1 and r ∈ N such that

(A + BK) r W ⊆ αW, α(1 -α) -1 Z r (K) ⊆ B n p ( ).
Under these conditions, the set

(1-α) -1 Z r (K) is an RPI ( , p)-outer approximation of Z ∞ (K).
Hence, for a given > 0 one can compute an RPI ( , p)-outer approximation of the mRPI by incrementing r until there exists α satisfying the above conditions.

Image sets of an initial RPI set

The second method is based on the computation of the successive image sets of an initial RPI set. Let Ω 0 be an initial RPI set for the system (1.7a) and constraints (1.7b). The one-step image set of Ω 0 , denoted Ω 1 , is given by

Ω 1 = (A + BK)Ω 0 ⊕ W.
Since Ω 0 is an RPI set, we have by definition Ω 1 ⊆ Ω 0 ⊆ X K , and Ω 1 is also an RPI set for the system (1.7a) and constraints (1.7b). This construction can be iterated, thus defining a decreasing set sequence of RPI sets for the system (1.7a) and constraints (1.7b), defined as Ω i+1 = (A + BK)Ω i ⊕ W, ∀i ∈ N. It can be shown that

Ω i = (A + BK) i Ω 0 ⊕ Z i (K).
he above decreasing RPI set sequence can be used as an outer approximation of the mRPI set.

Theorem 1.3 (RPI ( , p)-outer approximation of the mRPI) [Rakovic 2005a] Let > 0. If Ω 0 is a compact set, then there exists r ∈ N such that

(A + BK) r Ω 0 ⊆ B n p ( ).
Under this condition, the set

Ω r = (A + BK) r Ω 0 ⊕ Z r (K) is an RPI ( , p)-outer approximation of Z ∞ (K).
Any compact RPI set Ω 0 can be used as the initial RPI set for the above method. An admissible candidate is an element of the sequence (1

-α 0 (r)) -1 Z k (r) | r ∈ N .
Another choice for the initial RPI set which does not require the computation of a finite Minkowksi sum is introduced below.

Theorem 1.4 (Initialization of a decreasing RPI set sequence) [START_REF] Olaru | [END_REF]] Let (A+BK) = RΛR -1 be the Jordan decomposition of A+BK and consider a bounding box for the set W, as in

W = w ∈ R n | w -w ≤ w M -w M , w M ∈ R n .
Then, for any δ > 0 n the polytope given by

ξ ∈ R n | R -1 -R -1 ξ ≤ (I n -|Λ|) -1 |R -1 |w M + δ (I n -|Λ|) -1 |R -1 |w M + δ
is an RPI set for the system (1.7a) and constraints (R n , W).

We have presented two methods from the literature for the computation of RPI ( , p)-outer approximations of the mRPI set for a linear system and a linear control law.

Remarks on the computational complexity of mRPI approximations

The two methods presented above rely on the computation of the r-step image set of an initial set Ω 0 , chosen as the origin {0 n } (first method) or as an RPI set (second method). In both cases, the r-step image set uses the partial sum Z r (K) as follows

Ω r = (A + BK) r Ω 0 ⊕ r-1 i=0 (A + BK) i W = (A + BK) r Ω 0 ⊕ Z r (K).
In this section, we assume that the disturbance set W is a polytope. Under this assumption, the partial sum Z r (K) is also a polytope, and is given by

Z r (K) = Co r-1 i=0 w[i] | w[i] ∈ V ((A + BK) i W), i = {0, 1, ..., r -1} .
The following result provides an asymptotic order of magnitude of the V-complexity of this polytope.

Proposition 3 (V-complexity of partial sums) [Gritzmann 1993] Let W be a polytope of R n of V-complexity v. Then, the V-complexity of the set Z r (K) is of order r n-1 v 2(n-1) .

The V-complexity increases significantly with several parameters, such as the dimension of the state n (exponentially) and the V-complexity of the disturbance set W. For this reason, the computation of outer approximations of mRPI sets is difficult for systems of high dimension. In practice, these approximations can be computed in a reasonable time for n < 4.

The computation of mRPI set approximations is of particular interest, as the mRPI set is a set representation of the disturbance rejection property of the linear control law. When the control law is is not fixed, it can be used as a design parameter for the design of invariant sets (e.g. the mRPI set Z ∞ (K) is parametrized by the feedback gain K ∈ R m×n ). This is the topic of invariant sets for open-loop systems.

Open-loop systems

In this section, we consider the open-loop system (1.1) and constraints (1.2). When the control law is not pre-imposed, we define invariant sets as follows.

Definition 1.11 (Robust Control Invariant set) A set Z is a Robust Control Invariant (RCI) set for the system (1.1) and constraints (1.2) if Z ⊆ X and for all ξ ∈ Z, there exists u ∈ U such that ξ + = Aξ + Bu + w ∈ Z, ∀w ∈ W.

A trajectory starting in an RCI set remains in the RCI set regardless of the disturbances provided an adequate choice for the control action.

Naturally, there is a relationship between RCI sets and RPI sets. For a given RCI set Z for the open-loop system (1.1) and constraints (1.2) there exists at least a control law ν : R n → R m such that Z is an RPI set for the closed-loop system (1.5) and constraints (1.6). Likewise, if Z is an RPI set for the closed-loop system (1.5) and constraints (1.6), then Z is an RCI set for the open-loop system (1.1) and constraints (1.2). In these definitions we stress the dependency on the considered system and constraints. In the presentation of our subsequent results we shorten the notation if the dependence is clear in the context.

We can use the concept of one-step image set to define RCI sets. The set Z is an RCI set for the system (1.1) and constraints (1.2) if and only if there exists a control law ν :

R n → R m such that Z ⊆ X , ν(Z) ⊆ U, Image(Z, ν) ⊆ Z.
The computation of a control law that has the best disturbance rejection properties relates to the construction of an RCI set that is as small as possible. The definition of minimal RCI sets is discussed in the following section.

Minimal RCI set in terms of set inclusion

Note that, differently from RPI sets, the intersection of RCI sets that contain the origin is not an RCI set in general.

Example -intersection of RCI sets

Consider the system ξ + = u + w, where -1 ≤ w ≤ 1. The sets [-0.5, 1.5] and [-1.5, 0.5] are RCI sets (RPI sets for the control law ν(ξ) = 0.5 and ν(ξ) = -0.5 respectively). However, their intersection [-0.5, 0.5] is not an RCI set.

For this reason, defining a set-theoretic measure of the impact of the disturbances when the control input is a design variable, by means of a minimal RCI set in terms of set inclusion, introduces several technical difficulties. In the following, we denote R(X , U, W) the collection of RCI sets for the system (1.1) and constraint sets (1.2) that are bounded. We restrain to RCI sets that are bounded because infinite RCI sets represent null disturbance rejection.

Let us introduce the following definitions from order theory.

Definition 1.12 (Partial order) The binary relation ≤ is a partial order of the set A if it is reflexive, anti-symmetric and transitive. That is, for all (a, b, c

) ∈ A 3 , a ≤ a, (reflexive) if a ≤ b and b ≤ a, then a = b, (anti-symmetric), if a ≤ b and b ≤ c, then a ≤ c, (transitive).
Definition 1.13 (Total order) A partial order under which every pair of element is comparable is called a total order.

Set inclusion is not a total order over R(X , U, W), as certain pairs of elements are not comparable (see the above example). However, set inclusion can be used to establish the following definition regarding minimal RCI sets.

Definition 1.14 (minimal Robust Control Invariant set) A set Z is a minimal Robust Control Invariant (mRCI) set for the system (1.1) and constraint sets (1.2) if it is bounded, if it is an RCI set for the system (1.1) and constraint sets (1.2), and if it does not contain (strictly) any other RCI set for the system (1.1) and constraint sets (1.2).

Note that the above definition is not entitling to strong result regarding the uniqueness of mRCI sets in general. In the following, we introduce another binary relation over R(X , U, W) that allows to compare every pair of elements.

Minimal RCI set in terms of matrix-norm

As a first step, we use the radius of the minimal p-norm ball containing an RCI set to quantify its size, yielding

|Z| p inf ∈ R + | Z ⊆ B n p ( ) .
(1.9)

= max{|ξ| p | ξ ∈ V (Z)}.

Theoretical background on invariant set and Tube Based MPC

The criterion (1.9) is the supremum of the p-norm evaluated on all elements ξ ∈ Z, given by

|ξ| p = ( n i=1 |ξ i | p ) 1/p .
This criterion gives the same weight to all the components ξ 1 , ...ξ n of the state from the definition of the p norm.

For quadrotor applications, the state of the system includes position ξ 1 , velocity ξ 2 , and attitude ξ 3 of the quadrotor. Depending on the specifications on the mission, we could be interested in minimizing the impact of the disturbances either with regards to the position ξ 1 (when flying close to obstacles), or with regards to the attitude ξ 3 (when using vision-based methods for navigation), instead of over the three components with equal weight.

In the context of general system as in (1.1) subject to the constraints (1.2) where the state constraints are defined by the polyhedral set

X = {ξ ∈ R n | Hξ ≤ 1 h } , H ∈ R h×n ,
one could be interested in computing an RCI set Z that is as far as possible from the state constraints. That is, to compute an RCI set Z such that HZ is contained in a minimal ∞-norm ball (|HZ| ∞ minimal).

Matrix-norm criterion

More generally, the criterion we want to minimize could relate to the image of the set Z by a matrix H ∈ R h×n , h ∈ N + . For this reason, we introduce the following measure of minimality of bounded sets with regards to a norm p and a matrix

H ∈ R h×n , h ∈ N + , |HZ| p min ∈ R + | HZ ⊆ B h p ( ) .
(1.10) Likewise, we have|HZ| p = sup {|Hξ| p | ξ ∈ Z} = sup | ξ| p | ξ ∈ HZ , and if the Z is a polytope, then we have |HZ| p = max{| ξ| p | ξ ∈ HV (Z)}. In the following, we use the term matrix-norm criterion to refer to the criterion (1.10).

Note that the matrix-norm criterion (1.10) can be extended to sets Z ⊆ R n such that HZ is bounded.

Remark 1.2 We are interested here in the image of a bounded set via the linear operator

R n → R h , (1.11a) ξ → Hξ (1.11b)
which maps the state space of dimension n to a vector space of dimension h.

(R n ) .
This total preorder is used to compare bounded RCI sets, in particular when they are not comparable with respect to the set inclusion.

Sublevel sets of the matrix-norm criterion

For a positive scalar c ≥ 0, define

H p (c) = {ξ ∈ R n | |Hξ| p ≤ c} .
(1.12)

A vector ξ ∈ R n is contained in the set H p (c) if and only if its image by the linear operator (1.11), namely Hξ ∈ R h , is contained in the (c, p)-norm ball, that is

Hξ ∈ B h p (c).
The set H p (c) is a sublevel set of the matrix-norm criterion (1.10). Remark 1.3 In the particular case p = ∞, the sublevel set H p (c) as in (1.12) is a polytope. From the linearity of the p norm, we have ∀c ∈ R + , H p (c) = cH p (1). Thus, all the sublevels of the matrix-norm criterion have the same geometrical structure, as they differ only from their scaling, namely

∀c 1 > 0, ∀c 2 ≥ 0, H p (c 2 ) = c 2 c 1 H p (c 1 ).
(1.13)

In particular, sublevel sets can be compared using set inclusion.

Let Z be a bounded set, and consider the sublevel set H p (|HZ| p ). This sublevel set is the smallest sublevel set in terms of set inclusion that contains Z. To make the notation less cluttered, we denote H p (Z) = H p (|HZ| p ).

We have the following results regarding the sublevel sets of the matrix-norm criterion.

Proposition 5 Let H ∈ R h×n , p ∈ N+ . The bounded sets Z 1 and Z 2 satisfy Z 1 ≤ H,p Z 2 if and only if H p (Z 1 ) ⊆ H p (Z 2 ).
Proof: This result comes from (1.13).

Proposition 6 Let Z 1 and Z 2 be such that Z 1 ⊆ Z 2 and there exists a vector ξ ∈ Z 1 contained in the frontier of

H p (Z 2 ), that is ξ ∈ δH p (Z 2 ). Then, H p (Z) = H p (Z 2 . Proof: The vector ξ satisfies |Hξ| p = |HZ| p . Thus, |HZ 1 | p ≥ |Hξ| p = |HZ 2 | p ,
and H p (Z 2 ) ⊆ H p (Z 1 ). The equality comes from the set inclusion Z 1 ⊆ Z 2 and Proposition 5.

Theoretical background on invariant set and Tube Based MPC

Matrix-norm minimal RCI sets

All the elements are now present to give a formulation of the goal of this manuscript. Let us define the infimum

h * (H, p) = inf {|HZ| p | Z ∈ R(X , U, W)} .
(1.14)

We introduce the following definition of matrix-norm minimal RCI sets, using the criterion (1.10).

Definition Both Definitions 1.14 and 1.16 deal with the notion of minimal RCI sets. The difference between these definitions is that, in the first case, the minimality is related to a partial order while, in the second case, a scalar index and a matrix are considered to define minimality with regards to a total preorder. The matrix-norm definition does not ensure uniqueness, nor existence, of (H, p)-mRCI sets. A proof of the existence of (H, p)-mRCI sets is provided in Chapter 6 under the assumption that the constraint sets X and U are compact. Under this assumption, the infimum h(H, p) is attained (we are dealing with a minimum). The goal of computing (H, p)-mRCI sets is formulated in the following problem.

Problem 1.1 For given system (1.1), constraint sets (1.2), matrix H ∈ R h×n , and scalar p ∈ N+ , compute a bounded RCI set Z such that |HZ| p = h(H, p).

We express Problem 1.1 in terms of an optimization problem as follows, minimize

Z |HZ| p , (1.15a) subject to Z ∈ R(X , U, W). (1.15b)
This optimization problem cannot be solved analytically as we do not have a characterization of the collection of bounded RCI sets R(X , U, W). The computation of RCI sets relates to the computation of RPI sets for control laws ν : R n → R m , which allows to rewrite the optimization problem (1.15) as follows minimize

ν,Z |HZ| p , (1.16a) subject to Aξ + Bν(ξ) ⊆ Z W, ∀ξ ∈ Z, (1.16b) ν(ξ) ∈ U, ∀ξ ∈ Z, (1.16c) ξ ∈ X , ∀ξ ∈ Z, (1.16d) Z ∈ 2 R n , (1.16e) ν ∈ (R n ) (R m ) . (1.16f)
In a similar fashion, the optimization problem (1.16) cannot be solved analytically in general.

Our goal is to establish conditions on the system (1.1), constraints (1.2) and parameters (H, p) to provide a formal characterization of (H, p)-mRCI sets. In case when such a formal characterization cannot be obtained, we are interested in methods for the computation of candidate solutions. That is, the computation of bounded RCI sets Z such that |HZ| p is as close as possible to the infimum h * (H, p).

Computation of candidate solutions

A difficulty in the computation of minimal RCI sets comes from the fact that both the set Z in (1.16e) and the control law ν in (1.16f) are optimization variables.

There exists methods in the literature for the computation of candidate solutions where the criterion for minimality is either chosen as in (1.10) with H = I n [Raković 2005c, Rakovic 2010], or as the volume of the RCI set [START_REF] Cannon | [END_REF], Blanco 2010]. These methods rely on a parametrization of the control law, chosen either as linear or piece-wise affine, and/or of the RCI set, chosen polytopic or ellipsoidal. An overview of these methods is provided in Chapter 4. We present here how some of these parametrizations can be used to reformulate Problem 1.1.

Linear parametrization of the control law

A first parametrization is to impose a linear structure to the control law ν(ξ) = Kξ, K ∈ R m×n . With this parametrization, we define

h * L (H, p) = inf |HZ ∞ (K)| p | K ∈ R m×n , A + BK ∈ S n , Z ∞ (K) ⊆ X K . (1.17)
This consideration yields the following alternative problem.

Problem 1.2 For given system (1.1), constraint sets (1.2), matrix H ∈ R h×n , and scalar p ∈ N+ , compute a stabilizing feedback gain K ∈ R m×n that verifies |HZ ∞ (K)| p = h * L (H, p), and whose mRPI set satisfy the state and input constraints.

The optimization problem corresponding to Problem 1.2 reads minimize

K∈R m×n |HZ ∞ (K)| p , (1.18a) subject to Z ∞ (K) = ∞ i=0 (A + BK) i W, (1.18b) Z ∞ (K) ⊆ X , KZ ∞ (K) ⊆ U, (1.18c)
A + BK ∈ S n .

(1.18d)

Polytopic parametrization of the RCI set

The second approach relies on the polytopic parametrization of the RCI set. We denote T (X , U, W) the subset of the collection of RCI sets that includes polytopes. Define

h * P (H, p) = inf {|HZ| p | Z ∈ T (X , U, W)} . (1.19)
The alternative problem is formulated below.

Problem 1.3 For given system (1.1), constraint sets (1.2), matrix H ∈ R h×n , and scalar p ∈ N+ , compute a polytopic RCI set Z such that |HZ| p = h * P (H, p).

Assuming the sets X and U are convex, we can exploit the polytopic structure of the RCI set to reduce the constraints and cost function of the optimization problem (1.15) to the vertices of Z. With this consideration, the optimization problem reads minimize ν,(V,v),(G,g,l),r r,

(1.20a)

subject to |Hξ| p ≤ r, ∀ξ ∈ V, (1.20b) G(Aξ + Bν(ξ)) ≤ g -g(W), ∀ξ ∈ V (Z), (1.20c) ν(ξ) ∈ U, ∀ξ ∈ V, (1.20d) ξ ∈ X , ∀ξ ∈ V, (1.20e) {ξ ∈ R n | Gξ ≤ g} = Co(V ), (1.20f) V ∈ (R n ) v , v ∈ N + , (1.20g) G ∈ R l×n , g ∈ R l , l ∈ N + . (1.20h)
This optimization problem has a solution if and only if the infimum (1.19) is attained. The polytope Z is parametrized using both its V-representation (1.20g) and its H-representation (1.20h), where the constraint (1.20f) ensures equality of both representations. The optimization problem has non-linear constraints, as in (1.20c), and is a mixed-integer since the constraints (1.20g) and (1.20h) introduce integer variables. This optimization problem can be simplified by imposing an equality (or inequality) constraint on the H or V complexity (respectively l and v) of the polytope.

In general, the infimums (1.14) (1.17) and (1.19) are not equal. The parametrizations lead to optimization problems of lower complexity, at the expense of conservatism, that can be used to compute candidate solutions.

Objective of this work

This manuscript aims at solving Problem 1.1, that is to compute matrix-norm RCI sets. Chapter 5 provides a sufficient condition on the matrix H for the formal characterization of a solution to Problem (1.1). Chapters 5 and 6 propose methodologies to synthesis candidate solutions when we do not have formal characterization of the solution. These methodologies rely on the above parametrization of the RCI set and/or of the control law.

In Chapter 3, we compute minimal RCI sets in the larger goal of designing a robust control law for the stabilization of a quadrotor in view of its practical implementation. Here, the model parameters A ∈ R n×n , B ∈ R n×m and W are identified using experimental data in the goal of computing minimal RCI set. The robust control law we chose to implement is Tube Based MPC, and the RCI set is a key element in its synthesis [START_REF] Langson | [END_REF], Mayne 2005]. The principles of Tube Based MPC are presented in the following section.

Tube Based MPC of constrained linear discrete-time systems

There exist many developments related to Tube Based MPC, especially in the recent literature. This section does not aim at providing an exhaustive overview of these developments. Instead, we focus on the theoretical basis of this control technique in view of its practical implementation. We follow a mathematical formulation of Tube Based MPC applied to a constrained linear discrete-time system as introduced in [Mayne 2005].

Preliminaries

Consider the discrete-time linear system (1.1) and constraints (1.2). This system is called the uncertain system, or actual system. For a control law ν(.) ∈ X → U, an initial condition ξ[0] = ξ 0 and the disturbance realizations {w[k]} k∈N ∈ W ∞ , the closed-loop trajectory of the actual system is given by

ξ[k + 1] = Aξ[k] + Bν(ξ[k]) + w[k], ∀k ∈ N.
(1.21)

Theoretical background on invariant set and Tube Based MPC

The design objective is to compute a control law ν(.) such that the closed-loop trajectory (1.21) satisfies the state and input constraints regardless of the disturbance realizations. That is,

ξ[k] ∈ X , ν(ξ[k]) ∈ U, ∀k ∈ N. (1.22)
In addition, the control objective is to steer the initial state ξ 0 to a neighborhood of the origin. The performance of the control law is the measure of this neighborhood, which we want to be as small as possible. This neighborhood is the limit set of the trajectories, and is characterized in terms of RCI sets.

Definition 1.17 Consider the system (1.1) and constraints (1.2). We say that a control law ν : X → U ensures robust recursive constraints satisfaction over the state space region X 0 ⊆ R n if for all initial condition ξ[0] ∈ X 0 and disturbance realizations {w[k]} k∈N ∈ W ∞ , the closed-loop trajectory (1.21) satisfies (1.22).

Tube of trajectories

Consider the system (1.1) without disturbance w,

ξ+ = A ξ + B ū. (1.23)
The evolution of the state of this system is deterministic. This system is called the nominal system. The evolution of the error between the nominal system state and the uncertain system state, denoted z = ξ -ξ and called state error, is given by

ξ+ = A ξ + B(u -ū) + w, w ∈ W.
(1.24) For this error system, we consider the following control policy

u -ū = κ(z), (1.25)
where κ : R n → R m is further called the local controller. From (1.25), we have u = ū + κ(z). In the following, we denote X κ = ξ ∈ X | κ( ξ) ∈ U . With (1.25) in (1.24), we obtain

ξ+ = A ξ + Bκ(z) + w, w ∈ W. (1.26)
The main goal of these translations and manipulations is to show that the state error ξ -ξ can be recursively bounded using invariant set theory.

Proposition 7 [Mayne 2005] (Recursive bounding) Consider system (1.1), its disturbance free counterpart (1.23) and the control policy (1.25). Let Z be an RPI set for the closed-loop system (1.26) and constraints (X κ , W).

If ξ ∈ { ξ} ⊕ Z, then ξ + ∈ { ξ+ } ⊕ Z, ∀w ∈ W.
This proposition states that the control policy (1.25) allows to maintain the state of the uncertain system in a neighborhood of the state of the nominal system, characterized by the RPI set Z.

For an initial condition ξ[0] = ξ0 , the trajectory of the nominal system verifies

ξ[k + 1] = A ξ[k] + B ū[k], ∀k ∈ N. (1.27)
With the control policy (1.25), the trajectory of the actual system as in (1.21) satisfies

ξ[k] ∈ ξ[k] ⊕ Z, u[k] ∈ ū[k] ⊕ κ(Z), ∀k ∈ N, ∀ {w[k]} k∈N ∈ W ∞ .
We define the following sets

X = X Z, Ū = U κ(Z).
(1.28)

Note that these sets are not empty provided Z ⊆ X κ . The main elements are available in order to state the following result concerning the constraints satisfaction for the uncertain system in the presence of constraints.

Proposition 8 (Constraints satisfaction) Consider a system (1.1), its disturbance free counterpart (1.23) and the control policy (1.25). Let Z be an RPI set for (1.26) and constraints (X κ , W). Let ξ and ξ be such that ξ

∈ { ξ} ⊕ Z. Then, ξ ∈ X , ū ∈ Ū yields ξ ∈ X , u ∈ U.
This proposition states that trajectory of the actual system (1.21) with the control policy (1.25) satisfies (1.22) regardless of the disturbance realizations {w[k]} k∈N ∈ W ∞ provided the trajectory of the nominal system, given by (1.27), satisfies

ξ[k] ∈ X , ū[k] ∈ Ū, ∀k ∈ N, (1.29)
and the initialization of the nominal system state verifies ξ[0] ∈ {ξ[0]} ⊕ -Z.

The control policy (1.25) decomposes the original control design problem into two control design problems.

• The control of the error system (1.24) around the origin through the design of the local controller κ and the computation of an associated RPI set Z.

• The control of the nominal system (1.23) under the state and control input constraints (1.28).

To reduce conservatism, it is desirable to have an RPI set Z as small as possible. Indeed, it bounds the error state, and it is substracted (Pontryagin sense) to the actual system constraints to define the nominal system constraints in (1.28). If the nominal system trajectory satisfies ξ[k] → k→∞ 0 n , then the RPI set Z is the limit set of the closed-loop trajectories of the actual system. This further motivates us in our goal of computing RCI sets that are minimal, as discussed in Section 1.4.2.3.

The control of the nominal system, through the computation of the nominal control inputs {ū[k]} k∈N , can be done by any control technique taking into account the nominal constraints. The convergence can be measured by means of a cost function, and thus globally the design can be achieved by means of an optimization-based strategy. Two different formulations, as introduced in [Mayne 2005], of optimal control problems are presented below.

First formulation of the optimal control problem

A model predictive controller is considered for the control of the nominal system. The conventional optimal control problem P N ( ξ), in which ξ = ξ[k] is the current state of the nominal system and N is the length of the prediction horizon, has no uncertainties and has constraints that are tighter than the original ones for the uncertain system, as defined in (1.28). The optimization problem is defined by

P N ( ξ[k]) : minimize ū V N ( ξ0 , ū), (1.30a) subject to V N ( ξ0 , ū) = N -1 i=0 (|| ξi || Q + ||ū i || R ) + || ξN || P , (1.30b) ū = (ū 0 , ..., ūN-1 ) , (1.30c) ξi+1 = A ξi + B ūi , i = {0, ..., N -1}, (1.30d) ξi ∈ X , i = {1, ..., N -1}, (1.30e) ūi ∈ Ū, i = {0, ..., N -1}, (1.30f) ξ0 = ξ[k],
(1.30g) ξN ∈ Xf .

(1.30h)

The weight matrices Q, R and P are positive definite (R can be positive semidefinite).

The terminal weight matrix P and set Xf are chosen to satisfy the standard stability conditions, as presented in [Mayne 2000]. Namely, they are designed along with a linear state feedback gain K f ∈ R m×n such that

(A + BK f ) Xf ⊆ Xf , (1.31a) K f Xf ⊆ Ū, (1.31b) Xf ⊆ X , (1.31c) (A + BK f ) P (A + BK f ) + Q + K f RK f -P ≤ 0. (1.31d)
As discussed above, the stabilization around an equilibrium {ξ eq , u eq } consists of the stabilization around the origin of the translated system (1.3) and constraints (1.4). Due to the translation of the constraint sets X = X ⊕ {-ξ eq } and U = U ⊕ {-u eq }, the satisfaction of the set inclusion Xf ⊆ X and K f Xf ⊆ U becomes a sensitive point of the design. Namely, the terminal set is a function of the equilibrium target state and control input, Xf (ξ eq , u eq ). We have considered a linear state feedback control law in (1.31), however the theoretical results hold for a generic non-linear control law, provided the suitable modifications on the translations and topology of the sets are considered and under the necessary stability conditions.

The feasible region of the optimal control problem P N ( ξ) is denoted XN . It is given by the following recursion

X0 = Xf , (1.32a) Xi+1 = ξ ∈ X | ∃ū ∈ Ū, A ξ + B ū ∈ Xi , i ∈ N N -1 . (1.32b) For a given ξ[k] ∈ XN , we denote ū * ( ξ[k]) = ū * 0 ( ξ[k]), ..., ū * N -1 ( ξ[k]
) a solution the optimization problem P N ( ξ[k]). The first component of the optimal control sequence is applied to the nominal system, so that the closed-loop nominal system satisfies

ξ[k + 1] = A ξ[k] + B ū * 0 ( ξ[k]).
(1.33)

Using (1.25), the control action applied to the actual system is given by

u[k] = ū * 0 ( ξ[k]) + κ(ξ[k] -ξ[k]), (1.34)
yielding

ξ[k + 1] = Aξ[k] + B(ū * 0 ( ξ[k]) + κ(ξ[k] -ξ[k])) + w[k].
(1.35)

We have the following result regarding robust recursive constraints satisfaction.

Proposition 9 (Robust recursive constraints satisfaction) [Mayne 2005] Consider a system (1.1) and its disturbance free counterpart (1.23). Let Z be an RPI set for the closed-loop error system (1.24) and constraints (X κ , W). The control law as in (1.34) ensures robust recursive constraints satisfaction over XN ⊕ Z.

Proof: Let ξ[0] ∈ XN ⊕ Z. By definition of the Minkowski sum, there exists

ξ[0] ∈ XN such that ξ[0]-ξ[0] ∈ Z.
With this choice for the initialization of the nominal system state, the closed-loop nominal trajectory (1.27) satisfies (1.29). From Proposition 8, the closed-loop trajectory of the actual system (1.21) with the control law (1.34) satisfies (1.22) regardless of the disturbance realizations {w

[k]} k∈N ∈ W ∞ .
Consider the following definition regarding the stability of systems.

Definition 1.18 (Exponential stability of the origin with a region of attraction) The origin is exponentially stable (Lyapunov stable and exponentially attractive) for the autonomous system ξ[k + 1] = f (ξ[k]) with a region of attraction X if there exist c > 0 and 0 < γ < 1 such that for any ξ

[0] ∈ X , it holds |ξ[i]| ≤ cγ i |ξ[0]|, ∀i ∈ N.

Theoretical background on invariant set and Tube Based MPC

The following result is established regarding the closed-loop stability of the nominal system (1.33).

Proposition 10 [ Mayne 2005] (Exponential stability) Consider a system (1.1) and its disturbance free counterpart (1.23). Let Z be an RPI set for the closed-loop error system (1.24) and constraints (X κ , W). Under the assumption that the set XN is bounded, the origin is exponentially stable for the closed-loop nominal system (1.33) with a region of attraction XN .

Here, the value function V * N cannot be used as a Lyapunov function to prove asymptotic stability of the origin with respect to the closed-loop actual system (1.35). However, if the nominal state ξ converges towards the origin, then the RPI set Z is the limit set of the closed-loop trajectories of the actual system, as we have ξ ∈ { ξ} ⊕ Z.

The following section presents an alternative Tube Based MPC scheme where the value function can be used as a Lyapunov function for the actual system, as introduced in [Mayne 2005].

Second formulation of the optimal control problem

The model predictive controller presented here uses the solution of an optimal control problem in which the initial state of the nominal system ξ[k] is also a decision variable. This choice is possible because the state of the nominal system is not linked to a physically related trajectory. Instead, the state of the nominal system is an additional parameter of the control law. For a given initial state ξ[k] of the uncertain system, the optimal control problem is defined by

P N (ξ[k]) : minimize ξ0 ,ū V N ( ξ0 , ū), (1.36a) subject to V N ( ξ0 , ū) = N -1 i=0 (|| ξi || Q + ||ū i -u eq || R ) + || ξN || P , ū = (ū 0 , ..., ūN-1 ) , ξi+1 = A ξi + B ūi , i = {0, ..., N -1}, ξi ∈ X , i = {1, ..., N -1}, (1.36b) ūi ∈ Ū, i = {0, ..., N -1}, (1.36c) ξ0 ∈ {ξ[k]} ⊕ -Z, (1.36d) ξN ∈ Xf .
The only difference between the optimal control problems P N ( ξ[k]) and

P N (ξ[k])
is that the state of the nominal system ξ[k] is an optimization parameter. The constraint (1.36d) ensures the state error belongs to the RPI set Z.

The feasible region of the optimal control problem P

N (ξ) is X N = XN ⊕ Z. For ξ[k] ∈ X N , we denote ( ξ * (ξ[k]), ū * (ξ[k]
)) the solution of the above optimization problem.

The resulting control law for the uncertain system is given by ∀ξ

[k] ∈ X N , u[k] = ν(ξ[k]) = ū * 0 (ξ[k]) + κ ξ[k] -ξ * (ξ[k]) .
(1.37)

The closed-loop dynamics of the actual system are given by

ξ[k + 1] = Aξ[k] + B(ū * 0 (ξ[k]) + κ(ξ[k] -ξ * (ξ[k]))) + w[k]. (1.38)
Here, the control law uses the first term of the optimal control sequence ū * 0 (ξ[k]), along with the optimal initial state ξ * (ξ[k]). An interesting property of the control law (1.37) is the following.

Proposition 11 (Trivial solution) [Mayne 2005] Consider a system (1.1) and its disturbance free counterpart (1.23). Let Z be an RPI set for the closed-loop error system (1.24) and constraints (X κ , W). For any state ξ ∈ Z, we have ξ * (ξ) = 0, ū * (ξ) = {0 m , ..., 0 m } and u = κ(ξ).

Once the actual system state ξ has reached the RPI set Z, the optimal control problem (1.36) is trivial. The optimal nominal system state ξ * (ξ) is chosen as the origin, and the control law (1.37) satisfies u(ξ) = κ(ξ). Consequently, the set Z is an RPI set for the closed-loop system (1.38) and constraints (X κ , W).

Remark 1.5 Here, the state of the nominal system is given by ξ

[k] = ξ * (ξ[k]).
In a similar manner, the stability conditions (1.31) are used to prove that if ξ

[k] ∈ X N , then ξ[k +1] ∈ X N .
Hence, the control law (1.37) ensures recursive feasibility of the optimal control problem (1.36). The following result can be established regarding robust recursive constraints satisfaction.

Proposition 12 [Mayne 2005] (Robust recursive constraints satisfaction) Consider a system (1.1) and its disturbance free counterpart (1.23). Let Z be an RPI set for the closed-loop error system (1.24) and constraints (X κ , W). The control law (1.37) ensures robust recursive constraints satisfaction over X N .

The value function is used as a Lyapunov function to prove stability of the closedloop actual system (1.38).

Definition 1.19 (Robust exponential stability of a set) A set Z is robust exponentially stable for the autonomous system ξ

[k + 1] = f (ξ[k], w[k]), w ∈ W with a region of attraction X if there exist c > 0 and 0 < γ < 1 such that for any ξ[0] ∈ X , it holds d(ξ[i], Z) ≤ cγ i d(ξ[0], Z), ∀ (w[0], ..., w[i -1]) ∈ W i , ∀i ∈ N.
Theorem 1.5 Consider a system (1.1) and its disturbance free counterpart (1.23). Let Z be an RPI set for the closed-loop error system (1.24) and constraints (X κ , W). The set Z is robust exponentially stable for the system (1.1) and constraints (1.2) with the control law (1.37) with a region of attraction X N .

Theoretical background on invariant set and Tube Based MPC

For the proof, the reader is referred to [Mayne 2005].

We have presented two Tube Based MPC schemes, as introduced in [Mayne 2005], each using the solution of an optimal control problem. The resulting control laws ensure robust recursive constraints satisfaction over the set X N . Moreover, both control laws have stability properties, either with regards to the closed-loop nominal system or with regards to the closed-loop actual system. In both cases, the RPI set Z is the limit set of the closed-loop trajectories of the actual system initialized in X N .

Practical implementation of Tube Based MPC

We now discuss the practical implementation of these Tube Based MPC schemes.

Both schemes require to solve an optimal control problem at each time-step, respectively (1.30) and (1.36). If the sets X , U, Z and Xf are polytopes, then these optimization problems are QP (quadratic cost and linear constraints).

We recall that the terminal set in these optimization problems is a function of the equilibrium state and control input Xf (ξ eq , u eq ). For our application, our mission requires to stabilize the quadrotor around successive equilibrium states and inputs (e.g. waypoint navigation). This requires the ability to compute a terminal set at each change of equilibrium.

A valuable advantage of the first scheme is that the successive optimal control problems P N ( ξ[k]) can be pre-computed offline. Indeed, the trajectory of the nominal system is fully determined by the initial condition ξ[0], and is not affected by the disturbance sequence. Thus, it is possible to compute beforehand the control input sequence {ū[k]} k∈N and the associated state sequence { ξ[k]} k∈N (offline trajectory planning for the nominal system). The remaining component of the control law for the actual system as in (1.34), namely κ(ξ[k] -ξ[k]), which successive uncertain system states {ξ[k]} k∈N , is to be computed online. This term ensures that the trajectory of the uncertain system remains in the tube centered on the nominal system trajectory, namely ξ

[k] ∈ { ξ[k]} ⊕ Z, ∀k ∈ N.
The second scheme has theoretical advantages, namely the set Z is robust exponentially stable for the closed-loop actual system. However, its implementation is more computationally demanding because of the additional constraint ξ

[k] ∈ {ξ[k]} ⊕ -Z.
If the set Z is a polytope of H complexity l z , this additional constraint consists of l z linear inequalities. Moreover, the dimension of the optimization variable is increased by n, due to ξ[k] being an optimization parameter. These computational considerations are to be taken into account for the real-time implementation of a Tube Based MPC law.

As a concluding remark regarding the practical implementation of these Tube Based MPC schemes, we stress that the theoretical properties (robust recursive constraint satisfaction and stability) rely on the assumption that the disturbances realizations {w[k]} k∈N are all contained in the disturbance set W used for the computation of the RPI set W. This assumption is not restrictive in a simulation setup, but is an additional challenge in an experimental setup. The choice of the model parameters A, B and W have to account for all the possible disturbances that will affect the system.

A simulation example of the implementation of the Tube Based MPC schemes is provided in the following section.

Simulation analysis of the impact of the Tube Based MPC parameters

In this section, we provide a brief simulation analysis of the impact of the Tube Based MPC parameters on the closed-loop performances. We consider a discretetime linear system as in (1.1) with n = 2, m = 1 and

A = 1 0.1 0 1 , B = 0.005 0.1 .
The choice of a double integrator is motivated by existing discretized and linearized models of quadrotors translational dynamics, as discussed in Chapter 2. We assume that the system is subject to state and control input constraints (1.2), characterized by the following polytopes

X =        ξ ∈ R 2 |     1 0 -1 0 0 1 0 -1     ξ ≤     1 0.5 1 1            , U = u ∈ R | 1 -1 u ≤ 4.5 4.5 .
The Figure 1.1 shows the state-space representation of the disturbance set W ⊆ R 2 . It is worth noting that these sets satisfy Assumption 1.1. This section discusses the main challenges of the implementation of the Tube Based MPC laws presented above for the stabilization to the origin of this system.

Theoretical background on invariant set and Tube Based MPC

The basic parameters of the Tube Based MPC are chosen as N = 10, Q = 10 0 0 1 and R = 0.1. The impact of the weighting matrices Q and R, and of the prediction horizon length N are not detailed here. The remaining parameters of the Tube Based MPC schemes presented above are

• the local controller κ and the associated RPI set Z.

• the linear state feedback gain K f , and the associated terminal set Xf and weighting matrix P .

Let us recall that the RPI set Z is the set representation of the disturbance rejection of the control law, as it bounds the difference between the nominal and the uncertain systems states. Moreover, the nominal system state constraint set X is defined as the Pontryagin difference between the uncertain system state constraint set X and the RPI set Z. Finally, it is the limit set of the trajectories initialized in XN .

For these reasons, we are interested in finding an RCI set Z that is as small as possible.

Design of the local controller

This section does not aim at providing methods for the computation of minimal RCI sets as discussed in Section 1.4.2.3. It rather aims to provide an illustration of the impact of the choice of the local controller on the size of the resulting mRPI set.

We impose a linear structure to the local controller κ(ξ) = Kξ, K ∈ R 1×2 , and we compute a range of feedback gains K(i), i ∈ {1, 2, 3} using a pole placement strategy. This choice for the selection of the local controller is arbitrary, as the goal is to highlight the discrepancy between the resulting mRPI sets. The value of the feedback gains K(i), and the associated set of closed-loop poles, are shown in the following table .   i K(i) poles

1 56 12.2 {0.7, -0.2} 2 12 7.40 {0.8, 0.4} 3 36 13.2 {0.3, 0.2}
We compute outer-approximations of the mRPI sets Z ∞ (K(i)), i ∈ {1, 2, 3} using the "Image sets of an initial RPI set" method presented in 1.4.1.3. For a feedback gain K(i), the initial RPI set Ω 0 (K(i)) is computed using Theorem 1.4, and we chose to compute the 25 step image set Ω 25 (K(i)), which we further denote Z(K(i)) for clarity of presentation. The Figure 1.2 shows the sets Z(K(i)) for the feedback gains K(i), i ∈ {1, 2, 3}.

The set Z(K(2)) is significantly larger than the other two, and contains the set Z(K(3)). However, it is not possible to order the sets Z(K(1)) and Z(K(3)) using the set inclusion criterion, as it is a partial order. We use a total preorder introduced 1.6. Simulation analysis of the impact of the Tube Based MPC parameters 41

Figure 1.2 -State space representation of outer approximations of the mRPI for the linear gains K(i).

in Section 1.4.2.3, with H = 1 0 and p ∈ N+ . With this choice for the matrix H,

we have h(Z ( K(1)), H, p) < h(Z(K(3)), H, p). Another related set of interest is K(i)Z(K(i))
, which represents the control input required to maintain the state error ξ -ξ in the RPI set Z(K(i)). This set directly impacts the optimization problem of the Tube Based MPC through the constraint

ū ∈ Ū(i) = U K(i)Z(K(i)). These sets are respectively K(1)Z(1) = [-2.4, 2.4], K(2)Z(2) = [-5.6, 5.6], K(3)Z(3) = [-2.34, 2.34].
Note that we have K(2)Z(K(2)) ⊆ U, which further eliminates the feedback gain K(2). In the following we retain the feedback gain K(1), defining the local controller κ(ξ) = K(1)ξ.

The Figure 1.3 gives the state space representation of the successive i step image sets {Ω i (K(1))} i∈N 25 , and the relation between the volume, the H-complexity and the V-complexity of the successive iterations. Each successive image step further approximates the mRPI set at the expense of computational complexity. Note that the volume does not decrease significantly beyond the 5 th iteration, while the complexity of the polytope keeps increasing. Hence, we retain Z = Ω 5 (K(1)) in the following.

Terminal set design

The last parameters to be designed are the terminal set Xf , the terminal cost P , and the associated terminal feedback gain K f ∈ R m×n . These parameters have to satisfy the stability conditions (1.31). Recall that the choice of the terminal set impacts the feasible region of the optimization problems XN and X N as in (1.32).

In a similar manner, we consider a range of feedback gain K f (i), i ∈ {1, 2, 3} computed using pole placement strategy. This choice for the selection of the local controller is arbitrary, as the goal is to highlight the discrepancy between the resulting terminal sets, and thus the feasible regions. The value of the feedback gains i K f (i) poles 1 2 2.9 {0.9, -0.8} 2 12 6.4 {0.7, 0.6} 3 42 10.9 {0.4, 0.3}

K f (i),
For each terminal feedback gain K f (i), i ∈ {1, 2, 3}, the terminal set Xf (i) is computed using the Maximal Output Admissible Set algorithm [Kolmanovsky 1998a], which is not detailed here. We then compute the set X10 (i) from Xf (i) as in (1.32) with N = 10. We choose the terminal cost P (i) as the solution of the Riccati equations

(A + BK f (i)) P (i)(A + BK f (i)) -P (i) + Q + K f (i) RK f (i) = 0.
Disregarding the impact of the terminal cost, the criterion for the selection of K f (i) is the size of the associated feasible region X10 (i). The Figure 1.4 presents the terminal sets Xf (i) and the feasible regions XN (i), N = 10, relative to the three linear feedback gains. Given that the set X10 ( 1) is larger than the others, we choose

K f = K f (1) in the following.
1.6. Simulation analysis of the impact of the Tube Based MPC parameters 43

Closed-loop performances

We now compare the closed-loop performances of the two Tube Based MPC schemes using the same above parameters. For clarity of presentation, we will refer to the first Tube Based MPC as TBMPC, and the second, that uses initial state as an optimization variable, as ETBMPC (enhanced Tube Based MPC).

We compute the trajectory of the uncertain state {ξ[k]} k∈N using both control laws (1.34) and (1.37). Two scenarii for the disturbances are considered. In the first one, the disturbance realizations {w[k]} k∈N are generated randomly in the set W.

In the second, the disturbance realizations {w[k]} k∈N are constant and chosen as a vertex of the disturbance set W, namely w[k] = -0.006 -0.12 , ∀k ∈ N. In both scenarii, the equilibrium state and inputs are {ξ eq , u eq } = {0 2 , 0}, and the trajectory is initialized as ξ[0] = 0.7 0.6 . Note that for the TBMPC, we initialize the nominal system state as ξ[0] = ξ[0]. In total, we compute four trajectories starting from the same initial condition, with two disturbances scenarii and two control laws.

The strong theoretical results of Tube Based MPC requires the state error

ξ[k] -ξ[k] to remain in the RPI set Z.
It is guaranteed here given that the disturbances realizations are chosen in the set W. The Figure 1.5 shows the state space representation of the RPI set Z, along with the state error ξ

[k] -ξ[k].
We note that with the ETBMPC law, the state error takes extreme values in Z in both disturbances scenarii. With the TBMPC law, the same behavior happens only when the disturbance is persistently chosen as a vertex of the disturbance set (second disturbance scenario).

The Figure 1.6 shows the time evolution of the control input. We also present the two components of the control input, namely the first element of the optimal control sequence and the feedback term

K(ξ[k] -ξ[k]
). This figure illustrates the Proposition 11, which states that the optimization problem of the ETBMPC law becomes trivial (u * (ξ[k]) = 0) once the uncertain system state ξ reaches the set Z. From this point, the nominal system remains to the origin, and the control law satisfies u

[k] = Kξ[k].
This figure also shows that the nominal input for TBMPC, given by ū = u * ( ξ), does not depend on the disturbances that impact the actual system. Indeed, this control action is the same in both disturbance scenarii.

We note that the control input related to the ETBMPC law is closer to saturation in both disturbance scenarii, in particular in the early phase of the trajectory. This is due to the nominal system state being an optimization variable of the optimization problem. This leads to a faster convergence of the system toward the neighborhood of the equilibrium, as illustrated in Figure 1.7. The feedback term in the ETBMPC not only rejects the disturbances but also contributes in steering the uncertain system toward the reference.

The Figures 1.8 presents the state space representation of these four trajectories, along with the trajectory of the nominal state ξ[k]. These figures illustrate the recursive state and control input constraints satisfaction of the nominal and the un- certain system despite the presence of (random or consistent) additive disturbances. To conclude on the comparison, the ETBMPC is better for its stability properties with regards to the actual system, and its enhanced closed-loop performances. However, the real-time computation of the solution of the ETBMPC optimization problem is more demanding, as the dimension of the optimization variable is increased by n (dimension of the state), and an additional constraint is used. This constraint consists of l z linear inequalities, where l z is the H-complexity of the polytopic RPI set Z.

Conclusion

This chapter introduced the definitions, tools, and algorithms required for the setwise characterization of the impact of the disturbances on the system, using invariant set theory. We have set up the general framework for the implementation of a Tube Based MPC law for the stabilization of a discrete-time linear system subject to bounded additive disturbance, as originally presented in [Mayne 2005]. This control scheme uses an RCI set as a design parameter. Its impact on the closed-loop performances is: it bounds the error between the actual system and the nominal system trajectories ;it is substracted to the actual constraint sets to define constraints in the optimal control problem of the nominal system, and thus impacts its feasible region ; it is the limit set of the actual system closed-loop trajectories.

These considerations motivate us in computing RCI sets that are as small as possible. Given that set inclusion is not a partial order, and thus cannot be used to compare every pair of RCI sets, we introduced a criterion to measure the impact of the disturbances that uses the image of an RCI set by a matrix that defines directions of interest (e.g. direction defined by the presence of polytopic state constraints). The computation of RCI sets that are minimal with regards to this criterion, where either the control law or the model are the design parameters, is the focus of this manuscript.

We provided a simulation example of the implementation of a Tube Based MPC law to illustrate the main challenges. The practical implementation of a Tube Based MPC law for the stabilization of a quadrotor introduces additional issues, among which

• the experimental characterization of the disturbance set, as part of a larger system identification procedure.

• the computation of terminal constraints at each change of desired equilibrium.

• the satisfaction of real-time constraints with regards to the online resolution of optimization problems.

