Joseph Julien

René Nationalité

M Robert Beauwens

Dr Richard Peretz

Ir Raymond Hanus

Robert Van

Hauvermeiren Cursus

Conditioned Switching in Fault-Tolerant Control

Je voudrais, avant tout, remercier les membres de mon jury qui ont pris le temps d'examiner ce travail et de se déplacer, venant parfois de très loin. En particulier, je remercie très vivement

Organisation du document

Je décris dans ce document mes activités d'enseignant-chercheur depuis mon intégration au Centre de Recherche en Automatique de Nancy (CRAN, CNRS UMR 7039) en septembre 2005. Les travaux de recherche que j'ai mené depuis lors s'inscrivent principalement dans le thème de la commande tolérante aux défauts. Mes travaux antérieurs à mon arrivée au CRAN ne seront décrits que sommairement dans la première partie «Notice d'Activités » de ce document.

La structure du document fait apparaître trois parties distinctes. Ainsi, dans la première partie (Notice d'Activités), je présente une synthèse de mes activités d'enseignement, de recherche, d'encadrement de chercheurs et de participation à des projets de recherche. Cette présentation s'associe à une description des différentes responsabilités administratives et scientifiques.

La deuxième partie du manuscript, rédigée en anglais, est le mémoire de recherche qui porte sur la synthèse de mes activités de recherche en commande tolérante aux défauts. Cette partie est constituée de cinq chapitres dont le premier est un chapitre introductif qui présente brièvement les sujets sélectionnés traités dans le mémoire et les contributions apportées au domaine. Le chapitre 2 propose une synthèse des méthodes de commande tolérante aux défauts développées dans le paradigme comportemental. Le chapitre 3 traite d'une question annexe importante en commande tolérante aux défauts. Il s'agit du problème du transfert sans-à-coup lors des commutations de régulateurs qui se pose lors de la reconfiguration de commande suite à une défaillance du processus piloté. On présente dans ce chapitre deux approches, l'une dans le paradigme comportemental et l'autre dans un cadre plus classique. Le chapitre 4 et le chapitre 5 présentent les travaux de recherche effectués dans le paradigme à base de modèles et portent pour le premier sur la commande tolérante aux défauts par des techniques de commande optimal à horizon glissant et pour le deuxième sur la détection et le diagnostic de défauts.

La troisième partie du manuscript présente mes prospectives de recherche sur le court et moyen terme sous forme de deux projets de recherche. Bref résumé de la carrière hors Université de Lorraine J'ai débuté ma carrière en Janvier 1987 par un poste d'enseignant en automatique dans les grandes écoles d'ingénieurs de l'Institut National Polytechnique (INP) de Yamoussoukro en Côte d'Ivoire. Après dix années d'enseignement avec une lourde charge statutaire de 384h/an (cours magistraux, travaux dirigés, travaux pratiques et bureau d'études) en formation initiale à l'INP et des activités de formation continue et de consultance pour l'industrie, j'ai obtenu un financement belge pour la préparation de ma thèse de doctorat en sciences appliquées que j'ai soutenu en 2001 à l'Université Libre de Bruxelles. De 2000 à 2005, j'ai occupé un poste de chercheur sous contrat au Service d'Automatique et d'Analyse des Systèmes (SAAS) de l'Université Libre de Bruxelles. Durant ma carrière à l'INP de Yamoussoukro, J'ai eu l'entière responsabilité de la conception et de la mise en oeuvre des programmes d'enseignement en automatique pour les filières d'ingénieurs généralistes, électrotechniciens, électroniciens et informaticiens. J'ai mis en place des procédés pilotes pour les travaux de laboratoire (système de régulation de niveau avec des composantes d'instrumentation industrielle, asservissement de moteur à courant continu, maquette de commande de four, installation d'ordinateurs (PC) munis de cartes E/S analogiques et numériques pour la commande en temps réel). L'effectif globale d'élèves ingénieurs sur les 2 dernières années du 2ème cycle (2A, 3A) de l'ensemble des filières (généralistes, électrotechniciens, électroniciens et informaticiens) était de l'ordre de 120 en moyenne par an. J'ai encadré et supervisé une dizaine de stages de fin d'études en relation avec l'industrie (Raffinerie -Groupe TOTAL, CI-Télécom, Brasserie-Groupe Belge Stella-Artois, Industries sucrières, Société des Tabacs, Agro-industries-groupe Unilever). J'ai été membre de la commission d'évaluation des dossiers de recrutement des enseignants nationaux et français (coopérants et volontaires du service national français (VSN)). Ce parcours d'enseignant à temps plein à l'INP de Yamoussoukro m'a permis d'acquérir une expérience pédagogique importante avant d'entamer mes travaux de recherche pour le doctorat en sciences appliquées de l'Université de Bruxelles en 1997. Mes travaux ont porté sur l'analyse mathématique et la commande des systèmes échantillonnés sous leur aspect hybride (continu/discret), puis dans le cadre de deux contrats de projets successifs, mes activités de recherche ont été réorientées (mobilité thématique) vers la commande en logique floue dans un premier temps, puis vers le domaine de la régulation tolérante aux fautes.

Table des matières

Synthèse des activités pédagogiques

Responsabilités et animation d'équipes pédagogiques, création de modules d'enseignement

• Responsabilité du portage du Master EEA-Parcours CEE pour la nouvelle accréditation 2024

• Responsable de la spécialité ATIE (automatique et Traitement de l'information embarqués) et direction de l'équipe pédagogique ATIE du Master ISC (2009)(2010)(2011)(2012)(2013).

(Effectifs de 15 étudiants/an en moyenne avec une équipe pédagogique de huit Enseignants-Chercheurs permanents pour la spécialité ATIE).

-Animation de l'équipe pédagogique ATIE et suivi pédagogique des étudiants -Gestion des stages des étudiants de la spécialité ATIE : stages de Master Recherche et stage en entreprise : recherche de stage, signature des conventions de stage, gestion des visites -Préparation des jurys, évaluation des dossiers de candidature en M1 et M2

-Participation au montage des dossiers d'habilitation 2009-2013 et 2013-2017 (définition de la maquette pédagogique spécialité ATIE : contenu des enseignements, volumes, modalités d'évaluation, ...).

• Référent M1 du parcours automatique du Master IS-EEAPR (2005-2009) (17 étudiants en M1 en moyenne) A ce titre, j'ai coordonné les différentes UEs de ce parcours. J'ai été l'interlocuteur et le conseiller des étudiants tout en servant de relais entre le directeur des études et les enseignants impliqués dans le parcours automatique.

• Encadrement de moniteurs et ATERs (2005-2011) J'ai encadré dans les UEs d'automatique du Master 1 et de licence 1 les doctorants-moniteurs et ATERs intervenant dans les séances de TDs et TPs de ces UEs : Samir Aberkane, Fateh Guenab, Abdelfetah Hocine, Simona Dobre, Rodolfo Orjuela, Boulaïd Boulkroune.

• Gestion et suivi des stages M2 en entreprise De 2009 à 2013, j'ai géré les stages M2-ATIE en entreprises (conventions de stages, affectation des enseignants au suivi des étudiants, planification des visites en entreprise, préparation des soutenances, organisation des jurys). Chaque année, je me charge du suivi de stage d'un ou de deux étudiants en moyenne comme tuteur académique avec une visite sur sites.

• Encadrement de projets (apprentissage par projets) en Master 1 et Master 2 En moyenne, j'encadre 4 binômes d'étudiants par an sur des projets variés en automatique/systèmes embarqués et sûreté de fonctionnement mettant en oeuvre les différents aspects de la conduite d'un projet allant de l'analyse du besoin, la formulation des exigences, jusqu'à l'étape de réalisation et de déploiement de la solution.

• Evaluation de dossiers de candidatures d'entrée en Master 1 et Master 2 (IS-EEAPR, ISC)

• Responsable d'Unités Enseignements sur les deux cycles de formation (Licence/Master) Je suis responsable de plusieurs Unités d'Enseignements (UEs) de Licence SPI, de Masters ISC (Ingénierie des systèmes complexes) et EEA-CEE (Automatique -Contrôle pour l'Efficacité Energétique). J'ai monté la quasi-totalité de ces UEs en définissant clairement les syllabus associés indiquant les objectifs de l'UE, les méthodes d'évaluation, une bibliographie, les techniques d'enseignement utilisées, les techniques d'apprentissage conseillées, mais aussi les compétences attendues de l'apprentissage. Une liste des principales UEs et leurs contenus sont détaillés dans la section I.

Thématiques enseignées et pratiques pédagogiques

Mes enseignements portent sur les thématiques suivantes qui couvrent une large gamme d'enseignements fondamentaux du 1er cycle (Licence) et du 2nd cycle (Master) : Systèmes cyber-physiques (M2), Diagnostic des systèmes (M2), Commande avancée (M2), Modèles pour la sûreté de fonctionnement (M1), Modélisation des systèmes (M1), Automatisation-Supervision (M1), Surveillance et analyse des données (M1), Fondements des systèmes temps-réel (M1), Mécanique des fluidessystèmes pluritechnologiques (L3), Systèmes logiques et séquentiels (L1 et L2), Concepts fondamentaux de l'automatique (L1), Méthodologie du travail universitaire (L1, L2, L3-Génie Civil).

Pour des raisons de service, je suis aussi intervenu de manière non pérenne en L3 dans les cours d'Analyse numérique (responsable de l'UE : Frédéric Hamelin) et de Traitement du signal (responsable de l'UE : Alain Richard). Les enseignements se distinguent par une évolution du degré de spécialisation au fil des deux cycles universitaires (Licence & Master) ainsi que par une écoute et une attente différentes des étudiants. Les cours dispensés en Licence 1 et 2 sont essentiellement des UEs de formation générale aux études universitaires (e.g., méthodologie du travail universitaire) et d'introduction à l'automatique (concepts fondamentaux de l'automatique). Les modules dispensés en Master sont pour certains des modules de base pour les futurs spécialistes (modélisation des systèmes dynamiques, sûreté de fonctionnement des systèmes) et pour d'autres plus spécifiques (systèmes cyber-physiques, diagnostic des systèmes, commande avancée).