These topics are studied in Chapter 3, where we provide the details of the design and implementation of a Tube Based MPC law for the horizontal position stabilization of the Parrot AR.Drone 2.0 quadrotor.

Part I

Implementation of a robust MPC scheme for the stabilization of a quadrotor UAV

Introduction

This chapter presents an overview of the available studies on the application of MPC for the stabilization of a quadrotor that account for the presence of disturbances.

In particular, we focus on the applications to the Parrot AR.Drone 2.0, as it is the platform we are using, along with researches that provide experimental results. The design of Model Predictive Control laws requires a model of the quadrotor dynamics, which is the topic of the next section.

Quadrotor dynamics modelling

This section gives a general model of the dynamics of a quadrotor, and the simplified models used for the Parrot AR.Drone 2.0. applications

General model of the quadrotor dynamics

Let us consider a fixed inertial frame J and the body frame B, as seen in Figure 2.1.

The inertial frame is defined by the origin O = 0 0 0 and the set of orthonormal vectors {e i , e j , e k }, where e k is directed upwards along the local vertical. The body frame is defined by the quadrotor center of mass, denoted G = x y z in J , and the set of vectors {e x , e y , e z } defining respectively the longitudinal, transverse, and vertical axis of the quadrotor. The quadrotor speed in the frame

J is denoted v = v x v y v z .
We use Euler angles to describe the orientation of the quadrotor, and the roll-pitch-yaw vector and denote them as φ θ ψ .

A quadrotor is controlled by the angular velocity of its four rotors, denoted {w r 1 , w r 2 , w r 3 , w r 4 }. Each motor produces a thrust and a torque, and their combination generate the total thrust and torque acting on the quadrotor. The total force generated by the quadrotor propellers is assumed to be directed along the e z axis, and we denote F ∈ R + its magnitude. The magnitude of the total thrust is given by F = 4 i=1 c F w 2 i . The total torque can also be expressed as a function of the rotors angular speed, yielding

F τ =     c F c F c F c F 0 0 -dc F dc F -dc F dc F 0 0 c τ c τ -c τ -c τ         w 2 1 w 2 2 w 2 3 w 2 4     .
The parameters c F and c τ are respectively the thrust and torque coefficient, and d is the distance between the center of gravity and the propellers. The above square matrix is invertible. Hence, the angular velocities, which are positive, can be obtained from the desired thrust magnitude F ∈ R + and torque τ ∈ R 3 . For this reason, we consider in the following that the thrust magnitude F ∈ R + and the torque τ ∈ R 3 are the quadrotor inputs.

The forces due to external disturbances are denoted f = f x f y f z . The dynamic model of a quadrotor is given by the following equations

ẋ ẏ ż = v x v y v z , (2.1a) m v = -mge k + R(φ, θ, ψ)F e k + f, (2.1b) Ṙ(φ, θ, ψ) = R(φ, θ, ψ)Q(ω), (2.1c) J ω = -ω × Jω + τ, (2.1d)
where m is the mass of the quadrotor, ω = ω x ω y ω z its angular velocity, J its inertia matrix, g the gravity constant,

R(φ, θ, ψ) =   c(ψ)c(θ) c(ψ)s(θ)s(φ) -s(ψ)c(φ) c(ψ)s(θ)c(φ) + s(ψ)s(φ) s(ψ)c(θ) s(ψ)s(θ)s(φ) + c(ψ)c(φ) s(ψ)s(θ)c(φ) -c(ψ)s(φ) -s(θ) c(θ)s(φ) c(θ)c(φ)   ∈ SO 3
with c(.) = cos(.) and s(.) = sin(.), is the orientation matrix from J to B, and

Q(ω) =   0 -ω 3 ω 2 ω 3 0 -ω 1 -ω 2 ω 1 0   .
The quadrotor dynamics are highly non-linear, in particular the orientation dynamics (2.1c), (2.1d). A quadrotor is an under-actuated system, as it has 4 inputs (F ∈ R + and τ ∈ R 3 ) and 6 degrees of freedom (6 for the position, 6 for the attitude). We choose in the following to control the translational position {x, y, z} and the yaw angle ψ to reference values {x r , y r , z r , ψ r }.

Note that the orientation matrix simplifies greatly under the assumption of small angles (i.e. φ ≈ θ ≈ ψ ≈ 0), which corresponds in practice to quasi-stationnary flight assumptions, yielding

R(φ, θ, ψ) ≈   1 -φ θ ψ 1 -φ -θ φ 1   . (2.2) With (2.2) in (2.1a
) and (2.1b), we have

ẋ = v x , m vx = F θ + f x (2.3a) ẏ = v y , m vy = F φ + f y (2.3b) ż = v z , m vz = -mg + F + f z (2.3c)
A cascaded design of the position and attitude controllers around the desired references is based on the following blocks

• A block for the computation the scalar F ∈ R + by considering it as control input for (2.3c) (linear dynamics), applications • A block for the computation of the reference pitch and roll angles {θ r , φ r } as virtual inputs for (2.3a) and (2.3b) (altitude control),

• A block for the computation of the torque τ ∈ R 3 for the stabilization of {φ, θ, ψ} to the references {φ r , θ r , ψ r } from (2.1c) and (2.1d).

Such hierarchical controller is based on the assumption that the controllers are tuned such that the closed-loop orientation dynamics are faster than the position dynamics.

Modelling of the Parrot AR.Drone 2.0

The Parrot AR.Drone 2.0 is a quadrotor system that has four control inputs. Each of these inputs leads to a variation in the four rotors angular speed to obtain a desired movement. These four movements can be described as

• Roll -a rotational movement around the longitudinal axis e x , leading to a translational movement along the transverse axis e y .

• Pitch -a rotational movement around the transverse axis e y , leading to a translational movement along the longitudinal axis e x .

• Yaw -a rotational movement around the vertical axis e z .

• Throttle -a translational movement along the vertical axis e z .

The Figure 2.2 illustrates these four movements.

The four inputs of the AR.Drone 2.0 represent reference values for the roll and pitch angles, the yaw rate and the vertical speed. In the following, they are denoted as φ r θ r v r z ψr . Each component of this vector has values in the interval [-1, 1] that can be interpreted as percentages of maximal configured values [ [START_REF] Bristeau | [END_REF]]. The control laws embedded in the Parrot AR.Drone 2.0 allow to regulate the roll and pitch angles, the yaw rate, and the vertical speed to these references. These control laws cannot be bypassed nor modified, and are to be accounted for in the modelling.

The decoupling of these four movements allows to design four separated control laws for the stabilization of the longitudinal position x, the transverse position y, the yaw angle ψ and the vertical position z with regards to the user chosen references {x r , y r , z r , ψ r }. The control laws are computed in a PC station and transmitted to the UAV by WiFi. The Figure 2.3 illustrates the communications between the User, the PC station, a Motion Capture that provides measurements of the system (position, speed and attitude) and the Parrot AR.Drone 2.0.

Here, we focus in particular on the horizontal dynamics. The internal regulation of the pitch and roll angles can be modelled as the first order system [START_REF] Engel | [END_REF]]

θ = -C θ (θ -θ r ) + δ θ , φ = -C φ (φ -φ r ) + δ φ ,
with C θ > 0 and C φ > 0 and where δ θ and δ φ represent perturbations accounting for external disturbances and errors in the model. Under the assumption that the vertical speed v z has small variations around zero, the magnitude of the force F in (2.3a) and (2.3b) can be considered constant.

Remark 2.1 Small variations of the vertical speed correspond to an horizontal motion on a plane z = z r .

From the above considerations, the horizontal dynamics of the Parrot AR.Drone applications 2.0 can be modelled by the following continuous-time linear systems

ξx = A cx ξ x + B cx u x + d x ,
(2.4a)

ξ x =   x v x θ   , A cx =    0 1 0 0 0 F m 0 0 -C θ    , B cx =   0 0 C θ   , d x =     0 f x m δ θ     , (2.4b) ξy = A cy ξ y + B cy u y + d y ,
(2.4c)

ξ y =   y v y φ   , A cy =    0 1 0 0 0 F m 0 0 -C φ    , B cy =   0 0 C φ   , d y =     0 f y m δ φ     , (2.4d)
where u x = θ r and u y = φ r are two of the AR.Drone 2.0 inputs.

Here, the horizontal dynamics are modelled as two 3 dimensional linear systems, assuming small roll, pitch and yaw angles. This modelling approach is used in [START_REF] Hernandez | [END_REF]] [Krajník 2011[START_REF] Engel | [END_REF]] [Santoso 2016[START_REF] Hernandez | [END_REF].

A different choice of model is considered in [START_REF] Armah | [END_REF]] and [START_REF] Santana | [END_REF]]. In these works, the user inputs {φ r , θ r } of the Parrot AR.Drone 2.0 are instead identified as proportional to respectively a lateral and a longitudinal acceleration,

C y φ r = F φ m , C x θ r = F θ m .
This choice of modelling consists in neglecting the roll and pitch angles dynamics, and leads to

ξx = A cx ξ x + B cx u x + d x ,
(2.5a)

ξ x = x v x , A cx = 0 1 0 0 , B c = 0 C x , d x =   0 f x m   , (2.5b) ξy = A cy ξ y + B cy u y + d y , (2.5c
)

ξ y = y v y , A cy = 0 1 0 0 , B c = 0 C y , d y =   0 f y m   , (2.5d)
Therefore, with u x = φ r and u y = θ r , two models of the horizontal dynamics of the Parrot AR.Drone 2.0 exist in the literature. These models are linear in the state and the user inputs. The selection of a model, its discretization and the identification of its parameters are discussed in depths in Chapter 3.

It is important to note at this point that several researches conclude in the existence of a delay in the communication of the input from the PC station to the AR.Drone. The value of this delay is time-varying, and ranges between 0.2s [START_REF] Hernandez | [END_REF]] to 0.4s [START_REF] Engel | [END_REF][START_REF] Armah | [END_REF].

A state of the art on the application of MPC scheme for the stabilization of quadrotors is presented in the next sections. The models discussed above are the basis of these MPC schemes. Before presenting the MPC schemes, we first discuss the advantages and drawbacks of the models when employed in a TBMPC.

The general nonlinear model (2.1) has a state dimension of 12, namely

ξ = x y z v x v y v z φ θ ψ ω x ω y ω z ,
and an input dimension of 4 (F, τ ). The main advantage of this model is that it has the least modelling errors (no small angle assumption, no error due to hierarchical controller structure). However, the nonlinearities and the high state dimension raises issues with regards to the complexity of the MPC optimization problem.

The design of a hierarchical controller allows to decouple the high dimension model into two submodels of reasonable dimension, respectively a state dimension of 6 (x, y, z, v x , v y , v z ) and 6 (φ, θ, ψ, ω x , ω y , ω z ) and an input dimension of 4 (F, φ r , θ r , ψ r ) and 3 (τ ). Moreover, assuming small angles, the position dynamics can be linearized and decoupled, as shown in (2.3). The MPC optimization problem benefits from direction decoupling, the low state and input dimension and the absence of nonlinearities in the position dynamics (quadratic programming). We will not detail the attitude control of the hierarchical controllers discussed below, as our study focuses on the control of the horizontal position dynamics.

The presented models of the Parrot AR.Drone 2.0 horizontal dynamics are a particular case of a hierarchical control structure, where an internal controller stabilizes the orientation dynamics and the directions are decoupled. The models have a small state (3 and 2 respectively) and input dimension (1), and the dynamics are linear. The reduced complexity of these models are well-suited for MPC schemes. The control law is computed on a PC station, which has more computational capacity than quadrotor platforms. However, the control law is transmited by WiFi to the quadrotor, which induces a delay in the communication of the input. The handling of this delay in the implementation of a Tube Based MPC scheme is discussed in Chapter 3.

We present in the following sections a state of the art on the application of Model Predictive Control strategies that account for the presence of disturbances for the control of quadrotors.

Simulation application of a robust MPC law

As a first step, we review some existing works on the application of a MPC law for the control of a quadrotor that do not provide experimental results. More particularly, we focus on the MPC schemes that account for the presence of external disturbances or modelling errors.

A hierarchical controller is proposed in [Raffo 2010], where an H ∞ controller and an MPC are designed respectively for the attitude and the position stabilization. applications

The structure of the hierarchical controller is similar to the one described in the previous section, with the ramarkable difference that the author does not consider the small angle assumption. The three directions x, y and z are considered decoupled, and their dynamics are given by

ẋ = v x , m vx = F (c(ψ)s(θ)c(φ) + s(ψ)s(φ)) + f x (2.6a) ẏ = v y , m vy = F (s(ψ)s(θ)c(φ) -c(ψ)s(φ)) + f y (2.6b) ż = v z , m vz = F c(θ)c(φ) + f z -mg (2.6c)
The author introduces the virtual inputs

u x = c(ψ)s(θ)c(φ) + s(ψ)s(φ), u y = s(ψ)s(θ)c(φ) -c(ψ)s(φ).
(2.7)

The MPC controllers use the disturbance-free counterpart of (2.6) with (2.7), yielding

ẋ = v x , m vx = F u x (2.8a) ẏ = v y , m vy = F u y (2.8b) ż = v z , m vz = F c(θ)c(φ) (2.8c)
A first Model Predictive Controller computes the desired value of F from the timevarying model (2.8c) with the current measurements of φ and θ. Then, two Model Predictive Controllers compute the desired value of u x and u y using the time-varying models (2.8a) and (2.8b) with the previously computed value of F . The value of u x and u y are used to compute the reference roll and pitch angles φ r and θ r from (2.7). This choice of using time-varying models, and not assuming small angles, leads to reduced modelling errors. In addition, the MPC scheme in [Raffo 2010] uses a model with augmented state which includes the integral of the position error in order to achieve a null steady-state error. We do not present this state augmentation here for clarity of presentation, but we note that this consideration can reduce significantly the impact of a persistent disturbance on the closed-loop system. However, the MPC schemes do not consider any state or input constraints in their implementation, which could lead to violation of physical constraints (obstacles, saturation of the actuators,...).

Two robust Model Predictive Controllers are proposed in [Köhler 2019] and [START_REF] Hu | [END_REF]] that are based on the full dimensional nonlinear model (2.1). The MPC schemes are developed using robustness considerations for the general class of nonlinear systems, and we discuss the principles in the following.

In [START_REF] Hu | [END_REF]], the author considers the following class of continuous-time nonlinear systems

ξ(t) = f (ξ(t), w(t)) + u(t),
subject to polytopic state constraints ξ(t) ∈ X and bounded input and disturbances

u(t) ∈ U, w(t) ∈ W. The set of convex sets of X is denoted C(R n ).
For a given state feedback control law µ ∈ X U and a given initial state ξ 0 = ξ(0), the collection of all possible state realizations at time T is denoted X(T, ξ 0 , µ). An MPC scheme that ensures constraints satisfaction over the horizon [0, T ] is given by inf

µ∈X U T 0 L(X(t, ξ, µ)) (2.9a) subject to X(t, ξ 0 , µ) ⊆ X , ∀t ∈ [0, T ], (2.9b)
where L : C(R n ) → R denotes a stage cost, and ξ 0 is the current state measurement.

In most cases, such a formulation of the receding optimization problem is intractable.

It can be rewritten using the concept of Robust Forward Invariant Tubes (RFIT), which relates to the concept of RCI sets. A set-valued function

X(t) : R → C(R n ) is a T -RFIT if there exists a feedback control law µ ∈ X U such that ∀t ∈ [0, T ], ξ 0 ∈ X(0) X(t, ξ 0 , µ) ⊆ X(t).
The closed-loop trajectory remains in the RFIT set X(t) regardless of the external disturbances. Note that every RFIT set is induced by at least one feedback law µ.

The author considers an ellipsoidal parametrization of RFIT sets as follows

X(t) = ξ ∈ R n | (ξ -q(t) )Q(t)(ξ -q(t)) ≤ 1 , (2.10)
where ∀t ∈ [0, 1], Q(t) ∈ R n×n is a definite positive matrix and q(t) ∈ R n . The choice of ellipsoidal sets, as opposed to polytopic sets, induces more conservatism with regards to the size of the RFIT set, but benefits from their lower complexity (n + (n + n 2 )/2 design parameters).

The author selects the T -RFIT ellipsoidal set that minimizes the cost function (2.9a). A sufficient condition on Q(t) and q(t) for X(t) in (2.10) to be a T -RFIT set is established. This sufficient condition consists in non-linear constraints on Q(t), q(t), and additional decision variables, whose detail presentation are eluded here. A tractable method for the handling of these constraints in the optimization problem is detailed in [START_REF] Hu | [END_REF]]. Due to the assumption and the structure of the solution of the optimization problem, the approach yields a sub-optimal T -RFIT set, principally related to the ellipsoidal parametrization and the sufficient invariance condition. The associated state feedback control law µ is deduced from the RFIT set, and is used at a higher sampling-rate to control the system.

The robust MPC scheme uses the solution of a nonconvex nonlinear optimization problem. In comparison to a nominal MPC scheme, the state dimension is increased by n + (n + n 2 )/2 (dimension of q(t) and Q(t)), and the input dimension by m + mn + 2 (dimension of the additional decision variables). For the quadrotor applications application, this equals respectively to 90 and 54. This is particularly problematic for gradient-based solvers, which rely on an initial guess (candidate solution to the optimization problem). Nevertheless, the subsequent optimization problems are neighboring, which can be exploited to compute the successive initial guesses.

In a similar fashion to the Tube Based MPC scheme in Section 1, a terminal constraint is added to the optimization problem to ensure recursive feasibility of the successive optimization problems. Hence, this controller includes strong theoretical robustness considerations with respect to performances, thus ensuring recursive constraints satisfaction as long as the nonlinear optimization problems can be solved. However, ensuring closed-loop stability for nonlinear MPC schemes is more complex, and is beyond the scope of the work presented in [START_REF] Hu | [END_REF]]. An extension that provides closed-loop stability is to be sought to improve the robustness of the MPC scheme.

A discrete-time non-linear model is considered in [Köhler 2019], given by

ξ + = g(ξ, u) + w(ξ, u, d w ), ξ ∈ R n , u ∈ R m , d w ∈ D ⊆ R q ,
where the model mismatch w is a function of the current state ξ and control input u, and a disturbance term d w ∈ R q . This modelling of the disturbances is of particular interst for quadrotor applications. Several disturbance sources depend on the drone state, such as the aerodynamic effects (flying close to obstacles, regions with wind gust, etc). The development considers joint state and input constraints (ξ, u) ∈ Ω where Ω ⊆ R n+m is compact. Here, we present the development for a polytopic set

Ω = (ξ, u) ∈ (R n × R m ) | G ξ u ≤ 1 p ,
where G ∈ R p×(n+m) . The nominal system is defined by ξ+ = g( ξ, ū).

An MPC approach, over a prediction horizon N , is developed where the decision variables are the nominal control inputs ū0 , ..., ūN-1 . The trajectory of the actual system differs from the closed-loop trajectory of the nominal system due to the presence of disturbances. Instead of an open loop prediction, with u k = ūk , the author introduces the control policy u k = µ(ξ k , ξk , ūk ), where µ : R n × Ω → R m . The function µ is a mapping to be designed, and plays a similar role to the local controller ν in the Tube Based MPC scheme in Section 1.5.

With the above control policy, the predicted trajectories of the nominal and uncertain systems are given by ξ0 = ξ, ξk+1 = g( ξk , ūk ), ∀k ∈ N N -1 .,

(2.11)

ξ 0 = ξ, ξ k+1 = g(ξ k , µ(ξ k , ξk , ūk )) + w(ξ k , µ(ξ k , ξk , ūk ), d w,k ), ∀k ∈ N N -1 . (2.12)
The author uses a function (2.14) to bound the error between the nominal and the uncertain state and inputs. Here, the tube of trajectories are defined as level sets of the function V . This function depends on the selection of a function µ, and has to satisfy a set of conditions which are not recalled explicitly but briefly discussed as a concept. These conditions allow to construct tightened constraints for the nominal system, such that if the closedloop trajectory of the nominal system satisfies the tightened constraints, than the closed-loop trajectory of the actual system satisfies the actual constraints.

V : (R n × Ω) → R + , (2.13) (ξ, ( ξ, ū)) → V (ξ, ξ, ū),
The constraint tightening technique requires the knowledge of an upper bound, further denoted c k > 0, of V (ξ k , ξk , ūk ) for all k ∈ N N -1 for all possible disturbance realizations d w,k . The following recursion is obtained from the conditions on the function V ,

c k+1 = ηc k + w( ξ, ū, c k ), k ∈ N N -1 ,
where 0 ≤ η < 1, and w is a function that depends on the choice made for V (., ., .).

The MPC scheme proposed in [Köhler 2019], as function of the current system state ξ, reads minimize ū0 ,...,ū N -1

N -1 k=0 l( ξk , ūk ) + l f ( ξN ) (2.15a) subject to ξ0 = ξ, c 0 = 0, (2.15b) ξk+1 = g( ξk , ūk ), k ∈ N N -1 , (2.15c) c k+1 = ηc k + w( ξ, ū, c k ), k ∈ N N -1 , (2.15d) G ξk ūk ≤ 1 -c k g, k ∈ N + N -1 , (2.15e) c k ≤ c, k ∈ N + N , (2.15f) ( ξN , c N ) ∈ X f . (2.15g)
The optimization problem (2.15) corresponds to the computational demand of a nominal MPC scheme with an augmented state of dimension n + 1 (dimension of ξ plus dimension of c ∈ R) and an additional nonlinear dynamic equation (2.15d). This state augmentation allows to avoid regions where larger uncertainties (large w( ξ, ū, c)) are encountered by adding the constraint (2.15f), where c is a design parameter. The tightened constraints depend linearly on this additional state (2.15e), through a vector g ∈ R p . The resulting closed-loop system is given by

ξ + = g(ξ, u * (ξ)) + w(ξ, u * (ξ), d w ),
where u * (ξ) is the first element of the optimal control input sequence ū0 of (2.15). The functions V (., ., .) and w(., ., .), the control law µ(.), the scalar η and the vector g are parameters of the overall design, and are computed offline. The proposed controller requires the online computation of the solution to the optimization applications problem (2.15) at each time-step, where ξ is chosen as the current state of the system ξ[k]. The terminal cost l f and the terminal constraint set X f are constructed to ensure recursive feasibility of the optimization problem, and the asymptotical stability of the closed-loop system. Their construction is not detailed here.

The main advantage of this robust MPC controller is that it provides strong theoretical properties, such as closed-loop stability and robust recursive constraints satisfaction in the presence of disturbances for the given model. The optimization problem avoids regions with high disturbances if the parameter c is well tuned. However, it requires to solve in real-time an optimization problem with additional nonlinear dynamics (2.15d), and the tuning of the design parameters discussed above is not obvious, since it involves design parameters whose impact on the closed-loop performances of the system is complex to estimate.

The controllers proposed in [START_REF] Hu | [END_REF]] and [Köhler 2019] are tested in simulation with the full dimensional model of the quadrotor dynamics (2.1). Instead of considering the additive force disturbance f in (2.1b), they consider an additive disturbance on the speed, yielding

ẋ = v x + 0.1w x , ẏ = v y + 0.3w y , ż = v z + 0.5w z ,
where w x w y w z 2 2 ≤ 1. A simulation example showcases the recursive constraint satisfaction of both controllers. The computation time of the controller in [START_REF] Hu | [END_REF]] averages 82ms, which is reasonable for quadrotor application. The computation time in [Köhler 2019] is not provided. However, the MPC scheme is less computationally demanding, given that the dimension of the augmented state is n + 1 in [Köhler 2019] compared to the n + (n + n 2 )/2 of [START_REF] Hu | [END_REF]]. This reduced computational complexity is at the expense of conservatism. These papers can be considered as the state of the art in terms of robust constraints handling with MPC for quadrotor stabilization, but do not provide experimental results. We chose to implement a robust MPC scheme as in Section 1.5 instead of one of these MPC schemes as it is less computationally demanding, the tuning is more intuitive, and it can lead to a reliable real-time implementation.

As a transition toward the experimental application of Model Predictive Control, [Adhau 2019] presents an Hardware In the Loop (HIL) benchmark of nonlinear programming problem (NLP) solvers. The author considers three case studies of nonlinear systems, among which a simplified version of (2.1) with 9 states and 4 inputs. A nominal nonlinear Model Predictive Controller is considered, with convex constraints on the input and a quadratic cost function.

The average computation time and memory requirement is provided for a range of NLP solvers (qpOASES, qpDUNES, HPMPC) and a range of embedded platforms (Atmel ARM, PYNQ FPGA, Rasberry Pi) in HIL simulations. The average computation time were satisfactory for the PYNQ FPGA and Rasberry Pi (< 4ms)

Experimental application of MPC for the robust control of quadrotors 63

with an horizon prediction of length 5. The proposed nonlinear MPC is not robust to external disturbances and does not consider terminal constraints. The robustification and stability enhancement of this MPC scheme, along with an increase of the horizon prediction, could lead to an prohibitive increase in computation time. Moreover, there is a risk of the solver not converging toward a feasible solution, in which case the nonlinear MPC cannot be computed. This HIL analysis reinforces our choice to consider a linear model and a linear MPC, which has guaranteed convergence with a reasonable computation time, as presented in Chapter 3.

Experimental application of MPC for the robust control of quadrotors

We propose here an overview of existing works on experimental applications of robust MPC laws for the stabilization of a quadrotor. These researches differ in particular with the choice of the quadrotor platform, and the modelling choices.

Robustness in the choice of the model

The MPC scheme presented in [Alexis 2012], [Bouffard 2012] and [START_REF] Abdolhosseini | An efficient model predictive control scheme for an unmanned quadrotor helicopter[END_REF]] uses linearized models of the quadrotor dynamics.

The platform used in [START_REF] Abdolhosseini | An efficient model predictive control scheme for an unmanned quadrotor helicopter[END_REF]] is the Qball-X4. The model uses the small angles assumption and assumes that the dynamics in the directions x, y and z are decoupled, and thus is similar to the one in (2.3). The main contribution is to carry out a set of simulation and experiments to determine the validity domain of these assumptions. This allows to determine an upper bound on the admissible values of θ and φ to validate the small angles assumption. These values are then used as additional constraints in the MPC scheme to ensure θ and φ remain within the validity domain. Note that these values are specific to the quadrotor used in their experiments, namely the Qball-X4, and the MPC scheme does not account for the presence of external disturbances in its design. The author states that an experiment with an MPC scheme that uses the general nonlinear model could not be carried out due to the limited computational capacity available on the platform. This proves the advantage of linearized models for quadrotor applications, as the reduced computational complexity is adapted to the limited embedded computational capacity of these platforms.

Piece-wise affine models of the attitude, horizontal and vertical dynamics are considered in [Alexis 2012]. The attitude model will not be detailed below, as we focus on the control of the position dynamics. The discretized models, with sampling applications time T e of the horizontal and vertical dynamics are given by

z v z + = A z v z + B(c(θ), c(φ))u z , A = 1 T e 0 1 , B(c(θ), c(φ)) =   0 T e m c(θ)c(φ)   (2.16a)     x v x y v y     + = A 0 2,2 0 2,2 A     x v x y v y     + B(u z ) 0 2,1 0 2,1 B(u z ) u x u y , B(u z ) =        0 0 T e m u z 0 0 0 0 T e m u z       
.

(2.16b)

These models are piece-wise affine as B(c(θ), c(φ)) in (2.16a) derives from the current measurements of the roll and pitch angles, while B(u z ) in (2.16b) depends on u z , the control input u z computed from (2.16a). Onboard sensors are used to estimate the current value of θ and φ. The value of u x and u y are used to compute the reference roll and pitch angles of the attitude controller. The control inputs u x , u y and u z are computed using a standard MPC scheme with quadratic cost and polytopic state and input constraints. The proposed cascaded MPC schemes were tested experimentally, using UPATcopter, a prototype quadrotor of the University of Patras. The sampling time T e is chosen as 0.03s, and several scenarii were considered (hovering, trajectory tracking, aggressive maneuvering, with and without presence of a directional wind gust). The experimental results showcase the good performances of the approach compared to a linearized model, as the PWA approach reduces the modelling errors. However, the computation of multiple MPCs over the family of PWA systems does not have theoretical properties with regards to the stability of the general nonlinear system, and the MPC scheme does not account for the presence of external disturbances.

The dynamics in the directions x, y and z are considered as decoupled in [Bouffard 2012]. We present below the model of the dynamics in the direction x, where the state is chosen as ξ = x v x θ ω x . It is assumed that the nominal dynamics are linear, and given by

ξ + = Aξ + Bu,
where u is the input of the quadrotor relative to the direction x. The matrices A and B are computed using experimental data and classical identification techniques. A learning-based model is used, based on the above nominal model, and given by

ξ + = (A + F (β))ξ + (B + H(β))u + ζ(β),
(2.17)

Experimental application of MPC for the robust control of quadrotors 65

where F, H and ζ are linear in the components of β ∈ R 12 . The parameter β is computed jointly with estimates of the state ξ using a variant of the Extended Kalman Filter. Hence, the value of the model parameters F (β) and H(β) are constantly updated, using the measured state and inputs. The principle of the computation of β will not be detailed here. A standard MPC scheme with a quadratic cost and polytopic state and input constraints is used to compute the input u k using the model (2.17) with the current estimation of the parameter β = β k .

Experimental results are provided, using the Ascending Technologies Pelican quadrotor platform and a motion capture system that provides an accurate measurement of the state. The experimental data provided illustrates the benefits of model learning when the quadrotor is commanded to hover near the ground. In this situation, a component of the parameter β changes to account for ground effect after 1s. With this change in β, the learning-based MPC is able to maintain the quadrotor at the desired altitude.

In these three researches, the use of standard MPC schemes fails to provide robust recursive constraints satisfaction in the presence of external disturbances and modelling errors. However, these disturbances and modelling errors are accounted for in their choice of modelling, either by estimating the validity conditions of modelling assumptions [START_REF] Abdolhosseini | An efficient model predictive control scheme for an unmanned quadrotor helicopter[END_REF]], by using piece-wise affine models [Alexis 2012], or by updating the model [Bouffard 2012]. These modelling choices reduce the impact of the disturbances and modelling errors on the closed-loop performances of the quadrotor.

Robustness in the control design

In [Alexis 2016], a linearized model is considered with decoupled longitudinal, lateral and altitude dynamics. The model of the longitudinal and lateral dynamics is composed of four states (position, speed, associated angle and angle rate), and one input. The models have been identified using the methods presented in [Tischler 2006] and [Ljung 1995]. To account for external disturbances affecting the system, a minmax MPC scheme is introduced, where the optimal control input sequence minimizes the worst-case scenario along the predicted trajectories initialized with the current state ξ. The metric of minimality used here is the Minimum Peak Performance Measure, given by

V N (ξ, u 0 , ..., u N -1 ) = max {|ξ k | ∞ , k = {1, 2, ..., N }} ,

State of the art on the application of robust MPC in quadrotor applications

where ξ 0 = ξ, and ξ k+1 = Aξ k +Bu k +w k . The optimization problem can be written as minimize

u 0 ,u 1 ,...,u N -1 maximize w 0 ,w 1 ,...,w N -1 V N (ξ, u 0 , ..., u N -1 ) subject to V N (ξ, u 0 , ..., u N -1 ) = max {|ξ k | ∞ , k = {1, 2, ..., N }} , u k ∈ U, ∀k ∈ N N , w k ∈ W, ∀k ∈ N N , ξ k ∈ X , ∀k ∈ N + N +1 , ξ k+1 = Aξ k + Bu k + w k , ∀k ∈ N + N +1 .
The main drawbacks of such robust MPC schemes are the conservatism induced by the open-loop handling of the disturbances, and the fact that the optimization problem is rather intractable, especially for high horizon prediction length (N ) and system dimension (n, m).

For tractability of the optimization problem, they consider a feedback prediction strategy. Feedback prediction is a method that encodes the knowledge that a min-max receding horizon method is used to reduce the conservatism by forcing a parametrization of the decision variable (here, the control sequence {u 0 , ..., u N -1 }). The feedback prediction proposed in [Alexis 2016] is

u k = v k + k-1 i=0 L(k, i)w i .
The control sequence is parametrized in the previous uncertainties (w 0 , ..., w k-1 for u k ), and the matrices L(k, i) ∈ R m×n describe how the control actions use the disturbance vector. Note that the v k , k = {0, ..., N -1} are the newly introduced decision variables. This parametrization of the decision variable reduces the degrees of freedom, and the tuning of the matrices L(k, i) is pivotal.

An obstacle avoidance strategy is proposed for polytopic obstacles by introducing hard non-convex state constraints in the optimization problem. For real-time computation, the authors use an explicit MPC approach. It consists in dividing the state-space in polyhedral subregions P (j), j = {1, ..., r}, where each region corresponds to a piece-wise affine control law of the form u = K(j)ξ + u(j), where u(j) ∈ R m and K(j) ∈ R m×n . This subregion division and these control laws are such that if ξ ∈ P (j), then u * 0 (ξ) = K(j)ξ + u(j). Hence, instead of solving an optimization problem the control action is computed by searching which subregion contains ξ, and applying the associated piecewise affine control law. Note that the partition in subregions and the associated affine laws are computed offline.

Compared to implicit MPC, explicit MPC potentially reduces the online computation time but increases extensively the memory requirements. Indeed, explicit MPC requires to store the partition of the state-space, along with the associated affine control laws, on the embedded device.

Experimental application of MPC for the robust control of quadrotors 67

This robust MPC is applied to two vehicles, the ASLquad and the UPAT-TTR in a set of scenario involving disturbances (wind) and obstacles. They provide experimental results which showcase the experimental constraints satisfaction, along with obstacle avoidance, in presence of external disturbances. However, the proposed MPC scheme does not have theoretical recursive feasibility nor stability properties (terminal set/decrease of the cost function). The lack of stability can be an issue given that the horizon length is chosen as N = 6 with a sampling time of approximately 0.1s (trajectory predicted over 0.6s), which point to the computational limitations.

The system considered in [START_REF] Kamel | [END_REF]] is an hexacopter. However, the proposed MPC controller handles actuator failure, and could be applied to a quadrotor. Nonlinear models of the position and orientation dynamics are considered. A cascaded control strategy is implemented, where the outer loop aims to control the vehicle position and the inner loop controls the attitude. The design of the outer loop relies on standard LQR techniques, whereas a nonlinear MPC scheme is considered for the inner loop. The cost function of the optimization problem is chosen as the difference between the predicted attitude matrices and a reference attitude matrix. Box shaped input and angular rate constraints are considered due to physical and sensor restrictions. The resulting optimization problem is highly non-linear due to the nonlinearities in the orientation dynamics. A Sequential Quadratic Programming (SQP) technique is used where the successive QPs are solved by active set method. The optimization problem, and the technique used to solve it, do not include recursive feasibility ingredients. Hence, in case of infeasibility, slack variables are introduced to relax all constraints.

Simulation and experimental results, using a FireFly hexacopter, in closed-loops with the proposed controller are provided. The sampling time is chosen as T e = 5ms and the horizon prediction length to N = 10 (time horizon of N T e = 50ms). The low values of these parameters compared to the other experimental applications discussed here can be explained by the fact that in this case the MPC is used for the control of the orientation dynamics, which are significantly faster than the position dynamics. The computation time during the simulations averages 1ms and does not exceed 4ms on a 2.8 GHz Intel Core i7. The computation is not provided for the experimental results, where the control law is computed on a MasterMind embedded 1.8GHz Core2Duo processor. A motion capture system is used for position and attitude measurements.

The proposed control strategy exhibits satisfactory experimental trajectory tracking performances. However, the MPC scheme is used for the control of the orientation dynamics, and thus is not capable of handling constraints on the position and velocity of the hexarotor. In addition, the MPC controller does not have stability properties nor recursive feasibility, and its implementation does not consider external disturbances. applications 2.4.3 Focus on the Parrot AR.Drone 2.0

We now present the MPC schemes that use the Parrot AR.Drone 2.0 quadrotor.

A key aspect in the design of robust MPC schemes is the identification of the disturbance bounds affecting the system. Several UAV applications use a fleet of several agents, and their interaction leads to additional source of disturbances. Namely, if two quadrotors cross horizontally at different altitudes, the lower one is affected by the downwash perturbation generated by the upper one. The handling of this particular phenomenon by the control law is studied in [Suyama 2017]. The 6 dimensional discrete-time linear model, as in (2.5), is used to model the horizontal and vertical dynamics of the Parrot AR.Drone 2.0. The state ξ is composed of the positions and velocities and the disturbance w is assumed to be matched, yielding

ξ + = Aξ + B(u + w), ξ = x v x y v y z v z ∈ R 6 , u ∈ R 3 , w ∈ R 3 .
The inverse model used to estimate the perturbation is obtained using the pseudoinverse matrix of B, denoted B + ,

w = B + ξ + -B + Aξ -u.
The disturbance is assumed to be a function of the relative position between the two quadrotors, denoted δx, δy, δz. A disturbance grip map is generated by applying a Gaussian filter to experimental data and by calculating weighted averages of these disturbances on the edges of the grid. This disturbance grid map gives the disturbance that is most likely to affect the system, which we denote w(δx i , δy j , δz k ), i ∈ {1, ..., N x }, i ∈ {1, ..., N y }, i ∈ {1, ..., N z }, where {x i , y j , z k } are the edges of the grid. For a given relative position δx, δy, δz, the author uses an interpolation based method for the computation of w(δx, δy, δz).

The Model Predictive Control law uses the knowledge of the disturbance map to modify the model at each sampling time. The resulting optimization problem for the current state ξ and current relative position δx δy δz reads minimize

u 0 ,u 1 ,...,u N -1 N k=0 (ξ k Qξ k + u k Ru k ) + ξ N P ξ N ,
(2.18a)

u k ∈ U, ∀k ∈ N N -1 , (2.18b) ξ 0 = ξ, (2.18c) ξ k+1 = Aξ k + B(u k + w k (δx, δy, δz)), ∀k ∈ N N -1 . (2.18d)
The proposed approach is implemented on a Parrot AR.Drone 2.0, and CVXGEN is used for the real-time computation of the optimization problem. The experimental data showcases the better performances of the proposed MPC (2.18) compared to a PID (with and without the disturbance map) and an SMC controller. This MPC scheme does not provide robust recursive constraints satisfaction nor recursive feasibility of the optimization problem due to several reasons. First, the 2.4. Experimental application of MPC for the robust control of quadrotors 69 optimization problem does not consider any terminal elements (such as terminal set), or implicit Lyapunov candidates related to the cost function. Second, the model in (2.18d) changes at each time-stamp, as it is a function of the current relative position δx, δy, δz. However, from a practical perspective, the procedure for the construction of a disturbance mapping is promising, in particular in a case of persistent disturbances leading to a steady-state error.

In [Dentler 2016b], the state vector consists of the position x, y, z in the inertial frame, the yaw angle ψ, and the velocities v B

x , v B y , v B z in the body-frame. This choice of modelling induces nonlinearities in the model. A nonlinear MPC scheme is proposed with quadratic input constraints and a quadratic cost function

V N (ξ) = N -1 k=0 (ξ k Qξ k + u k Ru k ).
(2.19)

For real time constraints, a Condensed Multiple-Shooting Continuation Generalized Minimal Residual Method (CMSCGMRES) solver is considered, as presented in [Shimizu 2009]. The sampling time is chosen as 0.1s and the prediction horizon as 10 steps (1s). The main advantage of this solver is that, with the proposed choice of solver parameters, the computation time remains below 1ms. This range of computation time is suitable for quadrotor applications, and showcases good closed-loop performances. However, this MPC scheme does not handle state constraints.

An extension of this work that takes into account the presence of state constraints is presented in [Dentler 2016a], where a scenario of obstacle avoidance is considered. To each obstacle O 1 , ..., O L is associated a sigmoid function sig 1 (ξ), ..., sig L (ξ) that measures the distance of the quadrotor to the obstacles. The sigmoid functions are penalty functions added to the cost function of the optimization problem (2.19) as an incentive for the UAV to avoid the area of the obstacles, yielding

V N (ξ) = V N (ξ) + l i=1 sig i (ξ).
The parameters of the sigmoids are tuned with respect to the obstacles shape and size. Constraints handled in the cost function via such penalty term are called soft constraints, as opposed to hard constraints, because they are allowed to be violated. The main advantage of soft constraints is that they do not alter the feasibility region of the optimization problem, and represent a reduced computational burden with respect to the hard constraints. However, the predicted trajectories are not guaranteed to avoid the obstacles.

Experimental results are provided to illustrate the performance of the obstacle avoidance technique. The maximal computation time of the optimization problem is 1.6ms, which is satisfactory for the control of a quadrotor. However, the controller fails to generate a solution that satisfies the input constraints in presence of applications disturbances. Indeed, the nonlinear MPC implemented is not robust with regards to the external disturbances faced in the experiment, and does not ensure robust recursive constraints satisfaction. [START_REF] Hernandez | [END_REF]] considers the 3 dimensional linear model of the AR.Drone 2.0 longitudinal and lateral dynamics (2.4). The disturbance d is modelled as a colored white noise resulting of a filtered uncorrelated white noise. A nominal MPC controller with a quadratic cost function, and which uses the knowledge of the filter to compute an estimation of the future disturbance realizations is proposed. A trajectory tracking experimental illustration is provided to compare the performances of the MPC controller and a PD controller. The MPC controller experiences less deviation to the desired trajectory. However, the MPC scheme does not handle state and control input constraints, and does not have theoretical stability properties in presence of disturbances.

Conclusion

An overview of the main models of the quadrotor dynamics was presented, with a special focus on the Parrot AR.Drone 2.0. These models are pivotal elements of the design and implementation of an MPC scheme. We provided a state of the art on the application of MPC schemes for the control of quadrotors that account for the presence of external disturbances or modelling errors, where we analyzed the disturbance handling techniques and the constraint satisfaction properties. We centered around experimental applications, and in particular on the MPC schemes implemented on the Parrot AR.Drone 2.0. The controllers discussed in this Chapter either did not ensure robust constraint satisfaction and stability of the closed-loop system, or did not provide experimental validation. In the majority of the works presented in this chapter, the model parameters are computed using classical system identification methods before the control design, and the identification step does not account for the specificity of the subsequent MPC design. In contrast, we use and provide experimental validation of a controller that ensures robust constraints satisfaction along with closed-loop stability, and we use the identification procedure as a design parameter of the control law.

In the next chapter, we implement and validate experimentally the Tube Based MPC scheme described in Chapter 1. This MPC scheme was chosen because the translational dynamics of the AR.Drone 2.0 can be linearized, and it can be synthesized with appropriate theoretical properties, among which robust recursive constraints satisfaction and closed-loop stability. Thus, we provide the first experimental application of a robust MPC scheme for quadrotor control that ensures these theoretical properties. In particular, we document all the successive design and implementation steps.

A similar robust MPC scheme has been implemented and validated experimentally in other fields of application, namely autonomous ground vehicle [Gao 2014] and mobile robots [González 2011]. The authors used classic LQR techniques for the design of the feedback gain K, and classic identification methods for the computation of the matrices A, B and disturbance set W. In both papers, the authors acknowledge the key role of the RCI set, and note that more sophisticated techniques for the identification of the model parameters and the design of the control law could lead to smaller RCI sets.

For this reason, we focus in particular on the identification of the model parameters and disturbance bounds. An additional contribution is the joint system identification and control design with the global goal of computing an RCI set that is as small as possible. The RCI set is a key element of the MPC scheme, and the system identification is used as an additional tool to compute an RCI set that is as small as possible.

Introduction

This chapter details the design and implementation of a Tube Based MPC law for the stabilization of the horizontal dynamics of the Parrot AR.Drone 2.0 quadrotor. As a first step, we choose a model of the horizontal dynamics among those presented in the preceding chapter. The principles of the Tube Based MPC scheme were recalled in Chapter 1. The main challenges are the identification of the model parameters, especially disturbance set characterization such that it contains all possible disturbance realizations encountered during the flight, and the tuning of the local controller to design an RPI set that is as small as possible (according to appropriate measure). Here, the control law is not the sole design parameter of the RCI set. The local controller is tuned experimentally in closed-loop within a range of stabilizing linear control laws. Practically, it is chosen as the linear control law that exhibits the most satisfying closed-loop performances, in terms of deviation around the equilibrium. We propose an identification method using experimental data that seeks to compute the model parameters leading the the smallest mRPI set for the chosen local controller.

Our experimental scenario requires the stabilization around successive waypoints in the presence of hard state and input constraints. A method for the real-time computation of the terminal set at each change of reference is proposed. The Tube Based MPC control law is implemented and experimentally validated in flight. The results of the validation flights are presented, illustrating the properties of robust recursive state and control input constraints satisfaction, and closed-loop stability.

The horizontal dynamics modelling is presented in Section 3.2. The handling of delay in the implementation of the Tube Based MPC is discussed in Section 3.3. The experimental tuning of the local controller is presented in Section 3.4. A novel approach for system identification adapted to the computation of optimized invariant sets is proposed in Section 3.5. The remaining parameters of the control law, and details of its implementation are introduced in Section 3.6. The validation flight of the Tube Based MPC law is shown in Section 3.7.

Modelling of the horizontal dynamics

We will concentrate on the Parrot AR.Drone 2.0 quadrotor, used for the implementation of a Tube Based MPC scheme. In particular, we are interested in the control of the horizontal dynamics.

Two continuous-time models of the horizontal dynamics were presented in Chapter 2. In the following, we retain the model (2.4), where the state comprises of the position, speed, and associated Euler angle, because the model (2.5) does not account for the dynamics of the roll and pitch angles. Recall that the continuous-time model is given by

ξx = A cx ξ x + B cx u x + d x ,
(3.1a)

ξ x =   x v x θ   , A cx =    0 1 0 0 0 F m 0 0 -C θ    , B cx =   0 0 C θ   , d x =     0 f x m δ θ     , ξy = A cy ξ y + B cy u y + d y ,
(3.1b)

ξ y =   y v y φ   , A cy =    0 1 0 0 0 F m 0 0 -C φ    , B cy =   0 0 C φ   , d y =     0 f y m δ φ     ,
where u x = θ r and u y = φ r are two of the AR.Drone 2.0 inputs.

The parameter F in the above equations is constant if the vertical position z is constant. Moreover, these equations are decoupled provided the yaw angle ψ remains constant. For these reasons, we have implemented PID controllers for the control of the position z and yaw angle ψ around references, chosen respectively as 1.5m and 0 • . The variations of z and ψ around their respective references lead to modelling errors. These modelling errors are considered as a source of additive disturbances in the identification of the disturbance bounds.

Model discretization

MPC in genera, and Tube Based MPC in particular, is a discrete-time strategy as presented in Section 1.5. Thus, we discretize (3.1) to obtain

ξ x [k + 1] = A x ξ x [k] + B x u x [k] + w x [k],
(3.2)

ξ y [k + 1] = A y ξ y [k] + B y u y [k] + w y [k], (3.3)
where

ξ x [k] = ξ x (kT e ) is the state, u x [k] = u x (kT e ) is the control input, w x [k]
is the additive disturbance, T e > 0 is the sampling period and k ∈ N is the discrete time index. The sampling time is to be chosen as small as possible to limit the discretization error, while being larger than the online computation time of the control law. We chose a sampling time of T e = 50ms.

Thanks to the symmetry of the quadrotor, it is assumed that the dynamics in the directions x and y are governed by the same model. For clarity of presentation, the theoretical developments consider the following linear discrete-time system

ξ k+1 = Aξ k + Bu k + w k .
(3.4)

The experimental results will be presented for both directions using the notation in (3.2) and (3.3). stabilization of quadrotor horizontal dynamics

For model indentification purposes, we impose the following structure to the matrices A and B to obtain a model that preserves the physical continuous-time structure of the model,

A =   1 a 12 a 13 0 a 22 a 23 0 0 a 33   ∈ R 3×3 + , B =   b 1 b 2 b 3   ∈ R 3×1 + . (3.5)
It is worth noticing that the supradiagonal elements of the matrices are chosen nonnegative to preserve the physical structure of the model. For instance, a 12 > 0 as a positive velocity at time [k] leads to an increase in position. The elements a 22 , a 33 are not chosen unitary to account for aerodynamic forces (e.g. drag).

The model identification problem studied below consists of obtaining the parameters of the linear difference equation, A and B, and the bounds on the additive disturbances, W, using experimental data.

Disturbance set characterization from experimental data

We perform identification flights where the states and control inputs are measured over a time window of L + 1 sampling instants,

{ξ m k , u m k } k∈N L+1 . (3.6)
For a given set of experimental data, and a given pair of matrices (A, B) ∈

R 3×3 + × R 3×1 +
, the disturbance realizations are defined as the mismatch between the one-step prediction of the model and the measured system state and control input,

w m k = ξ m k+1 -Aξ m k -Bu m k , ∀k ∈ N L . (3.7)
Following Assumption 1.1, we look for a convex set of disturbances in view of the implementation of a Tube Based MPC scheme as presented in Section 1.5. The simplest possible convex set we can obtain from experimental data is the convexhull of the disturbance realizations, namely

W = Co {w m k } k∈N L = Co ξ m k+1 -Aξ m k -Bu m k k∈N L . (3.8)
Recall that the disturbances realizations encountered in flight with the Tube Based MPC law have to be contained in the disturbance set W used for its design to benefit from its theoretical properties (robust recursive constraints satisfaction and closed-loop stability).

Remark 3.1 For preimposed matrices A and B, a partial order on the disturbance set W leads to a partial order on the resulting RPI set Z.

For this reason, the conditions have to be identical during the identification flights and the validation flights, to not face new sources of disturbances that were not accounted for (wind gust, flying close to obstacles, modelling errors).

A source of modelling errors is the presence of delay in the transmission of the control input from the PC station to the quadrotor platform.

Delay in the control

The WiFi communication introduces a time-varying delay in the transmission of the input {θ r , φ r } = {u x , u y }, whose magnitude is difficult to track. In the literature, it ranges between 0.2 [START_REF] Hernandez | [END_REF]] and 0.4 ms [START_REF] Engel | [END_REF][START_REF] Armah | [END_REF]]. We propose below two modelling choices to account for the presence of delay in the transmission of the input.

Consider a general discrete-time linear system subject to a delay of r ∈ N steps in the control and additive bounded disturbances,

ξ k+1 = Aξ k + Bu k-r + w a,k , w a,k ∈ W a , ξ ∈ R n , u ∈ R m , k ∈ N.
(3.9)

An augmented state space representation of the above system is given by the n+rmdimensional linear discrete-time system

    ξ k+1 u k ... u k-r+1     = A 0 n,m(r-1) B 0 mr,n J m,r     ξ k u k-1 ... u k-r     +   0 n v k 0 m(r-1)   + w a,k 0 mr , (3.10)
where v k = u k is the control action computed at time k, and

J m,r =        0 m,m 0 m,m ... 0 m,m 0 m,m I m,m 0 m,m ... 0 m,m 0 m,m 0 m,m I m,m ... 0 m,m 0 m,m ... 0 m,m 0 m,m ... I m,m 0 m,m        ∈ R rm×rm , (m, r) ∈ N 2 .
With this model, the state dimension is augmented by rm. This state augmentation increases the dimension of the Tube Based MPC optimization problems (1.30) and (1.36), which can be problematic for real-time constraints. Moreover, the relation between the complexity of the mRPI approximations and the state dimension was discussed in Section 1.4.1.3. It is worth to be mentioned that the value of r is time-varying and ranges between r = 4 [START_REF] Hernandez | [END_REF]] to r = 8 [START_REF] Engel | [END_REF][START_REF] Armah | [END_REF]] for our quadrotor application with the sampling time T e = 50ms.

Another modelling choice consists of considering the delay in the control as an additional source of additive disturbance. Indeed, with u k-r = u k + wd,k in (3.9), stabilization of quadrotor horizontal dynamics we obtain

ξ k+1 = Aξ k + Bu k + w k , (3.11a
)

w k = w d,k + w a,k , w d,k = -B(u k -u k-r ). (3.11b)
Here, the model considers delay in the control as an additional source of disturbance in w d,k , and does not require a state augmentation. Note that this disturbance is proportional to the term u k -u k-r , and thus depend on the choice for the control law.

Here, we choose this second model that does not increase the state dimension to implement the Tube Based MPC scheme. Hence, the quadrotor horizontal dynamics are modelled as in (3.11), and the delay in the transmission of the input is considered as an additional source of disturbances.

However, this choice of model raises two main issues for the implementation of the Tube Based MPC scheme, namely

• a local controller that stabilizes the system (3.11) is not guaranteed to stabilize the delayed input system (3.9).

• we have to bound the disturbance term w k , which is a function of

w d,k = B(u k -u k-r ), in a set W.
These topics are discussed in the following section.

3.3 Input delay handling in the implementation of the Tube Based MPC

Simulation example

We give a simulation example to illustrate the impact of time-delay in the transmission of the input on the design of a local controller and the characterization of the disturbance bounds. Consider the discrete-time linear system (3.9) with n = 2, m = 1 and a delay value of r = 1,

x[k + 1] = Ax[k] + Bu[k -1] + w a [k],
(3.12)

A = 1 0.1 0 1 , B = 0.005 0.1 .
We assume that the disturbance affecting the system satisfies w a

[k] ∈ W a , ∀k ∈ N.
The model with a disturbance term due to the delay in the control, as in (3.11), is

x[k + 1] = Ax[k] + Bu[k] + w[k],
(3.13a) For a given feedback control law ν : R 2 → R, the closed-loop system is given by

w[k] = w a [k] + w d [k], w d [k] = -B(u[k] -u[k -1]). ( 3
x[k + 1] = Ax[k] + Bν(x[k]) + w[k], (3.14a
)

w[k] = w a [k] + w d [k], w d [k] = -B(ν(x[k]) -ν(x[k -1])). (3.14b)
Let {x[k]} k∈N be a trajectory of (3.14). The disturbance realizations {w[k]} are the one-step prediction error, given by

w[k] = x[k + 1] -Ax[k] -Bν(x[k]).
And we define the disturbance set W as the convex-hull of these disturbance realizations as in (3.8). This set depends on

• The initial condition x[0].
• The exogenous disturbance realizations {w a [k]} k∈N .

• The control law ν(.).

An illustration of this dependency is studied in the following. We compute closedloop trajectories over an horizon of 100 steps for a range of control laws and initial conditions, and we compare the resulting disturbance sets W. All the simulations are run with the same exogenous disturbance sequence {w a [k]} k∈{0,...,99} . Figure 3.1 shows the set W a and the exogenous disturbance realizations w a [k] used for the simulation.

We arbitrarily choose three linear state feedback control laws

ν i (x) = K i x, i = {1, 2, 3}, in (3.14) with K 1 = -4 -2 , K 2 = -3 -6 , K 3 = -10 -10 .
The matrices A + BK i , i = {1, 2, 3} are Schur. The initial conditions are chosen as

x[0] = x j [0], j = {1, 2, 3}
, where Remark 3.2 In a scenario of stabilization at successive equilibrium points (i.e. waypoint navigation for the quadrotor), the initial condition corresponds to the difference between the current state of the system and the next equilibrium state at the instant of waypoint switch.

x 1 [0] = 1 0 , x 2 [0] = -1 0 , x 3 [0] = 5
We present in Figure 3.2 the state space trajectories of the closed-loop system with the above control laws and the initial condition x 1 [0] = 1 0 . The control law ν 3 does not stabilize the system (3.12). This illustrates that the presence of delay can produce instability, thus underlying the importance of a degree of robustness w.r.t the delay for the stabilizing controller. The closed-loop trajectories with the control laws ν 1 and ν 2 converge toward a neighborhood of the origin. We denote W i,j , i ∈ {1, 2, 3}, j ∈ {1, 2, 3} the disturbance sets obtained with the control law ν i and the initial condition x j [0]. Figure 3.3 shows a state space representation of the disturbance sets.

By comparing the size of the disturbance sets obtained with the initial conditions

x 1 [0] and x 3 [0] (compare figures on the top and on the bottom), we see that the size of the disturbance set increases with the magnitude of the initial condition. The Figure 3.3 illustrates that, for the same initial condition, the disturbance set depends on the choice of the control law. In particular, the disturbance set associated to the control law ν 3 (x) are sensibly larger (the scales are not the same for all plots). This later case is illustrative on the impact of additive disturbances as long as the identified disturbance set is related to a divergent trajectory. This simulation example highlights the sensibility of the model (3.11), in which delay is considered as an additional disturbance term. Indeed, the disturbance realizations, and thus the disturbance set W identification, depend on the choice for the control law, initial condition, and the exogenous disturbance realizations.

Experimental data generation for system identification and experimental design of a Tube Based MPC law

In control engineering, the system identification step usually precedes the control design. For instance, we used the model to tune the local controller in the Tube Based MPC simulation example in Section 1.6. This is also the usual approach in Design and Flight Experiments of a Tube Based MPC for the stabilization of quadrotor horizontal dynamics the researches that provide experimental application of Tube Based MPC [Gao 2014, González 2011], and in studies that focus on the computation of optimized invariant sets, where the model and disturbance set are fixed and the local controller is the only design parameter. A notable exception is [Chen 2018], where the model parameters are also used as design parameters in the computation of optimized invariant sets.

Here, the model parameters A and B in (3.4) and the disturbance set W are unknown. We recall the disturbance set W must contain the disturbance realizations that will be encountered by the closed-loop system with the Tube Based MPC law that we want to implement. That is, the experimental data used for the identification of (A, B, W) must contain experimental data from flights with conditions (initial conditions, distance between subsequent waypoints, and external disturbances) similar to the future Tube Based MPC law. Experimental data with a distinct set of initial conditions will lead to disturbance realizations that might not occur with the Tube Based MPC law, and thus lead to conservatism representations of the disturbance set W as illustrated in the previous section. A conservative disturbance bound will be in direct relationship with the conservatism of the Tube Based MPC law.

To conclude, the generation of experimental data for the identification of the model parameters requires the knowledge of the Tube Based MPC law, which itself requires to have identified the model parameters. However, we note that the Tube Based MPC law becomes the local controller once the nominal system has converged to the equilibrium. Hence, it can be beneficial to first tune the local controller, and use the experimental data generated to identify the model. The choice of tuning the local controller prior to identifying the model parameters is further motivated by the presence of time-varying delay (see the control law ν 3 in the simulation example in the previous section). Finally, the disturbance mitigation of the Tube Based MPC depends on the local controller. Here, we choose experimentally the local controller that leads to the least deviation around the equilibrium in stationary flights (smallest experimental invariant set).

We propose the following approach, which comprises of six steps, for the tuning of the Tube Based MPC law.

Proposed practical system identification and Tube Based MPC design procedure

• (1 -Calibration flights) Perform flights with a collection of linear local controllers ν i (ξ) = K i ξ, K i ∈ R 3×1 , i ∈ N.
• (2 -Local controller tuning) Choose the control gain K = K i that leads to the smallest deviation in stationary flight.

• (3 -System identification) Identify a model and disturbance set using the experimental data generated with the chosen local controller K.

• (4 -Tube Based MPC implementation) Implement a Tube Based MPC law with the identified system and disturbance set.

• (5 -Identification flights) Perform identification flights with the Tube Based MPC law.

• (6 -System identification refinement) Identify a model and disturbance set using the experimental data generated with the Tube Based MPC law.

The steps 4, 5 and 6 are to be repeated until an implemented Tube Based MPC law from step 6 is validated in flight. The validation requires that the disturbance realizations {w m k } k∈N are contained in the theoretical disturbance set W used for the implementation of the Tube Based MPC law.

Contrarily to classical approaches, we tune experimentally the local controller and we use the model parameters A and B, and the associated disturbance set W, as design parameters of the invariant set Z. The experimental tuning of the local controller is presented in the following section.

Experimental tuning of the local controller

A key parameter of the Tube Based MPC is the local controller ν(.). As discussed and illustrated in Section 1.5 the Tube Based MPC law is equivalent to the local controller once the nominal system has converged toward the reference. This motivates the selection of the unconstrained linear feedback gain K with regards to the closed-loop performances of the actual system near equilibrium points.

The tuning is made by performing calibration flights with a collection of p ∈ N+ state feedback gains K i ∈ R 1×3 , i ∈ N p . The closed-loop performances are compared and we choose the gain K that led to the smallest variations around references. These variations represent the effect of the disturbances on the closed-loop system, defining an experimental invariant set. Indeed, the system has remained in this set despite the presence of disturbances.

We chose arbitrarily to retain the feedback gain that best mitigates the impact of the disturbances on the position, which represent the ultimate safety constraint. More elaborate criterion are to be sought in case the speed or other limitations are to privileged.

The Figures 3.4 illustrate the time-evolution of the position in the model-free experiments with the linear control laws ν i (ξ) = K i ξ, i = {1, 2, 3, 4}, whose values are presented in the following table. The current position reference x r is indicated in red. We retain the gains that stabilize the real system based on simple tracking error, that is K 1 , K 3 and K 4 . We select ν(ξ) = K 4 ξ = -0.4 -0.5 -1 ξ for the local controller thanks to the verified experimental performances of the closed-loop system, as it exhibits the smallest position variations around the successive references. The main advantage of this experimental tuning of K 4 is that it guarantees good robustness performance of the Tube Based MPC law, recalling that the Tube Based MPC becomes the local controller once the nominal system has reached the equilibrium.

Our goal being characterization of the RPI set, the limitation of this approach is that the local controller cannot be used as design parameter and needs to pass by the choice of the model parameters and disturbance characterization before obtaining an expression of the RPI set Z. We present a novel system identification technique that uses the matrices A and B as design parameters of Z in the next section.

Closed-Loop System Identification and Invariant Set Design

The identification of the model (3.4) consists in finding the LTI system parameters, i.e. the matrices A and B, and the bounds of the additive disturbances, W, from experimental data. These two parts of the model identification can be mixed up in the global goal of obtaining the smallest possible invariant set, in view of its use in the Tube Based MPC.

The system identification methods presented below aim at finding the model (A, B, W) such that the associated mRPI set, given by

Z ∞ (A, B, W) = ∞ i=0 (A + BK) i W (3.15)
minimizes a criterion as in (1.10),

|HZ| p min ∈ R + | HZ ⊆ B h p ( ) , (3.16)
for a given feedback gain K ∈ R 3×1 , matrix H ∈ R h×n and integer p ∈ N+ . We recall that the matrices A and B have to satisfy the particular structural constraints (3.5) while the disturbance set W is given by (3.8) as the convex hull of generators.

We draw attention that our set perspective on the identification assumes that a potentially inaccurate model, a large disturbance set W, associated to a small mRPI set is suitable. For the purpose of Tube-Base MPC, this is indeed preferable to an accurate model, with a reduced disturbance set W, which is associated to a large mRPI set. With these considerations, we can formulate the problem as follows. The corresponding optimization problem is given by minimize

A,B |HZ ∞ (A, B, W)| p , (3.17a) subject to A =   1 a 12 a 13 0 a 22 a 23 0 0 a 33   , B =   b 1 b 2 b 3   , (3.17b) a 12 , a 13 , a 22 , a 23 , a 33 , b 1 , b 2 , b 3 ≥ 0, (3.17c) w m k = ξ m k+1 -Aξ m k -Bu m k , k ∈ N L , (3.17d) W = Co ({w m k } k∈N L ) , (3.17e) Z ∞ (A, B, W) = ∞ i=0 (A + BK) i W. (3.17f)
Remark 3.3 In section 1.4.1.3, the mRPI is denoted Z ∞ (K) because A, B and W are fixed and K is the design parameter. Here, the design parameters are A, B and W, while K is fixed. Thus, the mRPI is denoted Z ∞ (A, B, W). Problem 3.1 relates to Problem 1.1 for the design of (H, p)-mRCI sets, where A, B, W are given and the control law is the optimization parameter. Instead, here the control law is fixed and linear, and we search for the model A, B, W yielding an optimal mRCI set. stabilization of quadrotor horizontal dynamics Problem 3.1 cannot be solved directly because, in general, an explicit characterization of the sets Z ∞ (A, B, W) does not exist. We propose a method for the computation of candidate solutions using optimization-based techniques.

Computation of candidate solution

The mRPI set is an infinite Minkowski Sum of the set sequence {(A + BK) i W} i∈N . These sets are the product of the successive powers of the matrix (A + BK) and the disturbance set W. Here, both the successive powers and the disturbance set are function of the matrices A and B. The optimization of the infinite sum of these products is intractable.

We compute candidate solutions by minimizing a norm of the disturbance sequence, to have a set W as small as possible, while using the constraints to asymptotically bound the norm of the closed-loop matrix A+BK, relying on the inequality

|Z ∞ (A, B, W)| p ≤ ∞ i=0 |A + BK| i p |W| p .
A classical system identification method is based on a Least Mean Square approach, which consists in minimizing the square norm of the disturbance realizations signal {w m k } k∈N L , which is quadratic in the elements of A and B.

We use the Gelfand's formula [Gelfand 1941] to have an asymptotic equivalent of the p norm of the matrix A + BK using the spectral radius as follows,

|(A + BK) i | 1/i p → i→∞ ρ(A + BK).
We bound the spectral radius of the matrix A + BK by a chosen scalar ρ ∈ [0, 1[.

With these considerations, we propose to compute the matrices (A, B) that are the solution of the following optimization problem minimize

A,B L k=1 |w m k | 2 , (3.18a) subject to w m k = ξ m k+1 -Aξ m k -Bu m k , (3.18b) A =   1 a 12 a 13 0 a 22 a 23 0 0 a 33   , B =   b 1 b 2 b 3   , (3.18c) a 12 , a 13 , a 22 , a 23 , a 33 , b 1 , b 2 , b 3 ≥ 0, (3.18d) ρ(A + BK) ≤ ρ, (3.18e)
where ρ is a scalar parameter to be tuned over the interval [0, 1[, and L is the dimension of the time window. For the tuning of ρ, we perform a grid search, we compute (A(ρ), B(ρ)) as solutions of (3.18) and compare the size of the resulting mRPI sets Z ∞ (A(ρ), B(ρ), W(ρ)). This approach is computationally demanding, as the optimization problem is not convex due to the constraint (3.18e), and it requires to compute outer approximations of the mRPI sets.

To reduce the computational complexity, we bound the eigenvalues of the matrix A instead, using its upper-triangular structure. Note that for a matrix A as in (3.18c), we have λ(A) = {1, a 22 , a 33 }, and ρ(A) ≥ 1. We choose to bound the eigenvalues of A, namely the parameters a 22 and a 33 . The resulting optimization problem reads

(A(β), B(β)) = arg minimize A,B L k=1 |w m k | 2 , (3.19a) subject to w m k = ξ m k+1 -Aξ m k -Bu m k , (3.19b) A =   1 a 12 a 13 0 a 22 a 23 0 0 a 33   , B =   b 1 b 2 b 3   , (3.19c) a 12 , a 13 , a 23 , b 1 , b 2 , b 3 ≥ 0, (3.19d) 0 ≤ a 22 ≤ β 1 , 0 ≤ a 33 ≤ β 2 . (3.19e)
where the bounds β = β 1 β 2 is a vector of parameters that limits the eigenvalues of A. Note that (3.19) is an unconstrained Least Mean Square identification if β is large enough.

The solution of (3.19) is thus parametrized with β ∈ R 2 . For a given β, we compute A(β) and B(β) as in (3.19), then the disturbance set W(β) in (3.8), and eventually we define the mRPI set Z ∞ (A(β), B(β), W(β)) using (3.15). For convenience, we denote the corresponding mRPI set Z ∞ (β). We perform a search on the space of parameters β to compare the value of |HZ ∞ (K(β))| p . The comparison uses the computation of ( , p)-outer approximations of the mRPI sets, using methods presented in Section 1.4.1.3, and we denote the partial sum

Z k (β) = k-1 i=0 (A(β)+B(β)K) i W(β).
The approach is recalled in Algorithm 3.1, where we choose q values (grid search) for the parameter β, denoted β (j)

j∈N + q .
We present in the following the matrices, disturbance sets and invariant sets obtained for β = β (j) , j = {1, 2, 3, 4} where

β (1) = 2 2 , β (2) = 0.97 2 , β (3) = 2 0.94
, β (4) = 0.97 0.94 .

The resolution of the optimization problem (3.19) with the above parameters β (j) leads to the matrices A(β (j) ) and B(β (j) ) presented in the following table. stabilization of quadrotor horizontal dynamics Algorithm 3.1: Constrained Least Mean Square Identification .5 shows the disturbance set W(j) and outer approximations of the RPI sets Z(j). These outer approximations are computed using the method Scaling of the partial sums, initially proposed in [Rakovic 2005a]. From these figures we see that the smallest RPI set is obtained with β (4) . In particular, the set Z( 4) is smaller than the other RPI sets with regards to the partial order induced by set inclusion, and consequently with regards to the total preorder induced by the criterion |HZ| p for any choice of matrices H ∈ R h×n and integer p ∈ N+ . In comparison, an unconstrained Least Mean Square (that is, β = β (1) ) leads to an RPI set Z(1) whose volume is 16 time larger than Z(4).

Inputs: {ξ m k } k∈N L+1 , {u m k } k∈N L+1 , {β (j) } j∈N + q , H ∈ R h×n , p ∈ N+ , > 0 Outputs: A, B, W, Z 1 Set V alue = ∞ and j ← 0; 2 repeat 3 Set j ← j + 1; 4 Set β ← β (j) ;
8 if |H Z| p < V alue then 9 Set A ← Ã, B ← B, W ← W, Z ← Z, V alue ← |H Z| p ; 10 end 11 until j = q; j β (j) A(β (j) ) B(β (j) ) 1 2 2   1
We arbitrary chose to minimize the square norm of the disturbance realizations in (3.19). Other criterion could be considered, for instance other p-norms of the

|W| p = max ({|w m k | p } k∈N L ) .
The optimization problem remains LP/QP for p = 1, 2, ∞ with these choice of norms.

In the following section we propose an iterative procedure for the system identification that refines a pre-existing identification to further reduce the volume of the associated mRPI set.

Iterative refinement of the identification

Let us consider that we have performed an identification of the system (3.4) that led to the matrices A (0) , B (0) and the disturbance set W (0) . The idea is to find a pair of matrices (A, B) such that the associated disturbance realizations (3.7) satisfy

w m k = ξ m k+1 -Aξ m k -Bu m k ∈ γW (0) , ∀k ∈ N L , (3.20)
where 0 ≤ γ < 1. The disturbance set W associated to such matrices satisfies 0) , G (0) ∈ R l 0 ×n , the inclusion in (3.20) can be rewritten as a set of linear inequalities,

W ⊆ γW (0) ⊆ W (0) . For a polytopic set W (0) = w ∈ R n | G (0) w ≤ g (
G (0) w m k ≤ γg (0) , ∀k ∈ N L . (3.21)
To allow for more degrees of freedom, we consider a vectorial scaling factor γ = γ 1 ... γ l 0 ∈ R l 0 in (3.21), leading to

G (0) w m k ≤ diag(γ)g (0) , ∀k ∈ N L , (3.22a) 0 ≤ γ i ≤ 1, ∀i ∈ N l 0 . (3.22b)
If the pair (A, B) is such that the disturbance realizations w m k , k ∈ N L satisfy (3.22a) with γ ∈ R l 0 as in (3.22b), then the associated disturbance set satisfies W ⊆ W (0) .

Among the matrices (A, B) satisfying (3.22), we chose the pair that minimizes a chosen norm of the vectorial scaling factor γ. These considerations lead to the following optimization problem

(A (1) , B (1) ) = arg minimize A,B |γ| p , (3.23a) subject to G (0) w m k ≤ diag(γ)g (0) , ∀k ∈ N L , (3.23b) A =   1 a 12 a 13 0 a 22 a 23 0 0 a 33   , B =   b 1 b 2 b 3   , (3.23c) a 12 , a 22 , a 13 , a 23 , a 33 , b 1 , b 2 , b 3 ≥ 0, (3.23d) 0 ≤ γ i ≤ 1, ∀i ∈ N l 0 . (3.23e)
Note that this optimization problem is an LP/QP for p = 1, 2, ∞, and p = ∞ relates to a scalar scaling vector γ ∈ R. Moreover, the optimization problem has a non-empty feasible region, given that (A (0) , B (0) ) satisfy the constraints. We denote W (1) the disturbance set associated to A (1) and B (1) , and Z

(1) 1) the associated mRPI set.

∞ = ∞ i=0 (A (1) + B (1) K) i W (
With this method, we use an initial identification of the system and compute a novel identification whose disturbance set is included in the scaled initial disturbance set. The optimization problem seeks to minimize a norm of the vectorial scaling factor. This procedure aims at obtaining (A, B, W) whose associated mRPI is smaller than the initial one.

This procedure can be iterated to define a (theoretically) infinite sequence of matrices {A (i) , B (i) )} i∈N , disturbance sets {W (i) } i∈N and mRPI sets {Z

(i) ∞ } i∈N .
It is worth noting that the sequence of disturbance sets is decreasing with regards to the partial order induced by set inclusion, as we have

W (i+1) ⊆ W (i) , ∀i ∈ N.
However, the set inclusion W (i+1) ⊆ W (i) does not guarantee Z (i+1) ⊆ Z (i) . As a consequence, the sequence of mRPI sets is not guaranteed to be decreasing with regards to the matrx-norm criterion (3.16).

We compare ( , p) outer-approximations of the mRPI sets Z (i)

∞ , using the first method presented in Section 1.4.1.3 and proposed in [Rakovic 2005a], which we further denote Z (i) to have an outer bound of the criterion h(Z (i) ∞ , H, p). We retain the model parameters (A (i) , B (i) ) that minimize h(Z (i) ∞ , H, p), and such that A (i) + BK (i) is Schur (closed-loop stability).

Remark 3.4 We chose not to add the constraint ρ(A + BK) < 1 in (3.23) to preserve the LP/QP structure of the optimization problem, as this inequality is nonlinear in the elements of the matrices A and B.

The resulting algorithm is presented below. In practice, it is not possible to compute the infinite sequence. Instead, we compute the first q elements of the infinite sequence of matrices, {A (i) , B (i) )} i∈Nq .

The iterative procedure defined in Algorithm 3.2 is guaranteed to generate a sequence of disturbance sets that are decreasing with regards to the partial order induced by set inclusion. The model parameters are retained if and only if the criterion (3.16) is decreasing.

The optimization problem (3.23) was solved using the matrices A (0) = A 4 and B (0) = B 4 and the disturbance set W (0) = W 4 obtained in the previous section, and p = 2. With the choice = 0.1, we obtain the matrices 0) , W (0) , Z (0) , H, p, , q Outputs: A, B, W, Z 1 Initialize (A, B, W, Z) ← (A (0) , B (0) , W (0) , Z (0) ) and i ← 0; For comparison, the disturbance set W (1) , and the initial disturbance set W (0) = W 4 , are presented in Figure 3.6. The associated invariant set Z (1) , and the initial invariant set Z 4 are shown in Figure 3.7. In particular, we note that we have the set inclusion Z (1) ⊆ Z (0) = Z 4 . This procedure has improved the size of the resulting mRPI set.

A (1) =   1.
Inputs: {ξ m k } k∈N L+1 , {u m k } k∈N L+1 , K, A (0) , B (
The subsequent iterations failed to improve the RPI set, as we obtained Z (i) ⊆ Z (1) . In the following, we retain the parameters A (1) , B (1) , W (1) and Z (1) . It could be useful to consider additional constraints on the coefficients of the matrices A and B in (3.23) to bound the absolute value of the eigenvalues of the matrix A + BK, as in (3.18), with the disadvantage of a considerable increase in the complexity, or to bound the eigenvalues of the matrix A as in (3.19).

We have presented a novel approach for system identification that aims at obtaining the smallest possible invariant set, for a pre-imposed structure of the LTI dynamics and a pre-imposed linear feedback gain, which comprises of two steps. The first step uses a Least-Mean Square identification with constraints on the eigenvalues of the matrix A. The second step consists of iteratively refining the identification by further constraining the disturbance set.

We used this method to identify the parameters (A, B, W) of the Parrot AR.Drone 2.0 horizontal dynamics model, as in (3.5). The next section discusses the practical implementation of the Tube Based MPC law for the stabilization of the Parrot Ar.Drone 2.0.

Practical implementation of the Tube Based MPC

This section details the practical implementation of the Tube Based MPC scheme for the stabilization of the quadrotor horizontal dynamics using the notation introduced in Section 1.5 and the model identified precedently.

ξ[k + 1] = Aξ[k] + Bu[k] + w[k], ∀k ∈ N,
where the model parameters are chosen as A = A (1) , B = B (1) from Section 3.5.2. We recall that we assumed the dynamics in the directions x and y are decoupled, and their models (i.e. A, B, W) are identical. The state and input constraint sets X and U are chosen polytopic, and we denote

X = ξ ∈ R 3 | G x ξ ≤ g x , G x ∈ R lx×3 , g x ∈ R lx , l x ∈ N + , U = {u ∈ R | G u u ≤ g u } , G u ∈ R lu×1 , g u ∈ R lu , l u ∈ N + .
The disturbance set W is the set W (1) from Section 3.5.2.

The goal is to stabilize the quadrotor around a succession of waypoints, defined by their coordinates in the horizontal plane {x r , y r }. With matrices A and B as in (3.5), a state equilibrium ξ r x = x r 0 0 leads to an input equilibrium of u r x = 0. Here, the desired equilibrium is an user input defined in real-time, and it may not be known a priori for a prediction window. That is, at time k we choose the current reference ξ r [k] = ξ r , but we may not have the knowledge of the future references to come ξ r [k + i], i ∈ N + .

The disturbance free counterpart of the actual system is given by

ξ[k + 1] = A ξ[k] + B ū[k], ∀k ∈ N.
We use the local controller ν(ξ) = K 4 ξ from Section 3.4 to control the error state

ξ[k] -ξ [k], yielding globally a control signal u[k] = ū[k] + K 4 (ξ[k] -ξ[k]).
We compute ( , p)-outer approximations of the mRPI set Z ∞ (A, B, W) using the first method in Section 1.4.1.3. There is a trade-off between the quality of the approximation, quantified by the scalar , and the complexity of polytopes, quantified by their H or V complexity. Recall that this method requires the computation of partial sums Z r (A, B, W) = r-1 i=0 (A + BK) i W. The H and V complexity of these partial sums depend on the eigenstructure of A + BK and on the H and V complexity of the disturbance set W. Here, the disturbance set W has an H-complexity and V-complexity of respectively 120 and 64. Figure 3.8 shows the relation between and the H-complexity with p = ∞. As expected from the H-complexity of W, the polytopic approximations have a high H-complexity, even for high values of .

For a given RPI set Z, we define the nominal system constraint sets as follows

X = X Z, Ū = U KZ
It is desirable to have an RPI set Z as small as possible, as it is substracted to the actual constraint sets to define the constraint sets for the nominal system. These sets retain the polytopic structure of X and U, and have an H-complexity of at most l x and l u from (6.10), regardless of the H-complexity of Z.

Two different formulations of optimal control problems for the control of the nominal system subject to the nominal constraints were proposed in Section 1.5. The high H-complexity of the mRPI approximations increases extensively the computational complexity of the second optimal control problem where the initial state of the nominal system is an optimization variable (1.36). Indeed, the additional constraint ξ ∈ {ξ} ⊕ -Z consists of l z linear inequalities, where l z is the H-complexity of the polytopic RPI set Z. For this reason, we choose to implement the first Tube Based MPC scheme, in which the nominal state is not an optimization parameter. The complexity of the mRPI approximation does not impact the computational complexity of the optimization problems, and we can choose the approximation with the smallest , namely = 0.01.

The conventional optimal control problem P N ( ξ[k], ξ r ), in which ξ[k] is the current nominal system state, and ξ r the reference, reads

minimize ū V N ( ξ0 , ū, ξ r ), (3.24a) subject to V N ( ξ0 , ū, ξ r ) = N -1 i=0 (|| ξi -ξ r || Q + ||ū i || R ) + || ξN -ξ r || P ,(3.24b) ū = {ū 0 , ..., ūN-1 } , (3.24c) ξi+1 = A ξi + B ūi , i ∈ N N -1 , (3.24d) ξi ∈ X , i ∈ N N -1 , (3.24e) ūi ∈ Ū, i ∈ N N -1 , (3.24f) ξ0 = ξ, (3.24g) ξN ∈ Xf (ξ r ). (3.24h)
The terminal set is a function of the reference ξ r . We assume that the terminal sets are polytopic, and we denote Xf (

ξ r ) = ξ ∈ R 3 | G f (ξ r )ξ ≤ g f (ξ r ) , G f (ξ r ) ∈ R l(ξ r )×3 , g f (ξ r ) ∈ R l f (ξ r ) , l f (ξ r ) ∈ N + .
We discuss next the selection of a solver for the real-time computation of this QP, along with the real-time computation of the terminal set Xf (ξ r ) as a function of the desired reference ξ r .

Solving the MPC QP using CVXGEN in real time

We use CVXGEN [START_REF] Mattingley | [END_REF]] to solve in real-time the QP (3.24). CVXGEN generates code for small, QP-representable convex optimization problems.

CVXGEN relies beforehand on the generation of C code for optimization problems with fixed structure. Namely, the state and control input dimension n = 3 and m = 1, the H-complexity of the state, control input, and terminal constraint stabilization of quadrotor horizontal dynamics sets l x , l u and l f (ξ r ), and the length of the prediction horizon N . These dimensions cannot be changed during the flight. The C code solves in real-time the QP with the measured values of the parameters, such as the current nominal state ξ[k] and current reference ξ r [k], an H-representation of the state and control input constraint sets G x , G u , g x , g u , an H-representation of the terminal constraint set G f (ξ r ), g f (ξ r ), and the weighting matrices Q, R, P in (3.24).

Throughout the flight, we chose the weighting matrices Q, R and P , and the state and control input constraints, characterized set-wise by X and U, to remain constant. However, the terminal set Xf (ξ r ) has to be computed online, and must have the same H-complexity l f (ξ r ) = l f ∈ N + regardless of the desired reference ξ r . This issue is discussed in the following section.

Change of reference and terminal set computation

Let x r be a user-chosen waypoint with regards to the position x. We define the associated reference state ξ r x = x r 0 0 . In the following, we propose a method for the online computation of a terminal set X f (ξ r ). We have ξ r = Aξ r from the upper-triangular structure of the matrix A.

The terminal set Xf (ξ r ) has to satisfy the stability conditions (1.31), namely that there exists a state feedback gain

K f ∈ R 1×3 satisfying (A + BK f )( Xf (ξ r ) ⊕ {-ξ r }) ⊆ ( Xf (ξ r ) ⊕ {-ξ r }), (3.25a) Xf (ξ r ) ⊆ X , (3.25b) K f ( Xf (ξ r ) ⊕ {-ξ r }) ⊆ Ū. (3.25c)
Note that in order to satisfy (3.25a) the set Xf (ξ r )⊕{-ξ r } has to contain the origin. Hence, from (3.25b), the following set inclusion has to hold,

ξ r ∈ X = X Z, (3.26)
The terminal set, along with the constraint sets of the nominal system X and Ū, define the feasible region XN of the optimization problem (3.24), given by the following recursion

X0 (ξ r ) = Xf (ξ r ), (3.27a) Xi+1 (ξ r ) = ξ ∈ X | ∃ū ∈ Ū, A ξ + B ū ∈ Xi (ξ r ) , i ∈ N N -1 . (3.27b)
The optimization problem is feasible for the reference ξ r if and only if the current state of the system ξ is contained in the feasible region XN (ξ r ). For this reason, we are interested in computing a terminal set Xf that is as large as possible to have a feasible region that is as large as possible.

For a given state feedback gain K f , the largest set satisfying the constraints (3.25) is the Maximal Output Admissible Set (MOAS) O ∞ (ξ r ) [Kolmanovsky 1998a] for the system ξ+ = (A + BK f ) ξ and constraints ξ ∈ X ⊕ {-ξ r }, K f ξ ∈ Ū, given by the recursion

O 0 (ξ r ) = ξ ∈ R 3 | ξ ∈ X ⊕ {-ξ r }, K f ξ ∈ Ū , (3.28a) O i+1 (ξ r ) = ξ ∈ R 3 | ξ ∈ O i (ξ r ), (A + BK f ) ξ ∈ O i (ξ r ) , ∀i ∈ N. (3.28b)
Given that X and U are polytopes, the MOAS is also a polytope. The online computation of the MOAS at every change of reference is not possible given the real-time constraints. Moreover, the MOAS for two distinct references have, in general, not the same H-complexity. This is particularly problematic with regards to the use of CVXGEN.

For this reason, we propose a method for the computation of a terminal set Xf (ξ r ) for a reference ξ r ∈ int( X ) that satisfy (3.25) by avoiding real-time computation of complex set-theoretic operations (Minkowski sum and Pontryagin difference for instance).

Let K f ∈ R 1×3 be a linear state feedback gain such that A + BK f is Schur. Compute offline O ∞ (0 3 ), the MOAS for the reference ξ r = 0 3 given by a waypoint chosen as the origin x r = 0. This polytopic set satisfies (3.25) with ξ r = 0. In the following, we denote

Xf = ξ ∈ R 3 | G f ξ ≤ g f , G f ∈ R l f ×3 , g f ∈ R l f , l f ∈ N + .
This set is computed offline. Consider a reference ξ r = x r 0 0 ∈ R 3 satisfying (3.26). The terminal set Xf (ξ r ) for the reference ξ r will be chosen as a scaled and translated image of the terminal set computed for the origin Xf .

Let λ > 0 be the scaling factor. Define the set Ω = {ξ r } ⊕ λ Xf . This set satisfies

(A + BK f )(Ω ⊕ {-ξ r }) = λ(A + BK f ) Xf
. By construction of the set Xf , we have

(A + BK f ) Xf ⊆ Xf . Hence, (A + BK f )(Ω ⊕ -{ξ r }) ⊆ λ Xf ⊆ (Ω ⊕ -{ξ r }),
and Ω satisfies (3.25a). stabilization of quadrotor horizontal dynamics

The set inclusions Ω ⊆ X and K f (Ω ⊕ {-ξ r }) ⊆ Ū are equivalent to λ Xf ⊆ X ⊕ {-ξ r } and λK f Xf ⊆ Ū. These sets are compact and bounded, and contain the origin in their interior. Hence, there exists λ > 0 such that Ω = λ Xf ⊕ {ξ r } satisfies (3.25b) and (3.25c). Moreover, the set of admissible scalars λ is a closed interval whose upper-bound is denoted λ * (ξ r ).

The computation of λ * (ξ r ) consists of solving the following optimization problem

λ * (ξ r ) = arg minimize λ∈R + -λ (3.29a) subject to λ Xf ⊕ {ξ r } ⊆ X , (3.29b) λK f Xf ⊆ Ū. (3.29c)
Define the set Xf (ξ r ) = λ * (ξ r ) Xf ⊕ {ξ r }. This set satisfies (3.25), and is given by

Xf (ξ r ) = ξ ∈ R 3 | G f ξ ≤ λ * (ξ r )g f -G f ξ r .
Hence, the H-complexity of the sets Xf (ξ r ) is independent of the reference ξ r , that is

l f (ξ r ) = l f , ∀ξ r ∈ X .
The method presented here for the computation of the terminal set will be used for the online computation of the terminal set at every change of reference.

Remark 3.5 It is worth to be mentioned that the inclusion ξ ∈ XN (ξ r ), with XN (ξ r ) as in (3.27), is not guaranteed. Thus, the proposed approach does not guarantee feasibility of the optimization problem in case of a reference change. The implementation of a reference governor is to be sought in case of infeasibility during a change of reference.

The solution of the optimization problem can be obtained using basic mathematical operations, thus not increasing extensively the online computation load of the control law. A simulation illustration of the terminal sets obtained with this approach is provided in Section 3.7.

We denote V ( Xf ) the set of vertices of the polytope Xf . The constraints of the optimization problem (3.29) can be rewritten ∀ξ ∈ V ( Xf ),

G x (λξ + ξ r ) ≤ ḡx , (3.30) λG u K f ξ ≤ ḡu . (3.31)
The computation of λ * (ξ r ) consists of solving the following linear optimization problem

λ * (ξ r ) = arg minimize λ∈R + -λ, (3.32a) subject to G x ξ G u K f ξ λ ≤ ḡx -G x ξ r ḡu , ∀ξ ∈ V ( Xf ).(3.32b)
The scalar λ * (ξ r ) can be obtained via basic mathematical operations, as detailed in Algorithm 3.3 Algorithm 3.3: Online tuning of the scalar parameter λ Inputs:

G x , ḡx , G u , ḡu , K f , V ( Xf ), ξ r Outputs: λ * (ξ r ) 1 Set {ξ 1 , ..., ξ f } the elements of V ( Xf ); 2 Initialize λ ← 0 and i ← 0; 3 repeat 4 Set i ← i + 1 and ξ ← v i ; 5 Initialize j ← 0; 6 repeat 7 Set j ← j + 1; 8 Set λ ← max(λ, ḡx -G x ξ r ḡu j G x ξ G u K f ξ j ); 9 until j = l x + l u ; 10 until i = f ; 11 Set λ * (ξ r ) ← λ;

Structure of the code

We present here a description of the code we are running on the PC station for the stabilization of the Parrot AR.Drone 2.0. The Motion Capture consists of the Motive Software with OptiTrack cameras, providing an accurate measurement of the actual system state {ξ[k]} k∈N , that comprises of the position, speed and attitude, at each time-step. The initial state of the nominal system is initialized with the first measurement of the Motion Capture system, yielding ξ[0] = ξ[0]. The function CVXGEN, which solves the QP, is also initialized with the state and input constraint sets X , U, the weighting matrices Q, R, P and the model parameters A, B.

At each time-step kT e , k ∈ N, the current nominal system state ξ[k] is updated in the CVXGEN C code. The optimization problem is solved, yielding ū [k]. The difference between the current nominal system and uncertain system states is multiplied by the feedback gain K, yielding K(ξ[k] -ξ[k]). These two control laws are added, defining the control input u

[k] = K(ξ[k] -ξ[k]) + ū[k]
which is transmitted to the Parrot AR.Drone 2.0. The function Nominal system updates the state of the nominal system, given by ξ

[k + 1] = A ξ[k] + B ū[k].
The user can choose to change the reference for the stabilization, by modifying

x r and/or y r . The function Update Terminal Constraints solves the LQ as in (3.32) to compute the scalar λ * (ξ r ). We use this scalar and the equilibrium ξ r [k] to update the CVXGEN code accordingly. The structure of the controller is presented in Figure 3.9 and corresponds to stabilization of quadrotor horizontal dynamics

Experimental results

This section presents the experimental results obtained from validation flights with the Tube Based Model Predictive Controller described in the previous section. We use the Robot Operating System (ROS) framework for the communication between the Motion Capture system, the PC station and the Parrot AR.Drone 2.0. In particular, the communication with the quadrotor uses the ardrone_autonomy package [START_REF] Monajjemi | [END_REF]].

Tube Based MPC parameters

The system and the flight zone are subject to physical constraints, such as the presence of obstacles or actuator saturation. Here, the flight zone is defined by the square -1.2m ≤ x, y ≤ 1.2m, and the actuator saturation by -1 ≤ u x , u y ≤ 1.

We introduce additional constraints to ensure that the system evolves within the domain of validity of theoretical assumptions. In Section 3.2, we assume small roll and pitch angles. For this reason we introduce the following constraints |θ|, |φ| ≤ 0.3rad to ensure that we remain within the validity conditions of these assumptions.

We finally introduce arbitrary constraints whose purpose is to illustrate the recursive constraints satisfaction property of Tube Based MPC. We consider tighter Design and Flight Experiments of a Tube Based MPC for the stabilization of quadrotor horizontal dynamics constraints on the control input, given by -0.6 ≤ u x , u y ≤ 0.6, along with constraints on the horizontal speed, -1m.s -1 ≤ v x , v y ≤ 1m.s -1 . All the constraints presented here lead to the following polytopes

X = ξ ∈ R 3 | G x ξ ≤ g x , U = {u ∈ R | G u u ≤ g u } , (3.33) G x =          1 0 0 0 1 0 0 0 1 -1 0 0 0 -1 0 0 0 -1          , g x =          1.2 1 0.3 1.2 1 0.3          , G u = 1 -1 , g u = 0.6 0.6 . (3.34)
Note that the sets X , U contain the origin in their interior, thus satisfying the Assumption 1.1. The constraints on the nominal system are given by the Pontryagin differences

X = X Z, Ū = U KZ. (3.35)
With our choice for the RPI set Z and local controller ν(ξ

) = K 4 ξ, we have X = ξ ∈ R 3 | G x ξ ≤ ḡ , and Ū = {ū ∈ R | G u ū ≤ ḡu } , with ḡ = 0.6 0.35 0.1 0.6 0.35 0.1 , ḡu = 0.3 0.3 .
The terminal set is selected according to the discussion in Section 3.6.2 with K f = K 4 = -0.4 -0.5 -1 . We present in Figure 3.10 the terminal sets Xf (ξ r (i)) = λ * (ξ r (i))O ∞ ⊕ {ξ r (i)}, i = {1, 2, 3}, for the following choice of equilibrium ξ r (i), i = {1, 2, 3} ξ r (1) = 0 0 0 , ξ r (2) = 0.4 0 0 , ξ r (3) = 0.65 0 0 .

The solutions of the associated LP (3.29) are

λ * (ξ r 1 ) = 1, λ * (ξ r 2 ) = 0.8709, λ * (ξ r 3 ) = 0.1539.
The weighting matrices of the optimization problem are arbitrarily chosen as

Q =   0.8 0 0 0 1.2 0 0 0 1   and R = 5.
The terminal weighting matrix P is chosen as the solution of the Riccati equation CVXGEN is adapted to optimization problem of small size, with Karush-Kuhn-Tucker (KKT) matrices of less than 4000 non-zeros entries [START_REF] Mattingley | [END_REF]]. The H-complexity of the state, control input and terminal constraint sets are respectively l x = 6, l u = 2 and l f = 66. With these constraint sets, a prediction horizon length of N = 30, which corresponds to 1.5s, leads to a KKT matrix of 3349 non-zeros entries.

(A + BK f ) P (A + BK f ) -P + Q + K f RK f = 0.
The parameters of the Tube Based MPC law are recalled in Table 3.1. The disturbance set W and the RPI set Z are shown in Figure 3.6 and 3.7.

Closed-loop horizontal trajectory

The Figure 3.11 presents the closed-loop trajectory of the actual system in the horizontal plane (x, y), along with the desired waypoints (x r , y r ). The quadrotor flies from the neighborhood of a waypoint to the neighborhood of the next waypoint.

The distance between two successive waypoints is chosen as 0.3m to ensure the feasibility of the optimization problem. The implementation of a reference governor is to be sought to further increase the feasibility region of the control law. However, the introduction of a reference governor modifies the control law. Hence, it requires to update the matrices A and B and the disturbance set W obtained from identification flights performed with the chosen reference governor, as discussed in Section 3.3.

Nominal system trajectory

At every time-step, the optimization problem of the MPC (3.24) is solved for the current nominal system state ξ[k] and reference ξ r . The resolution of this QP yields Design and Flight Experiments of a Tube Based MPC for the stabilization of quadrotor horizontal dynamics N 30 N T e 1.5s

G x I 3 -I 3 g x 1.
2 1 0.3 1.2 1 0.3 ḡx 0.7 0.321 0.094 0.7 0.321 0.094

G u 1 -1 g u 0.6 -0.6 ḡu 0.18 0.18 K -0.4 -0.5 -1 K f -0.4 -0.5 -1 Q diag(0.8, 1.2, 1) R 5
Table 3.1 -Parameters of the Tube Based MPC Law.