Outre ces enseignements classiques (CM,TD,TP), je suis fortement investi dans l'encadrement de projets d'ingénierie système individualisée et collective, ces projets portant sur des sujets relatifs à l'automatique ou à la sûreté de fonctionnement des systèmes. A travers ces projets, les étudiants développent des compétences en acquérant à la fois des savoirs et des savoir-faire qu'ils mobilisent pour arriver à résoudre des problèmes complexes. A cet effet, les plateformes mises en place dans le cadre des travaux de recherche au CRAN (plateforme pont-roulant, plateforme Eco-sûr) servent de support à ces projets d'ingénierie système individualisée et collaborative en M1 et M2. Afin de rendre nos formations plus attractives en automatique dans la nouvelle accréditation du master EEA (orientation Contrôle pour l'Efficacité Energétique -CEE), j'ai mis en place 10 kits d'ingénierie Arduino comprenant trois projets de pointe basés sur la plateforme OpenSource Arduino et l'environnement MATLAB/SIMULINK. Cet environnement Arduino/MATLAB/SIMULINK permet aux étudiants d'acquérir des compétences en prototypage rapide et déploiement de code pour la commande automatique en temps réel des processus physiques. En Master 1 ISC, compte tenu du retour d'expérience (anciens étudiants et stages master), je me suis totalement investi dans la transformation de l'UE de Sûreté de fonctionnement en intégrant dans sa partie pratique l'utilisation du progiciel SIMFIA de Apsys-Airbus qui permet d'automatiser les études de sûreté de fonctionnement d'un système. Les étudiants acquièrent ainsi une compétence dans un usage avancé et spécialisé d'outils informatiques très demandée en ingénierie de la sûreté de fonctionnement.

Présentation synthétique des enseignements

Une vue générale de mes principaux enseignements dans les différents niveaux du L1 au M2 et des volumes horaires avec les publics concernés, sur les dix dernières années, est synthétisée dans le tableau suivant. Je n'ai reporté dans ce tableau que les enseignements en formation initiale et en présentiel. Les heures affectées à l'encadrement des étudiants sur les projets n'étant pas représentatives de l'investissement associé, elles ne sont pas prises en compte dans ce tableau.

Mon service d'enseignement représente un volume horaire annuel moyen de 238 heures-équivalent TD (HETD). Le tableau ci-dessous reprend mon bilan de service explicite sur les dix dernières années où les volumes enseignés sont exprimés en heures, excepté dans la dernière colonne où ils sont donnés en équivalent heures TD (HETD).

Tableau de bilan de service (sur les 10

Syllabus (contenu résumé) des modules d'enseignements principaux

Je présente le contenu de quelques uns de mes enseignements et les supports de cours réalisés sous forme de polycopiés de cours, transparents, et sujets de travaux pratiques. Ces documents ont fait l'objet de remise à jour de manière continue afin de les adapter aux approches pédagogiques mises en oeuvre.

UE -Systèmes cyber-physiques -Master 2 -(16h CM, 8h TD, 6h TP)

Ce module est enseigné aux étudiants du Master 2 EEA-CEE et aux étudiants de Polytech Nancy inscrits au Master ISC -recherche C'est un cours avancé que j'ai monté sur ce sujet d'actualité que constituent les systèmes cyberphysiques. Ce cours est construit à partir d'une synthèse des connaissances en automatique acquises en master 1 et présente des exemples types de systèmes cyber-physiques (contrôle de trafic, bâtiments intelligents, smart grids, gestion intelligente de l'énergie, . . .). Les outils d'analyse et de synthèse de ces systèmes sont abordés et s'articulent autour des points suivants :

• Modélisation des systèmes cyber-physiques : dynamique continue/dynamique discrète • Réalisation des sytèmes déterministes (Kalman-Ho) et application à la modélisation pilotée par les données J'ai conçu de nouveaux travaux pratiques, essentiellement sous forme de bureau d'études dans l'environnement Matlab/Simulink, en m'attachant particuliérement à illustrer la démarche de l'automaticien par la mise en oeuvre de la modélisation, des techniques d'analyse et de commandes automatiques sur des systèmes rencontrés dans la pratique industrielle (machines électriques, systèmes mécaniques, systèmes hydrauliques, systèmes thermiques, bâtiments).

UE -Modèles pour Sûreté de fonctionnement -Master 1 -(16h CM, 14h TD, 15h TP)

L'objectif du cours est de présenter les concepts généraux de la sûreté de fonctionnement (SdF) ainsi que les grandeurs fondamentales des attributs de la SdF d'un système (fiabilité, maintenabilité, disponibilité) et d'amener l'étudiant à maîtriser les lois de la fiabilité utiles pour les analyses quantitatives. Le contenu de l'enseignement porte sur les points suivants :

• Notions de systèmes, composants, défaillances

• Concepts principaux (fiabilité, maintenabilité, disponibilité, sécurité, notion de risque,...)

• Classification des défaillances, modes de défaillances Cette UE porte sur la thématique de la commande robuste et H ∞ . L'objectif que je me suis assigné dans ce cours est d'apporter aux étudiants les bases de la démarche méthodologique pour la mise en oeuvre d'une commande robuste via les outils de CAO disponibles (Robust Control Toolbox de MATLAB/SCILAB). Aprés une introduction sur les bases mathématiques (espaces L 2 , opérateurs, normes, SVD), la question de la robustesse des systèmes asservis est posée via la problématique de la stabilité et de la performance de ces systèmes en présence d'incertitudes de modèle et l'étude de certains transferts caractéristiques (fonction de sensibilité, fonction complémentaire de sensibilité). La formulation du problème de commande H ∞ et sa solution sont abordées dans le cadre de la structure LFT (Linear Fractional Transformation).

•

UE -Systèmes de commande temps réel et embarqués -Master 1 -(6h CM, 6h TD)

Cette UE porte sur les mécanismes de base des systèmes temps réel et multitâches. J'y développe particuliérement les techniques d'ordonnancement temps réel. Je me suis attaché à exposer les techniques d'ordonnancement temps réel utilisables dans l'industrie pour traiter les applications à contraintes temporelles strictes avec une articulation autour des points suivants :

• Problématique de l'ordonnancement temps réel, Politiques d'ordonnancement classiques

• Eléments de taxonomie de l'ordonnancement

• Faisabilité, optimalité, ordonnançabilité

• Modélisation des tâches (périodiques, apériodiques, sporadiques)

• Algorithmes d'ordonnancement temps réel (Rate Monotonic, Deadline Monotonic, Earliest Deadline First)

• Problème de l'inversion de priorité

UE -Surveillance et analyse des données -Master 1 -(24h CM, 24h TD)

Ce module a pour objectif de présenter les concepts généraux de la surveillance des systèmes à partir du traitement statistique des données de mesures. Les tests statistiques sont étudiés pour la prise de décision (Bayes, Neyman Pearson,...). Les tests séquentiels (SPRT, Page-Hinckley) sont aussi étudiés. La maîtrise statistique des procédés est introduite comme application des tests statistiques.

Un deuxième aspect abordé dans l'UE concerne l'analyse des données (science des données) sous l'angle de l'apprentissage statistique. On y aborde l'apprentissage supervisé (régression et classification) et l'apprentissage non supervisé avec un accent particulier sur l'analyse en composantes principales.

Diffusion, rayonnement et activités internationales

Publications pédagogiques J'ai produit deux publications pédagogiques1 ([100], CESTIS) et ( [START_REF] Yamé | Teaching control valve friction using modeling and simulation[END_REF], ASEE -American Society for Engineering Education) l'une portant sur la plateforme expérimentale pont-roulant du CRAN comme support pour une démarche d'ingénierie système en commande automatique et l'autre sur la nécessité d'introduire dans l'enseignement de l'automatique à l'Université des notions pratiques sur les actionneurs courant de l'industrie que sont les vannes de réglages et les problèmes de friction auxquels elles sont sujettes et qui détériorent de manière très significative les performances des régulations classiques. Travaux antérieurs effectués à l'Université Libre de Bruxelles

Activités internationales

• Commande duale adaptative des systèmes stochastiques Historiquement, j'ai débuté mes activités de recherche lors de mes études post-graduées conduisant au grade d'Ingénieur en automatique (études de 3ème cycle à l'Université de Bruxelles) avec un travail de recherche portant sur la commande duale adaptative des systèmes stochastiques. Dans le contexte des systèmes stochastiques où les paramètres du système à régler sont inconnus et peuvent évoluer au cours du temps, l'insuffisance de l'information sur le système commandé nécessite une certaine combinaison de l'identification et de la commande du procédé. La commande présente alors une dualité qui consiste à assurer simultanément deux tâches étroitement liées mais différentes en principe : elle contribue d'une part à une meilleure connaissance du système par une action "exploratrice", et d'autre part elle élabore le signal correcteur proprement dit basé sur cette connaissance pour amener le système à l'état désiré. La résolution du problème de la commande duale est extrêmement difficile et cette difficulté a conduit au développement des méthodes approximatives (dites sous-optimales) bien connues en commandes adaptives et basées sur le principe de séparation ou sur le principe plus fort d'équivalence certaine. Malheureusement, ces commandes sous optimales sont des commandes neutres dans le sens où la neutralité indique l'absence de toute action "active" du régulateur en vue de la réduction de l'incertitude sur le système stochastique.