( ξ[k], ξ r ), ..., ξ * N ( ξ[k], ξ r ) with ξ * 0 ( ξ[k], ξ r ) = ξ[k], and ξ * i+1 ( ξ[k], ξ r ) = A ξ * i ( ξ[k], ξ r ) + B ū * i ( ξ[k], ξ r ), i = {0, .
.., N -1}. We present in Figure 3.12 the optimal state sequence obtained with the initial nominal state ξ[k] = ξ[0] = -0.4 0 0 and reference ξ r = -0.1 0 0 . We see that the state sequence satisfies the terminal constraint ξ * N ( ξk , ξ r ) ∈ Xf (ξ r ) = Xf (0.1). This set inclusion ensures that the optimization problem P N ( ξ[k + 1], ξ r ) is feasible. The optimization problems is recursively feasible as long as the reference ξ r is not changed.

The terminal cost P ensures that the cost function of the optimization problem decreases without change of reference. Namely, the optimal cost function

V * N ( ξ[k], ξ r ) = V N ξ[k], ū * ( ξ[k], ξ r ), ξ r satisfies V * N ( ξ[k + 1], ξ r ) ≤ V * N ( ξ[k], ξ r ) -|| ξ[k] -ξ r || Q + ||ū * 0 ( ξ[k], ξ r )|| R ,
Figure 3.13 illustrates the decreasing of the cost function for both directions x and y on each interval with constant references ξ r x and ξ r y respectively. Figure 3.14 shows the trajectory of the nominal system for both directions x and y. At every change of reference, the waypoints (x r , y r ) were chosen to ensure that the resulting optimization problems were feasible. As a consequence, the nominal system satisfies the tight state constraints ξ[k] ∈ X , ū[k] ∈ Ū throughout the flight.

Disturbance realizations and tube properties

We chose to implement a Tube Based MPC scheme for its strong theoretical property regarding robust recursive constraints satisfaction of the actual system. This property requires that the error states {ξ[k]-ξ[k]} k∈N remain in the RPI set Z. This Despite containing all the disturbance realizations, the disturbance set W is conservative, as it is larger than the disturbances encountered, in particular for negative position and angle and positive speed, and its origin-symmetric counterpart (bottom left and upper right in Figure 3.15). An approach to reduce conservatism would be to reiterate the system identification step using the experimental data generated here. stabilization of quadrotor horizontal dynamics

Here, the theoretical properties of Tube Based MPC are obtained at the expense of conservatism. Indeed, the RPI set Z, used to bound the error state, is much larger than the error state encountered in flight. The state error ξ[k] -ξ[k] approaches the boundary of the RPI set only if the previous disturbance realizations {w [k -1], w[k -2], ...} were persistent and chosen on the boundary of W. Figure 3.15 shows that the disturbance realizations are concentrated in a neighborhood of the origin, and only a few disturbance realizations approach the boundary of W.

These larger disturbances realizations are encountered after a modification of the reference ξ r , and are due to the unmodelled delay in the transmission of the inputs. Indeed, a change of the reference yields larger nominal control inputs {ū[k]}, and thus larger actual control inputs {u[k]}, and thus to larger disturbance due to the impact of the unmodelled delay

w d = -B(u[k] -u[k -d]) as in (3.11).
The disturbances realizations being contained in the disturbance set is a sufficient condition for the error state to be contained in the RPI set. However, it is not a necessary condition. For this reason, an alternative approach to further reduce conservatism is to define a practical RPI set Z p , defined as the convex hull of the measured state errors, as in

Z p = Co ({w m [k]} k∈N ) ,
instead of computing Z as a theoretical RPI set for the identified model parameters A, B and where the disturbance set W is defined as the convex-hull of the measured disturbances as in (3.8).

This notion can be seen as a version of probabilistic set invariance [Kofman 2012]. These theoretical notions are not detailed here, but we point to the interest of such practical invariance tools

Recursive state and input constraints satisfaction

Given that the error state remains in the RPI set Z, the control law ensures robust recursive state and control input constraint satisfaction of the actual system. Figure 3.17 shows that the trajectory of the uncertain system remains in the state constraint set X .

Figure 3.18 shows the time evolution of the position, speed, angle, reference, and control input of the nominal and uncertain systems along the trajectory for the x direction. This further illustrates the recursive constraint satisfaction guaranteed by the Tube Based MPC law. Similar results are obtained in the direction y.

Real-time computation

We now discuss the real-time implementability of this Tube Based MPC scheme for the control of a quadrotor UAV.

We present in Figure 3.19 the time evolution of the computation time of the Tube Based MPC law for both directions x and y. The computation time is defined Figure 3.17 -State space representation of the uncertain system trajectory for both directions x (blue) and y (red), along with the constraint set X (white). as the time required to perform a loop of the Algorithm 3.4, that is the steps 9 to 16. In particular, it comprises of the computation time of the MPC QP (3.24), and of the terminal constraint LP (3.29).

The maximal computation time is 2ms for each direction x and y, or 4ms in total. A box plot of the computation time is presented in Figure 3.19. For both directions the median is around 0.9ms, and 90% of the time lower than 1.2ms. This computation time is satisfactory, as the sampling time was chosen as T e = 50ms. Here, the computation are performed on a PC station, that has better computational capacity than most quadrotor platforms. However, the computation time of the proposed Tube Based MPC are suitable for modern quadrotor systems.

Conclusion

We have implemented and validated experimentally a Tube Based MPC law for the horizontal stabilization of a quadrotor UAV. The system considered here is the Parrot AR.Drone 2.0. The horizontal dynamics modelling benefits from the internal controller that decouples the horizontal, vertical, and yaw dynamics. The modelling assumptions (small angles and zero vertical speed and yaw angle) lead to a three dimensional linear discrete-time system subject to additive disturbances. The command is sent to the system with WiFi, which induces delay in the control. The model chosen does not account for the delay to avoid a state augmentation which would increase the computational complexity of the Tube Based MPC law, especially the computation of mRPI set approximations. This choice for the modelling prevented us from using existing methods for the computation of RCI sets, where the model is fixed and the control law is the optimization variable, as the delay is an additional source of disturbances.

Instead, we proposed a data-driven procedure for the computation of minimal RCI sets, where the model parameters are the design variables. This procedure consists first in selecting the local controller from calibration flights, by comparing the deviation of the closed-loop system for a range of linear control laws. Then, the model parameters and disturbance bounds are identified using experimental data.

We proposed an identification method that aims at minimizing the associated mRPI set, which consists in bounding the eigenvalues of the closed-loop matrix while minimizing a norm of the disturbance realizations. The advantages of the proposed identification method compared to a classical unconstrained Least Mean Square approach, in terms of size of the resulting RPI set, were illustrated using experimental data from identification flights. More sophisticated identification methods are to be sought to further reduce the size of the RPI set (as for example a maximum likelihood estimations with an mRPI objective). We detailed the practical implementation of a Tube Based MPC scheme where the nominal system state is not a design variable, due to the high H-complexity of the mRPI set approximations. We chose CVXGEN to solve in real-time the Tube Based MPC QP. A method for the real-time computation of the terminal constraints that is adapted to the use of CVXGEN is proposed. The computation of the terminal set proposed here is the solution of a linear optimization problem, which is compatible with the real-time constraints of our system. A less conservative approach is to be sought to compute a larger terminal set, as the proposed method seeks the largest scaling of the terminal set associated to the origin.

We provided the experimental results of the implementation of the Tube Based MPC law. The closed-loop trajectories presented in this chapter suffers from the low distance, approximatly 0.3m, between successive waypoints (or references). This distance ensures that the optimization problems remain feasible despite a change of reference. Indeed, the former reference must be contained in the feasible region of the optimization problem with the new reference to ensure the optimization problem is feasible. There are other manners to increase the maximal distance between references. The first is to choose a larger prediction horizon length N , which increases the computational complexity. Here, we were limited by the use of CVXGEN. The second is the implementation of a reference governor. However, a change in the control law requires to reiterate the identification procedure, as it modifies the disturbances induced by the unmodelled delay in the transmission of the input. The third is to enlarge the nominal system constraint sets X and Ū. Given that the sets X and U are fixed by the environment and the quadrotor, an enlargement of the nominal constraint sets consists in computing a smaller RPI set Z, which was the underlying goal of this Chapter.

An extension of this work is to implement the Tube Based MPC scheme on a quadrotor platform with embedded computational capacity to not suffer from delay in the transmission of the input. Such experimental setup mitigates the disturbances encountered in flight, and allows to use the control law as an additional design parameter of the invariant set. An overview of the existing methods for the computation of optimized RCI sets where the control law is the only design variable is presented stabilization of quadrotor horizontal dynamics in Chapter 4, while Chapter 5 and Chapter 6 present our research on this topic.

Part II

Introduction

This chapter proposes an overview of the existing researches on the computation of optimized RCI sets for constrained discrete-time linear systems where the structure of the model and the parameters are fixed while the control law is the only design variable. Consider a linear discrete-time system subject to additive disturbances as in (1.1),

ξ + = Aξ + Bu + w, (4.1)
where ξ ∈ R n is the current state, u ∈ R m is the control input, and w ∈ R n is an exogenous disturbance. We assume that the system is subject to constraints as in (1.2), characterized set-wise by

(ξ, u, w) ∈ (X × U × W) . (4.2)

Theoretical background -invariant set design

In the following, we assume that the sets X , U and W are polytopic of respective H complexity l x , l u and l w , and respective minimal H representation

X = {ξ ∈ R n | G x ξ ≤ g x } , U = {u ∈ R m | G u ξ ≤ g u } , W = {w ∈ R n | G w ξ ≤ g w } .
As opposed to Chapter 2, we assume here that the model parameters (A, B) and the disturbance set W are known.

A set Z ⊆ X is RCI if and only if there exists a control law ν : R n → R m such that ∀ξ ∈ X , ∀w ∈ W, we have ν(ξ) ∈ U and Aξ + Bν(ξ) + w ∈ Z. The computation of a control law that has the best disturbance rejection properties relates to the construction of an RCI set that is as small as possible. There are two main difficulties for the computation of optimized RCI sets.

The first comes from the fact that both the set Z and the control laws u = ν(ξ) are design parameters. The direct computation of a set Z and ν that satisfy the above constraint is complex. Regarding computational tractability, two types of convex sets have been proposed as candidate RCI sets, namely ellipsoidal and polytopic sets. In the following, we do not discuss the methods that use an ellipsoidal parametrization as in [Nazin 2007, Nguyen 2014]. In a similar fashion, a linear parametrization of ν is often considered in the computation of optimized RCI sets to reduce the computational complexity at the expense of conservatism. The second is that set inclusion is not a total order over the collection of RCI sets. Alternative criteria are to be considered to measure the size of RCI sets. In the literature, the measure is either chosen as the radius of the minimal p norm ball containing the RCI set, as in [Rakovic 2010], the volume of the RCI set, as in [START_REF] Cannon | [END_REF], Blanco 2010], or the size of the RCI set in a predifined direction of dimension 1. The first and last criterion relate to the criterion (1.10) where the matrix H ∈ R h×n is chosen respectively as the identity matrix I n or as a row vector (i.e. h = 1). The main advantage of such criterion is that, for a polytopic RCI set Z, the criterion is linear (respectively quadratic) with regards to the vertices of Z for p = 1, ∞ (respectively for p = 2).

In this chapter, we denote G ∈ R l×n and g ∈ R l the matrices defining a minimal H representation of the polytope Z, where l is its H-complexity, as in

Z = {ξ ∈ R n | G z ξ ≤ g z } . (4.3)
Its minimal V representation is given by the sequences of vertices denoted {v 1 , ..., v f }, where f ∈ N + is its V complexity. Some approaches presented next rely on the infinite refinement of an initial polytopic RCI set, through the construction of polytopic RCI set sequences. For a sequence of polytopes {Z(i)} i∈N of respective H and V complexity {l(i)} i∈N and {f (i)} i∈N , we denote the matrices G(i) ∈ R l(i)×n and g(i) ∈ R l(i) and the vertices {v 1 (i), ..., v f (i) (i)}.

There are three main approaches in the recent literature for the computation of optimized RCI sets.

• The first uses a linear parametrization of the control law ν(ξ) = Kξ and the characterization of the mRPI set as infinite Minkowski sums.

• The second proposes candidate solutions by restraining the control law to be linear and the RCI set to be polytopic.

• The third characterizes families of polytopic RCI sets obtained with a class of piece-wise affine control laws.

These three classes of approaches are detailed in the next sections.

Linear parametrization of the control law

A first approach in the computation of optimized RCI sets is to impose the control law to have a linear feedback structure, and to study the size of the resulting mRPI sets. Recall that for a feedback gain K ∈ R m×n , the mRPI set is given by

Z ∞ (K) = ∞ i=0 (A + BK) i W.
In [Wang 2011], the author uses the notion of support function. The support function of a bounded set Z is given by

h Z : R n → R, (4.4) a → sup{a ξ | ξ ∈ Z}. (4.5)
If the set Z is a polytope, we can use its vertex representation to have h

Z (a) = max{a ξ | ξ ∈ V (Z)}. Moreover, we have ∀a ∈ R n , ∀Z 1 ⊂ R n , ∀Z 2 ⊂ R n , h Z 1 ⊕Z 2 (a) = h Z 1 (a) + h Z 2 (a).
The support function can be used to evaluate the size of a bounded set with regards to a chosen direction a ∈ R n . The author is interested, for a given direction a ∈ R n , to compute the stabilizing feedback gain K leading to an mRPI set that is as small (or large) as possible. This problem is related to the computation of (H, p)-mRCI sets discussed in Section 1.4.2.3, where H ∈ R 1×n is a row vector, with a linear parametrization of the control law.

The resulting optimization problem reads minimize

K∈R m×n h Z∞(K) (a), (4.6) subject to Z ∞ (K) = ∞ i=0 (A + BK) i W. (4.7)
The direct optimization of the feedback gain is not possible, as the mRPI set is an infinite Minkwoski sum. The author uses the fact that the partial sums Theoretical background -invariant set design Z r (K) = r-1 i=0 (A + BK) i W can approximate with arbitrary accuracy the mRPI set to show that h Z∞(K) (a) (and its derivative) can be approximated with arbitrary accuracy by h Zr(K) (a) = r-1 i=0 sup{a (A + BK) i w | w ∈ V (W)}. This approach suffers from the fact that the support functions of the partial sums have a nonlinear dependency on the feedback gain K for r > 2.

The support function is used as a criterion to measure the size of the mRPI set with respect to a predefined direction. This approach uses the expression of the mRPI set as an infinite Minkwoski sum, and approximates its support function by the support function of its partial sums.

Joint parametrization of the RCI set and the control law

A second class of approach uses a parametrization of both linear control laws and a polytopic RCI sets. These approaches rely on the determination of computationally tractable (necessary and) sufficient conditions for a polytopic set to be RPI for a linear control law. Let us introduce the following definition.

Definition 4.1 (Linear RCI set) A set Z is linear RCI for the system (4.1) and constraints (5.4) if and only if Z ⊆ X and there exists K ∈ R m×n such that KZ ⊆ U and (A + BK)Z ⊕ W ⊆ Z.

Extended Farkas Lemma

In [START_REF] Bitsoris | Positively invariant polyhedral sets of discrete-time linear systems[END_REF]], a necessary and sufficient condition for the existence of a linear control law ν(ξ) = Kξ, K ∈ R m×n such that a polytope Z which contains the origin in its interior is positively invariant (PI) for linear discrete-time systems (without external disturbances) is presented. This result is extended to the whole polytope class in [Hennet 1991], to the presence of polytopic state and input constraints in [START_REF] Vassilaki | [END_REF]], and to the presence of additive bounded disturbances in [Blanchini 1990]. This necessary and sufficient condition is recalled below.

Proposition 13 (Extended Farkas Lemma) Given Z = {ξ ∈ R n | G z ξ ≤ g z }, Z
is linear RCI if and only if there exist a feedback gain K ∈ R m×n and matrices

S ∈ R l×l + , S x ∈ R l×lx + and S u ∈ R l×lu + , S w ∈ R lz×lu + such that G(A + BK) = SG, G = S w G w , Sg + S w g w ≤ g, (4.8a) G u K = S u G, S u g ≤ g u , (4.8b) G x = S x G, S x g ≤ g x . (4.8c)
The equations (4.8a), (4.8b) and (4.8c) are respectively equivalent to (A + BK)Z ⊕ W ⊆ Z, KZ ⊆ U, and Z ⊆ X .

This provides a necessary and sufficient condition for a polytope Z to be linear RCI. A major advantage of this invariance condition is the fact that it uses only one representation of the polytope, namely the H-representation. Note that if Z is fixed (i.e. G and g are fixed), the constraints are linear in the matrices S, S x , S u and S w and on the feedback gain K, and leads to particularly advantageous control design to achieve Robust Control Invariance of the set. These conditions are bilinear if the matrix G and g, which describe the polytope, are also considered as design parameters. As a consequence, the joint optimization on both the set Z and the feedback gain K is complex (bilinear optimization).

Recent approaches based on the simplification of the geometrical structure of Z are presented below.

Hyper-rectangles

The author of [Tahir 2010] considers G = I n , thus imposing the polytopes to be hyper-rectangles. The only design parameter of Z is the vector g ∈ R n which defines the right-hand side of the linear inequalities in (4.3). The size of Z is defined as the sum n i=1 g i . A similar structure is assumed for U and W. Note that no state constraints are considered, but the results are easily generalizable provided X is an hyper-rectangle. Using the particular geometric structure of hyper-rectangles, [Tahir 2010] presents a necessary and sufficient condition for Z to be linear RCI without bilinearities, based on the Extended Farkas Lemma. A semi-definite optimization problem is providedin view of computing the minimal (or maximal) linear RCI hyper-rectangle Z.

This approach benefits from the convexity of the optimization problem, but the restriction to hyper-rectangles is very conservative.

Low H complexity polytopes

Low H complexity polytopes are characterized by n linear inequalities, where n is the dimension of the state, defining a square and invertible matrix G ∈ R n×n , and where g = γ1 n . The matrix G is assumed to be invertible, and the set Z to be symmetric. It is thus given by

Z = ξ ∈ R n | G -G ξ ≤ γ1 2n .
A first advantage of low-complexity polytopes is the reduction on the number of design variables of Z. Moreover, [START_REF] Cannon | [END_REF]] observes that the volume of Z is (2γ) n /det(G), which is inversely proportional to |det(G)|. The approaches presented in this section aim at computing the minimal (or maximal) linear low-complexity polytopic RCI set, where the criterion for minimality is its volume.

The computation of the linear invariant low-complexity polytope with minimal (resp. maximal) volume and an associated feedback gain requires the minimization Theoretical background -invariant set design (resp. maximization) of a non-convex cost function (the volume) with bilinear constraints (as in (4.8)). Several iterative approaches exist in the literature for the computation of candidate solutions [START_REF] Cannon | [END_REF]], [Blanco 2010], [START_REF] Tahir | [END_REF]], and [Gupta 2017]. These methods have in common the construction of infinite sequences of convex optimization problems, denoted {P (k)} k∈N . The solution of a convex optimization problem P (k) yields a linear RCI low-complexity polytope Z(k), and its associated feedback gain K(k). Moreover, the convex optimization problems are designed such that Z(k) is in the feasible domain of P (k + 1), thus ensuring recursive feasibility. [START_REF] Cannon | [END_REF]] also shows that low complexity polyopes have 2 n vertices that can be expressed as v j = γG -1 s j , where {s j } j∈N + 2 n is the collection of all possible n dimensional vectors whose elements are 1 or -1. By ordering the s j such that the first n are linearly independent, the vertices of Z can be expressed as linear combinations of the first n vertices, denoted v 1 , ..., v n . It is then shown that the volume of Z is inversely proportional to det v 1 ... v n . Hence, the maximization/minimization of the volume relates to the computation of n linear invariant vertices v 1 , ..., v n such that their determinant is minimal/maximal.

The author establishes a necessary and sufficient condition for linear invariance of the polytope using the vertices {v 1 , ..., v n } and a feedback gain K that accounts for the presence of input constraints. However, this condition is bilinear in the vertices {v i } i∈N + . Due to this bilinearity, the optimum cannot be computed analytically. Instead, the author introduces an infinite sequence of optimization problems, where every optimization problem is concerned with optimizing a single vertex, while the remainder are fixed at the values computed at the previous iteration. With this consideration, the constraints of the optimization problems are convex with regards to the vertex to optimize and the feedback gain K. Note that this approach requires the a priori computation of an initial linear RCI low complexity polytopic set Z(0), and depends on the choice of the initialization.

In [Blanco 2010] the successive polytopes are recursively parametrized by

G(k + 1) = G(k)X(k + 1)
, where X(k + 1) ∈ R n×n is a positive definite matrix. This recursive parametrization of the matrices G(k) allows to linearize the bilinear conditions of (4.8), and simplifies the cost function (volume). Likewise, this approach requires the knowledge of an initial linear low-complexity polytope RCI Z(0) to construct a sequence of RCI sets of increasing or decreasing volume. The second contribution of [Blanco 2010] is to provide an algorithm for the computation of the initial linear low-complexity RCI set Z(0) as the solution of an LMI.

A slack variable approach is adopted in [START_REF] Tahir | [END_REF]] to establish a sufficient condition for a low-complexity polytope to be linear RCI that is linear in the matrix G and the feedback gain K at the expense of conservatism. The volume of the polytope is lower and upper bounded relying on the inequality

S 1 ≤ G -1 G -1, ≤ S 2 ,
with S 1 and S 2 two positive definite matrices. These bounds can be expressed as LMIs, using a similar slack variable approach. The optimization problem then seeks to minimize (resp. maximize) det(S 2 ) (resp. det(S 1 )) to compute an outer (resp. inner) approximation of the minimal (resp. maximal) volume linear low-complexity polytopic RCI set. These approximations are refined iteratively by updating the slack variables that give sufficient condition for linear invariance, to define decreasing (resp. increasing) sequences of linear low-complexity polytopic RCI sets in terms of volume.

The slack variable approach proposed in [START_REF] Tahir | [END_REF]] is also used in [Gupta 2017]. However, a different approach is considered for the handling of the non-linearities in the cost function (volume). The author uses the state transformation θ = Gξ. In these coordinates, the polytope is an hyper-rectangle. This allows to establish a sufficient condition for invariance as an LMI in G and K, in a similar fashion as in [Tahir 2010]. A first algorithm provides a suboptimal linear low-complexity polytopic RCI set by imposing the matrix G to be symmetric (with which det(G) is a convex function). The conservatism of this algorithm comes both from G being symmetric, and the use of a sufficient condition for invariance. An iterative algorithm is then proposed, that computes a sequence of linear invariant polytopes Z(k) of increasing (or decreasing) volume, defined by the sequence {G(k)} k∈N . Instead of restraining the G(k) to be symmetric, the volume increase is guaranteed by imposing G(k + 1) G(k + 1) ≥ G(k) G(k). A sufficient LMI condition on G(k + 1) is derived from this quadratic inequality to ensure the volume increases (or decreases). The sufficient condition for invariance is also updated at each iteration to reduce conservatism, using the same approach as in [START_REF] Tahir | [END_REF]].

The iterative methods presented here allow for the computation of approximations of the minimal or maximal RCI set. The main limitation is the low-complexity polytopic structure assumed for the RCI set. A more complex structure is required to provide a better approximation of these sets, at the expense of an increase in computational complexity.

High H complexity polytopes

We consider high H complexity polytopic sets, that is we do not restrict G to be an invertible square matrix. Instead, G ∈ R l×n , where l ∈ N + . In [Liu 2015] and [START_REF] Gupta | [END_REF]], the matrix G is given by the product G = G 1 G 2 , where G 1 ∈ R l×n , G 2 ∈ R n×n and G 2 is assumed to be invertible. In both papers, G 1 is fixed while G 2 is the design parameter of the polytope. The matrix G 2 allows to scale and rotate the polytope defined by G 1 . As for low-complexity polytopes, the joint optimization of the polytope Z and the feedback gain K makes use of iterative optimizations for the handling of the bilinearities in the Extended Farkas Lemma.

Two measures of the size of a polytope are proposed in [Liu 2015] and [START_REF] Gupta | [END_REF]].

Theoretical background -invariant set design

In the former, the size is given by the radius of the maximal sphere contained in the set. In the later, the size is given by max(det(Q)) among the matrices Q ∈ R n×n such that the associated ellipsoid {ξ ∈ R n | ξ Qξ ≤ 1} is contained in Z. Both papers provide conditions for the inclusion of the respective set that are only sufficient, which induces conservatism.

In [START_REF] Gupta | [END_REF]], the parametrization of G is used to define the state transformation θ = G 2 ξ. In these coordinates, the polytope verifies

Z = {θ ∈ R n | G 1 θ ≤ 1 p } ,
which is a known set. This allows to express the set inclusions Z ⊆ X and KZ ⊆ U as affine constraints on K and G 2 . This choice for the handling of state and input constraints reduces the computational complexity of the iterative approach.

[Gupta 2018] provides a sufficient condition for Z to be linear RCI that consists in matrix inequalities. The only non-linearity is linearized by making use of a slack variable. The optimization problem is iteratively solved, with an update of the slack variable. The update is chosen to guarantee recursive feasibility, as in [Gupta 2017] and [START_REF] Tahir | [END_REF]]. This iterative procedure computes a sequence of polytopic sets of fixed dimensions, parametrized by a fixed matrix G 1 and a sequence of matrices G 2 (k), and of increasing (or decreasing) volume. The approach requires the computation of an initial linear polytopic RCI set Z(0) parametrized by G 1 and G 2 (0). This set is computed by initializing the slack variable chosen as proportional to the identity matrix, with a grid search over the scaling parameter until the optimization problem is feasible. Note that this approach is sensitive on the choice for the initialization (grid search over the scaling of the slack variable), and on the choice for the fixed matrix G 1 . This is less problematic for the design of approximation of the maximal RCI set, as the union of the sets obtained for a collection of initial conditions retains the RCI property. However, it is more problematic for the computation of minimal RCI sets, as the intersection needs to be used in view of the minimal objective. As the intersection of RCI sets does not retain the RCI property, the approach needs to be adopted with precautions in this case.

In [Liu 2015], the nonlinearities of the Extended Farkas Lemma are handled using the Elimination Lemma [START_REF] Boyd | [END_REF]]. This linearization leads to a very conservative sufficient condition for the set Z to be linear RCI. Moreover, it increases consequently the number of design parameters. The solution of the successive optimization problems P (i) is used to update the Elimination Lemma, which iteratively refines the linearization.

Polytopic sets of bounded V complexity

The previous approaches focused on the H complexity of the polytopic set Z. Here, we present an existing approach that focuses on the V complexity of the polytope, that is the number of vertices of its minimal V representation. [Athanasopoulos 2010] studies a linear discrete-time system with polytopic uncertainties on the matrices A and B. Note that the paper does not consider additive disturbances. The goal of this paper is to compute a CI (Control Invariant instead 4.3. Joint parametrization of the RCI set and the control law 123 of Robust Control Invariant, as there are no disturbances) set of maximal volume. The method is based on enlarging an initial CI set by incrementing the number of vertices.

Consider a polytope Z of V complexity f , whose vertices are {v 1 , ..., v f }. The authors show that the polytope characterized by {v 1 , ..., v f , v * } retains the RCI property provided v * ∈ X , and there exists u ∈ U, and non-negative scalars

(λ 1 , ..., λ f , λ * ) ∈ R f +1 + such that Av * + Bu = f j=1 λ j v j + λ * v * , f j=1 λ j + λ * ≤ 1.
(4.9)

This necessary and sufficient condition consists of linear equalities and inequalities, except for the term λ * v * . The set of vertices v * that satisfy the above constraints is possibly infinite, and the author proposes a selection by adding a linear inequality of the form G i v * ≥ (1 + )g i while maximizing , where G i is a chosen row of G. This optimization problem seeks to find the vertex v * that is as far away as possible from the facet G i ≤ g i of the polytope Z. This choice for the selection of vertex v * ensures that if the optimization problem is infeasible for every row of G, then Z is maximal.

The extension of this approach to linear discrete-time systems subject to bounded additive disturbances raises a different nonlinear structure in (4.9) with an additional complexity. For instance, if the disturbance is bounded by a polytope W, the term Av * + Bu is to be replaced by Av * + Bu + w for all vertices w of W to ensure the robust invariance. This requires the computation of non-negative scalars {λ 1 (w), ..., λ f (w), λ * (w)} for each vertex of W. The main issue raises from the card(V (W)) non-linear terms λ * (w)v * , which prevents from using a grid search approach.

This approach yields a sequence of polytope RCI sets of possibly increasing V complexity. Indeed, the maximal V complexity of the enlarged polytope has a complexity of at most f + 1, where f is the V complexity of Z. However, some vertices can become redundant with the addition of a new vertex.

In [START_REF][END_REF]], necessary and sufficient conditions are provided on the selection of the vertex v * such that the enlarged polytope has a specified V complexity. The outer region of Z is partitioned in subregions that consist of vertices v * such that, if added to the polytope Z, the resulting polytope is of specified V complexity. These subregions are characterized as intersection of families of polyhedral cones defined by {ξ ∈ R n | G i ξ ≥ g i }. This complements the previous iterative approach, where the vertex v * can be chosen at each iteration such that the V complexity of the resulting polytopic RCI sets does not exceed a predefined value f * .

Note that, contrarily to the other approaches in this section, the polytopic RCI sets computed here are not necessarily linear RCI sets. That is, this approach allows for non-linear control laws. However, this method can not be used to compute decreasing sequences of RCI sets.

Theoretical background -invariant set design

The approaches presented in this section use a parametrization of the RCI set to establish sufficient conditions for its invariance. They use the H representation of the polytope and a (linear) parametrization of the control law to make use of the Extended Farkas Lemma, which eventually yields a sufficient condition that is bilinear in the matrix G and the feedback gain K. Iterative methods are developed to cope with the bilinearities, defining sequences of candidate solutions of increasing (or decreasing) volume.

Parametrization of families of RCI sets for piece-wise affine control laws

The research initially presented in [Rakovic 2007], and extended in [Rakovic 2010], proposes a characterization of families of RCI sets that does not impose the control law to be linear. To introduce the construction of these families of RCI sets, we consider trajectories initialized at the origin for the discrete-time linear system

ξ[k + 1] = Aξ[k] + Bu[k] + w[k], w[k] ∈ W, ∀k ∈ N, ξ ∈ R n , u ∈ R m ,
(w[0], w[1]) = K(1)w[0] + K(0)w[1]. We obtain ξ[3] = Aξ[1] + B(K(1)w[0] + K(0)w[1]) + w[2]
. This can be rewritten

ξ[3] = (A 2 + ABK(0) + K(1))w[0] + (A + BK(0))w[1] + w[2].
Likewise, the control action is chosen as ν

(w[0], w[1], w[2]) = K(2)w[0]+K(1)w[1]+ K(0)w[2]
, and we have

ξ[4] =(A 3 + A 2 BK(0) + ABK(1) + K(2))w[0] +(A 2 + ABK(0) + ABK(1))w[1] +(A + BK(0))w[2] +w[3].
This procedure can be iterated to define

ξ[k] =(A k-1 + A k-2 BK(0) + ... + ABK(k -2) + BK(k -1))w[0] +(A k-2 + A k-3 BK(0) + ... + ABK(k -3) + BK(k -2))w[1] +... +(A + BK(0))w[k -2] +w[k -1].

Conclusion 125

With this choice for the control policy, the state ξ[k] depends linearly in the parameters {K(0), ..., K(k -1)}. In the following, we denote

K k = {K(j)} j∈N k-1 , ∀k ∈ N + , and A(K k , i) = A k-i + A k-i-1 BK(0) + ... + BK(k -i -1)), ∀k ∈ N + , ∀i ∈ N k-1 .
With these notations, we have

ξ[k] = k-1 i=0 A(K k , i)w[i].
By considering all possible disturbances realizations {w

[0], ..., w[k -1]} ∈ W k , we obtain ξ[k] ∈ k-1 i=0 (A(K k , i))W.
In the following, we denote this set R(K k ). These sets are finite Minkowski sums of polytopes provided W is a polytope. [Rakovic 2010] also provides a sufficient condition for a scaling of the set R(K k ) to be RCI as follows. If the feedback gains {K(j)} j∈N k-1 are such that A(K k , 0)W ⊆ αW with 0 ≤ α < 1, then the set (1 -α) -1 R(K k ) is RCI for the system (4.1) and constraints (R n , R m , W). This sufficient condition is linear in the parameters {K(j)} j∈N k-1 and in α. Moreover, this condition has at least one solution for any scalar 0 ≤ α < 1 when the index k is superior to the controllability index of the pair

(A, B).
For a fixed pair (k ∈ N, 0 ≤ α < 1), and for polytopic state and input constraints, the computation of K k such that (1 -α) -1 R(K k ) is RCI for the system (4.1) and constraints (X , U, W) can be achieved by solving an LP (provided X , U and W are polytopes). However, the LP can have an empty feasible domains, depending on the choice for the parameters k and α, and the sets X , U and W. The criterion This paper establishes the existence of families of RCI sets for which the corresponding control law is nonlinear (generally piecewise affine). An interesting property of these families of RCI sets is that its elements are parametrized linearly by a finite sequence of feedback gains {K(j)} j∈N k . This linear dependecy allows to compute RCI sets that satisfy state and input constraints by solving LPs when the parameters k ∈ N and α are fixed. The main limitation of this approach lies in the tuning of the parameters k and α such that the associated LP has a non-empty feasible domain.

|H(1 -α) -1 R(K k )| p is linear (quadratic)

Conclusion

We have presented an overview of the recent works on the three main approaches for the computation of optimized RCI sets. The first approach imposes the control law to be linear, and compares the size of the resulting mRPI sets using the support function. The second approach uses a parametrization of the RCI set, chosen polytopic, to establish sufficient condition for set invariance. In general, these conditions are bilinear in the design parameters of the polytope. They are locally linearized Theoretical background -invariant set design around successive candidate polytopic RCI set solutions to infinitely refine an initial polytopic RCI set. The main advantage is that the H or V complexity of the RCI set can be limited to reduce the computational complexity. The third and last approach allows for nonlinear control laws to characterize families of polytopic RCI sets, each parametrized by a finite sequence of feedback gains and a scaling factor. For a fixed scaling factor and sequence dimension, these sets depend linearly on the feedback gains, which allows for efficient optimization. However, this approach requires the tuning of these two parameters to obtain optimization problems that have non empty feasible domain in presence of state and input constraints.

Chapter 5 and Chapter 6 present our research on the topic of computing optimized RCI sets. In both chapters, we use the matrix-norm criterion as in (1.10) to measure the size of a bounded RCI set. In Chapter 5, we study the mRPI sets obtained with discrete-time sliding mode control laws with linear reaching laws, which is a particular case of linear control laws. This chapter uses the disturbance rejection properties of the SMC laws. In Chapter 6, we develop two iterative methods for the construction of optimized RCI sets which use respectively the concepts of reachable set and the Extended Farkas Lemma. The first allows for non-linear control laws, while the second imposes the control law to be linear, and both use a polytopic parametrization of the RCI set.

Introduction

This chapter proposes a method for the computation of minimal RCI sets for linear discrete time systems subject to bounded additive disturbances. Here, we assume that the model and disturbance bounds have been identified, and the control law is the only design parameter of the RCI sets. The approach developed here consists of analyzing the structure and size of the invariant sets obtained using sliding mode control, an approach known for its intrinsic disturbance rejection properties, especially in continuous-time.

Continuous-time sliding mode control (CSMC) consists of defining a new output (or sliding variable) y with the same dimension as the input, such that the zero dynamics associated to this output are stable. The subset of the state space characterized by the equation y = 0 is the sliding surface. The system defined by the new output has to be stabilized, defining the so-called reaching law. The reaching law steers and maintains the system on the sliding surface, and the entire mechanism guarantees stability of the closed-loop system. For more information about CSMC, the reader is referred to [Utkin 1993[START_REF] Decarlo | [END_REF], Edwards 1998, Utkin 2013, Bandyopadhyay 2015].

The first transpositions to discrete time Sliding Mode Control (DSMC) emerged in the 80's, see for example [Milosavljevic 1985], [START_REF] Sarpturk | [END_REF]], [Spurgeon 1992] and [Gao 1995]. As for CSMC, the design procedure of a sliding mode control law for a discrete time system (DSMC) is based on two steps. The selection of a new output with the same dimension as the input, defining the sliding surface, and the design of a reaching law. The first step is exactly the same as for the design procedure for CSMC. The design of a reaching law differs in that it is not possible in general to maintain the system on the sliding surface for a discrete time system subject to disturbances. Instead, the control law seeks to steer and maintain the system in a neighborhood of the sliding surface, named Quasi-Sliding Mode Band (QSMB). The QSMB can be characterized in terms of invariant sets, and depends on the choice of the reaching law. Several reaching laws have been considered in the literature, with several properties regarding both the reachability phase and the size of the QSMB. For a discussion on reaching laws, the reader is referred to [Monsees 2003].

We focus here on the construction of minimal RCI sets, and thus on the local behavior of the closed-loop system around the origin. More precisely, we study the properties of the mRPI sets obtained with DSMC laws. Hence, the criterion for the choice of the reaching law will not account for the closed-loop performances during the reachability phase. We first present the framework of DSMC, with the necessary mathematical definitions. Then, we study the geometrical structure of 5.2. Preliminaries 129 the invariant sets for DSMC laws, with a special focus on the mRPI set. We use the unique structure of the DSMC mRPI sets to provide sufficient condition for the formal characterization of mRCI sets for the criterion introduced in Section 1.4.2.2. In the absence of formal characterization, we propose methods for the computation of candidate minimal RCI sets.

Section 5.2 introduces the system and assumptions. The DSMC framework is described in Section 5.3. The characterization of QSMB as RCI sets is proposed in Section 5.4. The geometrical structure of mRPI sets for DSMC laws is studied in Section 5.5. The computation of matrix-norm mRCI sets is discussed in Section 5.6. We provide a simulation analysis of the presented results in Section 5.7. This chapter ends with a conclusion and perspectives for future work.

Preliminaries

In this chapter, we consider the general class of discrete time linear systems subject to additive bounded disturbances as in (1.1) given by

ξ + = Aξ + Bu + w, w ∈ W, (5.1)
where ξ ∈ R n is the state, u ∈ R m is the control input, and w ∈ R n is an exogenous bounded additive disturbance. We assume that the system (5.1) satisfies m ≤ n, and the following assumption. (5.2)

The system (5.2) is in regular form. In the following, we assume that the system (5.1) is already in regular form, that is B = 0 n-m,m I m . Using the structure of the matrix B, we partition the state ξ, the disturbance w, and the matrix A as follows

ξ = ξ B ⊥ ξ B , w = w B ⊥ w B , A = A 11 A 12 A 21 A 22 , (5.3)
where m) , and A 22 ∈ R m×m . The system (5.1) reads

ξ B ⊥ ∈ R n-m , ξ B ∈ R m , A 11 ∈ R (n-m)×(n-m) , w B ⊥ ∈ R n-m , w B ∈ R m , A 11 ∈ R (n-m)×(n-m) , A 12 ∈ R (n-m)×m , A 21 ∈ R m×(n-
ξ + B ⊥ = A 11 ξ B ⊥ + A 12 ξ B + w B ⊥ , ξ + B = A 21 ξ B ⊥ + A 22 ξ B + u + w B .
The results presented in this chapter are obtained under the following assumption.

Invariant set obtained with sliding mode control Assumption 5.2 (Controllability) The pair (A, B) is controllable.

This assumption implies that the pair (A 11 , -A 12 ) is controllable [Edwards 1995].

The system is subject to constraints as in (1.2), characterized set-wise by (ξ, u, w) ∈ (X × U × W) .

(5.4)

Assumption 5.3 (Polytopic disturbance set)
The disturbance set W is polytopic, bounded and contains the origin in its interior.

Here, the model parameters A, B and W are known. Our goal is to compute bounded RCI sets for the system (5.1) and constraints (5.4) that are minimal, either with regards to set inclusion, or to the matrix-norm criterion

|HZ| p min ∈ R + | HZ ⊆ B h p ( ) . (5.5)
for a given matrix H ∈ R h×n , and integer p ∈ N+ . Recall that a set is RCI if it is RPI for a control law ν. In this chapter, we study the mRPI sets obtained with discrete time sliding mode control laws, whose design is detailed below.

Discrete time sliding mode control

This section presents the details of the methodology and the framework for the design of a DSMC law.

Sliding variable, sliding surface, and change of coordinates

As a first step of the control design procedure, let us consider a matrix C = m) , and C B ∈ R m×m an invertible matrix. We define the sliding variable as s = Cξ ∈ R m . Note that this choice for the matrix C leads to a sliding variable s of same dimension as the input u.

C B ⊥ C B , with C B ⊥ ∈ R m×(n-
The sliding surface for a given matrix C is defined as the subspace where the sliding variable s = Cξ is equal to 0 m , namely

S(C) = {ξ ∈ R n | Cξ = 0 m } .
(5.6)

The invertibility of the block matrix C B allows to define the following change of coordinates

ξ = ξ B ⊥ s = Qξ, Q = I n-m 0 n-m,m C B ⊥ C B .
(5.7)

Discrete time sliding mode control 131

With (5.7) in ( 5.1), we obtain

ξ+ = Â ξ + Bu + Qw, w ∈ W, (5.8a)
 = QAQ -1 = Â11 Â12 Â21 Â22 , B = 0 n-m,m C B , Â11 = A 11 -A 12 C -1 B C B ⊥ , Â12 = A 12 C -1 B , (5.8b
)

Â21 = C B A 11 + C B ⊥ A 21 -(C B A 12 + C B ⊥ A 22 )C -1 B C B ⊥ , Â22 = (C B A 12 + C B ⊥ A 22 )C -1 B .
Note that Â21 Â22 = CAQ -1 . The equation (5.8a) reads

ξ + B ⊥ = Â11 ξ B ⊥ + Â12 s + I n-m 0 n-m,m w, w ∈ W s + = Â21 ξ B ⊥ + Â22 s + C B u + Cw, w ∈ W.
This state transformation is adapted to the two steps of the design of a DSMC law, namely

• The design of a sliding surface with stable internal dynamics (i.e. the dynamics of ξ B ⊥ are stable provided s = 0) through the choice of a matrix C,

ξ + B ⊥ = Â11 ξ B ⊥ + I n-m 0 n-m,m w, w ∈ W. (5.9)
• The design of the reaching law u = ν( ξ) that stabilizes the sliding variable dynamics

s + = Â21 ξ B ⊥ + Â22 s + C B ν( ξ) + Cw, w ∈ W.
(5.10)

The design of a stable sliding surface has been widely studied [Spurgeon 1992], and we refer to the main elements needed to the understanding of the principles. The following definition refers to the stable sliding surfaces for the system (5.8b).

Definition 5.1 For any pair of matrices A 11 ∈ R (n-m)×(n-m) , and A 12 ∈ R (n-m)×m as in (5.1), the set of matrices C that define stable sliding surfaces (5.6) is given by

K(A 11 , A 12 ) = C ∈ R m×n | C B ∈ GL n-m , A 11 -A 12 C -1 B C B ⊥ ∈ S m .
We give next a result on the existence of stable sliding surfaces.

Proposition 15 (Existence of stable sliding surfaces) Under Assumption 5.2 the set K(A 11 , A 12 ) is non-empty.

Proof: From Assumption 5.2, we have that the pair (A 11 , -A 12 ) is controllable. Hence, there exists a matrix L ∈ R m×(n-m) such that A 11 -A 12 L ∈ S m . Then, for any invertible matrix R ∈ GL m we have RL R ∈ K(A 11 , A 12 ), which proves that the set of stable sliding surfaces is not empty. This closed-loop system is stable provided the matrix A C4 is Schur. This state representation of the closed-loop system highlights the goal of the linear reaching law, which is to stabilize the dynamics of the sliding variable s, through the selection of a Schur matrix A C4 and the removal of the crossed terms in ξ B ⊥ (bottom-left block). The block triangular superior property of the state matrix will have a pivotal role in the design of minimal invariant sets.

The change of coordinates (5.7) and the associated gain (5.15) with A C4 = 0 m,m were discussed in [Spurgeon 1992] and [Hui 1999]. In these papers the goal was to define stability conditions with regard to the design of the sliding surface and to reject a constant disturbance. Other structure of reaching laws have been studied, see for example [Monsees 2003], where the focus is on the reachability phase. Our contribution is the study of the invariant sets obtained with DSMC laws, and the linear reaching law (5.14) leads to an mRPI set with a remarkable geometrical structure that leads to strong theoretical properties in terms of minimality. As a first step, we study a family of RPI sets for DSMC laws, namely the Quasi-Sliding Mode Bands (QSMB).

Quasi-sliding mode bands

Recall that with the linear reaching law (5.14), the closed-loop dynamics of the sliding variable s satisfy

s + = A C4 s + Cw, w ∈ W.
(5.17)

The presence of disturbance prevents us from maintaining the system on the sliding surface (i.e. s = 0). Instead, the reaching law steers and maintains the sliding variable in a neighborhood of the origin, called Quasi-Sliding Mode Band.

We propose in the following a characterization of QSMB that unifies the DSMC framework and invariant set theory. Definition 5.2 (Quasi-Sliding Mode Band) Consider the closed-loop system (5.16), with K as in (5.15)

, C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . A set S(C, V) = {ξ ∈ R n | s = Cξ ∈ V} ,
where V ⊆ R m is an RPI set for the system (5.17) and constraints (R m , W), is called a quasi-sliding mode band (QSMB) for the system (5.16).

In particular, we have the following relation CS(C, V) = V. We also have the following properties regarding the invariance property of a QSMB, and the intersection of QSMBs.

Proposition 16 (Set invariance of a QSMB) Consider the closed-loop system (5.16), with K as in (5.15), C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . If V is an RPI set for the system (5.17) and constraints (R m , W), then the QSMB S(C, V) is an RPI set for the closed-loop system (5.16) and constraints (R n , W).

Invariant set obtained with sliding mode control

Proof: Let ξ ∈ S(C, V). From the definition of a QSMB, we have s = Cξ ∈ V. Given that V is an RPI set for system (5.10), we have

Cξ + = s + = A C4 s + Cw ∈ V, ∀w ∈ W. Hence, ξ + ∈ S(C, V), ∀w ∈ W.
Proposition 17 (Intersection of QSMBs) Consider the closed-loop system (5.16), with K as in (5.15), C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . Let V 1 and V 2 be RPI sets for the system (5.17) and constraints (R m , W). The set

S(C, V 1 ∩ V 2 ) = {ξ ∈ R n | s = Cξ ∈ V 1 ∩ V 2 }
is a QSMB for the system (5.16).

Proof: It follows from the fact that the intersection of RPI sets is an RPI set. This allows us to introduce the concept of minimal QSMB in terms of set inclusion.

Definition 5.3 (minimal QSMB) Consider the closed-loop system (5.16), with K as in (5.15), C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . The minimal Quasi-Sliding Mode Band (mQSMB) is defined as the (possibly infinite) intersection of all the QSMBs, and is denoted S ∞ (C, A C4 ).

The mQSMB can be characterized set-wise using the mRPI set of the system (5.17).

Proposition 18 (Characterization of the mQSMB) Consider the closed-loop system (5.16), with K as in (5.15), C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . Let V ∞ (A C4 ) be the mRPI set for the system (5.17) and constraints (R m , W), given by the infinite Minkwoski sum

V ∞ (A C4 ) = ∞ i=0 A i C4 CW. The mQSMB satisfies S ∞ (C, A C4 ) = {ξ ∈ R n | s = Cξ ∈ V ∞ (A C4 )} .
We used the DSMC framework with linear reaching laws to characterize a family of unbounded RCI sets for the system (5.1) and constraints (R n , R m , W). These RCI sets are parametrized set-wise by the matrices C ∈ K(A 11 , A 12 ) ⊆ R m×n and

A C4 ∈ S m .
As a next step, we study the mRPI sets for linear DSMC laws.

Characterization of the mRPI set for DSMC law

In this section, we analyze the geometrical properties of another family of RCI sets, namely the mRPI sets obtained with linear DSMC laws. In particular, we study the image of these sets by the associated matrices C.

General expression

Let K ∈ R m×n be a DSMC feedback gain as in (5.15), with C ∈ K(A 11 , A 12 ) and

A C4 ∈ S m ⊂ R m×m .
The closed-loop system in the coordinates ξ = Qξ reads

ξ+ = Â11 Â12 0 m,n-m A C4 ξ + Qw, w ∈ W.
The mRPI set in these coordinates, denoted Ẑ∞ (K), satisfies

Ẑ∞ (K) = ∞ i=0 Â11 Â12 0 m,n-m A C4 i QW.
(5.18)

We have the following expression of the i th power of the above triangular matrix

Â11 Â12 0 m,n-m A C4 0 = I n , (5.19a)
Â11 Â12 0 m,n-m A C4 i = Âi 11 j=i-1 j=0 Âj 11 Â12 A i-j-1 C4 0 m,n-m A i C4 , ∀i ∈ N + . (5.19b)
These considerations lead to the following result on the geometrical structure of the mRPI sets obtained with linear DSMC laws.

Proposition 19 (Characterization of the mRPI set) Consider the closed-loop system (5.16), with K as in (5.15), C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . The mRPI set Z ∞ (K) is given by

Z ∞ (K) = W ⊕ Q -1 ∞ i=0 Âi 11 j=i-1 j=0 Âj 11 Â12 A i-j-1 C4 0 m,n-m A i C4 QW .
(5.20)

Proof: With (5.19) in (5.18) the mRPI set in the coordinates ξ is given by

Ẑ∞ (K) = QW ⊕ ∞ i=1 Âi 11 j=i-1 j=0 Âj 11 Â12 A i-j-1 C4 0 m,n-m A i C4 QW.
Hence, the mRPI set in the initial coordinates ξ is given by

Z ∞ (K) = Q -1 Ẑ∞ (K) = W ⊕ Q -1 ∞ i=1 Âi 11 j=i-1 j=0 Âj 11 Â12 A i-j-1 C4 0 m,n-m A i C4 QW .
(5.21)

As a direct consequence, we have the following result regarding the image of the mRPI set by the matrix C.

Proposition 20 (Image of the mRPI set) Consider the closed-loop system (5.16), with K as in (5.15), C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . The image of the mRPI set Z ∞ (K) by the matrix C is given by

CZ ∞ (K) = ∞ i=0 A i C4 CW. (5.22)
Proof: As a first step, we note that

CQ -1 = C B ⊥ C B I n-m 0 n-m,m -C -1 B C B ⊥ C -1 B = 0 n-m,m I m . (5.23)
With (5.23) in ( 5.20), we have

CZ ∞ (K) = CW ⊕ 0 n-m,m I m ∞ i=1 Âi 11 j=i-1 j=0 Âj 11 Â12 A i-j-1 C4 0 m,n-m A i C4 QW = CW ⊕ ∞ i=1 0 n-m,m A i C4 QW.
We conclude with

0 n-m,m A i C4 Q = 0 n-m,m A i C4 I n-m 0 n-m,m C B ⊥ C B = A i C4 C, ∀i ∈ N + .
This set is parametrized by the matrices involved in the control law, namely C ∈ K(A 11 , A 12 ) and A C4 ∈ S m . Furthermore, for a given matrix C ∈ K(A 11 , A 12 ), the set CZ ∞ (K) is minimal in terms of set inclusion for A C4 = 0 m,m . This choice of A C4 corresponds to

K = -(CB) -1 CA, (5.24)
and the following closed-loop dynamics of the sliding variable

s + = Cw, w ∈ W.
In the following, we call 0-DSMC a linear DSMC law as in (5.14) with the particular choice A C4 = 0 m,m , and the next section focuses on the mRPI properties in this case.

mRPI set for a 0-DSMC law

Let us consider the closed-loop system (5.16) with K as in (5.24) and C ∈ K(A 11 , A 12 ). The expression of the mRPI set in (5.20) is simplified due to the choice A C4 = 0 m,m . We have

j=i-1 j=0 Âj 11 Â12 A i-j-1 C4 = Âi-1 11 Â12 , ∀i ∈ N + .
(5.25)

With (5.25) in (5.20), the mRPI set is given by

Z ∞ (K) = W ⊕ Q -1 ∞ i=1 Âi 11 Âi-1 11 Â12 0 m,n-m 0 m,m QW = W ⊕ Q -1 ∞ i=1 Âi-1 11 Â11 Â12 0 m,n-m 0 m,m QW.
(5.26)

The choice A C4 = 0 m,m allows to express each term of the infinite Minkowski sum as Cartesian products

Â11 Â12 0 m,n-m 0 m,m QW = Â11 Â12 QW × {0 m }.
(5.27)

We note that the matrix Â11 Â12 corresponds to the first n -m rows of the matrix  = QAQ -1 , and

I n-m 0 n-m,m Q = I n-m 0 n-m,m
. With these considerations, we have

Â11 Â12 = I n-m 0 n-m,m QAQ -1 = I n-m 0 n-m,m AQ -1 .
By right multiplying this relation by the matrix Q, we obtain

Â11 Â12 Q = I n-m 0 n-m,m A = A 11 A 12 .
(5.28) With (5.27) and (5.28) in (5.26), we obtain

Z ∞ (K) = W ⊕ Q -1 ∞ i=1 Âi-1 11 A 11 A 12 W × {0 m } = W ⊕ I n-m 0 n-m,m -C -1 B C B ⊥ C -1 B ∞ i=1 Âi-1 11 A 11 A 12 W × {0 m } = W ⊕ I n-m -C -1 B C B ⊥ ∞ i=1 Âi-1 11 A 11 A 12 W = W ⊕ I n-m -C -1 B C B ⊥ ∞ i=0 Âi 11 A 11 A 12 W.
The mRPI set Z ∞ (K) is the sum of two terms. The first is the disturbance set W.

The second is the image of the infinite Minkowski sum ∞ i=0 Âi 11 A 11 A 12 W by the matrix

I n-m -C -1 B C B ⊥
. Note that this matrix satisfies the remarkable property

C I n-m -C -1 B C B ⊥ = C B ⊥ -C B ⊥ = 0 m,n-m .
(5.29)

Invariant set obtained with sliding mode control

The infinite Minkowski sum is the mRPI set of the closed-loop system

x + = Â11 x + A 11 A 12 w, w ∈ W.
(5.30)

We denote Z∞ ( Â11 ) = ∞ i=0 Âi 11 A 11 A 12 W in the following. The mRPI set for the closed-loop system (5.16) verifies

Z ∞ (K) = W ⊕ I n-m -C -1 B C B ⊥ Z∞ ( Â11 ).
(5.31)

The matrix C ∈ K(A 11 , A 12 ) plays two roles in the design of the mRPI set.

First, it defines the matrix

I n-m -C -1 B C B ⊥
which maps the infinite Minkowski sum, and orientates the mRPI set. Second, it impacts the size of the infinite Minkowski sum via the successive powers of the matrix

Â11 = A 11 -A 12 C -1 B C B ⊥ .
We use this double role of the matrix C for the formal characterization of minimal RCI sets in Section 5.6, both in terms of set inclusion and matrix-norm. In absence of a formal characterization, we provide methods for the computation of candidate solutions.

These methods require to evaluate the criterion (5.5) for a range of 0-DSMC mRPI sets as in (5.31), thus relying on the computation of RPI outer approximations.

Computation of RPI outer approximations of 0-DSMC mRPI sets

We presented methods from the literature for the computation of polytopic RPI ( , p) outer approximations of mRPI sets for linear control laws in Section 1.4.1.3. The computational complexity of these approximations, both in terms of computation time and V-complexity, scales exponentially with the dimension of the state (Proposition 3). In this section, we use the geometrical structure of mRPI sets obtained with 0-DSMC control laws to reduce this computational complexity. We consider a feedback gain K as in (5.24), with C ∈ K(A 11 , A 12 ). The associated mRPI set is given by (5.31), where Z∞ ( Â11 ) is the mRPI set of the closed-loop system (5.30), whose state is of dimension n -m. We use approximations of the mRPI set Z∞ ( Â11 ) to construct approximations of the mRPI set Z ∞ (K). Indeed, the computation of approximations of the former is less computationally demanding, as the dimension of (5.30) (n -m compared to n).

Let Z be an RPI set for the system (5.30), that is

Â11 Z ⊕ A 11 A 12 W ⊆ Z,
and define W ⊕ I n-m -C -1 B C B ⊥ Z.
The following proposition provides conditions for this set to be RPI for the closed-loop system (5.16) and constraints (R n , W).

Characterization of the mRPI set for DSMC law 139

Proposition 21 (Set invariance) Consider the closed-loop system (5.16), with K as in (5.24) and C ∈ K(A 11 , A 12 ). Let Z be an RPI set for the closed-loop system

(5.30) and constraints (R m-n , W). Then, the set

W ⊕ I n-m -C -1 B C B ⊥
Z is an RPI set for the closed-loop system (5.16) and constraints (R n , W).

Proof: Let ξ ∈ W ⊕ I n-m -C -1 B C B ⊥ Z.
By definition, there exists w ∈ W and

x ∈ Z such that ξ = w + I n-m -C -1 B C B ⊥
x. We have

ξ B ⊥ = I n-m 0 n-m,m w + x, ξ B = 0 m,n-m I m w + C -1 B C B ⊥ x, s = Cξ = Cw + C I n-m -C -1 B C B ⊥ x = Cw. Let w 2 ∈ W. Let us prove that ξ + = (A + BK)ξ + w 2 ∈ W ⊕ I n-m -C -1 B C B ⊥ Z. First, we note that s + = Cw 2 . The dynamics of the state ξ B ⊥ read ξ + B ⊥ = Â11 ξ B ⊥ + Â12 s + I n-m 0 n-m,m w 2 = Â11 ( I n-m 0 n-m,m w + x) + Â12 Cw + I n-m 0 n-m,m w 2 = Â11 x + ( Â11 0 n-m,m + Â12 C)w + I n-m 0 n-m,m w 2 . (5.32) With Â12 = A 12 C -1 B (from (5.8b)), we have Â12 C = A 12 C -1 B C B ⊥ A 12 . (5.33) With Â11 = A 11 -A 12 C -1 B C B ⊥ (from (5.8b
)) and (5.33), the equation ( 5.32) reads

ξ + B ⊥ = Â11 x + A 11 A 12 w + I n-m 0 n-m,m w 2 . We use ξ + B ⊥ and s + to evaluate ξ + B as follows using s + = Cξ + , ξ + B = C -1 B (-C B ⊥ ξ + B ⊥ + Cs + ) = -C -1 B C B ⊥ ( Â11 x + A 11 A 12 w) -( C -1 B C B ⊥ 0 n-m,m + C -1 B C)w 2 . The second term reads ( C -1 B C B ⊥ 0 n-m,m + C -1 B C)w 2 = ( -C -1 B C B ⊥ 0 n-m,m + C -1 B C B ⊥ I m )w 2 = 0 m,n-m I m w 2 .
The full state ξ + is given by

ξ + = I n-m -C -1 B C B ⊥ ( Â11 x + A 11 A 12 w) + I n-m 0 n-m 0 m,n-m I m w 2 .
The set Z is RPI for the system (5.30) and constraints (R n-m , W). Given that w ∈ W and x ∈ Z, we have

Â11 x+ A 11 A 12 w ∈ Z. Hence, ξ + ∈ W ⊕ I n-m -C -1 B C B ⊥ Z.
This result can be extended in the presence of polytopic state and input constraints as in (5.4), where the sets X and U are polytopic. We denote their minimal H representation as follows

X = {ξ ∈ R n | G x ξ ≤ g x } , G x ∈ R lx×n , l x ∈ N + , (5.34a)
U = {u ∈ R m | G u u ≤ g u } , G u ∈ R lu×m , l u ∈ N + .
(5.34b)

The closed-loop system (5.16) satisfies the constraints (5.4) provided ξ ∈ X K with

X K = ξ ∈ R n | G x G u K ξ ≤ g x g u . (5.35) We have W ⊕ I n-m -C -1 B C B ⊥ Z ⊆ X K if and only if I n-m -C -1 B C B ⊥ Z ⊆ X K W. Note
that the Pontryagin difference X K W is given by (see e.g. [Kolmanovsky 1998b])

X K W = ξ ∈ R n | G x G u K ξ ≤ g x g u -g X K (W) , (5.36)
where the i th row of g X K (W) ∈ R lx+lu is given by

(g X K ) i (W) = max w∈W G x G u K i w = max w∈V (W) G x G u K i w,
and

G x G u K i denotes the i th row of the matrix G x G u K . Define the polytope XK = x ∈ R n-m | H x H u K I n-m -C -1 B C B ⊥ x ≤ g x g u -g X K (W) .
(5.37)

These considerations allow us to provide the following result regarding set invariance and constraints satisfaction.

Proposition 22 (Constraints handling) Consider the closed-loop system (5.16), with K as in (5.24) and C ∈ K(A 11 , A 12 ). Consider the polytopic sets X K and XK as in (5.35) and (5.37). Let Z be an RPI set for the closed-loop system (5.30)

and constraints ( XK , W). Then, the set

W ⊕ I n-m -C -1 B C B ⊥
Z is an RPI set for the system (5.16) and constraints (X K , W).

Characterization of the mRPI set for DSMC law 141

To prove the result, one needs to use virtual polytopic state constraints on the closed-loop system (5.30) to ensure that the set

W ⊕ I n-m -C -1 B C B ⊥
Z satisfies the actual system state and input constraints.

As a last step, we are interested in the relation between the quality of the approximation of Z∞ ( Â11 ) and the quality of the resulting approximation of Z ∞ (K). For a given p ∈ N+ , let us introduce

η p = max I n-m -C -1 B C B ⊥ x p , x ∈ B n-m p (1) ∈ R + .
(5.38)

Theorem 5.1 (Construction of RPI outer approximation) Consider the closed-loop system (5.16), with K as in (5.24) and C ∈ K(A 11 , A 12 ). Consider the polytopic sets X K and XK as in (5.35) and (5.37). For p ∈ N+ , > 0 and η p defined by (5.38), let Z be an RPI η p , p -outer approximation of the mRPI set for the system (5.30) and constraints ( XK , W). Then the set W ⊕

I n-m -C -1 B C B ⊥
Z is an RPI ( , p)outer approximation the mRPI set for the closed-loop system (5.16) and constraints (X K , W).

Proof: The set inclusion W ⊕ I n-m -C -1 B C B ⊥ Z ⊆ X K comes from Proposition 22. The invariance property of W ⊕ I n-m -C -1 B C B ⊥
Z is inherited from the invariance property of Z, as shown in Proposition 21. From the definition of an η p , p -outer approximation, we have

Z∞ ( Â11 ) ⊆ Z ⊆ Z∞ ( Â11 ) ⊕ B n-m p η p . (5.39)
We can multiply this set inclusion by the matrix

I n-m -C -1 B C B ⊥ to obtain I n-m -C -1 B C B ⊥ Z∞ ( Â11 ) ⊆ I n-m -C -1 B C B ⊥ Z ⊆ I n-m -C -1 B C B ⊥ ( Z∞ ( Â11 ) ⊕ B n-m p η p ).
With (5.31), we can sum W to (5.39) to obtain

Z ∞ (K) ⊆ W ⊕ I n-m -C -1 B C B ⊥ Z ⊆ Z ∞ (K) ⊕ I n-m -C -1 B C B ⊥ B n-m p η p .
By definition of η p , we have

I n-m -C -1 B C B ⊥ B n-m p η p ⊆ η p η p B n p (1) ⊆ B n p ( ).
The previous result states that we construct an RPI ( , p) outer approximation of the mRPI set for a 0-DSMC control law for a system with a state dimension of n by computing an RPI η p , p -outer approximation of the mRPI set of a system with a state dimension of n -m. This allows to reduce significantly the computational complexity, as the V-complexity of the polytopic mRPI approximations computed with the methods presented in Section 1.4.1.3 scales exponentially with the state dimension. From a different perspective, if the control design has as ultimate objective the manipulation of the mRPI set of the closed-loop system, then the 0-DSMC design should be privileged at least for the computation advantages it offers for the characterization of the mRPI sets.

We studied the mRPI sets obtained with linear DSMC laws, and provided a novel expression of the mRPI set that showcases its unique geometrical structure. We then focused on the particular case of 0-DSMC laws, where the only design parameter is the matrix C. We provided an expression of the mRPI set that highlight the double role of the matrix C, which orientates and scales the set. The mRPI set is characterized using the mRPI set of a virtual system with a state dimension of n -m. This allows to reduce significantly the complexity in the computation of mRPI approximations. This reduction of computational complexity is illustrated in simulation in Section 5.7.2. This is of particular interest in the larger goal of computing (H, p)-mRCI sets, as discused in the following section.

Computation of minimal RCI sets using DSMC

The structures of the mRPI sets obtained with 0-DSMC laws presented in the previous section are exploited for our goal of computing minimal RCI sets. We provide conditions for the formal characterization of matrix-norm mRCI sets. In the absence of formal characterization, we propose methods for the computation of candidate solutions.

Minimal RCI sets for set inclusion

As a first step, we consider set inclusion as the criterion to measure the optimality of an RCI set. We recall below the definition of a minimal RCI set in terms of set inclusion.

Definition 5.4 (mRCI set) A set Z is a minimal Robust Control Invariant (mRCI) set for the system (5.1) and constraints (5.4) if it is an RCI set for the system (5.1) and constraints (5.4) and if it does not (strictly) contain any other RCI set for the system (5.1) and constraints (5.4).

We can establish the following result regarding the minimality of mRPI sets for 0-DSMC laws in terms of set inclusions.

Proposition 23 (mRCI sets in terms of set inclusion) Consider the closed-loop system (5.16), with K as in (5.24), C ∈ K(A 11 , A 12 ). The set Z ∞ (K) is an mRCI set for the system (5.1) and constraints (R n , R m , W).

Proof: Let Z be an RCI set for the system (5.1) and constraints (R n , R m , W), and suppose Z ⊂ Z ∞ (K). By definition, ∀ξ ∈ Z, there exists u ∈ R m such that

Aξ + Bu + w ∈ Z ⊆ Z ∞ (K), ∀w ∈ W.

This yields

CAξ + CBu + Cw ∈ CZ ⊆ CZ ∞ (K) = CW, ∀w ∈ W.
This inclusion can be rewritten {CAξ + CBu} ⊕ W ⊆ W. For every element ξ ∈ Z, the control input is given by u = -(CB) -1 CAξ = Kξ. That is, the set Z is an RPI set for the closed-loop system (5.16). Hence, it contains the mRPI set Z ∞ (K), and we have Z ∞ (K) = Z, which is a contradiction. We conclude that the RCI set Z ∞ (K) does not contain any other RCI set as a strict subset.

We have shown that the mRPI sets for 0-DSMC laws are minimal RCI sets for the system (5.1) and constraints (R n , R m , W) in terms of set inclusion. In addition, if the set Z ∞ (K) satisfies Z ∞ (K) ⊆ X K then it is an mRCI set for the system (5.1) and constraints (X , U, W).

Minimal RCI sets in terms of matrix-norm

The definition of minimality of RCI sets that uses set inclusion is not entitling to strong result regarding the uniqueness of mRCI sets in general. Indeed, set inclusion is a partial order on the collection of RCI sets, and thus some pairs of elements cannot be compared. For this reason, we introduced the matrix-norm criterion (5.5) to measure the size of bounded RCI sets with regards to a norm p and a matrix

H ∈ R h×n , h ∈ N + .
We recall below the definition of an (H, p)-mRCI set.

Definition 5.5 ((H,p) minimal Robust Control Invariant set) A set Z is an (H, p) minimal Robust Control Invariant ((H, p)-mRCI) set for the system (5.1) and constraints (5.4) if it is an mRCI set for the system (5.1) and constraints (5.4) and if it minimizes (5.5).

We partition the matrix H as -m) and

H = H B ⊥ H B , where H B ⊥ ∈ R h×(n
H B ∈ R h×m .
Consider the closed-loop system (5.16), with K as in (5.24) and C ∈ K(A 11 , A 12 ). With the above partition of H and (5.31), we have

HZ ∞ (K) = HW ⊕ H I n-m -C -1 B C B ⊥ ∞ i=0 Âi 11 A 11 A 12 W.
(5.40)

The results presented in the following depend on the choice for the matrix H.

Matrix H defining a stable sliding surface and formal characterization

The first results are obtained under the following assumption. Under this assumption, the matrix H can be used to define a stable sliding surface.

With C = H in (5.40), we have (5.41) This allows to establish the following result.

HZ ∞ (K) = HW ⊕ H I n-m -H -1 B H B ⊥ ∞ i=0 Âi 11 A 11 A 12 W = HW.
Proposition 24 (Formal characterization of (H, p)-mRCI sets) Consider Assumption 5.4 and the closed-loop system (5.16), with K as in (5.24) and C = H. The set Z ∞ (K) is an (H, p)-mRCI set for the system (5.1) and constraints (R n , R m , W).

Proof: From (5.41), we have HZ ∞ (K) = HW. Let Z be an RCI set for the system (5.1) and constraints (R n , R m , W). By definition of an RCI set, we have W ⊆ Z and thus HW ⊆ HZ. Hence, the set inclusion HZ ∞ (K) ⊆ HZ holds. We conclude with

|HZ ∞ (K)| p = sup {|Hξ| p | ξ ∈ Z ∞ (K)} = sup {|y| p | y ∈ HZ ∞ (K)} ≤ sup {|y| p | y ∈ HZ} = |HZ| p .
Hence, Z ∞ (K) minimizes the criterion |HZ| p . Moreover, Z ∞ (K) is an mRCI set for the system (5.1) and constraints (R n , R m , W) from Proposition 23.

We have established a criterion on the matrix H for the formal characterization of (H, p)-mRCI sets in the absence of state and input constraints. That result is particularly important as it characterize the conditions under which Problem 1.1 can be solved analytically for the system (5.1) and constraints (R n , R m , W) under Assumption 5.4. Note that the condition for formal characterization, and the analytical solution, do not depend on the choice for the norm p ∈ N+ .

In the presence of constraints, the set Z ∞ (K) has to satisfy the set inclusion Z ∞ (K) ⊆ X K . Hence, we have an analytical solution to Problem 1.1 for the system (5.1) and constraints (5.4) under Assumption 5.4 provided Z ∞ (K) ⊆ X K .

Computation of candidate mRCI sets

We do not have a formal characterization of (H, p)-mRCI sets when Assumption 5.4 does not hold. The fact that H ∈ K(A 11 , A 12 ) prevents us from using the feedback gain K as in (5.15) with C = H since the matrix A + BK is not Schur. Hence, the desirable property of existence of an analytical solution to Problem 1.1 is lost. The general approach is to consider an alternative problem, with a parametrization of the control law. In Section 1.4.2.3, Problem 1.2 consists of restraining the control law to be linear ν(ξ) = Kξ, K ∈ R m×n . Recall that the associated optimization problem reads minimize

K∈R m×n |HZ ∞ (K)| p , subject to Z ∞ (K) = ∞ i=0 (A + BK) i W, Z ∞ (K) ⊆ X , KZ ∞ (K) ⊆ U, A + BK ∈ S n .
A solution of this alternative problem leads to a candidate solution for the original problem of computing (H, p)-mRCI sets. Despite the compact formulation, this optimization problem remains intractable, due to Z ∞ (K) being an infinite Minkowski sum.

Here, we chose to further restrict the class of linear controllers to the subclass of 0-DSMC linear controllers ν(ξ) = -(CB) -1 CAξ, where the design parameter is the matrix C. The associated optimization problem reads minimize

C∈R m×n |HZ ∞ (C)| p , (5.43a) subject to Z ∞ (C) = W ⊕ I n-m -C -1 B C B ⊥ ∞ i=0 Âi 11 A 11 A 12 W, (5.43b) Z ∞ (C) ⊆ X , KZ ∞ (C) ⊆ U, (5.43c)
C ∈ K(A 11 , A 12 ), (5.43d)
Â11 = A 11 -A 12 C -1 B C B ⊥ . (5.43e)
Note that this optimization problem has the same dimension of the argument space. The optimization per se remains intractable despite the parametrization of the control law. In the following we propose methods for the computation of candidate solutions to the optimization problem (5.43) by upper-bounding the cost function, relying on the geometrical structure of the mRPI set (5.43b).

Matrix defining an unstable sliding surface

As a first step, we consider that the matrix H defines an unstable sliding surface.

HZ ∞ (K) = HW ⊕ H I n-m -H -1 B C B ⊥ ∞ i=0 Âi 11 A 11 A 12 W (5.44) = HW ⊕ (H B ⊥ -C B ⊥ ) ∞ i=0 Âi 11 A 11 A 12 W. (5.45) Recall that Â11 = (A 11 -A 12 H -1 B C B ⊥ ).
The term HW does not depend on the matrix C. To minimize |HZ ∞ (K)| p , the cost function of (5.43), we seek the matrix C that minimizes From the definition of matrix norms, we have ∀i ∈ N

(H B ⊥ -C B ⊥ ) ∞ i=0 Âi 11 A 11 A 12 W p . ( 5
Âi 11 A 11 A 12 W p = max Âi 11 A 11 A 12 w p | w ∈ W ≤ Âi 11 p max A 11 A 12 w p | w ∈ W ≤ Âi 11 p A 11 A 12 W p .
(5.47)

We use the Gelfand's formula to have an asymptotic equivalent of the p norm of Âi 11 using the spectral radius of Â11 as follows,

| Âi 11 | 1/i p → i→∞ ρ( Â11 ).
For this reason, we propose to compute candidate solutions for the minimization of (5.46) by

• upper-bounding the spectral radius of Â11 , thus asymptotically bounding (5.47),

• minimizing |H B ⊥ -C B ⊥ | p ,
relying on the inequality

(H B ⊥ -C B ⊥ ) ∞ i=0 Âi 11 A 11 A 12 W p ≤ |H B ⊥ -C B ⊥ | p ∞ i=0 Âi 11 A 11 A 12 W p
We now briefly discuss the upper-bounding of the spectral radius of the matrix Â11 . Let λ ∈]0, 1[, and define the matrix -m) .

M = λ -1 Â11 ∈ R (n-m)×(n
The matrix M is Schur if and only if ρ(M ) < 1, that is ρ( Â11 ) < λ.
Moreover, we have the following relationship on the spectral radius of the matrices Â11 and M ,

ρ( Â11 ) = λρ(M ).
That is, ρ( Â11 ) < λ if and only if the matrix M is Schur. We use the Schur-Cohn criterion (see [Vieira 1977] and [START_REF] Keel | [END_REF]) to express a necessary and sufficient condition for the matrix M to be Schur, as presented below. The characteristic polynomial of the matrix M ∈ R (n-m)×(n-m) is given by

det(qI n-m -M ) = q n-m + α 1 q n-m-1 + ... + α n-m-1 q + α n-m ,
where α j = (-1) j m j (M ), ∀j = {1, ..., n -m} and m j (M ) is the sum of the n j principal minors of order j of M . These principal minors are polynomial in the elements of C B ⊥ . Let us define the following matrix

Σ =     a 0 . . . . a n-m-1 . . a 0         a 0 . . a n-m-1 . . . . a 0     -     a n-m . . . . a 1 . . a n-m         a n-m . . a 1 . . . . a n-m     .
It can be shown that the matrix M is Schur if and only if the matrix Σ is positive definite [Vieira 1977]. Moreover, the matrix Σ is positive definite if and only if its nm principal leading minors are strictly positive [START_REF] Keel | [END_REF]]. These principal leading minors are polynomial in the m j (M ), j ∈ {1, ..., n -m}, and thus are polynomial in the coefficients of M , and thus on the elements of the matrix C B ⊥ . We denote p λ (C B ⊥ ) the n -m dimensional vector of those n -m polynomials. With these considerations, the matrix M is Schur if and only if

p λ (C B ⊥ ) > 0 n-m .
For a fixed scalar λ ∈]0, 1[, we propose to compute the matrix C B ⊥ that is the solution of the following optimization problem minimize

C B ⊥ ∈R m×(n-m) |H B ⊥ -C B ⊥ | p , (5.48a)
subject to

p λ (C B ⊥ ) > 0 n-m . (5.48b)
This optimization problem is tractable, as we have an explicit characterization of all the elements. However, the cost function and the constraints are possibly nonconvex in the elements of C B ⊥ . We denote C B ⊥ (λ) a solution of (5.48), and define

C(λ) = C B ⊥ (λ) H B , K(λ) = -(C(λ)B) -1 C(λ)A
, and Z ∞ (K(λ)) the associated mRPI set.

Note that the optimization problem (5.48) does not account for the presence of state or input constraints. In the original optimization problem, the constraints (5.43c) used the set Z ∞ (K), of which we do not have an explicit characterization.

The relation between the criterion to minimize, |HZ ∞ (K(λ))| p , and the scalar λ is complex (nonlinear, non-analytic and whose feasibility needs to be evaluated independently). We propose a grid search for the parameter λ over the interval 0 < λ < 1. For each scalar λ in this range, we compute an approximation of Z ∞ (K(λ)), and we choose value that minimizes the cost |HZ ∞ (K(λ))| p while satisfying the state and input constraints. The mRPI approximations are computed using the results presented in Section 5.5 to reduce the computational complexity. The overall approach is described by means of the Algorithm 5.1, where we choose q candidate values for the scalar λ, denoted {λ 1 , ..., λ q }.

To resume, we proposed a method for the computation of candidate (H, p)-mRCI sets under the assumption that the matrix H defines an unstable sliding surface. We restrain to 0-DSMC laws, where the design parameter is the matrix C B ⊥ ∈ R m×n . The matrix is chosen as the solution of an alternative optimization problem parametrized by a scalar λ that bounds the spectral radius of the matrix Â11 . We perform a grid search for the parameter λ, and compute ( , p)-outer approximations of the resulting mRPI sets. Then, we retain the approximation that is minimal with regards to the matrix-norm criterion (5.5).

The proposed approach is suitable for matrices H with h = m rows. This restriction is relaxed in the following section, where we allow the number of rows to be higher than m the dimension of the input.

Matrix of higher dimension

We now enlarge the scope and device a strategy for the computation of candidate (H, p)-mRCI sets for the case when the previous assumptions are relaxed and under the assumption that the number of rows of the matrix H is strictly superior to m. Compute C B ⊥ as the solution of (5.48);

Set C ← C B ⊥ H B and Â11 ← A 11 -A 12 H -1 B C B ⊥ ;
Compute η p as in (5.38);

Compute Z as an RPI η p , p outer approximations of Z∞ ( Â11 );

Compute K = -(CB) -1 CA and ZK as in (5.37); The proposed approach consists in exploiting the previous results for all the possible m × n submatrices of H, and to conclude by comparing the size of the associated sets with regards to the full matrix H.

if Z ⊆ ZK then 10 Compute Z ← W ⊕ I n-m H -1 B C B ⊥ Z; 11 if |HZ| p ≤ V
Let us introduce the following definition regarding k-combination.

Definition 5.6 (k-combination) A k-combination of a set I is a subset of k distinct elements of I.

For a set I containing p elements, there exists p k distinct k-combinations of the set I.

Here, we consider m-combinations over the rows of the matrix H, which consists of h elements. Hence, there exists h m of these m-combinations. In the following, we denote {σ r } r∈N + ( h m )

these m-combinations, and

{H(σ r ) ∈ R m×n } r∈N + ( h m )
the vertical concatenation of the m-combination of rows. We partition these matrices as follows,

H(σ r ) = H(σ r ) B ⊥ H(σ r ) B , ∀r ∈ N + ( h m )
, where -m) and

H(σ r ) B ⊥ ∈ R m×(n
H(σ r ) B ∈ R m .
We consider the matrices H(σ r ) with an invertible block H(σ r ) B .

More formally, we define the set

H = H(σ r ), r ∈ N + ( h m ) | H(σ r ) B ∈ GL m .
Every matrix H(σ r ) ∈ H satisfies either Assumption 5.4 (stable sliding surface) or Assumption 5.5 (unstable sliding surface). For each matrix, we compute an (H(σ r ), p)-mRCI set using the results in Section 5.6.2.1 (formal characterization) or the method presented in Section 5.6.2.2 (candidate solution).

The feedback gain K and the mRPI set Z ∞ (K) obtained for a particular arrangementH(σ r ) are respectively denoted K(σ r ) and Z ∞ (K(σ r )). We define the set

Z(H) = {Z ∞ (K(σ r )) | H(σ r ) B ∈ H} .
In the presence of state or control input constraints defined by the original sets X and U, we consider instead

Z(H, X , U) = {Z ∞ (K(σ r )) ⊂ R n , r ∈ N + ( h m ) | H(σ r ) B ∈ GL m , Z ∞ (K(σ r )) ⊆ X , K(σ r )Z ∞ (K(σ r )) ⊆ U}.
We then select the element of Z(H, X , U) that minimizes the criterion |HZ ∞ | p , defining

Z * (H, p) = arg minimize Z∈Z(H,X ,U ) |HZ| p .
The computation of Z * (H, p) requires to compute approximations of the mRPI sets Z ∞ (K(σ r )) ∈ Z(H, X , U) for 0-DSMC laws. The general procedure is recalled in Algorithm 5.2. This algorithm is based on Algorithm 5.1 for unstable sliding surfaces, and Proposition 24 for stable surfaces.

In this section we studied the minimality properties of the mRPI sets for 0-DSMC laws. Two minimality criteria were considered, namely set inclusion and the matrix-norm (5.5). We showed that the mRPI sets for 0-DSMC laws are mRCI set (minimal in terms of set inclusion). We proposed different approaches for the computation of (H, p)-mRCI sets depending on the choice for the matrix H. A formal characterization is provided under the assumption that the matrix H defines a stable sliding surface. If the assumptions leading to such formal characterization are not fulfilled, we proposed methods for the computation of candidate solutions among the collection of mRPI sets for 0-DSMC laws. These methods use the geometrical structure of the mRPI sets, and the double role of the matrix C.

The following section provides a numerical example for the construction of (Hp) mRCI sets for a range of matrices H.

Illustrative examples

We provide illustrative examples of the results presented in this chapter for the construction of (H, p)-mRCI sets using 0-DSMC laws.

We consider a three dimensional discrete time linear system (n = 3) with a scalar control input (m = 1),

ξ + = Aξ + Bu + w, w ∈ W.
c 1 c 3 c 2 c 3 1 ∈ K(A 11 , A 12
). We have

Â11 = A 11 -A 12 c 1 c 2 = 1 -0.375c 1 0.5 -0.375c 2 -0.25c 1 1 -0.25c 2 .
The characteristic polynomial of Â11 is given by t 2 -tr( Â11 )t + det( Â11 ), with

tr( Â11 ) = -0.375c 1 -0.25c 2 + 2, det( Â11 ) = -0.2813c 1 c 2 + 0.125c 1 -0.25c 2 + 1.
The determinant of this second order polynomial is given by ∆ = tr( Â11 ) 2 -4det(A 11 ), yielding ∆ = 0.1406c 2 1 + 0.625c 2 2 + 0.0938c 1 c 2 + 0.25c 1 + 0.5c 2 .

The spectral radius of Â11 is inferior to 1 provided det( Â11 ) < 1 and either

• ∆ ≤ 0 (complex conjugate eigenvalues).

• ∆ ≥ 0, |tr( Â11 )| < 2 (real eigenvalues).

Figure 5.1 shows the sets of matrices C = c 1 c 2 1 defining stable sliding surfaces, which is here a polytope. Every matrix C in this polytope can be used to define a 0-DSMC law ν(ξ) = Kξ, K = -(CB) -1 CA, and its associated mRPI set Z ∞ (K).

Our goal is to compute the matrix C leading to an mRPI set Z ∞ (K) that minimizes a matrix-norm criterion (5.5) for a given matrix H ∈ R h×n , and p ∈ N+ . The practical implementation of the methods presented in this chapter rely on the computation of ( , p)-outer approximations of mRPI sets.

Computation of mRPI approximations

The methods for the construction of mRPI approximations used below, and presented in Section 1.4.1.3, rely on the computation of partial sums.

Recall that for a 0-DSMC law, it is possible to construct RPI outer approximations of Z ∞ (K) by computing RPI outer approximations of the mRPI set Z∞ ( Â11 ) for the virtual system

x + = Â11 x + A 11 A 12 w, w ∈ W.
The benefits of this construction, in comparison to general linear control laws, is illustrated below.

We compare the V-complexity, and computation time, of the partial sums Z r (K) and Zr ( Â11 ) where

• The feedback gain K is computed using pole placement on the pair (A, B).

• The matrix Â11 = A 11 -A 12 c 1 c 2 is computed using pole placement on the pair (A 11 , -A 12 ).

The computation time and V-complexity of the partial sum depend on the poles selection. To remove bias, we use a Random Number Generator (RNG) to generate poles in the interval ] -1, 1[. Moreover, we consider 20 values of K and Â11 (that is, 20 sets of poles for each scenario). The results are presented for , 20, 30, 40, 50, 60, 70, 80, 90, 100}.

r ∈ {10
The computation time and V-complexity of the partial sums are presented in Figure 5.2, where we use the color blue for Z r (K) (general linear feedback gains) and the color red for Zr ( Â11 ) (0-DSMC feedback gains).

We see that the average V-complexity is smaller for 0-DSMC feedback gains by one order of magnitude (5 times smaller for low values of r, 15 times smaller for high values of r). The average computation time is significantly smaller for 0-DSMC feedback gains, by one order of magnitude for low values of r, and 3 orders of magnitude for high values of r. In particular, the computation time of the 100 step image set for 0-DSMC feedback gains is smaller than the computation time for the 20 step image set for general feedback gains.

These figures illustrate the computational advantage of 0-DSMC feedback gains. The computation of mRPI approximations is significantly less demanding than general linear control laws. The reduction in the computational complexity allows for a faster computation of mRPI approximations, which is of particular interest in the manipulation of mRPI sets for the computation of candidate (H, p)-mRCI sets.

Computation of minimal RCI sets using 0-DSMC laws

This section gives simulation examples for the computation of (H, p)-mRCI sets using the methods presented in Section 5.6. This section is divided into three subsections, each corresponding to a different class of matrix H.

The polytopic mRPI approximations used below are computed using the first method discussed in Section 1.4.1.3, and the Theorem 5.1 to reduce the computational complexity. Throughout the simulations, we arbitrarily choose = 0.01 and p = ∞.

Matrix defining stable sliding surfaces

The first class of matrices considered are matrices H that satisfy Assumption 5.4, that is matrices H defining stable sliding surfaces. Proposition 24 gives a formal characterization of a (H, p)-mRCI set for this case. We present below a simulation example for four different matrices H, denoted {H (i) } i∈N + 4 in the following table.

For each matrix H (i) , i = {1, 2, 3, 4}, we compute the feedback gain i) , and the RPI ( , p) outer-approximation of Z ∞ (K (i) ), which we denote Z (i) . We also compute

K (i) = -(C (i) B)C (i) A with C (i) = H (
H (i)
∞ (Z (i) ), the sublevel set of the of the matrix-norm criterion |H (i) Z| ∞ associated to Z (i) , as defined in (1.12).

The matrices H (i) and the resulting feedback gains K (i) are given in the following table .   i H (i) K (i) 1 0.4 0.2 1.0 -0.40 -2.20 -1.65 2 0.3 1.8 1.0 -0.30 -1.95 -1.56 3 5.0 4.0 3.0 -1.67 -2.17 -1.96 4 20 -19 1.0 -20.0 9.00 -3.75

Figure 5.3 shows the RPI set Z (i) , the sublevel set

H (i) ∞ (Z (i)
) along with the disturbance set W for the four matrices H (i) (top to bottom). The 2 dimensional representation of 3 dimensional polytopes is complex. Hence, we chose to provide two plots for each i = {1, 2, 3, 4}, with different angles of observations. Figures on the right hand side highlight that the disturbance set intersects the frontier of the sublevel set H (i) ∞ (Z (i) ). That is, there exists a vector w ∈ W such that w ∈ H (i) p (Z (i) ). From Proposition 6 and the set inclusion W ⊆ Z (i) , we have

H (i) (Z (i) ) = H (i)
∞ (W). The disturbance set W and the RCI set Z (i) relate to the same sublevel of the matrix-norm criterion with the particular choice H = H (i) . The invariant sets Z (i) presented in Figure 5.3 are all minimal for the matrixnorm criterion |HZ| p for their respective matrix H (i) regardless of the choice for p ∈ N+ . In particular, Z (4) is a (H (4) , p)-mRCI set.

However, this mRPI set is large in terms of projection on the states ξ 1 , ξ 2 and ξ 3 . Note that the axis limits on the figure for i = 4 (bottom) are 5 to 10 times larger than for i = {1, 2, 3}. The mRPI set is given by

Z ∞ (K (4) ) = W ⊕   1 0 0 1 -20 19   ∞ i=0 -6.5 7.625 -5
5.75 i 1 0.5 0.375 0 1 0.25 W.

The square matrix -6.5 7.625 -5

5.75 has a spectral radius of 0.866 (two complex conjugate eigenvalues). We have the following asymptotical equivalence of the p norm of its powers, The latter scenario is illustrated in the following section.

Matrix defining unstable sliding surfaces

The second class of matrices considered are matrices H that satisfy Assumption 5.5, that is matrices H defining unstable sliding surfaces. Within this class of matrices, one cannot choose C = H as the resulting matrix A + BK is not Schur and the associated mRPI set is infinite (unbounded).

We use the method presented in Section 5.6.2.2 for the computation of candidate (H, p)-mRCI sets, where the matrix C is designed to limit the spectral radius of the matrix Â11 . The approach is detailed in Algorithm 5.1.

Our approach consists of computing the matrix C that minimizes |H B ⊥ -C B ⊥ | p while upper-bounding ρ( Â11 ) by a chosen scalar λ. This results in the optimization problem (5.48), parametrized by λ. We perform a grid search over the scalar parameter 0 ≤ λ < 1. We chose to perform the grid search over the q = 10 values {λ j } j∈N + 10 shown in the following table. j 1 2 3 4 5 6 7 8 9 10 λ j 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

For each scalar λ j , we use the solution of the optimization problem (5.48) to define the matrices C(λ j ) = C B ⊥ (λ j ) H B , and the feedback gains K(λ j ) = -(C(λ j )B) -1 C(λ j )A. We then compute an RPI ( , p) outer approximation of Z ∞ (K(λ j )) which we further denote Z(λ j ). Among the RPI sets Z(λ j ), we chose the one that minimizes |HZ(λ j )| p .

We present the results for two matrices H, denoted {H (i) } i∈N + 2 in the following, and chosen as H (1) = -0.3 -0.5 1 , H (2) = 10 5 1 .

The value of the matrices {C(λ j )} j∈N + 10 and feedback gains {K(λ j )} j∈N + 10 obtained for the matrix H (1) are provided in the following table. j C(λ j ) K(λ j ) 1 7.8 -3.84 1.0 -7.84 -0.08 -2.98 2 6.5 -2.53 1.0 -6.50 -0.71 -2.80 3 5.1 -1.28 3.0 -5.12 -1.28 -2.60 4 3.9 -0.28 1.0 -3.92 -1.68 -2.40 5

2.9 0.48 1.0 -2.88 -1.92 -2.20 6 2.0 1.00 1.0 -2.00 -2.00 -2.00 7

1.4 1.18 1.0 -1.38 -1.87 -1.81 8

1.1 0.92 1.0 -1.12 -1.48 -1.61 9 0.8 0.62 1.0 -0.82 -1.03 -1.46 10 0.5 0.28 1.0 -0.48 -0.52 -1.25 both matrices H (1) (left) and H (2) (right). As expected, the more we constrain the spectral radius of the matrix Â11 (through the selection of the upper bound λ), the larger is

|C B ⊥ -H B ⊥ | ∞ .
Figure 5.4 also shows the relation between the parameter λ and the (H, p) minimality criterion |HZ(λ)| ∞ (blue). Note that this criterion is not minimal for the same value of λ for different values of the matrix H. In the following, the scalar λ that minimizes |HZ(λ) is denoted λ * . We have used the geometrical structure of the mRPI sets for 0-DSMC laws to compute candidate (H, p)-mRCI sets, where H defines an unstable sliding surface. The design of candidate solutions have to account for the double role of the parameter C.

We present here the application of Algorithm 5.2 for the computation of (H, p)-mRCI sets in the presence of state and control input constraints.

Matrix of higher dimension

In this simulation example we consider that the system is subject to the state and input constraints (5.49)

ξ ∈ X = ξ ∈ R n | H -H ξ ≤ 1 6 , u ∈ U = {u ∈ R m | G u u ≤ g u } ,
Figure 5.6 shows the state constraints X along with the disturbance set W. For a matrix K ∈ R 1×3 and set Z, the set inclusions Z ⊆ X and KZ ⊆ U can be written in terms of matrix-norm as follows

|HZ| ∞ ≤ 1, |H u KZ| ∞ ≤ 0.25.
The state constraint set X is a sublevel set of the criterion |HZ| p , namely

H ∞ (X ) = H ∞ (1) = {ξ ∈ R n | |Hξ| ∞ ≤ 1} .
Our goal is to compute an RCI set that is as far as possible from the state constraints. That is, we want to compute an (H, p)-mRCI set Z with H as in (5.49) and p = ∞. In particular, Z has to satisfies the state and input constraints. Given that the matrix H ∈ K(A 11 , A 12 ), we do not have a formal characterization of (H, p)-mRCI sets.

We compute a candidate (H, p)-mRCI set using the method presented in Section 5.6.2.4, and detailed in Algorithm 5.2. The 1-combination of the rows of H consists of 3 elements H σr = H r , r ∈ N + 3 . The matrices H(σ r ) satisfy {H(σ r ) B } r=1,2,3 = {3, 4, 3} ∈ GL 1 (invertible).

We have H(σ r ) ∈ K(A 11 , A 12 ) for r ∈ {1, 2}. For these matrices, that define stable sliding surfaces, we have a formal characterization of the (H(σ r ), p) mRCI set. We compute the feedback gains K(σ r ) = -(H(σ r )B) -1 H(σ r )A, and the polytopic RPI ( , p) outer approximation of the mRPI sets Z ∞ (K(σ r )), denoted Z(K(σ r )), for r ∈ {1, 2}.

The matrix H(σ 3 ) ∈ K(A 11 , A 12 ) defines an unstable sliding surface. As in the previous section, we solve the optimization problem (5.48) with H = H(σ 3 ) over the q = 10 values 0 < λ j < 1, j ∈ N + 10 , shown in the following table. j 1 2 3 4 5 6 7 8 9 10 λ j 0.01 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

For a given scalar λ j , we use the solution of the optimization problem to define the matrix C(λ j ) = C B ⊥ (λ j ) H(σ 3 ) B , and the feedback gain K(λ j ) = -(C(λ j )B) -1 C(λ j )A. We compute the polytopic RPI ( , p) outer-approximation of the mRPI set Z ∞ (K(λ j )), denoted Z(K(λ j )).

The constraints satisfaction is checked a posteriori and Figure 5.7 shows which values of λ j ensure state and input constraints satisfaction, namely

|HZ(λ j )| ∞ ≤ 1, |H u K(λ j )Z(K(λ j ))| ∞ ≤ 0.25.
We retain the parameter λ j that satisfies both the state and input constraints, and that minimizes |HZ(K(λ j ))| ∞ , namely λ j = 0.7. Note that here, we choose the scalar λ that minimizes the (H, p) criterion instead of the (H 3 , p) criterion. In the following, we denote Z(K(σ 3 )) = Z(K(0.7)).