J'ai apporté une solution très originale à ce problème en proposant une nouvelle stratégie de commande adaptative pour les systèmes stochastiques. Cette stratégie résulte de la formulation du problème de commande optimale en incluant une mesure d'information dans la fonction de coût qui prend en compte de manière explicite les deux rôles qui caractérisent la dualité de la commande.

Partant des éléments de la théorie de l'information, j'ai quantifié le degré d'incertitude d'un système dynamique par l'entropie de Shannon. J'ai ensuite élaboré un algorithme de commande adaptative, à caractère dual, sur la base de la minimisation d'un critère à deux composantes : une composante à minimum de variance et une composante d'excitation active représentée par un terme d'entropie. J'ai montré comment par le choix du facteur d'apprentissage pondérant le terme d'entropie dans la fonction de coût, on assurait une solution unique au problème de la commande optimale au sens du critère utilisé. J'ai en outre montré sous certaines conditions la convergence de la loi obtenue vers la loi de commande à variance minimale. L'approche s'est révélée fructueuse et conduit à des lois de commandes duales, ayant une forme analytique, faciles à implanter en temps réel. Ce travail a fait l'objet des publications aux conférences internationales [START_REF] Yamé | An adaptive PID controller with learning ability, Mathematical Theory of Networks and Systems International Conference[END_REF][START_REF] Yamé | Dual adaptive control of stochastic systems via information theory[END_REF]. En raison de l'absence d'un cadre statutaire pour la recherche lorsque j'ai rejoint l'INP de Yamoussoukro, je n'ai pas pu poursuivre ce travail de recherche.

• Théorie mathématique des systèmes échantillonnés C'est le thème des mes travaux dans le cadre de ma thèse de doctorat en sciences appliquées. Il s'agit d'une recherche fondamentale dont l'objectif est le développement d'une approche mathématique rigoureuse pour la modélisation, l'analyse et la formulation des problèmes de synthèse des systèmes échantillonnés. On désigne par système échantillonné, un système où apparaît en certains points des signaux à temps discret et en d'autres points des signaux à temps continu. Par opposition à un système discret ou à un système continu, un système échantillonné est donc caracterisé par sa nature hybride qui réside dans la mixité de la variable temporelle qui évolue par endroits dans un ensemble discret et ailleurs dans un continuum. Une telle approche, initialement motivée par les problèmes de commandes optimales et robustes H 2 /H ∞ , vise la prise en compte du comportement inter-échantillonage et donc du comportement exact de ces systèmes hybrides. Cette question du traitement exact des systèmes échantillonnés n'a pas de solutions triviales et en réalité la question a été éludée dans le passé en considérant ces systèmes hybrides comme des systèmes purement discrets, la difficulté essentielle résidant dans la différence des domaines temporels sur lesquels évoluent les sous-systèmes interconnectés.

L'idée directrice et originale de mes travaux a été d'exploiter la structure dynamique particulière de type périodique propre aux systèmes échantillonnés. J'ai montré que tout système continu périodique est équivalent dans un sens fort et exact à un système discret temps-invariant dont les signaux d'entrées et de sorties sont des suites discrètes à valeurs dans un espace de dimension infinie. Cette équivalence externe résulte d'une propriété de groupe de la structure temporelle des systèmes périodiques et permet de définir de manière rigoureuse, grâce à la théorie mathématique des représentations, la notion de réponse fréquentielle des systèmes périodiques [19]. Une digression dans le domaine du traitement du signal montre que cette approche de groupe permet une vision unificatrice de différentes procédures d'analyse des filtres multicadences [START_REF] Yamé | Spectral analysis of periodically-time varying digital filters[END_REF]. Adoptant un point de vue interne, j'ai établi pour ces systèmes périodiques des équations aux différences du premier ordre sur un espace Hilbertien de dimension infinie pour le système discret équivalent [START_REF] Yamé | An Infinite-dimensional representation of continuous-time periodic systems[END_REF]. La structure infini-dimensionnelle du modèle d'état discret alors obtenu pour les systèmes échantillonnés conduit d'une part à des résultats profonds qui permettent une relecture et une clarification des propriétés de ces systèmes hybrides (e.g. propriétés spectrales, propriétés structurelles qui revêtent ici un caractère topologique) [15], [START_REF] Yamé | On sampled-data system zeros[END_REF]- [START_REF] Yamé | Representation and stabilizability of continuous-time systems under digital controls[END_REF], et d'autre part à la mise en évidence de certaines erreurs fréquemment commises dans la littérature [15], [START_REF] Yamé | On sampled-data system zeros[END_REF]- [START_REF] Yamé | Representation and stabilizability of continuous-time systems under digital controls[END_REF]. Les techniques mathématiques utilisées relèvent essentiellement de la théorie des opérateurs sur les espaces fonctionnels de type Hilbertien.

Le cadre développé permet d'aborder des problématiques diverses dont la question des limites de performance des systèmes échantillonnés et celle des synthèses robustes échantillonnées. Un prolongement inattendu de ces travaux permet d'apporter une réponse concrète et originale à une question posée par Sz-Nagy sur l'équivalence de la similarité à une contraction des opérateurs à itérés uniformément bornés et la stabilité asymptotique de ces opérateurs [14].

• Sur l'équivalence analytique des commandes à logique floue

Ce thème a constitué le sujet de mes travaux dans le cadre du projet CELOFA sous la responsabilité du professeur R. Hanus. L'objectif poursuivi dans le cadre de cette thématique présentait deux volets : un volet fondamental visant à répondre à certaines questions théoriques et un volet plus concret visant à étudier les problèmes de mise en oeuvre des commandes floues en milieu industriel.

D'un point de vue théorique, j'ai basé mon approche sur une propriété de dualité que présente la logique floue : elle formalise le mode de raisonnement d'un opérateur humain expérimenté et permet de mettre en oeuvre son expertise et son savoir-faire pour mimer son comportement dans une boucle de régulation et d'autre part elle réalise des régulateurs non linéaires à partir d'une information heuristique. Ce dernier aspect amène à considérer les régulateurs à logique floue comme une forme spéciale de "paramétrisation" de régulateurs non linéaires. Par conséquent, au delà de la méthode de conception basée sur des règles linguistiques, il en résulte des régulateurs déterministes ayant une structure analytique parfaitement définie. Cette structure analytique sous-jacente aux commandes à logique floue n'est en général pas aisée à mettre en évidence. J'ai considéré les deux grandes classes de moteur d'inférence (Mamdani et Takagi-Sugeno) et les résultats de mes travaux ont montré que les structures analytiques des régulateurs flous de type PID sont extrêmement simples et se ramènent à l'interconnexion de blocs classiques de saturation, de gains statiques et de gains préprogrammés (lookup table) [16,[START_REF] Yamé | Conventional structure of a simple class of Takagi-Sugeno PI type controllers[END_REF].

Cette équivalence résulte directement d'un calcul à partir des caractéristiques (fonctions d'appartenance, base de règles, etc) du régulateur flou. Une approche plus formelle de l'équivalence des régulateurs flous de Takagi-Sugeno aux régulateurs conventionnels basée sur leur structure architecturale a permis de les identifier comme des régulateurs à transition ou commutation douce de modes. L'idée a été de considérer la base de règles comme un 'automate fini' dont les entrées et sorties sont des symboles, la qualification (en anglais, fuzzification) et la quantification (defuzzification) étant vues comme des interfaces (convertisseurs A/N et N/A généralisés) entre le domaine flou et le domaine réel. Cette approche formelle facilite l'analyse et la vérification des spécifications des systèmes de régulation à base de règles de Takagi-Sugeno. Ces travaux ont été appliqués à l'observation des systèmes non linéaires pour la détection de défauts et publiés dans la 2ème partie de l'article [18] en co-auteur avec M. Kinnaert.