Figure 5.8 presents for comparison the three sets Z(K(σ r )), r ∈ {1, 2, 3}, along with the state constraints X . In each figure we color the pair relative to the corresponding sub-matrix H(σ r ).

Eventually, Figure 5.9 presents the value of |HZ(K(σ r ))| ∞ and |H u K(σ r )Z(K(σ r ))| ∞ for r ∈ {1, 2, 3}. We also indicate the thresholds one needs to observe to satisfy the state and control input constraints. In particular, we note that Z(σ 2 ) ⊆ X . We see that the set that minimizes |HZ(K(σ r ))| ∞ is Z(K(σ 1 )).

We have provided simulation examples of the methods developed in this chapter for the computation of (H, p)-mRCI sets for the three classes of matrices H studied. We gave the minimal RCI sets whenever the formal characterization is available. When we do not have a formal characterization, we detailed the successive steps of the Algorithms 5.1 and 5.2 to compute candidate solutions.

Conclusion

We presented the framework of discrete time sliding mode control with linear reaching law for discrete-time linear systems. The design parameters of the control law are a matrix C that defines the sliding variable, and a matrix A C4 that defines the closed-loop dynamics of the sliding variable. We studied in depth the invariant sets obtained with linear DSMC laws, with a particular focus on the mRPI set. This set has geometrical properties inherited from the structure of the DSMC law. For the particular case A C4 = 0 m,m , the mRPI set is the image of the mRPI set for a virtual system by a matrix. The virtual mRPI set defines the size of the mRPI set, while the matrix defines the orientation of the mRPI set. These two elements are function of the only design parameter of DSMC law, namely the matrix C. This expression of the mRPI set allows to reduce the complexity of the mRPI set approximations.

We use the geometrical property of 0-DSMC laws for the computation of minimal RCI sets. As a first step, we note that mRPI sets for 0-DSMC laws are minimal in terms of set inclusion among the collection of RCI sets. As a second step, we considered the matrix-norm criterion for different classes of matrices H. We provided a formal characterization of (H, p)-mRCI sets for matrices H that define stable sliding surfaces. When we do not have a formal characterization, we developed methods for the computation of candidate (H, p)-mRCI sets among the collection of 0-DSMC laws. These methods use the geometrical structure of the mRPI set and the double role of the design parameter C.

This chapter is a first approach in the construction of (H, p)-mRCI sets. There are two main limitations in the methods proposed for the computation of candidate solutions. The first is that the constraints are not considered in the design of candidate solutions. Instead, the approach consists in checking a posteriori if the candidate solutions satisfy the state and input constraints. The second is the restriction a class of linear control laws, namely the 0-DSMC laws.

Introduction

In this chapter we focus on the use of set theoretic methods with the aim to construct minimal RCI sets, a topic which has been a subject of multiple studies due to the complexity of the underlying problem. This chapter composes of a preliminary section in which we present the class of system, assumptions, and notations. The following two sections present optimization problems in the quest for a feedback mechanism minimizing the impact of disturbances. These optimization problems yield decreasing sequences of RCI sets in terms of set inclusion. Some results regarding the topology of decreasing RCI set sequences are provided at the premises of this chapter, along with a proof of the existence of matrix-norm mRCI sets. We provide simulation examples in the last section to illustrate the benefits and limitations of the proposed iterative methods.

The method relates to the computation of mRPI set approximations, as in Section 1.4.1.3, with the notable difference that the control law is now considered as a 6.2. System description, assumptions and notation 167 design parameter. We use an initial RCI set and compute its image set using a control law that ensures the one-step image set is contained in the RCI set. With this choice for the control law, the image preserves the RCI property, thus allowing to iterate the procedure. This allows to define decreasing sequences of RCI in terms of set inclusion by recursively computing appropriate control laws and the associated one-step image sets. For a given RCI set, there exists possibly an infinite amount of control laws with such property.

We introduce the novel concept of selection function, that associates to all RCI set a unique control law such that the associate image set retains the RCI property. The selection function are used to define set mappings over the collection of RCI sets. We propose several selection functions using optimization-based strategies where the control law is chosen to minimize the size of the image set with regards to either set inclusion or the a chosen matrix-norm criterion. For tractability purposes, the iterative approach is restrained to polytopic RCI sets to reduce the computational complexity.

The limit sets of the set mappings are of particular interest, as they represent the infinite refinement of polytopic RCI sets towards their minimization. We provide a first characterization of the limit sets by studying the image of RCI sets contained in minimal Quasi-Sliding Mode Bands (mQSMB). Section 6.2 provides a description of the class of systems and disturbances considered in this Chapter. Topological results on decreasing sequences of RCI sets, leading to leading to the constructive proof of matrix-norm mRCI sets existence, are presented in Section 6.3. The construction of decreasing sequences of RCI sets, as image sets where the control law is a design parameter, is discussed in Section 6.4. The concept of selection function to define set mappings of RCI sets is discussed in Section 6.5. We propose several several selection functions using optimization-based methods in Section 6.6. Section 6.7 provides a connection between this chapter and the previous by studying the properties of the set mappings of RCI set contained in an mQSMB, thus providing a first characterization of the set mappings limit sets. A simulation study of the set mappings for the design of (H, p) minimal RCI sets is shown in Section 6.9.

System description, assumptions and notation

We consider a general discrete-time linear system as in (1.1) subject to bounded additive disturbances and state and control input constraints as in (1.2), ξ + = Aξ + Bu + w, (6.1)

(ξ, u, w) ∈ (X × U × W) ⊆ (R n × R m × R n ). (6.2)
For clarity of presentation, an RCI set denotes an RCI set for the above system dynamics and constraints. The collection of RCI sets is denoted R.

The goal is of this chapter is to compute (H, p)-mRCI sets, where H ∈ R h×n and p ∈ N+ are given. For a bounded set RCI Z, we define

|HZ| p = sup {|Hξ| p | ξ ∈ Z} , (6.3)
and we recall below the definition of matrix-norm mRCI sets. The existence of matrix-norm mRCI sets is not guaranteed, as h(H, p) is an infimum. This topic is discussed in the following section.

Existence of matrix-norm mRCI sets

This section studies the existence of matrix-norm mRCI sets, and the results presented in this section are obtained under the following assumption.

Assumption 6.1 (Compact constraint sets) The sets X and U are compact.

From the definition of an infimum, there exists an infinite sequence of RCI sets, which we further denote {Z i } i∈N , that verifies

lim i→∞ |HZ i | p = h(H, p).
Without loss of generality we can assume that |HZ i+1 | p ≤ |HZ i | p , ∀i ∈ N. That is, this sequence of RCI sets is (not strictly) decreasing in terms of the matrix-norm criterion (6.3).

In the following we provide a constructive proof of the existence of a matrix-norm mRCI set that builds on the sequence {Z i } i∈N . An intuitive idea is to consider ∞ i=0 Z i . However, this set is not guaranteed to be minimal with regards to the matrix-norm criterion and, more important, it is not RCI in general.

Let us introduce the following definition regarding set sequences. Definition 6.2 (Decreasing set sequence) The infinite set sequence

{Z i } i∈N is de- creasing if ∀i ∈ N, Z i+1 ⊆ Z i .
We have the following result regarding the set invariance property of the set

∞ i=0 Z i .
Proposition 25 (Intersection of compact RCI sets) Consider Assumption 6.1. If {Z i } i∈N is a decreasing sequence of compact RCI sets, then Z * = ∞ i=0 Z i is an RCI set.

Proof: The set inclusion Z * ⊆ X is immediate. Let ξ ∈ Z * , and let us prove that there exists u ∈ U such that, ∀w ∈ W, Aξ + Bu + w ∈ Z * . That is,

∀w ∈ W, ∀i ∈ N, Aξ + Bu + w ∈ Z i .
For all i ∈ N, there exists at least one control law, further denoted ν i , such that Z i is an RPI set for the control law ν i . We consider the infinite sequence {ν i (ξ)} i∈N ∈ U ∞ . From the compactness of U, this infinite sequence allows for a convergent subsequence. Let t : N → N be a strictly increasing function such that {ν t(i) (ξ)} i∈N is convergent, and denote u ∈ U its limit.

Let w ∈ W. The sequence {Aξ + Bν t(i) (ξ) + w} i∈N is convergent, verifies Aξ + Bν t(i) (ξ) + w ∈ Z t(i) , ∀i ∈ N, and its limit is Aξ + Bu + w. Let k ∈ N. Given that {Z i } i∈N is a decreasing set sequence, and that t : N → N is a strictly increasing function, we have

Aξ + Bν t(i) + w ∈ Z t(i) ⊆ Z t(k) , ∀i ≥ k.
Hence, {Aξ + Bν t(i) (ξ) + w} i∈N,i≥k is an infinite sequence of elements of Z k . From the compactness of Z k , we conclude that ∀w ∈ W, ∀k ∈ N, Aξ + Bu + w ∈ Z k .

The set ∞ i=0 Z i retains the RCI property provided the set sequence {Z i } i∈N is decreasing with regards to set inclusion, and the sets are compact. By definition of an RCI set, they are contained in the state constraint set X , which is compact. Thus, they are bounded, and they are compact if and only if they are closed.

In the following, we study the properties of the closure of an RCI set, both with regards to set invariance and with regards to the matrix-norm criterion. Proof: As a first step, we note that Cl(Z) ⊆ X from the compactness of X . Let ν : R n → R m be such that Z is an RPI set for the control law ν(.).

Let ξ ∈ Cl(Z). In the following, we prove that there exists u ∈ U such that ∀w ∈ W, Aξ + Bu ∈ Z. From the definition of set closure, there exists a sequence of elements of Z, further denoted {ξ j } j∈N ∈ Z ∞ , that converges toward ξ. Given that Z is an RPI set for the control law ν, for all element ξ j of this sequence we have ∀w ∈ W, Aξ j + Bν(ξ j ) + w ∈ Z. (6.4) Consider the associated infinite sequence of control inputs {ν(ξ j )} j∈N . Every element of this infinite input sequence is contained in the compact set U.

From the Bolzano-Weierstrass theorem, this infinite sequence allows for (at least) one convergent subsequence. Let t : N → N be a strictly increasing function such that the subsequence {ν(ξ t(i) )} i∈N is convergent, and denote u its limit. From the compactness of U, we have u ∈ U.

Let w ∈ W. From (6.4), we have Aξ t(j) + Bν(ξ t(j) ) + w ∈ Z, ∀j ∈ N.

The sequence {Aξ t(i) + Bν(ξ t(i) ) + w} i∈N is also convergent, as the sum of two convergent series, and its limit is Aξ + Bu + w. We conclude that Aξ + Bu + w ∈ Cl(Z), and Cl(Z) is an RCI set.

We have proven that the closure of an RCI set is also an RCI set. Define the set sequence {Cl(Z i )} i∈N . These sets are compact RCI sets, and they satisfy

∀i ∈ N, |HCl(Z i )| p = |HZ i | p .
In particular, we have

|HCl(Z i )| p i→∞ = h(H, p).
For this reason, we can consider without loss of generality that the sets Z i are compact. Thus, the set Z * = ∞ i=0 Z i is compact, as the infinite intersection of compact sets.

From the compactness of Z * there exists ξ * ∈ Z * such that

|HZ * | p = |Hξ * | p . Given that ξ * ∈ Z i , ∀i ∈ N, we have |HZ i | p ≥ |Hξ * | p = |HZ * | p , ∀i ∈ N. This yields h(H, p) ≥ |HZ * | p .
We have yet to prove that the set Z * is RCI. We cannot use directly the Proposition 25 as the sequence of compact RCI sets {Z i } i∈N is not decreasing.

We construct a decreasing sequence of compact RCI sets from the sequence {Z i } i∈N , based on the result presented below.

Proposition 28 Consider an infinite sequence of compact RCI sets

{Z i } i∈N such that |HZ i+1 | p ≤ |HZ i | p , ∀i ∈ N. For all i ∈ N, define Y i = ∞ j=i Z j , ∀i ∈ N. (6.5)
Then, ∀i ∈ N, we have

|HY i | p = |HZ i | p and Y i is RCI. Proof: Let i ∈ N.
The set Y i is RCI as the infinite union of RCI sets. The inequality

|HY i | p ≥ |HZ i | p is immediate as Z i ⊆ Y i .
The inequality |HZ i+1 | p ≤ |HZ i | p , ∀i ∈ N can be rewritten in terms of set inclusion as follows,

Z j ⊆ H p (Z i ), ∀j ≥ i. We conclude that Y i = ∞ j=i Z j ⊆ H p (Z i ), and thus |HY i | p ≤ |HZ i | p .
We conclude that Y i and Z i have the same value of the matrix-norm criterion.

The main elements are available in order to state the following result concerning the existence of matrix-norm mRCI sets. Define Y i = ∞ j=i Cl(Z j ), ∀i ∈ N. These sets are RCI, as the infinite union of RCI sets. From Proposition 28, they verify

|HY i | p = |HCl(Z i )| p , ∀i ∈ N.
From Proposition 27 and Proposition 26, {Cl(Y i )} i∈N is an infinite sequence of compact RCI sets that verifies |HCl(

Y i )| p = |HY i | p , ∀i ∈ N. Moreover, {Cl(Y i )} i∈N is a decreasing sequence of compact RCI sets, as we have Y(Z) i+1 ⊆ Y i , ∀i ∈ N by construction. Define Y(Z) * = ∞ i=0 Cl(Y i ).
From Proposition 25, the set Y(Z) * is RCI. From the compactness of Y(Z) * , and the above relations, we have

|HZ i | p = |HCl(Z i )| p = |HY i | p = |HCl(Y i )| p ≥ |HY(Z) * | p , ∀i ∈ N We conclude that lim i→∞ |HZ i | p = h(H, p) ≥ |HY(Z) * | p ,
and therefore the existence of an (H, p)-mRCI set.

We have presented a constructive proof of the existence of matrix-norm mRCI sets under the assumption that the state and input constraints sets are compact. The constructive proof relies on the construction of a decreasing sequence of RCI sets.

In the following sections, we propose methods for the construction of decreasing sequences of RCI sets, with the goal of computing matrix-norm mRCI sets.

Construction of decreasing sequence of RCI sets

Chapter 5 provided a condition on the matrix H for the formal characterization of matrix-norm mRCI sets. In this chapter we develop set-theoretic iterative methods for the computation of candidate (H, p)-mRCI sets in the absence of formal characterization, by constructing decreasing sequences of RCI sets. These methods rely on the concept of one-step image set.

One-step image set of an RCI set

The method for the computation of candidate solutions presented in this chapter relates to the method Image sets of an initial RPI set for the computation of mRPI outer approximations presented in Section 1.4.1.3, where we compute the successive image sets of an initial RPI set for a closed-loop system. In the respective context, we consider an open-loop system. Here, the control law is a design parameter, which allows for more freedom in the minimization of the impact of disturbances, and thus on the size of the resulting image sets of an initial RCI set.

Recall the following definition which relates to the one-step image of a set for a feedback control law. Definition 6.3 (One-step image set) For a non-empty set Z and a control law

ν : R n → R m such that Z ⊆ X ν = {ξ ∈ X | ν(ξ) ∈ U}, the one-step image set is Image(Z, ν) = {Aξ + Bν(ξ) + w | ξ ∈ Z, w ∈ W} (6.6a) = {Aξ + Bν(ξ) | ξ ∈ Z} ⊕ W. (6.6b)
If the set Z is RCI, then there exists at least one control law ν(.) that verifies Image(Z, ν) ⊆ Z, (6.7a)

Z ⊆ X ν . (6.7b)

For a given set Z, we denote N (Z) the collection of admissible control laws ν(.) satisfying (6.7). By definition, a set Z is RCI if and only if N (Z) is non-empty. We have the following propositions based on the above definitions.

Proposition 29 If Z is RCI and ν ∈ N (Z), then ν ∈ N (Image(Z, ν)).

Proof: From (6.7), we have Image(Z, ν) ⊆ Z ⊆ X ν . From the definition of the one-step image, we have Image(Image(Z, ν), ν) ⊆ Image(Z, ν). We conclude that ν ∈ N (Image(Z, ν)).

In particular, the set Image(Z, ν) is RCI. From an initial RCI set Z it is possible to compute at least one (not strictly) smaller RCI set, in terms of set inclusion, through the computation of a control law ν ∈ N (Z). The collection of control laws N (Z) is possibly infinite. The selection of the control law to minimize the size of the one-step image is a key aspect of this approach, and it is discussed in Section 6.6.

Given that the set Image(Z, ν) is also RCI, this construction can be repeated to construct a decreasing sequence of RCI sets {Z i } i∈N , where

Z 0 = Z, ∀i ∈ N, ν i ∈ N (Z i ), Z i+1 = Image(Z i , ν i ).
From Proposition 29, the control law at the iteration i remains feasible at the iteration i + 1, namely ν i ∈ N (Z i+1 ). However, we perform a novel selection of the control law at each iteration to further diminish the impact of the disturbances on the closed-loop system. This iterative procedure is detailed in Algorithm 6.1, where we perform q ∈ N iterations to construct the decreasing RCI set sequence {Z i } i∈Nq . Each step of this iterative procedure further refines the initial RCI set Z.

The computational aspect, lines 3 and 5 of the algorithm, is the core of this iterative approach, and it will be developed in the following section.

Convex RCI sets

A first step to reduce the computational complexity is to consider convex RCI sets. We denote C the collection of convex RCI sets. Remark 6.1 Note that the collection of RCI sets for the system (6.1) and constraints (6.2) depend on the model matrices A, B and the sets W, X , U. For simplicity of notation, we do not stress the dependency.

If Z ∈ C, we can exploit the extreme combination of disturbances to rewrite the constraint (6.7a) as follows

Aξ + Bν(ξ) + w ∈ Z, ∀w ∈ V (W), ∀ξ ∈ Z.
For a given ξ ∈ Z, the computation of a control action ν(ξ) satisfying the above condition has a convex feasible domain.

For a convex RCI set Z and ν ∈ N (Z), the set Image(Z, ν) is not convex in general. To retrieve the convexity property of the image set we consider instead the convex hull of the one-step image set, which has the following property. We conclude that ν ∈ N (Co(Image(Z, ν))).

By restraining to convex RCI sets we simplify the constraints (6.7) for the computation of a suitable control law ν ∈ N (Z). However, these constraints remain extensively complex in the context of an iterative approach.

Polytopic RCI sets

As a second step to reduce computational complexity, we consider polytopic RCI sets. We denote T the collection of polytopic RCI sets. Remark 6.2 Likewise, the collection of polytopic RCI sets for the system (6.1) and constraints (6.2) depend on the model matrices A, B and the sets W, X , U. For simplicity of notation, we do not stress the dependency.

For a given polytopic RCI set Z, we denote l z its H-complexity, V (Z) the set of vertices, and {ξ

∈ R n | G z ξ ≤ g z } , G z ∈ R lz×n , g z ∈ R lz a minimal H-representation.
In the same way as for convex sets, the one-step image of a polytopic RCI set Z by a control law ν ∈ N (Z) does not preserve the polytopic property of Z. For this reason, we consider the convexhull of the one-step image of the vertices, that has the following property.

Proposition 31 With Z ∈ T and ν ∈ N (Z), we have Co(Image(V (Z), ν)) ⊆ Co(Image(Z, ν)), (6.8) and Co(Image(V (Z), ν)) ∈ T .

Proof: The set inclusion (6.8) is immediate. However, it is not guaranteed that ν ∈ N (Co(Image(V (Z), ν))). In the following, we denote {v 1 , ..., v f } the vertices of Co(Image(V (Z), ν)).

Let ξ ∈ Co(Image(V (Z), ν)). We can rewritte ξ as a linear combination of the vertices, ξ = f i=1 λ i v i , where 0 ≤ λ i ≤ 1, ∀i = {1, ..., f }, and

f i=1 λ i = 1. With u = f i=1 λ i ν(v i ), we have u ∈ U and ∀w ∈ W, Aξ + Bu + w = f i=1 λ i (Av i + Bν(v i ) + w) ∈ Co(Image(V (Z), ν)).
We conclude that the set Co(Image(V (Z), ν)) is RCI, and it is a polytope as the convex hull of a finite set of points.

The computation of the image set is reduced to the computation of the convex hull of the image of V (Z). That is, we are only interested in computing the control law ν ∈ N (Z) on the vertices V (Z) instead of on all elements ξ ∈ Z.

The following sections discusses the computation of a control law ν ∈ N (Z) using either the vertex or the half-space representation of the polytope.

Set invariance condition using the V-representation

In the following, we provide a condition for ν ∈ N (Z) by computing a suitable control action ν(ξ) on every vertex ξ ∈ V (Z).

If ν ∈ N (Z), then

Aξ + Bν(ξ) ∈ Z W, ∀ξ ∈ V (Z), (6.9a
)

G u ν(ξ) ≤ g u , ∀ξ ∈ V (Z). (6.9b)
Note that Z W is given by [Kolmanovsky 1998b] 6.10) where the i th component of g z (W) is given by

Z W = {ξ ∈ X | G z ξ ≤ g z -g z (W)} , (
(g z (W)) i = max w∈W (G z ) i ξ = max w∈V (W) (G z ) i ξ. ( 6
.11)

Set invariance condition using the H-representation

We recall below the definition of a linear RCI set. Definition 6.4 (Linear RCI set) A set Z is linear RCI if and only if Z ⊆ X and there exists K ∈ R m×n such that KZ ⊆ U and (A + BK)Z ⊕ W ⊆ Z.

We use the Extended Farkas Lemma, recalled below, to provide a necessary and sufficient condition for set invariance.

Proposition 32 (Extended Farkas Lemma) The polytope Z is linear RCI if and only if there exist a feedback gain K ∈ R m×n and matrices

S z ∈ R lz×lz + , S x ∈ R lz×lx + , S u ∈ R lz×lu + and S w ∈ R lw×lz + such that G z (A + BK) = SG z , G z = S w G w , Sg z + S w g w ≤ g z , (6.14a)
G u K = S u G z , S u g z ≤ g u , (6.14b
)

G x = S x G z , S x g z ≤ g x . (6.14c)
These equations are respectively equivalent to

(A + BK)Z ⊕ W ⊆ Z, KZ ⊆ U, Z ⊆ X .
In a similar fashion, the third set inclusion is not necessary as we assume the polytope Z is RCI, and thus satisfies the state constraints. If the matrix G z and the vector g z are fixed, the constraints (6.14a) and (6.14b) are linear in the parameters (S z , S u , K), as they consist of n(l z + l u + l w ) linear equalities, l z + l u + l z (l z + l u + l w ) linear inequalities, and l z (l z + l u + l w ) + nm design parameters. The successive steps of the iterative approach using this set invariance condition is detailed in Algorithm 6.3, where we perform q ∈ N + iterations. Note that, despite only using the H-representation of the polytope in the invariance condition, it is necessary to compute the V-representation at every step to compute the image set as in (6.8).

The main advantage of this approach is that it requires to solve a unique LP (of higher dimension) instead of solving an LP at every vertex of the polytope. The main drawback of this set invariance condition is that it restrains the control law to be linear, i.e. ν(ξ) = Kξ, K ∈ R m×n . This is due to the fact that the Extended Farkas Lemma provides a necessary and sufficient condition for a polytopic set to be linear RCI (RPI for at least one linear control law). In the following, we denote T L the set of polytopic linear RCI sets, and for a given polytopic linear RCI set Z ∈ T L , we define For a given selection function µ as in (6.15), the collection of limit sets of the associated set mapping φ µ is given by

N L (Z) = K ∈ R m×n | (A + BK)Z ⊕ W ⊆ Z, KZ ⊆ U ,
φ ∞ µ = Y(Z) ∈ C | ∃Z ∈ T , s.t. Y(Z) = φ ∞ µ (Z) .
The characterization of the limit sets of a set mapping φ ∞ µ is an interesting research topic, as it represent all the infinite refinement of the collection of polytopic RCI sets for a given selection function.

The proposed iterative approach allows to refine initial polytopic RCI sets towards their minimization through the construction of decreasing sequences of RCI sets. Our goal is, if possible, to find a selection function µ and an initial polytopic RCI set Z such that φ ∞ µ (Z) is an (H, p)-mRCI set. We can not guarantee the convergence of the decreasing RCI set sequences towards an mRCI set (in terms of set inclusion or minimization of a matrix-norm criterion), however each iteration will further reduce the size of the RCI set (and thus converge towards a local minimum).

In the following section we give explicit characterization of selection functions µ as in (6.15) using optimization-based methods.

Optimization-based selection functions

This section details the construction of selection functions using optimization-based methods. The structure is divided according to the set invariance condition that is used, as presented in Sections 6.4.3 (V-representation) and 6.4.4 (H-representation).

Nonlinear control laws and V-representation

In this section we use the invariance conditions presented in Section 6.4.3, that use the V-representation of the polytope Z, to construct selection functions. These selection functions are defined using optimization-based strategies. We aim at selecting the control law that minimizes the size of the image set with regards to set inclusion, or a chosen the matrix-norm criterion as in ( 6.3).

Minimal scalar scaling factor

The first selection function µ 1 relies on the set inclusion ∀Z ∈ T , ∀ν ∈ N (Z),

Co(Image(V (Z), ν)) ⊆ α(Z, ν)Z, (6.17a) 0 ≤ α(Z, ν) ≤ 1. (6.17b) where α(Z, ν) = min {0 ≤ α ≤ 1 | Aξ + Bν(ξ) ∈ αZ W, ∀ξ ∈ Z} = max {min {0 ≤ α ≤ 1 | Aξ + Bν(ξ) ∈ αZ W} | ξ ∈ V (Z} .
To compute RCI sets that are as small as possible, our goal is to select a control law leading to a scaling factor that is minimal for a given polytopic RCI set Z.

Mathematically, we want to construct a selection function µ 1 as in ( 6.15) such that

∀Z ∈ T , α(Z, µ 1 (Z)) = min{α(Z, ν) | ν ∈ N (Z)}. (6.18)
From (6.10), we have

αZ W = {ξ ∈ R n | G z ξ ≤ αg z -g z (W)} . (6.19)
For a given ξ ∈ Z, the inclusion Aξ + Bν(ξ) ∈ αZ W consists of the l z linear inequalities

G z B -g z ν(ξ) α(ν) ≤ -G z Aξ -g z (W).
For every ξ ∈ Z, the control action µ 1 (Z, ξ) is chosen to minimize α where Aξ + Bµ 1 (Z, ξ) ∈ αZ W. The resulting linear optimization problem P 1 (Z, ξ) is given by

P 1 (Z, ξ) : minimize (u,α) α, (6.20a) subject to     G z B -g z G u 0 0 1,m 1 0 1,m -1     u α ≤     -G z Aξ -g z (W) g u 1 0     . (6.20b)
We have the following result regarding the feasibility of the above optimization problem.

directions defined by the lines of the matrix G z while ultimately minimizing a chosen norm p ∈ N+ of α. With a vectorial scaling factor, the constraint (6.20b) becomes

G z B -diag(g z ) u α ≤ -G z Aξ -g z (W),
where diag(g z ) ∈ R lz×lz is a diagonal matrix composed of the elements of the vector g z . The vectorial counterpart of the optimization problem (6.20), denoted P 2,p (Z, ξ), is formulated as

P 2,p (Z, ξ) : minimize (u,α) |α| p (6.22a) subject to     G z B -diag(g z ) g u 0 lu,lz 0 lz,m I lz 0 lz,m -I lz     u α ≤     -G z Aξ -g z (W) g u 1 lz 1 lz     (6.22b)
Note that with p = 1, ∞ (resp. p = 2), the above optimization problem preserves the LP (resp. QP) structure. The scalar version presented above corresponds in fact to the particuliar vectorial choice p = ∞. The above optimization problem consists in 3l z + l u linear inequalities, l z + m optimization variables, and a convex cost function. We have the following result regarding the feasibility of the above optimization problem.

Proposition 35 (Feasibility) Let Z ∈ T and let ξ ∈ Z. The optimization problem P 2,p (Z, ξ) has a non-empty feasible set.

Proof: Let ν ∈ N (Z). The constraints (6.22b) are feasible with u = ν(ξ) and α = 1 lz .

Similar to the previous subsection, uniqueness of the solution of P 2,p (Z, ξ) is not guaranteed. However, the optimal set is convex and closed, and all its elements lead to the same p norm of the vectorial scaling factor α, which we further denote α 2,p (Z, ξ). In a similar fashion, we define uniquely the control action µ 2,p (Z, ξ) as the solution of the following optimization problem

µ 2,p (Z, ξ) = arg minimize (u,α) |u| 2 2 , (6.23a) subject to G z B -diag(g z ) g u 0 lu,lz u α ≤ -G z Aξ -g z (W) g u , (6.23b
)

|α| p = α 2,p (Z, ξ). (6.23c)
The above optimization problem consists in l z + l u linear inequalities, 1 (possibly nonlinear) equality constraint, l z + m optimization variables and a convex cost function. We have the following result regarding the optimal set of this optimization problem.

Proposition 36 (uniqueness) Let Z ∈ T and let ξ ∈ Z. All the elements of the optimal set of the optimization problem (6.23) lead to the same control action.

Proof: The feasible set of the optimization problem (6.23) is the optimal set of P 2,p (Z, ξ), which is convex and closed while the 2-norm is strictly convex.

Any cost function that is strictly convex in u leads to an optimization problem that has a unique solution. For every Z ∈ T , ∀ξ ∈ Z, we define µ 2,p (Z, ξ) as the optimal control action computed from optimization problem (6.23). This uniquely defines the control law µ 2,p (Z).

We have presented a family of optimization-based selection functions µ 2,p as in (6.15), where p ∈ N+ , thus defining the set mappings φ µ 2,p as in (6.16). For a given Z, the computation of φ µ 2,p (Z) requires to solve the optimization problems (6.20) and ( 6.23) on the vertices of Z. Hence, it requires the resolution of 2card(V (Z)) linear optimization problems, with l z + m optimization variables, 3l z + l u and l z + l u linear inequalities respectively, 0 and 1 (possibly nonlinear) equality respectively, and l z + l u optimization variables. For tractability purposes, it is preferable to choose p = 1, ∞ (resp. p = 2) to preserve the LP (resp. QP) structure of the optimization problems.

Minimal matrix-norm scaling factor

In the two selection functions µ 1 and µ 2,p presented above, we aim at minimizing the scaling factor with regards to the directions spawned by the rows of G z . An alternative idea for the selection function µ 3,p is to minimize the scaling factor of the RCI set with regards to the directions defined by the rows of the matrix H that defines the matrix-norm criterion (6.3) we want to minimize.

For a given polytopic RCI set Z, we are interested in finding the control law ν ∈ N (Z) that minimizes |Image(Z, ν)| p . That is, for all its elements ξ ∈ Z, the control action ν(ξ) such that max {|H(Aξ + Bν(ξ) + w)| p | w ∈ W} is minimal.

Remark 6.4 In addition, the control action ν(ξ) has to satisfy Aξ+Bν(ξ) ∈ Z W, and ν(ξ) ∈ U, because ν ∈ N (Z).

These considerations lead to the following optimization problem,

P 3,p (Z, ξ) : minimize (u,α) α, (6.24a) subject to G z B 0 lz,l g u 0 lu,l u α ≤ -G z Aξ + g z -g z (W) g u , (6.24b
) 6.24c) This optimization problem consists in m + 1 optimization variables, l z + l u linear inequalities (6.24b), card(V (W)) convex (and possibly nonlinear) inequalities as in (6.24c), and a linear cost function. With p = 1, ∞ (resp. p = 2) this optimization problem preserves the MPLP (resp. MPQP) structure. We have the following result regarding the feasibility of the above optimization problem.

|H(Aξ + Bu + w)| p ≤ α, ∀w ∈ V (W). (
Proposition 37 (Feasibility) For all Z ∈ T and all elements ξ ∈ Z, the optimization problem P 3,p (Z, ξ) has a non-empty feasible set.

Proof: Let ν ∈ N (Z), and define α * = max {|H(Aξ + Bν(ξ)) + w| p | w ∈ W}. The constraints (6.24b) and (6.24c) are satisfied with u = ν(ξ) and α = α * .

Uniqueness of the solution of the optimization problem (6.24) is not guaranteed. The optimal set is convex and closed, and all its elements lead to the same unique minimal scaling factor that we further denote α 3,p (Z, ξ). To uniquely define the control action µ 3,p (Z, ξ), we consider the following optimization problem

µ 3,p (Z, ξ) = arg minimize u |u| 2 2 , (6.25a) subject to G z B 0 l g u 0 l u α 3,p (Z, ξ) ≤ -G z Aξ -g z (W) + g z g u , (6.25b
)

|H(Aξ + Bu + w)| p ≤ α 3,p (Z, ξ), ∀w ∈ V (W). (6.25c)
The above optimization problem consists in l z + l u linear inequalities, card(V (W)) convex (and possibly nonlinear) inequality constraint, m + 1 optimization variables and a convex cost function (strictly convex with regards to u). We have the following result regarding the optimal set of this optimization problem.

Proposition 38 (uniqueness) Let Z ∈ T and let ξ ∈ Z. The optimization problem (6.25) has a unique solution.

Proof: The feasible set of the optimization problem (6.25) is the optimal set of P 3,p (Z, ξ), which is convex and closed while the 2-norm is strictly convex. Remark 6.5 It is worth noticing that any cost function in (6.25) that is strictly convex in u leads to an optimization problem that has a unique solution.

For every Z ∈ T , ∀ξ ∈ Z, we define µ 3,p (Z, ξ) as the unique solution of the optimization problem (6.25). This uniquely defines the control law µ 3,p (Z).

For a given polytopic RCI set Z, the computation of φ µ 3,p (Z) requires to solve the optimization problems (6.24) and (6.25) on the vertices of Z. Hence, it requires to solve 2card(V (Z)) optimization problems, with m + 1 optimization variables, l z + l u linear inequalities, and card(V (W)) (possibly nonconvex) equalities. This choice for the selection function is to be considered with p = 1, ∞ (resp. p = 2) to preserve the LP (resp. QP) structure of the optimization problems.

The selection functions presented above require to solve an optimization problem for each vertex of the polytopic RCI sets, and is summarized in Algorithm 6.4. This approach can be computationally demanding, as the V and H complexity of the polytopic sets is likely increasing at each iteration. In particular for µ 2,p , where the number of optimization variables scales linearly with the H-complexity of the polytopic set. Algorithm 6.4: Iterative refinement of an initial polytopic RCI set with selection functions

Inputs: A, B, X , U, W, Z 0 and q ∈ N + Outputs: A decreasing sequence of RCI sets

{Z i } i∈Nq 1 Initialize i ← 0; 2 repeat 3 Compute (G z , g z ) as a minimal H-representation of Z i ; 4 Compute V (Z) the minimal V-representation of Z i ; 5
Set f the V-complexity of Z i , and {ξ 1 , ..., ξ f } its vertices;

Linear control laws and H-representation

In this section, we use the Extended Farkas Lemma, recalled in Proposition 32, to have a necessary and sufficient condition for a polytope to be linear RCI by obtaining an associated linear feedback gain.

This allows to construct selection functions that restrain to linear control laws,

μ : T L → R m×n , (6.26a) Z → K ∈ N L (Z). (6.26b)
These selection functions are a particular case of the selection function in (6.15).

The selection functions presented in this section are the linear counterpart of those presented above.

P2,p (Z) : minimize

(S,Su,Sw,K,α) |α| p , (6.30a) subject to G z (A + BK) = S z G z , G z = S w G w , (6.30b)
S z g z + S w g w ≤ diag(α)g z , (6.30c)

G u K = S u G z , (6.30d)
S u g z ≤ g u , (6.30e)

K ∈ R m×n , (6.30f) S z ∈ R lz×lz + , S u ∈ R lu×lz + , S w ∈ R lz×lw + , (6.30g
)

0 lz ≤ α ≤ 1 lz , (6.30h)
Compared to the optimization problem P1 (Z), P2,p (Z) has l z -1 additional optimization variables (α is of dimension l z instead of 1), and 2(l z -1) additional linear inequalities (6.30h). The LP (QP) structure of the optimization problem is preserved for p = 1, ∞ (resp. p = 2).

We have the following result regarding the feasibility of P2,p (Z).

Proposition 41 (Feasibility) For all Z ∈ T L , the optimization problem P2,p (Z) has a non-empty feasible set.

Proof: Let Z ∈ T L , and let (S, S u , S w , K, α) be the solution of P1 (Z). Then, (S, S u , S w , K, α1 lz ) satisfy the constraints of P2,p (Z).

Uniqueness of the solution is not guaranteed, but the optimal set is closed and convex for p = 1, 2, ∞ (LP/QP), and any element of the optimal set leads to the same minimal p norm of the vectorial scaling factor, further denoted α2,p (Z). An additional optimization problem to uniquely define the feedback gain μ2,p (Z) is to be considered, such as (., ., ., μ2,p (Z), .) = arg minimize

(S,Su,Sw,K,α) n i=1 m j=1 K 2 i,j , (6.31a
) subject to G z (A + BK) = S z G z , (6.31b
)

G z = S w G w , (6.31c)
S z g z + S w g w ≤ diag(α)g z , (6.31d)

G u K = S u G z , (6.31e) S u g z ≤ g u , (6.31f) K ∈ R m×n , (6.31g) S z ∈ R lz×lz + , S u ∈ R lu×lz + , ( 6
z (A + BK) = S z G z , G z = S w G w , (6.33b)
S z g z + S w g w ≤ g z , (6.33c)

G y H(A + BK) = S z G z , G y = S w G w , (6.33d) S z g z + S w g w ≤ αg y , (6.33e
)

G u K = S u G z , (6.33f)
S u g z ≤ g u , (6.33g)

K ∈ R m×n , (6.33h) S z ∈ R lz×lz + , S u ∈ R lu×lz + , S w ∈ R lz×lw + , (6.33i
)

S z ∈ R ly×lz + , S w ∈ R ly×lw + , (6.33j) 0 ≤ α ≤ 1. (6.33k)
The constraints (6.33b), (6.33c), (6.33f) and ( 6.33g) ensure that the set (A+BK)Z ⊕ W is RCI, while the constraints (6.33d) and (6.33e) ensure the set inclusion (6.32).

Compared to the optimization problem (6.28), the above optimization problem has l y (l z + l w + 1) additional inequalities (6.33e) and (6.33j), l y (l z + l w ) additional optimization variables, and l y (n + h) equality constraints (6.33d). This optimization problem is LP, as the cost function is linear and the inequality and equality constraints are linear in the optimization variables. Remark 6.7 A vectorial scaling factor α ∈ R ly + could be considered here instead of a scalar scaling factor to allow for more degree of freedoms at the expense of computational complexity (more optimization variables).

We have the following result regarding the feasibility of P3 (Z).

Proposition 43 (Feasibility) For all Z ∈ T L , the optimization problem P3 (Z) has a non-empty feasible set.

Proof: Let Z ∈ T L , and let K ∈ N L (Z). By definition, we have (A + BK)Z ⊕ W ⊆ Z. As a direct consequence, H((A+BK)Z ⊕W) ⊆ HZ. From Proposition 32, there exists (S z , S z , S u , S w , S w , K) satisfying the constraints of P3,p (Z) with α = 1.

Uniqueness of the solution is not guaranteed. However, the optimal set is closed and convex from the LP structure of the optimization problem, and any elements of the optimal set leads to the same minimal scaling factor α3 (Z). To uniquely define the feedback gain μ3 (Z), we consider Proof: We prove this result by contradiction. Suppose the matrix CB is not full rank. Then, there exist m scalars λ 1 , ..., λ m such that 6.35) where at least one scalar λ i , i = {1, ..., m} is not equal to 0. Thus, given that the c i , i = {1, ..., m} are linearly independent, we have c = 0 n . Let ξ ∈ Z. From the RCI property of Z, there exists u ∈ U such that Aξ + Bu ∈ Z W. We multiply this inequality by c to obtain (6.37) This equation is verified for all ξ ∈ Z. Under the assumption that Z contains the origin in its interior, we conclude that c A = 0 1,n .

c B = (λ 1 c 1 + ... + λ m c m )B = 0 1,m (
c Aξ + c Bu ∈ ( m i=1 λ i c i )(Z W). ( 6
Consider the controllability matrix B AB ... A n-1 B . We multiply the controllability on the left hand side by c , leading to

    c B c AB ... c (A n-1 )B     = 0 1,mn .
Given that c = 0 n , the controllability matrix is not full rank. Thus, the system is not controllable, which is a direct contradiction of Assumption 6.5. We conclude that the matrix CB ∈ R m×m is full rank, and thus invertible.

We have shown that, under the assumption that the system is controllable, if an RCI set that contains the origin in its interior is such that Z W is of dimension n -m, then there exists a full rank matrix C ∈ R m×n such that CZ = CW and the matrix CB is full rank. From 45, we conclude that the set Z is contained in a mQSMB. This provides a necessary condition for a polytopic RCI set that contains the origin in its interior to be contained in a mQSMB.

Let us prove that the above necessary condition is also sufficient. Let Z be an RCI set is contained in a mQSMB. From Proposition 45, we have CZ = CW and the polytope Z W is of dimension at most n -m. We prove that the dimension of the polytope is exactly n -m by contradiction. Suppose it is of dimension strictly inferior to n -m. Then, there exists at least n + m + 1 linearly independent vectors c 1 , ..., c n+m+1 of R n that satisfy c i Z = c i W, ∀i = {1, ..., m + 1}.

Meanwhile, the matrix B ∈ R m×n has a kernel of dimension n -m. The vectorial subspace generated by the vectors c i is of dimension m + 1. We conclude that the two linear subspaces have a non-empty intersection. That is, there exists λ 1 , ..., λ m+1 such that

B c = B (λ 1 c 1 + ... + λ m+1 c m+1 ) = 0 m .
The transpose of this equation gives c B = 01, m. With the same reasoning as in the proof of Proposition 48, we have c A = 0 1,n . If Z contains the origin in its interior, then the controllability matrix is not full rank, which contradicts the assumption 6.5. We conclude that the set Z W is exactly of dimension n -m.

The above developments are condensed in the following theorem.

Theorem 6.2 (Necessary and sufficient condition) Consider Assumption 6.5. Let Z be a polytopic RCI set that contains the origin in its interior. The set Z is contained in a mQSMB if and only if the set Z W is of dimension n -m.

This provides a computationally efficient necessary and sufficient condition to know wether an RCI set is contained in an mQSMB, along with the matrix C. As a first step, we compute the Pontryagin difference .38) We compute a (if possible the minimal) V-representation of Z W, and we denote {v 1 , ..., v f }, f ∈ N + its vertices. We define the matrix V = v 1 ... v f ∈ R n×f , and we compute its rank. The set Z is contained in an mQSMBIf if and only

Z W = {ξ ∈ X | G z ξ ≤ g z -g z (W)} . ( 6
if rank(V ) = n -m. In that case, a matrix C =   c 1 ... c m   such that CB is full rank and CZ = CW is obtained by computing m linearly independent vectors (c 1 , ..., c m ) ∈ (R n ) m such that V c i = 0 f , ∀i = {1, ..., m}.
We have proposed a computationally efficient necessary and sufficient condition for the existence of a mQSMB that contains a fixed polytopic RCI set. In addition, this condition gives the matrix C that fully defines the mQSMB. This necessary and sufficient condition has a key role in the iterative approach presented in this chapter for the computation of minimal RCI sets, as the optimization problem are greatly simplified for RCI sets contained in a mQSMB.

We derive a necessary condition for the set to be contained in an mQSMB from the above result and the Proposition 1. Define g z (W) as in (6.10). The Pontryagin difference Z W contains the origin in its interior if and only if g z -g z (W) > 0 lz . In this case, the Pontryagin difference is of dimension n > n -m. We conclude initial polytopic RCI set Z, as it constructs a sequence of RCI sets that is decreasing for set inclusion. In the following section, we propose a different approach for the iterative refinement of an initial polytopic RCI set where the set inclusion constraint is relaxed.

Set mappings of linear RCI sets with relaxation of the set inclusion constraints

The approach presented in this section consists in constructing a sequence of RCI sets that is decreasing with regards to the matrix-norm criterion (6.3) while not constraining the sequence to be decreasing in terms of set inclusion.

Matrix-norm decreasing sequence of RCI sets

We are interested in constructing set mappings over the collection of polytopic RCI set, Note that the set mappings φ µ as in (6.16) are a particular case of the above set mappings, as the set inclusion φ µ (Z) ⊆ Z leads to (6.41). The notable difference is that the sets mappings ψ do not force ψ(Z) to be a one-step image set of Z in (6.40), and thus do not force the set inclusion ψ(Z) ⊆ Z.

ψ : T → T , (6.40a)
Z → ψ(Z), (6.40b
The set mapping ψ as in (6.40) can be iterated from any polytopic RCI set Z ∈ T , to define the set sequence Ψ(Z) = ψ i (Z) i∈N where

ψ 0 (Z) = Z, ψ i (Z) = ψ(ψ i-1 (Z)), ∀i ∈ N + .
Here, we do not have a result regarding the monotonicity of the sequence Ψ(Z) with regards to set inclusion. In particular, we do not have result regarding the convergence of this set sequence towards an RCI set. However, each iteration further reduce the size of the RCI set with regards to the matrix-norm criterion from (6.41).

The inequality (6.41) can be rewritten in terms of set inclusion using the concept of sublevel sets of the matrix-norm criterion as follows ∀Z ∈ T , H p (ψ(Z)) ⊆ H p (Z).

(6.42)

In particular, the set sequence H p (ψ i (Z)) i∈N is decreasing with regards to set inclusion (decreasing set sequence).

Set mappings of linear RCI sets with relaxation of the set inclusion constraints 201

The following sections detail the construction of two set mappings ψ using optimization-based methods. The construction of the two set mappings follow the same steps.

• The characterization of a family of polytopes obtained by non-uniformly scaling the initial polytopic RCI set. The non-uniform scaling involve two scaling factor. The first, chosen sub-unitary, contracts the set with regards to the directions defined by the matrix H. The second is not necessarily sub-unitary, and allows the set to dilate with regards to the other directions.

• The restriction to the elements of the family that are RCI.

• The selection of the RCI element that corresponds to the maximal contraction with regards to the matrix-norm criterion, along with the computation of an associated control law.

• The computation of the one-step image of the selected element.

As a first step, we consider scalar parameters for the joint dilatation and contraction of the RCI set.

Scalar dilatation and contraction of the RCI set

The set mapping ψ presented in this section relies on the joint contraction of the RCI set Z in the direction defined by the matrix H and dilatation on the other directions, where we consider scalar scaling factors.

Parametrization of candidate sets

To define the contraction with regards to the directions defined by the matrix H, we use the sublevel set H p (Z). For an RCI set Z and positive scalars α ≥ 0 and

0 ≤ β ≤ 1, we consider Z p (α, β) = αZ ∩ βH p (Z).
(6.43)

The set Z p (α, β) is dilated in the directions defined by the matrix G z , and contracted in the directions defined by the matrix H.

It satisfies |HZ p (α, β)| p ≤ β|HZ| p ≤ |HZ| p .
We have proposed a parametrization of a family of sets, and any element of this family has a matrix-norm criterion smaller than the initial polytopic RCI set Z. However, these elements are not polytopic, as the sublevel set H p (Z) is not a polyhedron in general. The sets Z(α, β) retain the polytopic structure of Z provided the choice p = 1 or p = ∞, as the intersection of a polytope and a polyhedron. The developments presented below are presented for the particular choice p = ∞. We leave to further study the remaining choice leading to polytopic sets, namely p = 1.

These constraints uses the (possibly non-minimal) H representation of the set Z(α, β) given in (6.44).

To reduce the computational complexity of the optimization problem, and avoid redundant constraints, it is preferable to compute a minimal H-representation of the polytope Z(α, β) to define optimization problems of smaller dimension both with regards to the number of optimization variables and the number of linear inequalities and linear equalities. However, this gain in terms of computational complexity is at the expense of the computation of the minimal H-representation of Z(α, β), which can be computationally demanding.