Un avant-projet de transfert de ces résultats en milieu industriel a été réalisé en collaboration avec le groupe sidérurgique Cockerill-Usinor pour la commande floue d'un train finisseur de laminoir à chaud. Une solution innovante dans le contexte des commandes de laminoirs, a consisté à exploiter le modèle du système inter-cage des laminoirs sous une forme quasi-linéaire grâce à un paramétrage incluant la position angulaire des boucleurs. La synthèse du régulateur flou sur la base de ce modèle quasi-linéaire permet alors d'offrir des garanties sur les propriétés globales du système de régulation malgré les phénomènes de frottements secs susceptibles de dégrader la stabilité des boucleurs [116]. Références associées : [11,12,70,73,76,[START_REF] Yamé | On control performance specification in network-based control systems[END_REF][START_REF] Sauter | Yamé -Sampling Rate Optimization for Fault Diagnosis of Autonomous Systems sharing a Single Communication Channel[END_REF] Toujours dans le cadre du projet SafeNECS, la thèse de Kamel Menighed [Men10] dont j'ai formulé le sujet et que j'ai co-encadré avec Christophe Aubrun se rattache à l'aspect «spatial» de la FTC des systèmes contrôlés en réseau. La thématique a porté sur les commandes coopératives embarquées et tolérante aux défauts pour des systèmes de grande dimension ou des systèmes physiquement couplés. Cette thématique, motivée par le volet «embarqué» du projet Safe-NECS, s'intéresse à l'architecture des systèmes de contrôle orientée vers une distribution coopérative ou collaborative des algorithmes de commande et de diagnostic par opposition aux systèmes traditionnellement réalisés à partir d'un calculateur central qui collecte l'ensemble des informations relevées globalement sur le procédé, puis les traite pour élaborer un ensemble de commande qui est appliqué au procédé (voir figure 1). Les résultats de ces travaux ont fait l'objet de nombreuses publications dans des revues internationales [START_REF]Réalisation des systèmes déterministes -Data-driven modelling (Transparents)[END_REF][START_REF]Modèles pour la sûreté de fonctionnement des systèmes (Transparents)[END_REF][START_REF]Fiabilité des systèmes avec redondance passive (standby) par graphes de Markov, 19 pages 9[END_REF]10] et en conférences internationales [35,38,[START_REF] Jain | A data-driven approach to real-time controller reconfiguration for fault tolerance[END_REF][START_REF] Jain | A trajectory-based real-time controller synthesis with bumpless interconnection mechanism[END_REF][START_REF] Yamé | A data-driven bumpless switching approach for energy efficiency of a multiple evaporator cooling system[END_REF][START_REF] Jain | Trajectory-based fault accommodation in winder turbine systems[END_REF][START_REF] Jain | On finite-time stability and frequency spectrum analysis of the trajectory-based active fault-tolerant control[END_REF][START_REF] Jain | A novel trajectory-based active fault-tolerant control : application to a wind turbine system[END_REF][START_REF] Yamé | On some features of the canonical controller of the behavioral approach[END_REF][START_REF] Jain | A novel online controller redesign approach to fault accommodation in wind turbine systems[END_REF][START_REF] Jain | A real-time projection-based approach for fault accommodation in NREL's 5MW Wind Turbine Systems[END_REF][START_REF] Jain | Trajectory-based online controller design approach to fault accommodation in wind turbine systems[END_REF][START_REF] Jain | Online redesign based approach for fault accommodation in wind turbines[END_REF][START_REF] Yamé | A note on the canonical controller in the behavioral system-theoretic approach[END_REF][START_REF] Jain | Case study on the behavioral approach to fault-tolerant control : Application to an electrical circuit[END_REF][START_REF] Jain | Time-trajectory based active fault-tolerant control[END_REF][START_REF] Jain | Role of performance evaluator in data-driven fault tolerant control[END_REF][START_REF] Jain | On implementing on-line designed controller for smooth interconnection in the behavioral framework[END_REF][START_REF] Jain | On active fault-tolerant control in behavioral context[END_REF][START_REF] Jain | PAPYRUS Workshop on Fault Diagnosis and Fault Tolerant Control in large scale processing industries[END_REF][START_REF] Jain | Synergy of canonical control and unfalsified control concept to achieve fault tolerance[END_REF]] et compte tenu du caractère novateur de cette approche à la commande tolérante aux défauts, une monographie sur le sujet a été publiée chez l'éditeur springer [17]. Les contributions de ce travail portent sur différents aspects de la commande efficiente des bâtiments. En commande centralisée, une commande MPC tolérante aux défauts est réalisée par une reconfiguration en ligne des contraintes suite à une défaillance des équipements terminaux VAV (variable-air-volume) du système HVAC. Les modifications en ligne des contraintes du problème MPC sont rendues possibles par un nouvel algorithme de détection et de diagnostic des défauts basé sur la conception d'un filtre particulier pour l'estimation des défauts de blocage des vannes de réglage de débit d'air de l'HVAC [36,37]. Sous l'angle de la fiabilité/disponibilité des actionneurs et de leur coût de maintenance, nous avons introduit une nouvelle fonctionnelle de coût économique régularisée induisant une parcimonie dans les signaux de commande [34]. Cette parcimonie de la commande a un effet extrêmement bénéfique sur la sollicitation des actionneurs VAV et par conséquent sur leurs durées de vie. Pour les structures de commandes distribuées dans les bâtiments non-résidentiels de grande dimension, une nouvelle approche de commande distribuée coopérative a été développée sur la base d'un algorithme de coordination de régulateurs distribués résultant d'une technique d'optimisation mathématique similaire à une relaxation Lagrangienne modifiée [START_REF] Yamé | Optimality condition decomposition approach to distributed model predictive control[END_REF].

Sur
Fort des premiers résultats obtenus sur la FTC basée le contrôle prédictif économique, j'ai développé une partie de ces travaux en collaboration avec Tushar Jain de l'Indian Institute of Technology sur une autre cible applicative, c'est-à-dire ici, des éoliennes. Le coût économique considéré pour ces éoliennes vise à atteindre deux objectifs principaux : la maximisation de la puissance et la réduction des phénomènes de fatigue de l'éolienne, éventuellement sous l'effet de défauts de biais du couple dans les convertisseurs de puissance. Le problème ainsi formulé n'est malheureusement pas convexe au regard des variables de décision et une procédure de « convexification » du problème est nécessaire pour sa résolution par des algorithmes efficaces [3,31,32].

Autres références associées : [2, [START_REF] Subramaniam | Fault-tolerant pseudo-decentralized eMPC for energy efficiency in a multi-zone building[END_REF][START_REF] Subramaniam | Decentralized eMPC based fault tolerant control for energy efficiency in a multi-zone building[END_REF]30].

Sur la détection et l'estimation de défauts à base d'observateurs

Hormis dans l'approche comportementale de la FTC, mes travaux en détection et estimation de défauts vont de pair avec la commande tolérante aux défauts car l'étape de diagnostic est nécessaire pour une estimation du défaut à renseigner dans le module de commande. Je renvoie le lecteur ici aux références [11,[START_REF] Sharan | Diagnosis of unbalanced and open-circuited faults in 3-phase uncontrolled rectifier as a part of the grid-tied wind turbine[END_REF]36,37,39,[START_REF] Menighed | Aubrun C -A slightly improved robust adaptive observer for actuator fault detection[END_REF]67,68] pour les techniques FDD (Fault Detection and Diagnosis) dont certaines ont développées conjointement avec les commandes tolérantes aux défauts mentionnées dans les sous-sections précédentes. Les algorithmes FDD issues de ces travaux sont tous construits à partir d'observateurs utilisant les modèles des systèmes sous-jacents.

Je ne résume ici que les travaux très récents en FDD issus de la thèse (PhD) de Mona Subramaniam A [Sub21] que j'ai co-dirigé avec Tushar Jain à l'Indian Institute of Technology (IIT), Mandi, Inde. Pour situer le contexte de ce travail, je mentionnerai que l'idée d'une supervision conjointe d'un doctorant est née lors du séjour au CRAN de Tushar Jain que j'ai accueilli comme chercheur invité en Juin et Juillet 2017. Le sujet de thèse que j'ai suggéré à Tushar de proposer à son premier doctorant à l'IIT se situe dans le contexte du Projet européen EiT (Energy-in-Time) dont j'étais le responsable scientifique. Ce sujet porte sur un aspect important que je n'avais pas abordé dans le cadre de la thèse de Tejaswinee Darure . Contrairement aux techniques de diagnostic et de FTC, développées dans la thèse de T. Darure, basées sur des modèles linéarisés des bâtiments/HVAC, l'idée est d'attaquer directement le problème sans linéarisation du modèle mais en exploitant le caractère intrinsèquement bilinéaire des modèles dynamiques des bâtiments équipés de dispositifs HVAC-VAV (variable-air-volume). Ce faisant, les techniques de diagnostic et de commande FTC développées restent valables dans une très grande plage de fonctionnement et donc peu sensibles aux taux d'occupation des zones et aux grandes variations de température extérieure et des conditions environnementales. La co-supervision a été actée en date du 06/03/2018 par le Président (Chairperson) de l'Indian Institute of Technology (IIT) Mandi.

Parmi les résultats marquants de ces travaux, je mentionnerai un nouvel algorithme de diagnostic de défaut (FD) basé sur un observateur bilinéaire construit à partir du modèle bilinéaire de la dynamique réelle thermique du bâtiment, rendant l'algorithme applicable à un large éventail de conditions de fonctionnement imprévisibles (dynamique météorologique, température de l'air extérieur, profil d'occupation des zones). Une fonction importante de l'observateur bilinéaire est l'estimation des températures des murs, qui sont généralement des variables d'état essentielles mais non mesurées en ce qui concerne le contrôle du confort dans les bâtiments. Une approche particulièrement simple et unique utilisant les estimations des températures des murs et la dérivée des mesures de température des zones est alors conçue pour estimer les positions des vannes bloquées des VAV. En particulier, j'ai suggéré une approche originale utilisant des observateurs à très grand gain (High gain observers) pour calculer avec précision en temps réel les dérivées des mesures de température des zones [START_REF]Modélisation des systèmes dynamiques continus (Transparents)[END_REF]. D'autres travaux sont en cours pour « robustifier » l'algorithme par rapport à la présence de bruits de mesure et de perturbations non mesurées.

Références associées : [START_REF]Modélisation des systèmes dynamiques continus (Transparents)[END_REF]20,22,[START_REF] Subramaniam | Diagnosis of actuator faults in VAV-HVAC system using a bilinear observer[END_REF][START_REF] Subramaniam | Output injected nonlinear observer for diagnosing faults in multi-zone building[END_REF]121]. -Titre de la thèse : Commandes coopératives embarquées et tolérantes aux défauts.

Encadrement de la recherche

-Co-encadrants (taux de co-encadrement) : Joseph YAMÉ (50%) -Christophe AUBRUN (50%)

-Financement : Projet ANR SafeNECS • Tejaswinee DARURE (01/10/2014-18/10/2017).

-Titre de la thèse : Contribution à l'optimisation de la performance énergétique des bâtiments de grande dimension : Une approche intégrée diagnostic/commande économique et coopérative à horizon glissant.

-Co-encadrants (taux de co-encadrement) : Joseph YAMÉ (50%) -Frédéric HAMELIN (50%)

-Financement : Projet européen Energy-In-Time.

- -Titre : Commande prédictive distribuée : Application à la commande tolérante aux défauts des systèmes interconnectés contrôlés en réseau.

-Situation après le master : Abderrhamane a enseigné à l'Ecole Marocaine des Sciences de l'Ingénieur (EMSI -Casablanca) et a préparé une thèse de doctorat au CRAN sous la direction de Dominique Sauter, thèse soutenue le 13 Juillet 2020.

• Nabontana YEO (01/03/2012-31/07/2012).

-Titre : Etude d'approches sans modèle en commande : Application à la commande tolérante aux défauts et mise en oeuvre sur la plateforme pont-roulant du CRAN.

• Hanane LOUAJRI (01/03/2011 -31/07/2011).

-Titre : Synthèse en ligne de contrôleurs LQR tolérants aux défauts par l'approche comportementale.

-Situation après le master : Hanane a obtenu un contrat doctoral au Département Informatique Automatique, Mines Douai.

• Hanping QIAO (01/03/2010 -31/07/2010).

-Titre : Etude comparative des techniques de transfert sans-à-coup pour les systèmes tolérants aux défauts à base de multi-contrôleurs.

-Situation après le master : Hanping a été recruté comme Ingénieur de développement chez Concretio Services à Paris.

• Hernando BERNAL (01/03/2007-31/07/2007).

-Titre : Caractérisation de la performance des boucles de commande des systèmes contrôlés en réseau

Production scientifique -bilan quantitatif

Ci-dessous, une vision synthétique et quantitative de ma production scientifique :

Publications et communications

Liste complète et classée par type de publications

Revues internationales avec comité de lecture

Part II

Mémoire

The best work in systems and control theory makes it clear that the creative modeling of phenomena and the conceptualization of systems using mathematics and the subsequent mathematical analysis of these models to get a deeper understanding of the phenomena are what is central to the field.

Sanjoy K. Mitter 3

Foreword

In this "mémoire d'habilitation à diriger des recherches", I have chosen to present my research activities which are related to the field of fault tolerant control and some related topics. I have therefore left aside some works carried out before joining the Centre de Recherche en Automatique de Nancy (CRAN), in particular those topics related to my doctoral research and to previous research, which are no longer related to my research topics as a member of a well-defined thematic department of the CRAN. The exposition of the work on the research topics dealed with in this manuscript is done through a bi-paradigmatic perspective. This perspective consisting of two basic paradigms, namely the model-based paradigm and the data-based paradigm, can be considered as a reference frame that filters our vision of the main methodological approaches we have used to deal with fault-tolerant control problems. The choice of such an outlook of our research is made for the following reasons: The first reason is that the topics I have worked on since joining the CRAN are diverse and a taxonomy of the solutions to the issues addressed in these research topics, while under the umbrella of fault tolerant control, naturally falls into model-based approaches and data-based approach which could be qualified also as model-free. The second reason is to produce a manuscript under a unifying title given the different issues addressed which, at first glance, are not necessarily related to each other in terms of the mathematical tools used. The narrative I have adopted in the manuscript is almost chronological in terms of the order of beginning my research on the selected topics dealt with along these several years, although in reality some temporal overlaps exist between all these research activities.

This part of the document is specifically devoted to the synthesis of my research work that I have chosen to present in the form of a memoir. This memoir contains five chapters and a short conclusion. My research prospectives, stated as a research agenda, is described in the last part of the overall document and not as a chapter of this synthesis of my research activities. The first chapter is a short introduction to fault-tolerant control and some of its basic issues. It introduces the selected topics covered in this memoir and outlines the contributions made to the field. The second chapter focuses on the behavioral paradigm that we have bring in as the mathematical framework of choice for data-driven fault-tolerant control (FTC). After a brief introduction to the basic concepts of behavioral system theory, we discuss two fault-tolerant control schemes that we have initiated and which are based on system trajectories: a projection-based FTC approach and a direct online synthesis approach. The third chapter deals with an important ancillary problem which is that of smooth control authority transfer during controller switchings, such switchings usually occur in fault tolerant systems during controller reconfiguration following system failures. Two solutions to this problem are provided, one in the behavioral framework and the other in a more classical framework. The last two chapters concern our contributions in the model-based paradigm. In particular, the fourth chapter deals with fault tolerant control based on receding horizon optimal control techniques, the so-called model predictive control (MPC) strategies. This work has been essentially conducted on application targets, namely buildings and wind turbines, and has left little room for purely theoretical considerations. The topics discussed are economic MPC based FTC on these target applications and distributed model predictive control. The last topic presented in the fifth chapter is model-based fault detection and diagnosis geared to the monitoring of building-HVAC systems and their fault-tolerant control which requires precise diagnosis of occurring faults. The subject is interesting because it involves a bilinear model and the approach developed could be extended to other similar dynamics other than buildings-HVAC systems. A short conclusion closes this memoir.

Chapter 1

Introduction

Abstract

This chapter is a concise introduction to the basics of fault-tolerant control and it provides the formulation of the fault-tolerant control (FTC) problem in its general form with a highligth of some of the challenges in solving this problem, particularly those issues addressed in our research. A brief overview of the main approaches in FTC is given and the topics covered in the remainder of the manuscript are introduced.

Issues in Fault-Tolerant Control

Fault-tolerant control (FTC) deals with the issue of controlling dynamic systems subject to faults in operational phases, where, in a broad sense, a fault is understood as any kind of malfunction of the actual dynamical system. Such malfunctions cause changes in the structure or parameters of the system and ultimately might lead to performance degradation or loss of system functions.

If the system has some kind of redundancy, either in physical or analytical form, then it may be possible to change the control law/structure to close the feedback loop. Basically, fault-tolerant control is an extension of conventional feedback control. Indeed, conventional automatic control deals primarily with feedback control of systems in a nominal mode in which all system components behave satisfactorily and provide their specified service to meet the closed-loop system performance while FTC considers non-nominal modes resulting from the malfunction of some parts of the system. The generic structure of fault-tolerant control systems is depicted in figure 1.1 and exhibits two layers, viz., an execution layer and a supervisory layer. The execution layer is nothing more than the conventional feedback control loop and consists of a controlled dynamical system and a controller. The controlled dynamical system may be viewed as consisting of three main components or subparts, i.e., the actuators, the plant also called the process, and the sensors. The actuator is a device which physically actuates the plant based on the control signal u, and the sensor measures the plant output y which, in turn, is used by the controller to compute the control signal u so that the measured output y tracks the reference signal r. The supervisory layer builds upon the conventional feedback loop as an upper level and consists of two modules in series interconnection, i.e., a fault detection and diagnosis (FDD) module and a control reconfiguration mechanism (CRM).

The FDD module has the task of detecting the existence of a fault and identifying/estimating • plant faults, i.e., faults which change the dynamics of the plant

• sensor faults, i.e., substantial errors in the sensor readings

• actuator faults, i.e. faults which modifiy or break the influence of the controller on the plant Note that the sensors and actuators faults do not affect the plant dynamics. With regards to their mathematical representations, faults may be viewed as signals, i.e., vector functions of the time variable t, usually denoted by f (t), which affect the system subparts as unintended inputs.

From a purely abstract point of view, the fault-tolerant control problem is similar to a conventional feedback control problem , but with features that make its practical design much more difficult. Recall that the standard control aims at designing a control algorithm in an admissible given set of control laws U , such that the overall feedback system achieves a closed-loop specification or control objective O, while the dynamics of the plant satisfies a set of constraints P. Concisely, the standard control problem is characterized by some triplet (O, P (0) , U (0) ) where the set of constraints P (0) is related to the nominal or healthy plant dynamics which can be expressed in the form of mathematical representations such as, e.g., input/state/output equations, and U (0) is the set of admissible control laws for this nominal control problem. For solving this problem, the knowledge of the set of constraints P (0) is a prior which might result either from first principles modeling or from data-based experimental modeling of the plant or other knowledge-based approaches. Note that this modeling process is a non-trivial part of any standard control problem and the obtained model will only be an approximation of the dynamics of the real plant. The controller is thereafter designed offline and subsequently implemented practically on the real plant. In actual operation, the controlled system may be subject to faults that compromise the specified closed-loop performance and eventually the overall system stability. Let F be the set of all faults of interest for the plant under control. Such set of faults may have been derived from a failure mode effects and criticality analysis (FMECA) which is a mandatory step in reliability and dependability Figure 1.2: "States" in the fault-tolerant control process analysis. The occurrence of faults in F changes the set of constraints P (0) into another different set of constraints P (α) for some α ∈ I where I is an (possibly continuous) index set allowing the description of faults f α (t) ∈ F . If the objectives of the nominal feedback system are to be retained, as they specify what the users expect from the system, then the following new control problem1 (O, P (α) , U (α) ) should be solved autonomously at run-time of the feedback system to ensure fault tolerance. Typically, FTC systems aim to continue and complete safely a mission despite the presence of faults, and repairs which are necessary to restore the system to its original state are then done at system shutdown. In some faulty mode situations, where some trade-off between achievable performance and available subparts (e.g., actuators) capability is necessary, some performance degradation should be accepted for achieving fault-tolerance. In that case, the nominal objective should be transformed into a new relaxed objective, say O (α) , in order to ensures a solution to the problem (O (α) , P (α) , U (α) ) allowing the mission to complete, albeit with degraded performance.

The main steps in the process of achieving fault tolerance can be depicted from a finite state machine standpoint by considering fault occurrence, fault diagnosis, control reconfiguration and possibly repair as events causing transitions from one "state" to another. The system is initially in the nominal "state". The occurrence of a fault seen as an event ev f brings the system into the undesired <faulty_system> state till the fault is detected and diagnosed which is signaled by the event ev d . The system is then in the "state" where the faulty system is diagnosed (or identified), which allows the controller redesign process. When this later design process is completed, it is signaled by the event ev rc and leads the system to the desirable reconfiguration mode. This situation remains until the repair event ev r occurs, which makes the system recovers its initial <healthy> state. Note that the repair is usually not performed in run-time but rather when the system is stopped on the ground.