Necessary condition for feasibility

To further reduce the computational needs of the grid search approach, we search for necessary condition on α and β for the optimization problem P ψ (Z, α, β) to be feasible. A necessary condition for the set Z(α, β) to be linear RCI is the set inclusion W ⊆ Z(α, β), (6.49) relying on the set inclusion (A + BK)Z(α, β) ⊆ Z(α, β) W. Define Proof: The optimization problem P ψ (Z, α, β) is feasible if and only if the set Z(α, β) is linear RCI. We derive from (6.49) and (6.44) two necessary conditions for Z(α, β) to be linear RCI, namely 0 n ∈ αZ W, (6.51a)

α m (Z) = max (g z (W)) i (g z ) i | i ∈ N + lz , (6.50a
0 n ∈ βH ∞ (Z) W. (6.51b)
We have αZ W = {ξ ∈ R n | G z ξ ≤ αg z -g z (W)}. This polytope contains the origin if and only if αg z ≥ g z (W). If α < α m (Z), then αg z < g z (W) and the set inclusion (6.51a) does not hold. The set inclusion (6.51b) is satisfied if and only if ∀w ∈ W, |Hw| p ≤ β|HZ| p . With β < β m (Z), there exists w ∈ W such that |Hw| p = |HW| p = β m (Z)|HZ| p > β|HZ| p . We conclude that if β < β m (Z), then the set inclusion (6.51b) does not hold.

We conclude that the optimization problem P ψ (Z, α, β) is not feasible if α < α m (Z) or β < β m (Z).

Set mappings of linear RCI sets with relaxation of the set inclusion constraints 207

We can use this result to restrain the grid search on the parameters α and β respectively to [α m , ∞[ and [β m , 1]. Note that for a given polytope Z, the computation of α m (Z) and β m (Z) consists in basic operations.

The grid search approach for the computation of a candidate solution to the optimization problem P ψ (Z) is presented in Algorithm 6.8, where we perform a grid search on α and β over r ∈ N + and s ∈ N + fixed values. More sophisticated method for the choice of the grid search parameters α and β are to be sought (e.g. using dichotomy).

We only consider the parameters α and β that satisfy the necessary conditions presented in Section 6.8.2.3, namely α ≥ α m (Z) and β ≥ β m (Z). When this condition is satisfied, we compute the minimal H-representation of the associated set to reduce the computational complexity of the optimization problem P ψ (Z, α, β).

We stop the grid search once an optimization problem P ψ (Z, α, β) is feasible, as we the parameters β are ordered from the lowest to the highest. Moreover, we chose α 1 = β r = 1 to ensure that at least one of the optimization problem is feasible, and Algorithm 6.8 is guaranteed to provide a candidate solution. The family of sets proposed in this section are chosen as the intersection of a uniform scaling of the initial polytopic RCI set and a uniform scaling of its polyhedral sublevel set. In the next section we consider non-uniform scaling using vectorial scaling factors as in Section 6.6.1.2 and Section 6.6.2.2.

Vectorial dilatation and contraction of the RCI set

The developments presented below are an extension of the previous section, with the notable difference that the scaling factors are chosen vectorial. Similar to the previous section, the developments presented below are obtained with the choice p = ∞.

Parametrization of candidate sets

The dilatation of a polytopic RCI set by the vector a ∈ R lz is given by

{ξ ∈ R n | G z ξ ≤ diag(a)g z } ,
where the right-hand side of the i th linear inequality is multiplied by the i th element of the vector a. Meanwhile, the contraction of the sublevel set H p (Z) by a vector b ∈ R 2h is given by We have presented a parametrization of family of polytopic sets based on the joint contraction in the directions defined by the matrix G z by a vector a ∈ R lz and dilatation in the directions defined by the matrix H by a vector b ∈ R 2h . The elements Z(a, b) that verify |b| ∞ ≤ 1 lead to an RCI set whose matrix-norm criterion is smaller than the one of Z.

ξ ∈ R n | H -H ξ ≤ b|HZ| ∞ .
Among the elements of this family, we retain the polytopes that are RCI. We have the following result regarding the feasibility of P ψ (Z).

Proposition 51 (Feasibility) Let Z be a linear RCI set. The optimization problem P ψ (Z) has a non-empty feasible set.

Proof: We have Z(1 lz , |HZ| ∞ 1 2h ) = Z ∈ T L . Hence, the constraints of P ψ (Z) allow for a solution with the above values of a and b, and b = 1.

The optimization problem P ψ (Z) is not LP/QP due to the bilinear terms S z gz (a, b), S u gz (a, b), S x gz (a, b) in the constraints (6.54c), (6.46e) and (6.54g). The number of nonlinear terms increase linearly with the dimension of the vectorial scaling factors a ∈ R lz and b ∈ R 2h , and the dimension of a is equal to the Hcomplexity of the initial polytopic RCI set Z. For this reason, the scaling approach with vectorial scaling factors is suitable for polytopes of low H-complexity.

The computation of a candidate solution is even more complex than for the optimization problem P ψ (Z) in (6.46). The one-step image set in (6.55) is a suitable candidate for ψ(Z), as it satisfies the constraints of P ψ (Z), and thus

|Hψ(Z)| ∞ ≤ |b| ∞ |HZ| ∞ ≤ b|HZ| ∞ ≤ |HZ| ∞ .
Here, the computation of a candiate solution using a grid search approach is not available given the dimension of a and b. We use a nonlinear optimization solver to compute candidate solutions in Section 6.9. This strategy is viable for systems of low dimension (n small), and for polytopes of low complexity (l z small). Remark 6.13 The set mapping presented in this section contracts the RCI set with regards to the direction H at the expense of a dilatation in the other directions. Once the set mapping has led to an RCI set of desired size with regards to the directions defined by H, we can use one of the set mappings presented in Section 6.5 to reduce the size of the RCI set in the other directions. Indeed, these set mappings ensure that the resulting set sequence is decreasing in terms of set inclusion. A less computationally demanding approach is to iterate the one-step procedure with the last feedback gain, as it also generates a decreasing sequence of RCI sets.

The main advantage of the approach presented in this section is that it does not impose the sequence of RCI sets to be decreasing in terms of set inclusion. Indeed, we allow subsequent RCI sets to dilate via the introduction of a scaling factor. Meanwhile, we ensure the RCI set sequence is decreasing with regards to the matrixnorm criterion by minimizing its contraction with regards to the directions defined by the matrix H. This defines families of polytopic sets, and we used the Extended Farkas Lemma to establish a necessary and sufficient condition for the polytopic sets to be linear RCI. Among the scalar factors leading to linear RCI sets, we select one where the contraction regarding the directions defined by the matrix H is maximal. The selection of the optimal scaling parameters require to solve biliner optimization problems. We considered both scalar and vectorial scaling factors, the latter allowing for more liberty at the expense of additional bilinearities in the constraints of the optimization problem. In both cases, the computation of a candidate solution using a nonlinear optimization solver is to be sought. For scalar scaling factors, we propose a grid search approach for the computation of candidate solution when the nonlinear optimization solver fails to provide a satisfactory candidate solution.

We provide simulation examples of the use of the set mappings presented in this chapter in the next section.

Simulation results

The simulation use the following system,

Preliminaries

We give an illustration of the concepts of decreasing set sequences 6.9.1.1 Decreasing sequences of RCI sets

As a first step, we provide an illustration of RCI set sequences obtained with the set mappings φ µ proposed in (6.16). This illustrates the main limitation of the set mappings defined in Section 6.5, as it is reliant on the choice for the initial RCI set Z. A necessary condition for the set sequence {φ i µ 3 (Z} i∈N to converge towards an (H, p)-mRCI set is that the (H, p)-mRCI set is contained in the initial polytopic RCI set Z.

We now consider the matrix H = 0.6 0.8 , that also characterizes a stable sliding surface. The associated 0-DSMC feedback gain is K = -0.55 -1 . Figure 6.10 gives the decreasing RCI set sequence {φ i µ 3 (Z} i∈N 50 , along with the mRPI set Z ∞ (K). Despite having Z ∞ (K) ⊂ Z, the decreasing RCI set sequence does not converge towards the mRPI set. This illustrates that Z ∞ (K) ⊆ Z is a necessary condition, but it is not a sufficient condition for reaching the (H, p)-mRCI set.

To cope with this limitation, we proposed the set mapping ψ (6.40) where we We compute the first q = 50 elements of the RCI set sequence {ψ i (Z)} i∈N as detailed in Algorithm 6.7. We compute candidate solutions using the grid-search approach presented in Algorithm 6.8. The parameters of the grid search are chosen as follows, {α 1 , ..., α 5 } = {1, 5, 10, 100, 1000}, (6.56a) {β 1 , ..., β 9 } = {0.7, 0.75, 0.8, 0.9, 0.95, 0.97, 0.98, 0.99, 1}. (6.56b) Figure 6.11 and 6.12 show the resulting RCI set sequence obtained with H = 0.05 1 and H = 0.6 0.8 respectively. Tables 6.1 and 6.2 provide the values of α and β and the feedback gain K obtained for the first iterations.

For both values of H, the RCI set sequence {ψ i (Z)} i∈N 50 converges towards the (H, p)-mRCI set. In the first case, with H = 0.6 0.8 , the feedback gain has converged to the associated 0-DSMC feedback gain K = -0.49 1 after 4 iterations (ψ 3 (Z)). As a consequence, the subsequent iteration satisfies |Hψ 4 (Z)| = |HZ ∞ (K)| = 0.055. For H = 0.05 1 , the feedback gain has converged to the associated 0-DSMC feedback gain K = -0.049 1 after 5 iterations (ψ 4 (Z)).

It is worth to be mentioned that the first iterations of the set mapping lead to high values of α, ranging from 1 to 100. Thus, it leads to RCI sets that are very large in certain directions, although the size of the RCI sets decreases in the directions defined by the matrix H. This can be problematic in a scenario w ith state and/or input constraints.

We gave an illustration of the use of the selection functions in Section 6.6 and the associated set mappings to compute (H, p)-mRCI sets. The set mappings φ µ 1 and φ µ 2 are not adapted, since the matrix H is not taken into account in the design of the selection functions µ 1 and µ 2 . On the contrary, the selection function µ 3 , which aims at minimizing the (H, p) criterion of the one-step image, can lead to (H, p)-mRCI sets. However, it is reliant on the choice for the initial polytopic RCI set Z. The set mapping ψ proposed in Section 6.8 is less dependant on the choice for the initialization, as it does not require the (H, p)-mRCI set to be contained in The resulting RCI set obtained with this method is denoted Z ECC , and is shown in Figure 6.14 along with ψ 3 (Z). This set satisfies |HZ ECC | ∞ = 0.5. In particular, we have

|HZ ECC | ∞ = 0.5 ≥ 0.4667 = |Hψ 3 (Z)| ∞ .
The iterative approach using set mappings led to an RCI set with smaller matrixnorm criterion. This inequality is shown in terms of set inclusion of the sublevel sets in Figure 6.14.

We have provided several simulation examples of the application approach for the computation of minimal RCI sets presented in this chapter, based on the concept of set mappings. These examples highlight the main advantage of the approach where we allow the RCI set to dilate, as it is less dependent on the selection of an initial polytopic RCI set. We computed candidate solutions for the bilinear optimization solvers using either a grid search approach or a nonlinear solver. In both cases, the polytopic RCI set sequence is decreasing with regards to the matrixnorm criterion, and it converges towards a (H, p)-mRCI set when the matrix H is chosen to define a stable sliding surface. For other choices of the matrix H, we are not capable of determining if the set sequence has converged towards a (H, p)-mRCI set as we do not have a formal characterization. However, the approach shows better performances than the one presented in Chapter 5 in our example.

Conclusion

This section proposed algorithms in view of the computation of minimal RCI sets using the concept of set mappings. The first approach relates to the computation of outer approximations of an mRPI set for an autonomous system. Here, the approach differs from the use of the control law as a design parameter in the computation of the successive one-step image sets. The control law must be chosen such that the onestep image set retains the RCI property. We proposed optimization-based methods for the selection of the control law leading to the one-step image set whose size is minimal, either with regards to set inclusion or with regards to the matrix-norm criterion. This approach is dependent on the initialization, i.e. the choice of an initial RCI set, and results in sequences of RCI sets that are decreasing in terms of set inclusion. The study of the limit sets of these set mappings is of particular interest, as they represent the infinite refinement of RCI sets. This approach has a deep connection with the DSMC framework presented in Chapter 5. The inclusion of an element of the decreasing set sequence set to a mQSMB is a stopping criterion for our iterative algorithms, as it is guaranteed that the set sequence converges to the 0-DSMC mRPI set associated to the mQSMB.

The second method differs in that the set mappings allow for the dilatation of the RCI set as long as it is contracting in the directions defined by the matrix H. These set mappings use a parametrization of candidate RCI sets, using a pair of scaling factors chosen either scalar or vectorial. For both structures of scalar factors, the set mapping requires to solve a bilinear optimization problem.

Both methods were illustrated in simulation to showcase the properties established in this chapter. Their performances were compared with the method presented in the previous chapter that uses the DSMC framework.

Conclusion

Context of the study

This manuscript focused on the theoretical properties and the practical methods for the construction of RCI sets for constrained discrete-time linear systems subject to bounded additive disturbances. The motivation of this work arises from the control of a quadrotor system in the joint presence of disturbances and constraints. The construction of RCI sets is key in this field of application since they characterize safe flight zones where the quadrotor trajectories are guaranteed to respect the constraints regardless of the disturbances. As an additional motivation, RCI sets are design parameters of Tube Based MPC laws, a technique adapted to the constrained control of systems subject to disturbances.

In this context, two RCI sets are of particular interest. The first is the maximal RCI set, which represents the largest zone where the quadrotor can fly while respecting the constraints (e.g. avoid collision). The second is the minimal RCI set, which represents the maximal deviation of the quadrotor in stationary flight. This manuscript focuses on the computation of minimal RCI sets. The first difficulty arises from the definition of a criterion for the minimality of an RCI set, as set inclusion is a partial order on the collection of RCI sets.

RCI sets depend on the selection of both the nominal model and associated disturbance bounds, and the control law. In studies with simulation or experimental applications, it is generally assumed that the system identification step precedes the control design, and the control is the only design parameter of RCI sets. In this manuscript, we use both the system identification and the control design steps as design parameters. In Chapter 1, we propose a criterion, parametrized by a matrix defining directions of interest and a norm, to measure the minimality of a set that defines a total order on the collection of RCI sets. The Chapters 3, 5 and 6 focus on the computation of RCI sets that are minimal with regards to this criterion. This conclusion is divided according to the structure of the manuscript.

Part I -Implementation of a robust MPC scheme for the stabilization of a quadrotor UAV

Contributions

The first part of the manuscript focused on the computation of minimal RCI sets for an experimental application with the Parrot AR.Drone 2.0 platform, with the ultimate goal of the experimental validation of a robust MPC law. In this part, the nominal model and the disturbance bounds, obtained from experimental data, are the design parameters of the RCI set. Most system identification methods seek to fit the measurements better, and result in smaller disturbance bounds at the expense of larger RCI sets. In Chapter 3, we propose a novel system identification method that is adapted to the computation of minimal RCI sets.

An experimental validation of a Tube Based MPC law is provided for the stabilization of the horizontal dynamics of the Parrot AR.Drone 2.0, documenting all the design and implementation phases. In comparison to classical Tube Based MPC implementations, the tuning of the local controller precedes the identification of the model parameters. The local controller is tuned experimentally based on its disturbance rejection property in stationary flight.

The experimental results validate that the disturbance set used for the computation of the RCI set contain the disturbances in flight with the Tube Based MPC. This inclusion guarantees strong theoretical properties to the closed-loop system, such as robust recursive constraints satisfaction and stability of the closed-loop system. These developments have been presented in [START_REF] Michel | [END_REF]].

Further directions of research

The adopted system identification approach bounds the eigenvalues of the evolution matrix A in an LTI model while minimizing the least mean square value of the disturbance realizations. Our choice for the identification algorithm benefits from the LP/QP nature of the optimization problem, but requires a grid search on the maximal absolute value of the eigenvalues of A. More sophisticated techniques for the identification of the model that use the knowledge of the feedback gain K are to be sought.

The experimental validation of the Tube Based MPC was performed in a basic scenario of stabilization around a sequence of waypoints. More complex scenario are to be considered, for instance by

• Increasing the distance between successive waypoints. The maximal distance between two waypoints depends on the feasible region of the finite-time optimal control problem underlying the nominal MPC scheme. The feasible region is a function of the length of the prediction horizon, which also impacts the complexity of the MPC scheme.

• Considering a larger set of external disturbances by introducing waypoints close to obstacles (wall, ground, other UAVs, ...) or generating wind gusts. These additional disturbances are function of the state, which raise the more general problem of designing a Tube Based MPC law in presence of state dependent disturbances. Here, we chose to construct a disturbance set that contains all possible disturbances and which does not account for the possible state dependency.

• Extending the Tube Based MPC law for path following. A first step would be to consider stabilization around successive waypoints in a direction (x or y) with the Tube Based MPC while using another controller to perform trajectory tracking.

The implementation of a reference governor, robust interpolation-based control, or constraints softening strategies are to be sought whenever the constraints cannot be fulfilled. However, it is worth to be mentioned that any modification in the flight conditions (maximal distance between waypoints, presence of additional sources of disturbances) requires to update the identification of the model parameters and disturbance bounds.

Part II -Computation of minimal RCI sets for discretetime linear systems subject to bounded additive disturbances

In the second part, the research was based on the assumption that a nominal model and the disturbance bounds are known, and the control law is the only design parameter of RCI sets. When the control law is fixed, the minimal RCI set is called the mRPI. Thus, the computation of minimal RCI sets relates to the computation of the control law whose mRPI set is minimal. The results presented in this part are illustrated in simulation.

Contributions

In Chapter 5, we characterized the mRPI sets obtained with Discrete-time Sliding Mode Control, an approach known for its disturbance rejection properties in continuous-time. The DSMC mRPI sets are minimal with regards to our criterion for matrices defining stable sliding surfaces. At our best knowledge, this represents a first attempt at the formal characterization of minimal RCI sets. In the absence of formal characterization, we propose optimization-based methods for the computation of candidate solutions.

The first method seeks for the DSMC mRPI set that is minimal. This method benefits from the simplified characterization of DSMC mRPI sets, inherited from their two-steps design (selection of a stable sliding surface and a reaching law). Indeed, the choice of a stable sliding surface guides the mRPI. These contributions have been presented in [Michel 2018b] and [Michel 2018a].

The second method, presented in Chapter 6, is based on the one-step image of polytopic RCI sets with a well-chosen control law. We propose optimizationbased strategies for the systematic computation of the control law such that the one-step image is contained in the initial RCI set, at least in the directions of interest. We propose several approaches for the systematic computation of such control laws as the solution of optimization problems, defining selection functions. We use the selection functions to define mappings of polytopic RCI sets that can be iterated to construct decreasing sequences of polytopic RCI sets. The convergence of the decreasing sequences of RCI sets towards a global minimum is not guaranteed, however each iteration further reduces the size of the RCI sets. The choice for the initialization of the RCI set sequence is of particular importance, along with the characterization of the limit set. We give a first characterization of the limit sets of the set mappings, which represent the infinite refinement of RCI sets, using the DSMC framework. We provide a necessary sufficient condition to simplify the optimization problems, relative to the selection of a suitable control law, that uses the concept of minimal Quasi-Sliding Mode Band from DSMC. Part of these results have been presented in [Michel 2018a], and the submission of a journal publication with the remaining material is ongoing at the time of the writing of this manuscript.

Further directions of research

As a first perspective, we note that DSMC is a particular case eigenstructure assignment [Andry 1983]. We are interested in further studying the relationship between the eigenvalues and eigenvectors of the closed-loop matrix and the geometrical structure of the associated mRPI sets. This interest takes motivation both from the computation of minimal RCI sets and the deduction of possible simplifications of the optimization problems in our iterative approach. Indeed, the sufficient condition for these optimization problems to be trivial were obtained by studying the structure of the invariant sets associated to DSMC laws.

As a perspective for this part, we point to further studying the properties of the set mappings of RCI sets presented in Chapter 6. This consists of the characterization of novel selection functions, along with the study of their limit sets and the relationships between the initial RCI set and the limit of the decreasing sequences. Indeed, the selection functions proposed in this manuscript fail to account for the iterative aspect of the approach, as they seek to minimize the size of the one-step image. It could be of interest to construct selection functions that seek to minimize the size of the N -steps image set, thus defining set mappings that are more efficient for the computation of minimal RCI sets.

General discussion

The main perspective of this work is to develop a method that unifies both the theoretical and the practical approaches, where the system identification and control design steps are performed simultaneously (and not sequentially) with the global goal of computing minimal RCI sets.

The presence of delay in the communication of the input to the Parrot AR.Drone 2.0 quadrotor restrained us from using both the model parameters and the control law as design knobs in the computation of minimal RCI sets. With our modelling choices, the delay induces additional disturbances that is function of the control law. As a consequence, the system identification step is highly dependent on the experimental data. At first, we generated the experimental data with a large range of control laws. With this experimental data, we obtained RCI sets that were too large to implement within a Tube Based MPC regardless of the choice of the model parameters and local controller. Due to this practical complication, we generated data with a smaller range of control laws, and retained the data that presented the smallest deviation around the waypoints. This forces the local controller to be in this range of control laws, since another choice of local controller generates disturbances that are not accounted for in the characterization of the disturbance bounds. Thus, the model-free tuning of the local controller precedes the system identification step in our experimental developments. It is not possible to apply the theoretical developments of Chapters 5 and 6 for the computation of minimal RCI sets in our experimental setting, as these developments require the knowledge of the model parameters and disturbance bounds.

We also point to the implementation of Tube Based MPC on a quadrotor platform with

• Embedded computational capacity to not suffer from delay in the transmission of the input.

• Better performances in terms of stabilization of the orientation dynamics. Nous nous intéressons à la partie commande de l'automatisation d'un quadrotor dans une problématique de vol dans un milieu encom-bré et en présence de perturbations aérodynamiques. Ces perturbations, et les erreurs de modèles, peuvent nuire au succès de la mission de vol, et doivent donc être prises en compte dans l'élaboration de trajectoires « sûres ». Une manière d'évaluer l'impact des perturbations sur un système bouclé consiste à calculer des ensembles invariants. Notre but est de synthétiser une loi de commande qui permette de générer des trajectoires en boucle fermées qui respectent les différentes contraintes en les bornant à l'intérieur de zone de vol sûres, en bornant les trajectoires dans des ensembles invariants. En particulier, nous étudions la syntèse de lois de commandes menant à des ensembles invariants qui soient les plus petits possible.

Title: Invariant set design for the constrained control of a quadrotor Keywords: Quadrotor, Control, Disturbance, Constraints, MPC, Invariant sets Abstract: Unmanned Aerial Vehicles (UAVs) quadrotors are versatile platforms capable of agile motion and stable hovering. The use of drones in civil application and industry has considerably increased in the last years, and is foreseen to continue growing. The design of autonomous UAVs should take into account safety and technological constraints, such as distance to obstacles, actuator limitations or real-time computational constraints for embedded implementation.

Here we focus on quadrotor control for applications in a cluttered environment, where we want to account for the presence of external disturbances. External disturbances and modelling mismatches can affect the execution of a mission and its impact on the closed-loop trajectories must be assessed. A systematic way to assess the influence of disturbances is to compute invariant sets. The goal is to compute control laws that generate collision-free trajectories by bounding them within safe flight regions, characterized set-wise by invariant sets, where all constraints satisfaction is guaranteed. In particular, we study the design of control laws leading to invariant sets that are as small as possible.

Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

Introduction1

  Theoretical background on invariant set and Tube Based MPC for linear discrete-time systems 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notation and basic definitions . . . . . . . . . . . . . . . . . . . . . . 1.3 Definition and properties of polyhedra . . . . . . . . . . . . . . . . . 1.4 Invariant set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Closed-loop systems . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Open-loop systems . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Tube Based MPC of constrained linear discrete-time systems . . . . 1.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Tube of trajectories . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 First formulation of the optimal control problem . . . . . . . 1.5.4 Second formulation of the optimal control problem . . . . . . 1.5.5 Practical implementation of Tube Based MPC . . . . . . . . . 1.6 Simulation analysis of the impact of the Tube Based MPC parameters 1.6.1 Design of the local controller . . . . . . . . . . . . . . . . . . 1.6.2 Terminal set design . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Closed-loop performances . . . . . . . . . . . . . . . . . . . . 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I Implementation of a robust MPC scheme for the stabilization of a quadrotor UAV3 Design and Flight Experiments of a Tube Based MPC for the stabilization of quadrotor horizontal dynamics 73 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 Modelling of the horizontal dynamics . . . . . . . . . . . . . . . . . . 74 3.2.1 Model discretization . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.2 Disturbance set characterization from experimental data . . . 76 3.2.3 Delay in the control . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Input delay handling in the implementation of the Tube Based MPC 78 3.3.1 Simulation example . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.2 Experimental data generation for system identification and experimental design of a Tube Based MPC law . . . . . . . . 80 3.4 Experimental tuning of the local controller . . . . . . . . . . . . . . . 83 3.5 Closed-Loop System Identification and Invariant Set Design . . . . . 84 3.5.1 Computation of candidate solution . . . . . . . . . . . . . . . 86 3.5.2 Iterative refinement of the identification . . . . . . . . . . . . 90 3.6 Practical implementation of the Tube Based MPC . . . . . . . . . . 92 3.6.1 Solving the MPC QP using CVXGEN in real time . . . . . . 95 3.6.2 Change of reference and terminal set computation . . . . . . 96 3.6.3 Structure of the code . . . . . . . . . . . . . . . . . . . . . . . 99 3.7 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.7.1 Tube Based MPC parameters . . . . . . . . . . . . . . . . . . 100 3.7.2 Closed-loop horizontal trajectory . . . . . . . . . . . . . . . . 103 3.7.3 Nominal system trajectory . . . . . . . . . . . . . . . . . . . . 103 3.7.4 Disturbance realizations and tube properties . . . . . . . . . . 105 3.7.5 Recursive state and input constraints satisfaction . . . . . . . 108 3.7.6 Real-time computation . . . . . . . . . . . . . . . . . . . . . . 108 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Part II Computation of minimal RCI sets for constrained discrete-time linear systems subject to bounded additive disturbances 113 1 La trajectoire est contenue dans l'ensemble invariant et respecte les contraintes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 2 La trajectoire réelle ne satisfait pas les contraintes. . . . . . . . . . . xiv 3 La trajectoire réelle est contenue dans le tube de trajectoire qui satisfait les contraintes. . . . . . . . . . . . . . . . . . . . . . . . . . . xv 4 Le choix de la surface de glissement permet d'orienter l'ensemble mRPI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 5 Suite décroissante d'ensembles RCI obtenues par une méthode itérative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 6 The trajectory remains in the RPI set and avoids constraints violation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 The actual trajectory does not respect the constraints. . . . . . . . 5 8 The actual trajectory remains in the tube of trajectories and avoids constraints violation. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1 State space representation of the disturbance set W. . . . . . . . . . 39 1.2 State space representation of outer approximations of the mRPI for the linear gains K(i). . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.3 State space representation of the successive image setsΩ i (K(1)) (left). Volume, V-complexity, and H-complexity of the successive image sets Ω i (K(1)) (right). . . . . . . . . . . . . . . . . . . . . . . 42 1.4 State space representation of the state constraints X and X , and the terminal sets Xf (i) (left) and feasible regions X10 (i) (right) for i ∈ {1, 2, 3}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 1.5 State space representation of the RPI set Z along with the state errors using TBMPC (top) and ETBMPC (bottom), for the first (left) and second (right) disturbance scenario. . . . . . . . . . . . . . . . . . . 44 1.6 Time evolution of the control inputs with TBMPC (top) and ETB-MPC (bottom) for the first (left) and the second (right) disturbance scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.7 Comparison of the time-evolution of the states ξ 1 (top) and ξ 2 (bottom) for the first (left) and second (right) disturbance scenario with TBMPC and ETBMPC. . . . . . . . . . . . . . . . . . . . . . . . . 45 1.8 State space representation of the trajectory of the uncertain and nominal state with TBMPC (top) and ETBMPC (bottom) for the first (left) and second (right) disturbance scenario. . . . . . . . . . . . . 46 2.1 Inertial frame J (left) and body fixed frame B (right). . . . . . . . . 52 2.2 The Pitch, Roll, Yaw and Throttle movements of the Parrot AR.Drone 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 vi List of Figures 2.3 Communications between the User, the PC station, the Motion Capture system and the quadrotor UAV. . . . . . . . . . . . . . . . . . . 55 3.1 State-space of the disturbance set W a and the disturbance realizations w a,k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Trajectory of the closed-loop system with the linear control laws ν i , i = {1, 2, 3} (left to right) and x[0] = [1 0] . . . . . . . . . . . . 80 3.3 Disturbance sets W i,j computed for ν i (x), i = {1, 2, 3} (left to right) and x[0] = x j [0], j = {1, 2, 3} (top to bottom). . . . . . . . . . . . . . 81 3.4 Time-evolution of the position and reference in the direction x obtained in experiments with a linear state feedback control law ν i (ξ), i = {1, 2, 3, 4} (left to right, top to bottom). . . . . . . . . . . 84 3.5 State-space representation of the disturbance sets W(β (j) ) (left) and mRPI sets Z(β (j) ) (right), for j = {1, 2, 3, 4} (top to bottom). . . . . 89 3.6 Initial disturbance set W (0) (white) and refined disturbance set W (1) (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.7 Initial invariant set Z (0) (red) and refined invariant set Z (1) (white). 93 3.8 Evolution of the approximation quality with regards to the Hcomplexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.9 Structure of the Tube Based Model Predictive algorithm for the stabilization of the Parrot AR.Drone 2.0. . . . . . . . . . . . . . . . . . 100 3.10 State space representation of the state constraints of the nominal and uncertain systems, and the terminal sets for ξ r = ξ r i , i = {1, 2, 3}. . 103 3.11 Horizontal (x, y) trajectory of the uncertain system during the validation flight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.12 State space representation of the optimal state sequence ξ * ( ξ[k], ξ r ) and the terminal set Xf (ξ r ) with ξk = -0.4 0 0 and ξ r = -0.1 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.13 Time evolution of the Model Predictive Control cost function for both directions x and y. The nonlinearities occur at the time instant of waypoint reference changes. . . . . . . . . . . . . . . . . . . . . . . . 106 3.14 State space representation of the nominal system trajectory for both directions x (blue) and y (red), along with the constraint set X (white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.15 State space representation of the disturbance set W and the disturbance realizations {w x [k]} k∈N and {w y [k]} k∈N encountered during the validation flight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.16 State space representation of the RPI set Z and the state errors {ξ x [k] -ξx [k]} k∈N and {ξ y [k] -ξy [k]} k∈N during the validation flight. 107 3.17 State space representation of the uncertain system trajectory for both directions x (blue) and y (red), along with the constraint set X (white). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xii

  Figure 1 -La trajectoire est contenue dans l'ensemble invariant et respecte les contraintes.

24

  Figure 2 -La trajectoire réelle ne satisfait pas les contraintes.

  Figure 3 -La trajectoire réelle est contenue dans le tube de trajectoire qui satisfait les contraintes.

Figure 4 -

 4 Figure 4 -Le choix de la surface de glissement permet d'orienter l'ensemble mRPI.

Figure 5 -

 5 Figure 5 -Suite décroissante d'ensembles RCI obtenues par une méthode itérative.

Figure 6 -

 6 Figure 6 -The trajectory remains in the RPI set and avoids constraints violation.

Algorithm 0. 2 :4 5

 25 Model Predictive Control Inputs: A model for the system 1 Repeat 2 Measure the current state ξ[kT ] and update the constraints ; 3 Compute the finite open-loop control sequence {u * [(k + i)T ]} i∈{0,1,...,N -1} leading to the feasible trajectory {ξ * [(k + i)T ]} i∈{1,...,N } that optimizes the cost function; Apply the first element of the sequence as control action u[kT ] ← u * [kT ].; Increment k ← k + 1;

Figure 7 -

 7 Figure 7 -The actual trajectory does not respect the constraints.

Figure 8 -

 8 Figure 8 -The actual trajectory remains in the tube of trajectories and avoids constraints violation.

1. 5 .

 5 Tube Based MPC of constrained linear discrete-time systems 35 Remark 1.4

1. 6 .

 6 Figure 1.1 -State space representation of the disturbance set W.

  and the associated set of closed-loop poles, are shown in the following table.

Figure 1

 1 Figure 1.3 -State space representation of the successive image sets Ω i (K(1)) (left). Volume, V-complexity, and H-complexity of the successive image sets Ω i (K(1)) (right).

Figure 1 . 4 -

 14 Figure 1.4 -State space representation of the state constraints X and X , and the terminal sets Xf (i) (left) and feasible regions X10 (i) (right) for i ∈ {1, 2, 3}.

Figure 1 . 5 -Figure 1 . 6 -Figure 1 . 7 -

 151617 Figure 1.5 -State space representation of the RPI set Z along with the state errors using TBMPC (top) and ETBMPC (bottom), for the first (left) and second (right) disturbance scenario.

Figure 1 . 8 -

 18 Figure 1.8 -State space representation of the trajectory of the uncertain and nominal state with TBMPC (top) and ETBMPC (bottom) for the first (left) and second (right) disturbance scenario.

Figure 2 . 1 -

 21 Figure 2.1 -Inertial frame J (left) and body fixed frame B (right).

Figure 2

 2 Figure 2.2 -The Pitch, Roll, Yaw and Throttle movements of the Parrot AR.Drone 2.0.

Figure 2

 2 Figure 2.3 -Communications between the User, the PC station, the Motion Capture system and the quadrotor UAV.

  Figure 3.1 -State-space of the disturbance set W a and the disturbance realizations w a,k .

Figure 3

 3 Figure 3.2 -Trajectory of the closed-loop system with the linear control laws ν i , i = {1, 2, 3} (left to right) and x[0] = [1 0] .

3. 3 .

 3 Figure 3.3 -Disturbance sets W i,j computed for ν i (x), i = {1, 2, 3} (left to right) and x[0] = x j [0], j = {1, 2, 3} (top to bottom).

Figure 3

 3 Figure 3.4 -Time-evolution of the position and reference in the direction x obtained in experiments with a linear state feedback control law ν i (ξ), i = {1, 2, 3, 4} (left to right, top to bottom).

Problem 3. 1

 1 For given experimental data {ξ m k , u m k } k∈N L+1 , feedback gain K ∈ R m×n , matrix H ∈ R h×n and scalar p ∈ N+ , compute the matrices A ∈ R n×n and B ∈ R m×n as in (3.5) that minimizes |HZ ∞ (A, B, W)| p , with W as in (3.8).

5

  Set the model parameters ( Ã, B) as the solution of (3.19); 6 Compute the associated disturbance set W as in (3.8); 7 Compute Z as an ( , p)-outer approximation of Z ∞ ( Ã, B, W );

  Figure3.5 shows the disturbance set W(j) and outer approximations of the RPI sets Z(j). These outer approximations are computed using the method Scaling of

Figure 3 . 5 -

 35 Figure 3.5 -State-space representation of the disturbance sets W(β (j) ) (left) and mRPI sets Z(β (j) ) (right), for j = {1, 2, 3, 4} (top to bottom).

4 6 Compute

 6 Set the model parameters ( Ã, B) as the solution of (3.23) with W (0) = W; 5 Compute the associated disturbance set W as in (3.8); Z as an ( , p)-outer approximation of Z ∞ ( Ã, B, W); 7 if |H Z| p ≤ |HZ| p then 8 Set (A, B, W, Z) ← ( Ã, B, W, Z) 9 end 10 until i = q;

3. 6 .Figure 3 . 6 -

 636 Figure 3.6 -Initial disturbance set W (0) (white) and refined disturbance set W (1) (red).

Figure 3

 3 Figure 3.7 -Initial invariant set Z (0) (red) and refined invariant set Z (1) (white).

Figure 3

 3 Figure 3.8 -Evolution of the approximation quality with regards to the Hcomplexity.

Figure 3 . 9 -

 39 Figure 3.9 -Structure of the Tube Based Model Predictive algorithm for the stabilization of the Parrot AR.Drone 2.0.

Figure 3 .

 3 Figure 3.10 -State space representation of the state constraints of the nominal and uncertain systems, and the terminal sets for ξ r = ξ r i , i = {1, 2, 3}.

Figure 3

 3 Figure 3.11 -Horizontal (x, y) trajectory of the uncertain system during the validation flight.

Figure 3 .

 3 Figure 3.12 -State space representation of the optimal state sequence ξ * ( ξ[k], ξ r ) and the terminal set Xf (ξ r ) with ξk = -0.4 0 0 and ξ r = -0.1 0 0 .

Figure 3

 3 Figure 3.13 -Time evolution of the Model Predictive Control cost function for both directions x and y. The nonlinearities occur at the time instant of waypoint reference changes.

Figure 3 .

 3 Figure 3.14 -State space representation of the nominal system trajectory for both directions x (blue) and y (red), along with the constraint set X (white).

Figure 3 .

 3 Figure 3.15 -State space representation of the disturbance set W and the disturbance realizations {w x [k]} k∈N and {w y [k]} k∈N encountered during the validation flight.

Figure 3 .

 3 Figure 3.16 -State space representation of the RPI set Z and the state errors {ξ x [k] -ξx [k]} k∈N and {ξ y [k] -ξy [k]} k∈N during the validation flight.

Figure 3 .Figure 3

 33 Figure 3.18 -Time-evolution of the position, speed, angle and control input of the nominal and the uncertain systems, and the real and nominal constraints, during the validation flight in the direction x.

  where the control action at time k is a function of the previous disturbance realizations u[k] = ν({w[0], ..., w[k -1]}).The state at time 0 is initialized as ξ[0] = 0 n , and we have u[0] = 0 m . At the next time-step, the system is affected by an exogenous disturbance w[0], leading to ξ[1] = w[1]. The control action u[1] is chosen as ν(w[0]) = K(0)w[0], where K(0) ∈ R m×n . This yields ξ[2] = (A + BK(0))w[0] + w[1], and the current control action is chosen as ν

  in the parameter K k with p = 1, ∞ (resp. p = 2) for fixed values of k and α.

  5.7.3.3 Matrix of higher dimension . . . . . . . . . . . . 159 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Assumption 5 . 1 (

 51 Full rank input matrix) The input matrix B ∈ R m×n is full rank, i.e. rank(B) = m. Proposition 14 (Regular form) Consider a discrete time linear system (5.1) under Assumption 5.1. With an adequate change of coordinates, the system can be written ξ+ = Ãξ + Bu + w, B = 0 n-m,m I m .

Assumption 5 . 4 (

 54 Stable sliding surface) We have h = m and the matrix H satisfies H ∈ K(A 11 , A 12 ).

Assumption 5 . 5 (

 55 Unstable sliding surface) We have h = m, and the matrix H satisfies H B ∈ GL m and H ∈ K(A 11 , A 12 ).The underlying strategy is to obtain the matrix C ∈ K(A 11 , A 12 ) closest to the matrix H. In the following, we impose C = C B ⊥ H B . Thus the design parameters of the feedback gain K is C B ⊥ . Remark 5.1 Note that using C = C B ⊥ H B in (5.14) does not introduce conservatism with respect to the choice C = CB ⊥ CB with arbitrary CB since CB ⊥ = CB H -1 B C B ⊥ leads to the same gain K. Under this consideration, (5.40) reads

  .46) This set is the product of the matrix H B ⊥ -C B ⊥ and the infinite Minkowski sum ∞ i=0 Âi 11 A 11 A 12 W. These two elements depend on the matrix C. Let us focus on the term ∞ i=0 Âi 11 A 11 A 12 W. The p norm of this set can be upper-bounded as follows

5. 6 .

 6 Computation of minimal RCI sets using DSMC 149 Algorithm 5.1: Grid search for candidate (H, p)-mRCI set computation using DSMC lawsInputs: A, B, W, X , U, H ∈ R h×n , p ∈ N+ , > 0, and {λ j } j∈N + q Outputs: A set Z * Set j ← 0 and V alue ← ∞; repeat Set j ← i + 1 and λ ← λ j ;

Figure 5 . 1 -

 51 Figure 5.1 -Set of matrices C = c 1 c 2 1 defining stable sliding surfaces.

Figure 5 . 2 -

 52 Figure 5.2 -Comparison of the V-complexity (left) and computation time (right) of the r step image set for general feedback gain (blue) and 0-DSMC laws (red).

Figure 5 .

 5 Figure 5.3 -State-space representation of the ( , p) approximation of the mRPI sets Z (i) , the sublevel set H

∞

  (Z (i) ) and the disturbance set W for i = {1, 2, 3, 4} (top to bottom).

  As a consequence, the infinite Minkowksi sum on the right hand side is large.This highlights the double role of the matrix C in the design of a 0-DSMC law.First, it defines the matrixI n-m -C -1 B C B ⊥that multiplies the infinite Minkowski sum, and consequently orientates the mRPI set. Second, it impacts the size of the infinite Minkowski sum via the successive powers of the matrix Â11 = A 11 -A 12 C -1 B C B ⊥ (and their spectral radius).As a concluding remark, whenever it is possible to chose C = H we are not interested in the spectral radius of the Schur matrix Â11 , asH I n-m -C -1 B C B ⊥ = 0 m,n-m .However, when H does not define a stable sliding surface, HI n-m -C -1 B C B ⊥ = 0 m,n-m, and the spectral radius of Â11 is to be taken into account in the selection of the matrix C for the design of the 0-DSMC law.

Figure 5 .Figure 5 . 4 -

 554 Figure 5.4 illustrates the relation between the scalar λ and the criterion that the optimization problem 5.48 seeks to minimize, namely |C B ⊥ -H B ⊥ | ∞ (red), for

Figure 5 .

 5 Figure 5.5 presents the invariant sets Z(λ * ), the sublevel H ∞ (Z(λ * )), along with the disturbance set W. Here, we do not have HZ(λ * ) = HW, since the selection H = C is not available in the design of the 0-DSMC law. Instead, we have HW ⊂ HZ(λ * ), which leads to |HW| p < |HZ(λ * )| p . To provide an illustration of this inequality, we represent the sublevels sets H ∞ (Z(λ * )) and H ∞ (W).

Figure 5 . 5 -

 55 Figure 5.5 -State-space representation of the RPI set Z(λ * ), the sublevel sets H ∞ (Z(λ * )) and H ∞ (W), and the disturbance set W for H (1) (left) and H (2) (right).

Figure 5 . 6 -

 56 Figure 5.6 -State space representation of the state constraints X and the disturbance set W.

Figure 5

 5 Figure 5.7 -Value of the norm criteria |HZ(K(λ(j)))| ∞ and |G u K(λ(j))Z(K(λ j ))| ∞ , and the constraint thresholds.

Figure 5 . 8 -Figure 5 . 9 -

 5859 Figure 5.8 -State space representation of the sets Z(K(σ r )) and the state constraints X for i = 1 (top left), i = 2 (top right), and i = 3 (bottom)

Proposition 26

 26 If Z is an RCI set, then |HCl(Z)| p = |HZ| p . Proof: This result comes from the definition of set closure. The closure of an RCI set has the same value of the matrix-norm criterion as its closure set. Proposition 27 (Closure of an RCI set) Consider Assumption 6.1. If Z is an RCI set, then Cl(Z) is an RCI set.

Theorem 6 . 1 (

 61 Existence of matrix-norm mRCI sets) Consider Assumption 6.1. For any choice of matrix H ∈ R h×n , h ∈ N and p ∈ N+ , there exists at least one (H, p)-mRCI set. Proof: Let {Z i } i∈N be an infinite sequence of RCI sets that verifies lim i→∞ |HZ i | p = h(H, p), and |HZ i+1 | p ≤ |HZ i | p , ∀i ∈ N. From Proposition 27 and 26, {Cl(Z i )} i∈N is an infinite sequence of compact RCI sets that verifies |HCl(Z i )| p = |HZ i | p , ∀i ∈ N.

Proposition 30

 30 With Z ∈ C, and ν ∈ N (Z), we have Co(Image(Z, ν)) ⊆ Z and Co(Image(Z, ν)) ∈ C. Proof: The set inclusion Co(Image(Z, ν)) ⊆ Z is a direct consequence of the convexity of Z and the set inclusion (6.7a). Using this, we have Co(Image(Z, ν)) ⊆ X ν , Image(Co(Image(Z, ν)), ν) ⊆ Image(Z, ν) ⊆ Co(Image(Z, ν)).

Figure 6 .

 6 Figure 6.1 -Illustration of a decreasing RCI set sequence {φ i µ 1 (Z)} i∈N 50 and disturbance set W.

6

  Compute g z (W) as in(6.11); ξ j ) as the unique unique solution of the chosen pair of optimization problems ((6.20) and (6.21),(6.22) and (6.23), or (6.24) and (6.25)) with ξ = ξ j ;11 Set ξ j ← Aξ j + Bν(ξ j );12 until j = f ;13 Increment i by one;14Set Z i = Co({ξ j + w | j ∈ N + f , w ∈ W}); 15 until i = q;

  Assumption 6.5. Let Z be a polytopic RCI set that contains the origin in its interior. If a full rank matrix C m×n satisfies CZ = CW, then the matrix CB is full rank.

  .36) From CZ = CW, we have c i Z = CW, ∀i = {1, ..., m}. With this consideration and the equation (6.35), (6.36) reads c Aξ = 0 1,n .

  , |Hψ(Z)| p ≤ |HZ| p .(6.41)

  )β m (Z) = |HW| p |HZ| p = max {|Hw| p | w ∈ V (W)} max {|Hξ| p | ξ ∈ V (Z)} . (6.50b)We have the following property regarding the feasibility of P ψ (Z, α, β).Proposition 50 (Grid search feasibility) Let Z ∈ T L . If α < α m (Z) or β < β m (Z), then the optimization problem P ψ (Z, α, β) has an empty feasible set.

Algorithm 6 . 8 : 6 Initialize

 686 Computation of candidate solution with a grid search approachInputs: A, B, W, H, {α 1 = 1, ..., α s }, 0 ≤ β 1 ≤ ... ≤ β r =1 and a polytopic linear RCI set Z. Outputs: A candidate solution (S z , S w , S u , S x , K, α, β) of P ψ (Z) 1 Initialize j ← 0 and f lag ← f alse; 2 Compute α m (Z) and β m (Z) as in (6.50); 3 repeat 4 if β ≥ β m (Z) then 5 Set j ← j + 1 and β = β j ; Set i ← i + 1 and α = α i ; 9 if α ≥ α m (Z) then 10 Set Z ← Z(α, β); 11 Compute ( Gz , g) a minimal H-representation of Z (remove redundant constraints); 12 if P ψ (Z, α, β) is feasible then 13 Set (S z , S w , S u , S x , K) as a solution of P ψ (Z, α, β) (LP); = true or i = s; 17 end 18 until f lag = true; 208 Set mappings

  For a linear RCI set Z and two vectors a ∈ R lz and b ∈ R 2h such that 0 lz ≤ a, 0 2h ≤ b, we propose the following parametrization of polytopesZ(a, b) = ξ ∈ R n | Gz ξ ≤ gz (a, b) , l×n , gz (a, b) = diag(a)g z b|HZ| ∞ ∈ R l, l = l z + 2h. (6.52b)By construction, the matrix-norm criterion of the setZ(a, b) satisfies |HZ(a, b)| ∞ ≤ |b| ∞ |HZ| ∞ . Thus, for |b| ∞ ≤ 1 we have |HZ(a, b) ≤ |HZ| ∞ .Remark 6.12 A particular choice of this parametrization is a = 1 lz and b = |HZ| ∞ 1 2h , leading to Z(a, b) = Z which is RCI.