These events characterized by their occurrence times and the duration of sojourn in each of the represented "states" highlight the decomposition of the time interval between the time of occurrence of a fault and the time when the system is in its reconfigured mode. This timed event sequence and the sample path of the overall FTC process is shown in figure 1.3 where the time instants t f and t f d denotes respectively, the time of occurrence of a fault and the time of detection of the presence of the fault by the FDD module. The time instant t f dd is the time at which the fault Figure 1.3: Event sequence and "sates" sample path in the FTC process is identified or diagnosed by the FDD module and t rc is the time by which the system is in its reconfigured mode. In this figure, s 0 denotes the <nomimal mode> while the <faulty_system> state corresponds to the union of s 11 and s 12 . The state <faulty_system diagnosed> is labeled s 13 and corresponds to the phase in which a knowledge of the faulty system has been acquired, as e.g. through a post-fault model, which allows a controller design to be performed based on this knowledge of the faulty system. The label s 2 denotes the <reconfigured system> state in which the newly designed controller operates on the faulty system and achieves the specified objectives thereby ensuring the fault-tolerance of the feedback control loop.

From the above, some of the challenges raised in solving the fault tolerant control problem can now be highlighted. We will not be exhaustive here but rather focused on those issues that have driven our research described in this manuscript. First of all, it is noteworthy that the fault-tolerant control problem has an underlying feedback control problem as defined by some triplet (O, P, U ) to be solved in real time and completely autonomously. In contrast to conventional nominal feedback design where the triplet data of the problem is known at the outset, a first challenge in FTC is that the constraint P arising from a fault is seldom available. Clearly, this constraint will be dependent on the particular fault f α (t) ∈ F affecting the controlled system and thus yielding an unknown constraint P (α) . Hence the fundamental need to detect and identify first, in real time, the occurring fault through a fault detection and diagnosis algorithm.

(i-1) -The core of FDD is built upon model-based algorithms such as Kalman filtering, state or output observation algorithms, parity space techniques or system parameter identification/estimation. Because a FDD algorithm relies on a precise model of the actual operating plant, any discrepancy between this model and the real plant can lead to false alarms or false detection that may mislead the reconfiguration mechanism with unfortunate consequences on the feedback system. Because such model mismatch is rather inherent in any pratical modeling process, the model uncertainties in FDD module is thereof a non-trivial challenge in achieving fault-tolerant control.

(i-2) -The interconnected modules in the supervisory layer of an FTC system, viz. the FDD module and the control reconfiguration mechanism, are dynamically coupled systems. The reconfiguration being based on the information provided by the FDD algorithm, an important issue which is seldom addressed in the literature is the study of the "post-fault FDD algorithm". As the FDD should embed a precise knowledge of the actual operating mode whose constraint becomes P (α) under the occurrence of a fault f α (t) ∈ F , how should this FDD algorithm be made adaptive to post-fault models of the running controlled system? (i-3) -As shown in figure 1.3, the time-to-reconfiguration is the time-delay τ δ = t rc -t f which is the length of the time-interval [t f , t rc ] on which the system is faulty but still controlled by the nominal controller or a controller which is not matched to the actual plant dynamics. During this time interval, the controller is driving a system for which it is not designed for. This situation inevitably leads to a significant loss of performance and could ultimately result in a destabilization of the closed loop if such time interval is long enough with respect to the dynamics of the system. The challenges here are how to reduce this reconfiguration time-delay τ δ to avoid destabilizing the feedback system? (i-4) -As mentioned above, the FTC problem is genuinely a real-time problem with new ingredients (O, P (α) , U (α) ) to be solved autonomously. Solving this problem naturally leads to a real-time implementation of a different closed-loop dynamics with regards to the initial running feedback system that solved the nominal problem (O, P (0) , U (0) ). Since the overall FTC system operates continuously in time without a re-start of the system when the new fault-tolerant control is implemented, a dynamic switching between two different operating modes occurs almost instantaneously at the time of implementation of the new control. Such mode-switchings can strongly impair the transient performance and the stability of the overall system. Though being a serious issue which need to be addressed in the synthesis of fault-tolerant systems, this problem is often eluded in solving the fault-tolerant control problem.

The above challenges provide a mild setup of the topics, dealed with in this report, which will be outlined in section 1.3.

A Brief Overview of Fault-Tolerant Control

This section aims at giving a very brief overview of fault tolerant control methods. Its purpose is not intended to provide a general overview nor a bibliography of the field, but rather to give a quick overview of the taxonomy of these FTC methods with regards to the topics addressed in this report. For exhaustive reviews of the state-of-the-art in fault tolerant control, the reader is referred to the excellent surveys, tutorials and research monographs available on the subject and references therein [Bla+16; LR08; ZJ08b].

The general accepted taxonomy of fault tolerant control systems distinguishes between two possible approaches, the passive approach and the active approach, according to whether a fixed or reconfigurable controller is used. With reference to the generic structure of FTC systems in figure 1.1, a passive FTC control system is simply a control structure consisting only of the basic execution layer. It is a classical feedback loop in which a single fixed controller has been designed to be robust or insensitive as possible against a set of presumed faults f α (t) ∈ F viewed as uncertain signals acting on the plant. Therefore neither a FDD algorithm, nor a controller reconfiguration mechanism are needed under such FTC structure. From the fault-tolerant control problem as formulated in section 1.1, this means that the passive FTC control solution is assumed to be common to the problems (O, P (0) , U (0) ) and (O, P (α) , U (α) ) with α ∈ I . Such control solutions are known to be conservative and their main drawback is their inability to handle unexpected faults.

Active fault-tolerant control (AFTC) structures have all the functional blocks of the generic structure of figure 1.1, where both layers are present. Therefore active fault-tolerant control performs the two conceptual stages for providing systems with fault-tolerant capability:

1. online fault detection and diagnosis algorithm to monitor the controlled system and collect information on faults 2. selection or design of the required corrective control action through a reconfiguration mechanism to maintain acceptable post-fault closed-loop performance Thanks to their very structure, active fault-tolerant control systems are intrinsically adaptive in the sense that the control law changes according to the changes in the controlled system dynamics. Historically, the developments of AFTC has been initially driven by problems in aircraft control systems for which adaptive control were needed for automatically adjusting the controller parameters in the face of changing aircraft dynamics due to rapidly varying aircraft flight conditions during standard maneuvers and unexpected failures [START_REF] Hovakimyan | L1 adaptive control for safety-critical systems[END_REF]. At first glance, adaptive control has an architecture very similar to that of fault-tolerant control systems, if the FDD module is replaced by a block that identifies the current plant parameters and the controller redesign block adapts the controller parameters to the result of the identification, see figure 1.1. However, the distinguishing feature for fault-tolerant control is that, under severe and sudden faults, the magnitude of the changes of the plant behaviour is more significant and not restricted to slow parameter variations. Nethertheless, adaptive control provides a reasonable solution to the fault-tolerant control problem when the plant is subject to incipient faults that represent gradual component degradation. Such active FTC-based adaptive control is referred to as fault accommodation because the controller structure is not changed by the control reconfiguration mechanism (CRM), only its parameters are eventually modified.

Two approaches widely used to achieve AFTC, which characterize the way in which the post-fault performance is achieved, are the so-called projection methods and the online redesign methods.

The projection methods rely on real-time controller selection from a set of offline pre-computed controllers where each of these controllers has been designed for a particular fault that may occur in the controlled system. Typical methods in this class of projection approaches are multiple model FTC and controller switching based FTC. The online redesign methods involve synthesizing a new controller in response to the impairment of the controlled system components. This controller synthesis is performed autonomously in real time on the basis of the information provided by the FDD module. Representative methods falling under this online redesign approach are modelmatching/following, adaptive control, model predictive control, online optimization as e.g. control re-allocation, and machine learning based methods.

Fault detection and diagnosis is a crucial step in all active fault-tolerant control systems and techniques that seamlessly combine fault diagnosis and control reconfiguration are in fact difficult to achieve. Though, this issue is still considered unsolved, it has been tackled from various sides, for example, by taking into account the uncertainties in the fault diagnosis, or through an architecture for integrated residual filter design for fault diagnosis and control adjustment. Subspace predictive control combining parameter identification and controller synthesis bears also some potential for solving the simultaneous fault diagnosis and control redesign problem.The application of unfalsified control to fault-tolerant control as suggested in [START_REF] Yamé | Performance-Based Switching for Fault Tolerant Control[END_REF] provides a systematic means for getting rid-off this integration issue in projection-based methods by selecting one controller from a set of finitely many candidate controllers from a FDD module free supersivory layer.

Selected Topics and Contributions

This manuscript contains a selection of contributions made, over several years of research, on specific topics related to the fault-tolerant control problem and some related issues. The issues of interest raised by the fault-tolerant control problem discussed in section 1.1 have involved different "modeling" approaches that are best viewed from a paradigmatic perspective. This perspective, seen as a big picture, is in fact a bi-paradigmatic framework consisting of a data-driven paradigm on one hand and a model-based paradigm on the other.

In the data-driven paradigm, hereon qualified also as a model-free approach, we have put forward the so-called behavioral framework for the study of dynamical systems in the context of fault-tolerant control. At the time we began this research, the use of the behavioral paradigm in this context was new and risky as it was a departure from all previous approaches, and Willems' behavioral approach was still quite controversial. However, a close-up view of the behavorial approach to system theory, developped by J.C. Willems as an alternative approach to system modeling at the end of the eighties, lends itself very well and almost naturally to the problem of obtaining dynamical models from observed signals or time series. The shortcomings of the fault-tolerant control problem delineated in the first three points (i-1), (i-2), (i-3) are clearly rooted in the model-based approach to FDD and in the two cascaded stages (FDD and CRM) of the supervisory level. In particular, the need for an explicit real-time post-fault model for controller redesign is an additional difficulty in achieving, autonomously, system fault-tolerance. Furthermore, the online updating of the modelbased FDD algorithm once the system is reconfigured is not trivial either. Our thesis was that solving these issues requires a paradigm shift from a model-based approach to a data-driven approach with the underlying philosophy of letting the data speak rather than making assumptions on the plant model. In that way, it is hopefully that the real-time data generated by the running plant under its actual operating conditions enables the design of a controller matched to the plant as closely as possible. It was obvious at the outset, that such procedure avoids the need to accurately model the plant and is likely to be fairly robust to all hypothetical parasitic dynamics and operating conditions as it considers only those operating modes that actually occur. The notion of behavior as defined by J.C. Willems is in fact nothing more than the set of all trajectories that can be generated by a dynamic system. Therefore, the behavioral paradigm, which makes the behavior the central object associated with a dynamical system rather than its parametric model derived from first principles, is completely appropriate as a theoretical framework for the design of fault-tolerant systems. It is on the basis of these considerations that I tackled the fault-tolerant control problem in the behavioral paradigm, setting up this paradigm as a right mathematical framework for dealing directly with data or trajectories of a controlled plant for the online synthesis of controllers. Our main contributions to fault-tolerant control in the behavioral paradigm are threefold. The first contribution, which can be classified in the class of projection-based techniques, is related to the falsification of pre-designed controllers associated to each failure modes. In fact, this approach is hybrid in the sense that it uses both data-based and model-based paradigms. Indeed, the pre-design of the set of controllers is model-based, but under the occurence of failures at run time, the online selection of the feedback controllers is determined in real-time solely on the basis of data generated by the plant. The second contribution is genuinely a trajectory-based (or data-based) fault-tolerant control technique, inspired from the concept of canonical controller introduced by A. J. van der Schaft [START_REF] Van Der Schaft | Achievable behavior of general systems[END_REF]. In this scheme, the online synthesis of the controllers is performed, based only on the real-time data generated by the controlled plant, to satisfy the control objective O whatever the plant's nominal or faulty modes. The solution to this problem boils down to that of a controller identification from the observed trajectory and the constraint imposed by the control objective. Our third contribution to FTC in the behavioral paradigm addresses the ancillary but important issue of the control authority transfer problem during the process of reconfiguring the control-loop. This transfer of control authority is usually and instantly followed by serious transients that have a strong negative impact on the control performance if no provision is made to avoid their occurrence. A solution to this problem has been provided thanks to its clear characterization in the behavioral framework. Chapter 2 and the first part of Chapter 3 cover the synthesis of the above mentioned research activities within the behavioral paradigm. The second part of chapter 3 still deals with the issue of control authority transfer in fault-tolerant control systems. Though not using the behavioral paradigm, the proposed approach is plant-model independent and relies on a judicious re-parametrization of a set of heteregeneous linear controllers and on the concept of self-conditioning control structure.

In the model-based paradigm, our contributions to fault-tolerant control are based on receding horizon optimal control techniques, the so-called model-predictive control (MPC), which are known to have fault-tolerance capabilities. I initially became interested in this topic as part of the thesis of K. Menighed, that I supervized, on fault-tolerant control in networked and embedded systems [START_REF] Menighed | Commandes coopératives embarquées et tolérantes aux défauts[END_REF]. The research questions addressed in this thesis were rather generic in nature and concerned standard regulation and tracking problems without concrete applicative targets. To avoid being too exhaustive on this topic of FTC-based MPC, I will focus in this habilitation thesis, on the work resulting from the Energy-In-Time project and for which I have been interested in economic model predictive control as a baseline control for the FTC problem. This research topic has been driven by real economic objectives in the context of energy efficiency of buildings and later by the formulation of non-standard control problems in the operation of wind turbines. Our contributions have been mainly on the right formulation of control problems by establishing non-standard cost functions for the constrained optimization problems. In particular, for buildings, an innovative formulation of the economic control optimization problem has been stated with an 1 regularization term which suppresses the oscillations in the set-point trajectories provided to the actuators with benefits on the lifespan of these actuators. In the context of fault-tolerant control of wind turbines, the major issue in the economic model predictive problem addressed is its highly non-linear and non-convex nature which makes it very hard to solve, and even in the case of a solution, a global minimum could not be guaranteed. Our contribution towards solving this problem has been the development of a convex relaxation procedure for the original economic non-convex optimization problem. In these above fault-tolerant control approaches relying on model-based predictive control, fault tolerance was achieved by constraints reconfiguration following a fault estimate provided by a fault detection and diagnosis algorithm. Another contribution within the model-based predictive control paradigm is on the topic of distributed control which was motivated by the large-scale system nature of non-residential buildings that has been of interest in the Energy-In-Time project. This contribution relying on a decomposition of the large-scale system and on a modified Lagrangian relaxation of the optimization problem ensures that the performance of the distributed control system is almost equivalent to that of the (reference) centralized control system. Chapter 4 summarizes the main aspects of the above mentioned research activities within the model-based predictive control paradigm.

The last topic discussed in this memoir is fault detection and diagnosis in the model-based paradigm. Though, I have worked on different FDD algorithms concomitantly with the fault tolerant control problem, notably in the doctoral theses of K. Menighed [START_REF] Menighed | Commandes coopératives embarquées et tolérantes aux défauts[END_REF] and T. Darure [START_REF] Darure | Contribution à l'optimisation de la performance énergétique des bâtiments de grande dimension : une approche intégrée diagnostic/commande économique et coopérative à horizon glissant[END_REF], I will not discuss here these works carried out essentially in the linear framework. Instead, I will focus on more recent and ongoing work related to fault diagnosis in building HVAC-systems, started and carried out during the doctoral thesis of A. M. Subramanian [START_REF] Subramaniam | Fault diagnosis and fault-tolerant control for energy efficiency in a multi-zone building[END_REF] that I co-supervised with T. Jain of the Indian Institute of technology, Mandi, India. This work focuses on specific building HVAC equipments, in particular on VAV (variable air volume) terminal equipments, however, the techniques developed can be extended to systems having a similar mathematical structure as those equipments, i.e., systems with a bilinear structure. The main contribution to this topic is a bilinear observer-based FDD algorithm designed for precise estimation of VAV damper faults over a wide range of operating conditions.

Notes and references

This chapter has briefly presented the main issues in fault tolerant control and its main ingredients. An excellent and fairly complete presentation of fault-tolerant control is the monograph [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF], a recent monograph dealing with both model-based and data-driven approaches in diagnosis and fault-tolerant control is [START_REF] Steven | Advanced methods for fault diagnosis and fault-tolerant control[END_REF]. As this field of research is very active, one can find quite often in the literature state-of-the art on this subject, see for example the references [START_REF] Zhang | Bibliographical review on reconfigurable fault-tolerant control systems[END_REF], [START_REF] Lunze | Reconfigurable Fault-tolerant Control: A Tutorial Introduction[END_REF], [START_REF] Hwang | A Survey of Fault Detection, Isolation, and Reconfiguration Methods[END_REF], [START_REF] Jiang | Fault-Tolerant control systems: A comparitive study between active and passive approaches[END_REF], [START_REF] Yu | A survey of fault-tolerant controllers based on safety-related issues[END_REF], [START_REF] Amin | A review of Fault Tolerant Control Systems: Advancements and applications[END_REF]. This introductory chapter also highlighted the challenges that motivated my research activities in this area and outlined the main topics addressed in this habilitation thesis and the contributions to the field. These different contributions are presented under a bi-paradigmatic view that allows to classify these contributions under trajectory-based (or data-based) and modelbased approaches.

Chapter 2

Data-driven Fault-Tolerant Control : The Behavioral Paradigm Abstract This chapter presents my research activities related to the behavioral system theoretic approach to fault-tolerant control. It considers methods based on system trajectories for achieving fault tolerance under system faulty modes by recovering as much as possible the performance of the nominal closed-loop system. The chapter is divided into three parts. The first part briefly recalls the notion of system behavior as defined by J.C. Willems and it gives the minimal elements necessary for an understanding of the following sections. In the second part, I present the first method I initially developed for data-driven fault tolerant control using the concept of controller unfalsification originally proposed by M. Safonov. This approach actually mixes two paradigms, the model-based approach for setting a bank of pre-computed controllers, and the data-driven approach for real-time implementation to achieve fault-tolerance. This approach falls into the category of projection-based fault-tolerant control techniques. In the last part, I present an autonomous data-driven controller redesign approach to achieve fault tolerance. The theory is fully developped in the behavioral framework and uses the key concept of system interconnection which characterizes a feedback system as a system whose behavior has been restricted by the laws imposed by a controller.

Mathematical setup

This section give a short overview of the behavioral framework of J.C. Willems built around the triptych: behavior-behavioral equations-latent variables. Here, the presentation of these concepts is tailored to the needs of the modeling and control problems under investigation.

Behaviors

An intuitive view of a dynamical system is that of an objet which evolves in time, and interacts with its environment through time-dependent functions, called signals, trajectories or time series. Reflecting this general and broad view, Willems' definition of a dynamical system involves three ingredients.

Definition 2.1. A dynamical system Σ is a triple Σ = (T, W, B) where T is a subset of the real axis R, W a set called the signal space and B a subset of W T called the behavior. The set T is the time axis and the set W specifies the space in which the system time-signals take on their values. The behavior is simply a family of W -valued time trajectories and constitutes the essential feature of the dynamical system. The elements of B are precisely the signals w : T → W in the power set W T which are compatible with the laws or constraints governing the dynamical system Σ, such signals are also called admissible, while those outside of B cannot occur. Note that as defined, a dynamical system in Willems' framework does not bother about which variables are inputs or outputs, hence the system may be represented as a black box with a number of terminals associated with signals without any flow direction.