6. 8 .

 8 Set mappings of linear RCI sets with relaxation of the set inclusion constraints 209 6.8.3.2 Condition for set invariance Similar to the previous section, we use the Extended Farkas Lemma to have a sufficient condition for the elements to be RCI (necessary and sufficient condition to be linear RCI).The set Z(a, b) is linear RCI if and only if there exists matricesS z ∈ R l× l + , S w ∈ R l×lw + , S u ∈ R lu× l + , S x ∈ R lx× l +with non-negative elements and a feedback gain K ∈ R m×n such thatGz (A + BK) = S z Gz , (6.53a) S z gz (a, b) + S w g w ≤ gz (a, b),(6.53b)G u = S u Gz , (6.53c) S u gz (a, b) ≤ g u ,(6.53d)G x = S x Gz ,(6.53e)S x gz (a, b) ≤ g x .(6.53f)We have a necessary and sufficient condition on the vectors a and b for the set Z(a, b) to be linear RCI. However, the collection of all vectors a and b such that the above constraints are feasible is possible infinite. The selection of the best pair of scaling vectors is discussed below.6.8.3.3 Selection of the optimal vectorial scaling factorsFor a linear polytopic RCI set Z and vectors a ∈ R lz and b ∈ R 2h , we have the following upper-bound of the the matrix-norm of the set Z(a, b),|HZ(a, b)| ∞ ≤ |b| ∞ |HZ| ∞ .For this reason, we retain the vectorial parameters (a, b) such that the constraints (6.53) are satisfied and |b| ∞ is minimal. Note that minimizing |b| ∞ consists in minimizing b > 0 under the constraints b i ≤ b, ∀i ∈ N + 2h . The above considerations lead to the following optimization problem, P ψ (Z) : minimize (Sz,Sw,Su,Sx,K,a,b, b) BK) = S z Gz , (6.54b) S z gz (a, b) + S w g w ≤ gz (a, b), (6.54c)G u = S u Gz , (6.54d) S u gz (a, b) ≤ g u ,(6.54e)G x = S x Gz , (6.54f) S x gz (a, b) ≤ g x ,(6.54g)S z ∈ R l× l + , S w ∈ R l×lw + ,(6.54h) S u ∈ R lu× l + , S x ∈ R lx× l + , K ∈ R m×n , (6.54i) 0 ≤ b i ≤ b, ∀i ∈ N + 2h , b ≤ |HZ| ∞ , (6.54j) 0 ≤ a j , ∀j ∈ N + lz .(6.54k)

  Let (S z , S w , S u , S x , K, a, b, b(Z)) be a candidate solution of P ψ (Z), and define Image(Z(a, b), K) = (A + BK)Z(a, b) ⊕ W.(6.55)

Figure 6 .

 6 Figure 6.2 shows the state space representation of the disturbance set W.

Figure 6 .

 6 Figure 6.9 -Decreasing RCI set sequences {φ i µ 3 (Z)} i∈N 50 , the mRPI set Z ∞ (K) and disturbance set W with H = 0.05 1 .

Figure 6 .

 6 Figure 6.10 -Decreasing RCI set sequences {φ i µ 3 (Z)} i∈N 50 , the mRPI set Z ∞ (K) and disturbance set W with H = 0.6 0.8 .

  Value of the grid search parameters α and β, and the resulting optimal feedback gain K obtained with H = 0.05 1 .

Figure 6 .

 6 Figure 6.11 -RCI set sequence {ψ i (Z)} i∈N 50 , the mRPI set Z ∞ (K) and disturbance set W with H = 0.05 1 .

Figure 6 .

 6 Figure 6.12 -RCI set sequence {ψ i (Z)} i∈N 50 , the mRPI set Z ∞ (K) and disturbance set W with H = 0.6 0.8 .

Figure 6 .

 6 Figure 6.13 -Initial RCI set Z, RCI set sequence {ψ i (Z)} i∈N 3 , and their associated sublevel sets H(ψ i (Z)) for H = 2 2 -1 1 .

Figure 6 .

 6 Figure 6.14 -RCI set obtained with the DSMC approach Z ECC and ψ 3 (Z), along with their respective sublevel sets, for H = 2 2 -1 1 .

Titre:

  Construction d'ensembles invariants pour la commande d'un quadrotor sous contraintes Mots clés: Quadrotor, Commande, Perturbations, Contraintes, MPC, Ensembles invariants Résumé: Les quadrotors sont des plateformes versatiles, capable de manoeuvres agiles et de survol stables. Leur utilisation pour des applications civiles et industrielles a considérablement augmenté ces dernières années, et on estime que cette tendance ne devrait pas s'arrêter. La conception de quadrotors qui soient autonomes nécessite de prendre en compte des contraintes en termes de sûreté et d'embarquabilité, telles que l'évitement d'obstacles, la présence de saturation au niveau des actuateurs, ou encore les limitations en termes de temps de calcul disponible.

  Theoretical background on invariant set and Tube Based MPC 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

	Chapter 1
	Theoretical background on
	invariant set and Tube Based
	MPC for linear discrete-time
	systems
	34
	1.5.4 Second formulation of the optimal control problem . . . . 36
	1.5.5 Practical implementation of Tube Based MPC . . . . . . 38
	1.6 Simulation analysis of the impact of the Tube Based
	MPC parameters .

Contents 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Notation and basic definitions . . . . . . . . . . . . . . . . 14 1.3 Definition and properties of polyhedra . . . . . . . . . . 16 1.4 Invariant set theory . . . . . . . . . . . . . . . . . . . . . . 17

1.4.1 Closed-loop systems . . . . . . . . . . . . . . . . . . . . . 18 1.4.1.1 General definitions . . . . . . . . . . . . . . . . . 18 1.4.1.2 Linear control laws . . . . . . . . . . . . . . . . 20 1.4.1.3 Approximations of the mRPI set . . . . . . . . . 21 1.4.1.4 Remarks on the computational complexity of mRPI approximations . . . . . . . . . . . . . . . 23 1.4.2 Open-loop systems . . . . . . . . . . . . . . . . . . . . . . 24 1.4.2.1 Minimal RCI set in terms of set inclusion . . . . 24 1.4.2.2 Minimal RCI set in terms of matrix-norm . . . . 25 1.4.2.3 Computation of candidate solutions . . . . . . . 29 1.5 Tube Based MPC of constrained linear discrete-time systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5.2 Tube of trajectories . . . . . . . . . . . . . . . . . . . . . 32 1.5.3 First formulation of the optimal control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6.1 Design of the local controller . . . . . . . . . . . . . . . . 40 1.6.2 Terminal set design . . . . . . . . . . . . . . . . . . . . . . 41 1.6.3 Closed-loop performances . . . . . . . . . . . . . . . . . . 43

  Note that we have |Z| p = sup {|ξ| p | ξ ∈ Z} . If the set Z is closed and bounded, then it is compact and we have |Z| p = max {|ξ| p | ξ ∈ Z} . Moreover, if the RCI set is a polytope we have |Z| p

  We use the criterion (1.10) to define a binary relation ≤ H,p over 2(R n ) , the collection of bounded sets in R n . Two setsZ ∈ 2(R n ) and Y ∈ 2(R n ) satify |HZ| p ≤ |HY| p , ifand only if Z ≤ H,p Y. It can be shown that the binary relation ≤ H,p is reflexive, transitive, and every pair of element in 2(R n ) is comparable. However, this binary relation is not anti-symmetric. The relation ≤ H,p defines a total preorder over 2

	Definition 1.15 (Total preorder) A binary relation is a total preorder if it is re-
	flexive and transitive and if every pair of element is comparable.
	Proposition 4

  Practical implementation of the Tube Based MPC . . . 92 3.6.1 Solving the MPC QP using CVXGEN in real time . . . . 95 3.6.2 Change of reference and terminal set computation . . . . 96 3.6.3 Structure of the code . . . . . . . . . . . . . . . . . . . . . 99 3.7 Experimental results . . . . . . . . . . . . . . . . . . . . . 100 .8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

	Chapter 3
	Design and Flight Experiments of
	a Tube Based MPC for the
	stabilization of quadrotor
	horizontal dynamics
	3.7.1 Tube Based MPC parameters . . . . . . . . . . . . . . . . 100
	3.7.2 Closed-loop horizontal trajectory . . . . . . . . . . . . . . 103
	3.7.3 Nominal system trajectory . . . . . . . . . . . . . . . . . 103
	3.7.4 Disturbance realizations and tube properties . . . . . . . 105

Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 Modelling of the horizontal dynamics . . . . . . . . . . . 74 3.2.1 Model discretization . . . . . . . . . . . . . . . . . . . . . 75 3.2.2 Disturbance set characterization from experimental data . 76 3.2.3 Delay in the control . . . . . . . . . . . . . . . . . . . . . 77 3.3 Input delay handling in the implementation of the Tube Based MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.1 Simulation example . . . . . . . . . . . . . . . . . . . . . 78 3.3.2 Experimental data generation for system identification and experimental design of a Tube Based MPC law . . . 80 3.4 Experimental tuning of the local controller . . . . . . . . 83 3.5 Closed-Loop System Identification and Invariant Set Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.5.1 Computation of candidate solution . . . . . . . . . . . . . 86 3.5.2 Iterative refinement of the identification . . . . . . . . . . 90 3.6 3.7.5 Recursive state and input constraints satisfaction . . . . . 108 3.7.6 Real-time computation . . . . . . . . . . . . . . . . . . . . 108 stabilization of quadrotor horizontal dynamics 3

  (Matrix of high dimension) The matrix H = H B ⊥ H B has h rows, with h > m.

	150	Invariant set obtained with sliding mode control
	Assumption 5.6
	13	end
	14	end

alue then 12 Set V alue ← |HZ| p and Z * ← Z;

  Definition 6.1 (Matrix-norm mRCI set) A set Z is an (H, p)-minimal Robust Control Invariant ((H,p)-mRCI) set if it is an mRCI set and if |HZ| p = h(H, p), where h(H, p) = inf {|HZ| p | Z ∈ R} .

Table 6 .

 6 1 -Value of the grid search parameters α and β, and the resulting optimal feedback gain K obtained with H = 0.6 0.8 . allow for the RCI set to dilate in every direction, except for the direction of interest defined by the matrix H. Here, we consider scalar scaling factors, as presented in Section 6.8.2.

	Set	Z	ψ(Z)	ψ 2 (Z)	ψ 3 (Z)	ψ 4 (Z)	ψ 5 (Z)
	α	1	5	5	1	1	1
	β	0.7	0.7	0.7	1	1	1
	K	-0.06 -0.36	-0.59 -1.18	-0.55 -1	-0.55 -1	-0.55 -1	-0.55 -1
	|H(.)|	1.676	0.722	0.442	0.215	0.107	0.055

Theoretical background -invariant set design

1.4. Invariant set theory

Remerciements

Chapter 5

Invariant set for linear discrete time systems subject to bounded additive disturbance using sliding mode control

Invariant set obtained with sliding mode control

The computation of a stable sliding surface relates to the selection of a state feedback gain L = C -1 B C B ⊥ that stabilizes the pair (A 11 , -A 12 ). The choice of the matrix C -1 B C B ⊥ defines the internal dynamics (5.9), through the matrix Â11 = A 11 -A 12 C -1 B C B ⊥ . The second step of the design of a DSMC law, which consists of the selection of a reaching law, is discussed below.

Linear reaching law

The design of a reaching law consists in choosing a control law u = ν(ξ) = ν( ξ) that cancels the term Â21 ξ B ⊥ in (5.10), yielding 5.11) where the control law µ : R m → R m is a design parameter. With (5.11) in (5.10), we have

(5.12)

A Linear Reaching Law exploiting the structure of (5.12), by cancelling, or replacing, the dynamics of s is µ(s) = C -1 B (A C4 -Â22 )s, where the matrix A C4 ∈ R m×m is a parameter to be designed. The control law (5.11) satisfies the above structural properties, and leads to

(5.13)

In the original coordinates (ξ) the control law (5.13) is given by (5.14) therefore leading to a linear feedback gain

( 5.15) This linear structure of the reaching law represents the main difference with respect to the design of sliding modes in continuous-time. Under the control law (5.14), the system (5.1) satisfies ξ + = (A + BK)ξ + w, w ∈ W. (5.16) In the coordinates ξ, the state space representation of the closed-loop system reads ξ+ = Â11 Â12 0 m,n-m A C4 ξ + Qw, w ∈ W. Compute η p as in (5.38);

Compute Z as an RPI η p , p outer approximations of Z∞ ( Â11 );

Compute K ← -(CB) -1 CA and ZK as in (5.37); Compute C B ⊥ as the solution of (5.48) with H = H and

Compute η p as in (5.38);

Compute Z as an RPI η p , p outer approximations of Z∞ ( Â11 );

Compute K ← -(CB) -1 CA and ZK as in (5.37); Invariant set obtained with sliding mode control

The next chapter proposes methods for the computation of candidate solutions that allow for non-linear control laws, and whose design account for the presence of state and/or input constraints.

Chapter 6

Set mapping within the class of invariant sets for constrained linear discrete-time systems subject to bounded additive disturbance Algorithm 6.1: q-steps refinement of an initial RCI set Z Inputs: A, B, X , U, W, Z and q ∈ N + Outputs: A decreasing sequence of RCI sets {Z i } i∈Nq 1 Initialize i ← 0 and set Z 0 ← Z; 2 repeat 3 Compute ν(.) as in (6.7) with Z = Z i ;

4 Increment i by one;

Computational aspect

The set theoretic methods presented below use the solution of optimization problems which involve the input constraints as in (6.7b), and bounds on the disturbances as in (6.6b). The developments below use the following assumption to have tractable optimization problems. Assumption 6.2 (Polytopic constraint sets) We assume that the sets X and U are polyhedra, and the set W is a polytope.

We denote l x , l u and l w their respective H-complexity, and we have

We denote V (W) the set of vertices of the polytope W.

Each iteration of the refinement procedure presented above comprises two steps, namely the selection, or computation, of a suitable state feedback control law ν ∈ N (Z), and the computation of the one-step image set Image(Z, ν). The constraint (6.7b) can be rewritten

Note that, if the set Z is RCI, then the constraints G x ξ ≤ g x are automatically satisfied. Hence, the above constraints consist of l u linear inequalities for every element ξ in Z.

The constraint (6.7a) is more problematic given the structure of the set Z is not restricted to a specific class (convex, polytopic, etc). Without further assumption on this set, the computation of ν ∈ N (Z) is not tractable. The restriction to convex and polytopes are considered in the following.

Set mappings

In particular, Z W has an H-complexity of at most l z . With these notations, (6.9) reads .12) Note that the invariance condition (6.12) uses both the vertex (V (Z)) and the halfspace (G z , g z ) representations of the polytope Z. For a given vertex ξ ∈ V (Z), the computation of a suitable control action requires to solve an LP with (at most)

The convex-hull of the image set can be characterized as the convex-hull of a finite set of vectors as follows, 6.13) thus providing a (possibly not minimal) V-representation.

The successive steps of the iterative approach using this method for the computation of ν ∈ N (Z) is detailed in Algorithm 6.2, where we perform q ∈ N iterations. Algorithm 6.2: Iterative refinement of an initial polytopic RCI set: Vrepresentation Inputs: A, B, X , U, W, Z 0 and q ∈ N + Outputs: A decreasing sequence of RCI sets

Set f as the V-complexity of Z i , and {ξ 1 , ..., ξ f } its vertices;

The set invariance criterion presented here consists in computing a suitable control action for every vertex of the polytope. The following section provides a condition for the set invariance that requires to solve a unique optimization problem.

Set mappings Algorithm 6.3: Iterative refinement of an initial polytopic RCI set: Hrepresentation

Inputs: A, B, X , U, W, Z 0 and q ∈ N + Outputs: A decreasing sequence of RCI sets

Compute g z (W) as in (6.11); 6 Compute (S z , S u , S w , K) as a solution of the linear constraints (6.14);

7 Increment i by one;

the set of feedback gains K ∈ R m×n such that Z is RPI for the linear control law ν(ξ) = Kξ. In a similar fashion, Z is linear RCI if and only if N L (Z) is non-empty.

In this section, we used polytopic RCI sets to reduce significantly the computational complexity of the proposed approach for the iterative refinement of RCI sets, and have tractable computation of suitable control laws and one-step image set. Two approaches were considered to compute a suitable control law, either by solving a (reduced) LP for every vertex or by solving a unique, but larger, LP where the control law has a linear structure. A direct expression of a (not minimal) Vrepresentation of the image set is obtained by considering the image of the vertices. The approach requires to compute the minimal V and H representation at each iteration, which can be computationally demanding. Furthermore, the V-complexity of the polytopic RCI set sequence increases at each iteration (see Proposition 3). For this reason, the proposed approach to iteratively refine RCI sets is suitable for systems of low state dimension (n < 4).

As discussed previously, the set of control laws N (Z) is possibly infinite. The selection of the control law using the set invariance condition presented above is discussed in the following section.

Selection functions and set mappings of polytopic RCI sets

In general, there exists an infinity of admissible control laws ν ∈ N (Z) for a given RCI set Z. This section proposes optimization-based methods for the systematic selection of the control law, where we restrain to the class of polytopic RCI sets for computational purposes.

Selection functions and set mappings of polytopic RCI sets

A systematic method for the selection of a control law is to construct functions µ : T → X U , (6.15a)

Note that such functions µ exist, as the set N (Z) is non-empty for every polytopic RCI set Z ∈ T . For a given polytopic RCI set Z, the function µ selects a unique control law µ(Z) : X → U. For clarity of presentation, the control law µ(Z) evaluated at ξ ∈ Z is denoted µ(Z, ξ) to avoid the notation µ(Z)(ξ) or (µ(Z))(ξ) with successive parenthesis.

For a given selection function µ as in (6.15), we define a set mapping over the collection of polytopic RCI sets as follows φ µ : T → T , (6.16a)

The set mapping can be iterated from any initial polytopic RCI set Z to define the set sequence

where φ 0 µ (Z) = Z and φ i+1 µ (Z) = φ µ (φ i µ (Z)), ∀i ∈ N.

Lemma 6.1 Consider a function µ : T → X U as in (6.15). For all Z ∈ T , then Φ µ (Z) is a decreasing sequence of polytopic RCI sets.

Proof: Using Proposition 31, we have ∀i ∈ N, φ i+1 µ (Z) ⊆ φ i µ (Z).

Figure 6.1 shows an example of decreasing polytopic RCI set sequence {φ i µ 1 (Z)} i∈N 50 obtained in simulation. The selection function µ 1 is presented in Section 6.6.1, and the context of the simulation is detailed in Section 6.9.

We then define φ ∞ µ (Z) as the intersection of all the elements of the decreasing set sequence Φ µ (Z),

This set is of particular interest, as it represents the infinite refinement, or limit, of the initial polytopic RCI set Z for a selection function. From Proposition 25, φ ∞ µ (Z) is a compact RCI set, and it is convex as the infinite intersection of convex sets. However, it is not a polytope in general. Remark 6.3 For a given set polytopic RCI set Z and two selection functions µ and µ as in (6.15), we do not have φ ∞ µ (Z) = φ ∞ µ (Z) in general.

Set mappings

Proposition 33 (Feasibility) Let Z ∈ T and let ξ ∈ Z. The optimization problem P 1 (Z, ξ) has a non-empty feasible set.

Proof: Let ν ∈ N (Z). The constraints (6.20b) are feasible with u = ν(ξ) and α = 1.

We note that uniqueness of the solution of P 1 (Z, ξ) is not guaranteed. However, any element of the optimal set leads to the minimal scaling factor α 1 (Z, ξ), and the optimal set is convex and closed. To uniquely define the control action µ 1 (Z, ξ), we consider the following optimization problem (that can be seen as a regularization)

The constraints (6.21b) consists in l z + l u + 2 linear inequalities. We have the following result regarding the optimal set of this optimization problem.

Proposition 34 (uniqueness) Let Z ∈ T and let ξ ∈ Z. The optimization problem (6.21) has a unique solution.

Proof: The feasible set of the optimization problem (6.21) is the optimal set of P 1 (Z, ξ), which is convex and closed while the 2-norm is strictly convex.

Any cost function that is strictly convex in u leads to an optimization problem that has a unique solution. Here, the optimization problem (6.20) seeks to minimize the scaling factor in our goal of minimizing the size of the one-step image set. We chose to minimize the 2-norm of the control input as a way to have an RCI set Z such that |ν(Z)| 2 is minimal.

We define µ 1 (Z, ξ) as the unique solution of the optimization problem (6.21). Given that the optimization problem has an unique solution for every Z ∈ T , ξ ∈ Z, this uniquely defines the control law µ 1 (Z).

We have proposed a first optimization-based selection function µ 1 as in (6.15), thus defining the set mapping φ µ 1 as in (6.16). For a given Z ∈ T , the computation of φ µ 1 (Z) requires to solve the optimization problems (6.20) and ( 6.21) on the vertices of Z, namely V (Z). Hence, the computation of φ µ 1 (Z) requires the resolution of 2card(V (Z)) linear optimization problems, with 1 + m or m optimization variables, l z + l u + 2 linear inequalities and a linear or quadratic cost function.

Minimal vectorial scaling factor

In the context of polytopes, it is possible to replace the scalar scaling factor α in (6.17) with a vectorial counterpart α ∈ R lz , to scale with a different factor on the

Minimal scalar scaling factor

The first selection function μ1 relies on the set inclusion ∀Z ∈ T L , ∀K ∈ N L (Z), (6.27) where

Here, we are interested in computing the feedback gain K ∈ N L (Z) that leads to a scaling factor α(Z, K) that is as small as possible. The resulting linear optimization problem is given by

)

This optimization consists of n(2l z + l u ) linear equality constraints (6.28b) and (6.28d), l z + l u linear inequality constraints (6.28c) and (6.28e), l z (l z + l u + l w ) + 2 linear inequality constraints (6.28g) and (6.28h), l z (l z +l u +l w )+nm+1 optimization variables, and a linear cost function. Thus, it has an LP structure.

We have the following result regarding the feasibility of P1 (Z).

Proposition 39 (Feasibility) For all Z ∈ T L , the optimization problem P1 (Z) has a non-empty feasible set.

Proof: Let Z ∈ T L , and let K ∈ N L (Z). We have (A + BK)Z ⊕ W ⊆ Z. From Proposition 32, there exists (S z , S u , S w , K) satisfying the constraints of P1 (Z) with

Uniqueness of the solution is not guaranteed. However, any element of the optimal set leads to the minimal scaling factor, further denoted α1 (Z), and the optimal set is convex and closed as P1 (Z) is an LP. To uniquely define the feedback Set mappings gain µ 1 (Z), we consider the following optimization problem (., ., ., μ1 (Z)) = arg minimize

)

S u g z ≤ g u , (6.29e)

The above optimization problem consists in n(2l z + l u ) + 1 equality constraints (6.29b) and (6.29d), l z + l u linear inequality constraints (6.28c) and (6.28e), l z (l z + l u + l w ) linear inequality constraints (6.28g), l z (l z + l u + l w ) + nm optimization variables, and a quadratic cost function. We have the following result regarding the optimal set of this optimization problem.

Proposition 40 (uniqueness) Let Z ∈ T L . The optimization problem (6.29) uniquely defines the feedback gain μ1 (Z).

Proof:

The feasible set of the optimization problem (6.29) is the optimal set of P1 (Z), which is convex and closed while the 2-norm is strictly convex. Remark 6.6 Note that any cost function that is strictly convex in the elements of K leads to an optimization problem that uniquely defines the feedback gain K.

For every Z ∈ T L , we define μ1 (Z) as the unique solution of the optimization problem (6.29). Hence, the computation of μ1 (Z) requires to solve 2 optimization problems, which is considerably less than in Section 6.6.1. However, these optimization problems have significantly more optimization variables.

Minimal vectorial scaling factor

The second selection function μ2,p considers a vectorial scaling factor α ∈ R lz , as in (6.22). In a similar fashion, we seek to minimize the p norm of the vectorial scaling factor. The resulting optimization problem reads

Set mappings

Note that the LP (QP) structure of the optimization problem is preserved for p = 1, ∞ (resp. p = 2). For p = ∞, the constraint (6.31j) reads α i ≤ α2,p (Z), ∀i ∈ N + lz . Compared to (6.29), ( 6.31) has l z -1 additional optimization variables. The equality constraint (6.31j) is linear (quadratic) in α for p = 1, ∞ (resp. p = 2). We have the following result regarding the optimal set of the optimization problem (6.31).

Proposition 42 (uniqueness) Let Z ∈ T and let ξ ∈ Z. The optimization problem (6.31) uniquely defines μ2,p (Z) for p ∈ {1, 2, ∞}.

Proof: The feasible set of the optimization problem (6.25) is the optimal set of P2,p (Z), which is convex and closed. Moreover, the optimization problem (6.31) is QP while the cost function is strictly convex in the elements of K.

For every Z ∈ T L , we define μ2,p (Z) as the unique solution of (6.31). This feedback gain seeks to minimizes the scaling of the one-step image set with regards to the directions G z defining the polytope Z.

Minimal matrix-norm scaling factor

The third selection function μ3 seeks the feedback gain K that minimizes the size of the one-step image set with regards to the criterion (6.3). That is, for a given Z ∈ T L we seek the feedback gain K ∈ N L (Z) that minimizes |H((A + BK)Z ⊕ W)| p . Note that |H((A + BK)Z) ⊕ W| p = can be rewritten in terms of set inclusion as follows

The set B h p ( ) is not polytopic in general (except for particular choices as for exemple the one of the norm p = ∞). For this reason, we cannot use the Extended Farkas Lemma as in Proposition 32.

Instead, let us consider the polytope HZ. We seek the feedback gain K such that

where α is minimal. Note that the set inclusion (6.32) is independent of the choice of the norm p, and the sets are polytopes.

In the following, we denote l y the H-complexity of the polytope HZ, and we denote

Using the Extended Farkas Lemma, for a given gain K ∈ R m×n and scalar α ∈ R + , the set inclusion (6.32) holds if and only if there exist

The resulting optimization problem reads Set mappings (., ., ., ., ., μ3 (Z)) = (6.34a) minimize

)

)

S z g z + S w g w ≤ α3 (Z)g y , (6.34f)

S u g z ≤ g u , (6.34h)

)

)

Note that the LP structure of the optimization problem is preserved. Compared to P3 (Z), (6.34) has 1 less optimization variable and 1 less linear inequality. We have the following result regarding the optimal set of this optimization problem.

Proposition 44 (uniqueness) Let Z ∈ T L and let ξ ∈ Z. The optimization problem (6.34) has a unique solution.

Proof: The feasible set of the optimization problem (6.25) is the optimal set of P3 (Z), which is convex and closed. Moreover, the optimization problem is LP while the cost function is strictly convex in the elements of K.

For all Z ∈ T L , we define μ3 (Z) as the unique solution of (6.34). This choice for the feedback gain seeks to minimize the size of the one-step image with regards to the (H, p) criterion (6.3).

We presented three optimization-based selection functions that use the Hrepresentation of the polytope and the Extended Farkas Lemma as a condition for set invariance. The iterative approach using one of these three selection functions is summarized in Algorithm 6.5. The main advantage of these three selection functions, compared to those proposed in Section 6.6.1 is that it requires to solve a unique optimization problem, instead of solving one for each vertices of Z. However, the optimization problems have a larger number of optimization variables and equality constraints. Moreover, these selection functions force a linear structure to the control law, which induces conservatism compared to the selection function proposed in Section 6.6.1. Compute α as the optimal scaling factor of the optimization problem (6.28), (6.30) or (6.33);

8

Compute the unique feedback gain K as the solution of the additional optimization problem;

Compute (G z , g z ) as a minimal H-representation of Z i ;

13 until i = q;

We have introduced optimization-based methods for the construction of selection functions µ as in (6.15) and μ as in (6.26), using either the H or the V representation of the polytope. The choice for the design of the selection functions µ presented here consist in minimizing the size of the one-step image set. The main advantage of the selection functions presented here is the LP/QP structure of the optimization problems (with p = 1, 2, ∞). However, the computational complexity of the optimization problems increase with the V-complexity and/or H-complexity of the successive polytopic RCI sets.

It is worth to be mentioned that the choice of minimizing the one-step image set does not account for the iterative nature of the approach. More sophisticated selection functions with the goal of minimizing the size of the N -step image set are to be sought.

As discussed in Remark 6.3, for Z ∈ T L , the iterations of the set mappings may not converge toward the same limit. That is,

We are interested in characterizing the limit sets of the set mappings (6.16), which consists of the infinite refinement of polytopic RCI sets. The following section provides a result on such a characterization of these limit sets.

Set mappings and discrete-time sliding mode control

This section presents the particular relationship between the set mappings presented in this chapter and the 0-DSMC laws introduced in Chapter 5. As in Chapter 5, the results below are obtained under the following assumption.

Assumption 6.3 (Full rank input matrix) The matrix B is full-rank.

The following assumption relates to the necessary and sufficient conditions for a matrix C to define a sliding surface (stable or unstable).

Assumption 6.4 (Stable sliding surface) The matrix C ∈ R m×n is such that CB is invertible.

In the following, we denote K C = -(CB) -1 CA the associated linear state feedback gain, ν C (ξ) = K C ξ the 0-DSMC control law, and Z ∞ (C) = Z ∞ (K) the mRPI set.

Recall that the sliding surface is stable provided A+BK C = A-B(CB) -1 CA is Schur. For a matrix C as in Assumption 6.4, the minimal quasi-sliding mode band (mQSMB) is given by S

In the following, we study the properties of an RCI set contained in an mQSMB.

RCI set contained in an mQSMB

As a first step, let us provide a characterization of RCI sets contained in a fixed mQSMB.

Proposition 45 Let C be a matrix that satisfies Assumption 6.4. An RCI set Z satisfies Z ⊆ S ∞ (C) if and only if CZ = CW.

Proof: Let Z ⊆ S ∞ (K) be an RCI set. We have CZ ⊆ CW. Let ξ ∈ Z and let u ∈ U be such that Aξ + Bu + w ∈ Z, ∀w ∈ W. In particular, C(Aξ + Bu) ∈ (CZ CW). Hence, CZ = CW (else, if CZ ⊂ CW, then CZ CW = ∅, which is not possible). The reciprocal is immediate.

We use this result to have a characterization of the set N (Z) for an RCI set contained in an mQSMB.

Proposition 46 Let C be a matrix that satisfies Assumption 6.4. If an RCI set Z satisfies CZ = CW, then N (Z) = {ν C }.

Proof: Let ν ∈ N (Z), and let ξ ∈ Z. We have Aξ + Bν(ξ) ∈ Z W. We multiply this set inclusion by the matrix C and use Proposition 45 to obtain

If an RCI set Z is contained in an mQSMB S ∞ (C), then the only control law for which the set is RPI is the associated 0-DSMC law ν C . That is, for an RCI set Z contained in an mQSMB, N (Z) is a singleton set. All selection functions lead to the same control law, since N (Z) is a singleton set.

These results can also be used to reduce the computational complexity of the proposed iterative approach. Indeed, once an element φ k µ (Z) of the decreasing set sequence {φ i µ (Z)} i∈N is contained in an mSQMB S ∞ (C), then µ(φ j µ (Z)) = ν C , ∀j ≥ k, and the selection of the control law becomes trivial.

Characterization of polytopic RCI sets contained in an mQSMB

This section focuses on the establishment of a computationally efficient necessary and sufficient condition for a polytopic RCI set to be contained in an mQSMB.

Recall that an RCI set Z is contained in an mQSMB if and only if there exists a matrix C satisfying Assumption 6.4 and CZ = CW. This section provides a computationally tractable necessary and sufficient condition for the existence of such matrix obtained under the following assumption. Assumption 6.5 (Controllable system) The system (6.1) is controllable.

We have the following preliminary result regarding the existence of a full rank matrix C ∈ R m×n such that CZ = CW.

Proposition 47 Let Z be a polytopic set. If the polytope Z W is of dimension n -m, then there exists a full rank matrix C ∈ R m×n such that CW = CZ.

Let {c 1 , ..., c m } be a family of linearly independent vectors of (Z W) ⊥ , and

The matrix C is full rank by construction, and satisfies C(Z W) = 0 m , as the vectors c i are orthogonal to Z W.

Note that the above result does not require the set Z to be RCI. The next result uses the RCI property of the set to establish that the matrix CB is full rank.

Set mappings

with Theorem 6.2 that Z is not contained in an mQSMB. Note that this necessary condition requires to compute only g z (W). This term is used in the necessary and sufficient condition of Theorem 6.2 to have an expression of Z W. Algorithm 6.6: Characterization of an mQSMB containing a polytopic RCI set Inputs: An RCI set Z. The results presented in Section 6.7.1 are used to have a first characterization of the limit sets of set mappings in the following section.

Characterization of set mappings limit sets

The simplicity of the selection function for an RCI set contained in an mQSMB established in Corollary 6.1 can be used to characterize the limit sets of set mappings.

If an RCI set is contained in an mQSMB, all the selection functions lead to the associated 0-DSMC control law. That is, ∀Z ∈ S ∞ (C), a set mappings φ µ as in (6.16) with µ as in (6.15)

We obtain an expression of the successive elements of the sequence {φ i µ (Z)} i∈N and their limit φ ∞ µ (Z).

Corollary 6.2 Let C be a matrix that satisfies Assumption 6.4. Let Z be a polytopic RCI set such that CZ = CW. For all selection function µ as in (6.15), we have 6.7. Set mappings and discrete-time sliding mode control 199 ∀i ∈ N, 6.39)

Proof: Equation ( 6.39) is a consequence of Corollary 6.1 and the fact that φ j µ (Z) ∈ S ∞ (C), ∀j ∈ {0, 1, ..., i}. Provided the matrix (A + BK C ) is Schur and the set Z is bounded, we have

For this reason, the decreasing set sequence converges towards the mRPI set associated to the 0-DSMC law Z ∞ (K C ). This allows us to have a first result on the characterization of the limit sets of the set mappings. Theorem 6.3 Let C be a matrix that satisfies Assumption 6.4. If the mRPI set

for all set mapping φ µ as in (6.16), with µ as in (6.15).

By the construction of , Z + satisfies the state and input constraints. Hence, Z + is a polytopic RCI set. Moreover, this set satisfies

Let µ be a selection function as in (6.15), and let φ µ be the associated set mapping as in (6.16). Using Theorem 6.3, we have

We have shown that the mRPI sets obtained with 0-DSMC laws are elements of the limit sets of the set mappings provided they respect the state and input constraints. This represents a first attempt to characterize the limit sets of set mappings, and definitely does not establish a total relationship between the two classes of RCI sets. We leave for further research the complete study of the relation between the initial set Z and the limit sets φ ∞ µ (Z). The results presented in this section highlight a major limitation of the iterative approach presented in this Chapter. Namely, it is highly reliant on the choice for the

Set mappings

We denote Z(α, β) = Z ∞ (α, β). A (possibly not minimal) H-representation of this polytope is

(6.44b) Remark 6.8 A particular case of this parametrization is the choice α = β = 1, leading to Z(1, 1) = Z which is RCI.

We have proposed a family of polytopic RCI sets that are parametrized by the initial polytopic RCI set Z, its sublevel set H ∞ (Z), and two scalar scaling factors α, β. By construction, this family comprises of polytopes, and all the element have a matrix-norm criterion smaller than the initial polytopic RCI set Z. Among the elements of this family, we retain the ones that are RCI.

Condition for set invariance

Two conditions for a polytope to be RCI were presented in Sections 6.4.3 and Section 6.4.4. The first uses both the V and the H representations of the polytope, while the second uses only the H representation.

We have an expression of an H-representation of the polytopes Z(α, β) as in (6.44). However, we do not have an analytical expression of a V-representation of the polytopes as a function of the scalar scaling factors α and β. For this reason, we use the set invariance condition presented in Section 6.4.4, namely the Extended Farkas Lemma, to have a sufficient condition for the polytope to be RCI (necessary and sufficient condition to be linear RCI). Remark 6.9 Note that the choice of using the Extended Farkas Lemma restrains the control law to be linear, and thus induces conservatism compared to an approach that allows for nonlinear control laws.

The set Z(α, β) is linear RCI if and only if there exists matrices

with non-negative elements and a feedback gain K ∈ R m×n such that

S z gz (α, β) + S w g w ≤ gz (α, β), (6.45b)

S x gz (α, β) ≤ g x . (6.45f)

Set mappings of linear RCI sets with relaxation of the set inclusion constraints 203

We have a necessary sufficient condition on the scalar parameters α and β for the associated polytope Z(α, β) to be linear RCI. The collection of parameters α, β such that the above constraints are feasible is possibly infinite. Their unique selection is discussed in the next section.

Selection of the optimal scaling factors

Among the scalar scaling factors α, β such that the constraints (6.45), we retain the one leading to the polytope Z(α, β) whose matrix-norm criterion is minimal. Recall that |HZ(α, β)| ∞ ≤ β|HZ| ∞ . We seek the admissible scaling factors (α, β) such that the scalar β is minimal.

These considerations lead to the following optimization problem,

subject to

We have the following result regarding the feasibility of P ψ (Z).

Proposition 49 (Feasibility) Let Z ∈ T L . The optimization problem P ψ (Z) has a non-empty feasible set.

Proof: We have Z(1, 1) = Z ∈ T L . Hence, the constraints of P ψ (Z) allow for at least one solution with the particular choice α = β = 1.

Unlike the optimization problems presented in Section 6.6, P ψ (Z) is not LP/QP due to the bilinear terms S z gz (α, β), S u gz (α, β) and S x gz (α, β) in the constraints (6.46c), (6.46e) and (6.46g).

A direct consequence of the bilinearities is that the optimal set is not convex in general. Any element of the optimal set leads to the minimal scaling factor β(Z). However, we cannot use the convexity of the optimal set to uniquely define a solution to P ψ (Z) via the construction of an additional optimization problem with a quadratic cost in K and α as we did to construct selection functions in Section 6.6. This restrains us from uniquely defining a feedback gain K and a scalar factor α as the solution of a QP.

An additional difficult that raises from the bilinear constraints is that there is no simple way of computing a global minimum of P ψ (Z). However, any candidate solution (S z , S w , S u , S x , K, α, β(Z)) that satisfies the constraints of the optimization problem P ψ (Z) represents an refinement as we have β ≤ 1.

Let (S z , S w , S u , S x , K, α, β(Z)) be a candidate solution of P ψ (Z), and define Image(Z(α, β(Z)), K) = (A + BK)Z(α, β(Z)) ⊕ W ⊆ Z(α, β(Z)), (6.47) the one-step image of Z(α, β). The one-step image in (6.47) is a suitable candidate for ψ(Z) as it satisfies |Hψ(Z| ∞ ≤ β(Z).

Remark 6.10 The above definition of ψ(Z) is not unique as we have not uniquely defined α and K. However, any of the candidate solution of P ψ (Z) leads to a onestep image set that is suitable for ψ(Z).

The algorithm for the computation of the first q terms of the polytopic RCI set sequence {ψ i (Z)} i∈N is presented below, where we use candidate solutions of the optimization problem (6.45). An intuitive approach for the computation of candidate solution is the use of nonlinear solver (sub-optimal solution/local minimum). Note that an initial guess can easily be provided to nonlinear solver by considering the trivial case α = β = 1 (i.e. Z(1, 1) = Z). Algorithm 6.7: Matrix-norm decreasing sequence of polytopic RCI set using nonlinear optimization solver Inputs: A, B, W, H, X , U, a polytopic linear RCI set Z and a positive integer q ∈ N + .

Outputs: A sequence of polytopic RCI set ψ q (Z)

We present in the following a practical approach for the computation of a candidate solution when nonlinear solvers fail to provide a satisfactory candidate solution.

K ∈ R m×n . (6.48k)

Regardless of the choice for α, β and Z, the optimization problem P ψ (Z, α, β) is LP. Its feasible set is non-empty if and only if Z(α, β) ∈ T L . In this case, the optimal set is closed and convex, and all its elements lead to the same feedback gain, thus uniquely defining the feedback gain K(α, β). Remark 6.11 Any cost function that is strictly convex in the elements of K uniquely defines the feedback gain.

Analysis of the computational complexity

We now discuss the computational complexity of the optimization problem P ψ (Z, α, β). The cost function is quadratic, the optimization problem has ( l + l w + l u + l x ) l + nm optimization variables (the matrices S z , S w , S u , S x and K), and involves the following constraints

• n( l + l w + l u + l x ) linear equalities (6.48c), (6.48e) and (6.48g).

• ( l + l u + l x ) linear inequalities (6.48d), (6.48f) and (6.48h).

• ( l)( l + l w + l u + l x ) linear inequalities (6.48i), and (6.48j). Consider the feedback gain K = 0.04 0.28 , obtained with pole placement strategy with the pair of poles {0.9, 0.8}. We construct a polytopic RCI set Z using Theorem 1.4, as originally proposed in [Olaru 2006]. This set is shown in Figure 6.2. Note that this set is symmetric with regards to the origin.

Set mappings

In the following, we compute the first 50 elements of the RCI set sequences

Z)} i∈N using Algorithm 6.2. Figure 6.3 gives the state space representation of the RCI set sequences. As expected, these RCI set sequences are decreasing with regards to set inclusion.

Convergence of the decreasing set sequences

We now analyze the convergence properties of the decreasing RCI set sequences. Proposition 25 states that they converge toward an RCI set (possibly not polytopic).

We have an explicit characterization of this limit set provided there exists at least one element of the decreasing RCI set sequence that is contained in an mQSMB. Theorem 6.2 gives a necessary and sufficient condition for the existence of an mQSMB that contains a polytopic RCI set.

In our simulation example, the state and input dimensions are such that n -

Hence, the polytope is of dimension 1 if and only if there exist j ∈ {1, ..., l Z } such that (g Z ) j = (g Z (W)) j . Define

The scalar α(Z) illustrates the proximity of the polytope RCI set Z to the disturbance set W. The closer α(Z) is to 1, the closer Z is to W. In particular, if α(Z) = 1 and if Z is origin-symmetric, then the polytope Z W is of dimension n -m = 1.

That is, it is contained in an mQSMB. Figure 6.4 shows the value of α(.) for the first 10 elements of the decreasing RCI set sequences {φ i µ 1 (Z)} i∈N and {φ i µ 2,∞ (Z)} i∈N . For i ≥ 4 (resp. i ≥ 2), the RCI sets φ i µ 1 (Z) (resp. φ i µ 1 (Z)) are contained in an mQSMB. The two decreasing RCI set sequences converge towards mRPI sets for a 0-DSMC law, associated respectively to the matrices C µ 1 = 0.2885 1 and C µ 2,∞ = 0.2677 1 , and the feedback gains K µ 1 = -0.2551 -1 and K µ 2,∞ = -0.2361 -1 . Figure 6.5 shows the decreasing RCI set sequences along with the associated mQSMB.

These examples illustrate the basic concept of set mappings and decreasing RCI set sequences introduced in Section 6.5 with selection functions proposed in Section 6.6. These set mappings are used in the next section to design (H, p)-mRCI sets.

Design of matrix-norm RCI sets

The simulation results presented in this section are separated into two subsections, depending on the choice for the matrix H defining the minimality criterion. µ 1 (Z) and φ 50 µ 2 (Z).

Matrix characterizing a stable sliding surface

In this section we consider matrices H that characterize stable sliding surfaces (see Section 5.2). Chapter 5 gives an explicit characterization of an (H, p)-mRCI set for this class of matrices H, as the mRPI set of a 0-DSMC law. We analyze in this section the performances of the set mappings proposed in this chapter. Such matrices have 1 row, thus the criterion (6.3) does not depend on the scalar p. We use the polytopic RCI set Z introduced in the previous section and shown in Figure 6.2. We choose arbitrarily H = 0.31 0.95 . The associated 0-DSMC feedback gain is K = -0.2833 1 . The mRPI set Z ∞ (K) and the mQSMB S ∞ (C) are shown in Figure 6.6, along with the sets φ 50 µ 1 (Z) and φ 50 µ 2 (Z). Recall that Z ∞ (K) is an (H, p)-mRCI set for this choice of matrix H. regards to set inclusion. However, these sets are not minimal with regards to the criterion (6.3). This is to be expected, as the selection functions µ 1 and µ 2 do not account for the matrix H in their design.

The selection function µ 3 (6.24) is designed to minimize the (H, p) criterion of the successive iterations of the set mapping. We compute the first 50 elements of the decreasing RCI set sequence {φ i µ 3 (Z} i∈N . Figure 6.7 shows that |Hφ 50 µ 3 (Z)| converges towards |HZ ∞ (K)|. That is, the limit set φ ∞ µ 3 (Z) of the set mapping µ 3 initialized with Z is an (H, p)-mRCI set. The first 50 elements of the decreasing set sequence is shown in Figure 6.8, along with the mRPI set Z ∞ (K). We see that the limit set verifies φ ∞ µ 3 (Z) = Z ∞ (K). The decreasing RCI set sequence has converged to the mRPI set of the 0-DSMC law associated to the matrix H.

We now consider the matrix H = 0.05 1 , that also characterizes a stable sliding surface. The associated 0-DSMC feedback gain is K = -0.049 -1 . Figure 6.9 gives the decreasing RCI set sequence {φ i µ 3 (Z} i∈N 50 , along with the mRPI set Z ∞ (K).

For this matrix H, the mRPI set Z ∞ (K) is not contained in the initial polytopic the initial set Z.

In the following section we consider matrices H that characterize unstable sliding surfaces.

Matrices of higher dimension

Here, we consider matrices H that have more than one row, namely h > m = 1. for such matrices, we do not have an explicit characterization of an (H, p)-mRCI set.

We choose |HZ| ∞ with H = 2 2 -1 1 and p = ∞ in the following.

We use the polytopic RCI set Z introduced previously and shown in Figure 6.2 to initialize the set mapping approach with vectorial scaling factors presented in Section 6.8.3. We solve the successive bilinear optimization problems (6.54) using the fmincon solver. From the solutions, we compute the polytopic set sequence ψ i (Z). On average, the fmincon solver provides a solution to the bilinear optimization problem P ψ (Z) as in (6.54) in 30ms, with a maximum time of 46ms for the first iteration.

The first 4 values of the vectorial scaling factor b and the feedback gains K are provided in Table 6.3. As a first step, we note that the vectorial scaling factors b satisfy |b| ∞ ≤ 1. In addition, we note that the first and third entries of b, along with the second and fourth entries, are equal. This is due to the symmetry of the initial set Z, that is transmitted to the successive iterations of the set mappings ψ i (Z).

The feedback gain and the matrix-norm criterion remain constant after 3 iterations. Indeed, we obtain ψ i+1 (Z) = ψ i (Z) for i ≥ 3. The set mapping ψ has reached a fixed-point. Using the results presented in Section 6.7.2, we find that ψ 3 (Z) is contained in the mQSMB for C = 1.71 1 . More specifically, the fixed-point is the mRPI set for the associated 0-DSMC law Z ∞ (K C ), where K C = -0.9231 1 . That is, the set sequence {ψ i (Z)} i∈N has converged and attained the mRPI set Z ∞ (K C ). Figure 6.13 gives a state space representation of the first 4 iterations, namely {ψ i } i={0,1,2,3} .

From Table 6.3, we see that the matrix-norm criterion is decreasing. This decrease can be interpreted in terms of set inclusion as follows,

The sublevel set for the iteration i is contained in the sublevel sets of all previous iterations. This set inclusion is shown in Figure 6.13.

We compare the set obtained with this set mapping approach to the RCI set obtained with the DSMC approach presented in Chapter 5. We use Algorithm 5.2 to compute a candidate (H, p)-mRCI set given that the matrix H has more than m = 1 rows. We use the parameters and {λ j } j∈N + q presented in the simulations in Section 5.7.3.3. We do not discuss the successive steps of the algorithm as they were detailed in the above mentioned simulation section.