In the sequel we will consider continuous-time systems with T = R and W = R q , q ≥ 1. Hence, B ⊆ (R q ) T is a collection of q-vector valued trajectories. The properties of linearity and time-invariance of the dynamical system Σ are defined at the level of the trajectories, i.e., in the behavior B. The system Σ = (T, W, B) is linear if and only if B is a linear subspace of (R q ) T , i.e., and time-invariant if (w ∈ B) ⇒ (σ τ w ∈ B), where σ τ denotes the τ -translation (shift) operator, i.e., σ τ w(t) = w(t + τ ) for all t, τ ∈ T. The property of finite dimensionality of the dynamical system Σ is reflected in the fact that its behavior B can be described as the solution set of a system of differential equations. Such behavior B will be said to be differential. For our purposes, we only need the fact that this property of finite dimensionality is equivalent with the representability of B as the kernel of a polynomial matrix in the differentiation operator (d/dt). Therefore, we will deal with the family of continuous-time, linear, time-invariant and finite dimensional systems, which we denote as

L q ∆ = {Σ = (T, W, B) : B is linear, time-invariant, differential} (2.1.1)

Modeling dynamical system from data

At the level of behaviors, the modeling problem can be described as follows. Consider an observed (i.e., measured) finite set of trajectories, called the data set and denoted by D D = {w 1 , w 2 , , ..., w N } with w i : T → R q , i = 1, 2, .., N

Clearly, D ⊂ (R q ) T . The system Σ = (T, W, B) ∈ L q will be called an unfalsified model of D if and only if D ⊆ B.

Let Σ 1 , Σ 2 be unfalsified models of D. We say that Σ 1 is more powerful than Σ 2 , if B 1 ⊆ B 2 . The intuitive interpretation of this property is that: the more powerful a mathematical model is, i.e., the less its behavior explains, the more predictive power it has. This leads to the concept of the most powerful unfalsified model (MPUM). The dynamical system Σ * = (T, (R q ) T , B * ) is the most powerful model in L q , unfalsified by the data set D, if

(Σ ∈ L q and Σ is unfalsified by D) ⇒ B * ⊆ B It is shown in [Wil86a; Wil86b; Wil87]
, that given D, the MPUM in the class L q exists and is unique. Thus, the MPUM explains the given data set D and as little else as necessary so as to obtain a model in the desired class of linear systems in this case.

Behavioral equations

It is convenient to represent the trajectories making up the behavior of a dynamical system as solutions of appropriate equations. The second concept of the triptych is that of behavioral equations and the third are latent variables. Behavioral equations with latent variables involve the manifest (also called external) variables w and latent variables . There are many instances of behavioral equations among which the most used are : the kernel representation, called autoregressive (AR) equations, the image representation, called moving average (MA) equations and different first-order representations including the state variable representation, the input/state/output representation, and many other variants. The kernel representation involves only the manifest variables w whilst the image representation makes use of the manifest variables w and of the latent variables , called auxiliary or driving variables. State variable representations, in addition to w, make use of some latent variables x, which can be assigned the property of state, and are called state variables.

The class of continuous-time, linear, time-invariant and finite dimensional dynamical systems L q consists of those systems whose behavior is the solution set of a system of g differential equations

R 0 w + R 1 dw dt + ... + R L d L w dt L = 0 (2.1.2)
where the R i s,(i = 1, 2, ..., L) are real constant matrices belonging to R g×q . Equation (2.1.2) can be written compactly as

R d dt w = 0 (2.1.3) with R (ξ) = R 0 + R 1 ξ + ... + R L ξ ∈ R g×q [ξ]
where R g×q [ξ] denotes the set of real polynomial matrices with g rows and q columns, the symbol ξ being the indeterminate in a polynomial. Thus R (ξ) is a matrix of polynomials, d dt is the differentiation operator, and w : T → R q is the (manifest) signal that is being modeled by the behavioral equation (2.1.3). Such a representation is called a kernel representation, written B = kerR d dt , since the behavior B is the set of those signals w in (R q ) T which belong to the kernel, i.e., the null space, of the differential operator R d dt . We call the polynomial matrix R (ξ) the generator of the behavior B. In the ordinary sense of the concept of solution to the set of differential equations (2.1.3) it is obvious that the componenst of such solution w should be as often differentiable as required by the equation (2.1.3). Such solutions, defined as strong solutions, might be viewed as those signals w lying in the space C ∞ (R, R q ) of functions w : R → R q that can be differentiated arbitrarily often. We view, in that case, the polynomial matrix R d dt as the well-defined mapping

R d dt : C ∞ (R, R q ) → C ∞ (R, R g ).
Note however that other concept of solutions to (2.1.3) can be used, e.g., weak solutions which are signals that are locally integrable and which satisfy (2.1.3) in a distributional sense [START_REF] Polderman | Introduction to Mathematical Systems Theory: A Behavioral Approach[END_REF].

Dynamical systems Σ = (T, W, B) in L q (for short we say B ∈ L q ) admit many other representations, the most familiar of which are the input/output and input/state/output representations. It can be shown that the following statements hold:

1. B ∈ L q ⇔ ∃ m, p ∈ Z + , with m + p = q, polynomial matrices P (ξ) ∈ R p×p [ξ], Q (ξ) ∈ R p×m [ξ]
with detP (ξ) = 0, and a permutation matrix Π ∈ R q×q such that B consists of all w : R → R q for which there exist u : R → R m and y : R → R p such that

Q (ξ) -P (ξ) u y = 0 (2.1.4)
and Πw = u y . The (p × m)-matrix of rational functions G (ξ) = P -1 (ξ) Q (ξ) is called the transfer function of (2.1.4). Actually for a given B ∈ L q , it is always possible to choose Equations (2.1.4) and (2.1.5) are respectively called input/output (i/o) and input/state/output (i/s/o) representations of the corresponding behavior B ∈ L q . The variable x is called a state and it turns out to be a very useful latent variable that specifies the memory of the system [START_REF] Willems | The Behavioral Approach to Open and Interconnected Systems: Modeling by Tearing, Zooming, and Linking[END_REF][START_REF] Willems | Paradigms and puzzles in the theory of dynamic systems[END_REF]. It is easy to see that, the external behavior B ∈ L q of the system, i.e., the family of trajectories w = vec([u T , y T ]) consisting of input/output pairs defined by (2.1.5) is parameterized by a corresponding input u and an initial state x(0). Throughout, for notation convenience, we use the vectorization operation to obtain column vectors, i.e., the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another.

Π such that G (ξ) is proper. 2. B ∈ L q ⇔ ∃ m, p, n ∈ Z + , with m + p = q, matrices A ∈ R n×n , B ∈ R n×m , C ∈ R p×n , D ∈ R
At this stage, it might be useful to introduce some integer invariants associated to a behavior in L q . First, note that from (2.1.3), L q is parametrized by the set of polynomial matrices having q columns. The number of rows in (2.1.3) representing Σ ∈ L q will depend on the particular generator R(ξ) chosen to represent Σ. The kernel representation (2.1.3) will be called miminal if R(ξ) has the least number of rows. Let p denotes this least number, then it is clear that p = rankR(ξ). It is noteworthy that the generator R(ξ) of a behavior B ∈ L q is not unique. In particular, R 1 (ξ) and R 2 (ξ) are generators of two minimal kernel representations of the same behavior B if and only if there exists a polynomial unimodular matrix

U (ξ) ∈ R p×p [ξ], such that R 1 (ξ) = U (ξ)R 2 (ξ).
The integer p is therefore an invariant of the behavior B, and we write it as p(B) and we call it the output cardinality of the behavior. In fact, it is easy to see that in an input/output representation (see (2.1.4)), p is the number of output signals. The integer m(B) = q -p(B) is also an invariant of the behavior B called the input cardinality, it is the number of input signals in an input/output representation of the behavior. A third integer invariant is the state cardinality n(B) defined as the smallest number n of state variables over all i/s/o representations (2.1.5) of B.

Latent variables

When representations are directly written down from first principles such as the laws of physics, some internal variables arise naturally. The input/state/output representation is an example of representation with internal variables x. If we consider internal variables in their full generality, this leads to the so-called latent variable : T → R l resulting in the full behavior B full ∈ L q+l described by the joint equations governing the external variables w and latent variables

R( d dt )w = M ( d dt ) (2.1.6) with R(ξ) ∈ R •×q [ξ] and M (ξ) ∈ R •×l [ξ].
The latent variable signal "supports" w in the sense that, if (w, ) ∈ B full , then w is a possible manifest variable trajectory since can occur simultaneously with w. Representation (2.1.6) is also called an ARMA (Auto-Regressive Moving Average) representation of the dynamical system Σ. The latent variable signal in a sense "supports" w.

A special case of (2.1.6), called the dual pencil representation [START_REF] Kuijper | First-order Representations of Linear Systems[END_REF] or simply a state variable representation, is when this equation is first order in and order zero in w. In that case, the latent variable has the property of a state variable, i.e., it specifies the memory of the system. The full behavior in that case reads

E d dt = F + Gw (2.1.7)
where E, F ∈ R g×l and G ∈ R g×q . An important result, the elimination theorem [START_REF] Polderman | Introduction to Mathematical Systems Theory: A Behavioral Approach[END_REF], states that B full ∈ L q+l implies that the manisfest behavior B = w : T → R q ∃ : T → R l such that (2.1.6) holds belongs to L q , and therefore admits a kernel representation (2.1.3) of its own. An interesting class of representations is the class of ARMA representations (2.1.6) for which R(ξ) is the identity matrix

w = M ( d dt ) , M (ξ) ∈ R q×l [ξ] (2.1.8)
In other words, B = im(M ( d dt )) where im(•) denotes the image of the operator (•). Equation (2.1.8) is called an MA (Moving Average) representation or simply, for obvious reasons, an image representation of the behavior B. Image representations have the nice property that they completely characterize behaviors which are controllable in the sense of J.C. Willems. The time-invariant behavior B is said to be controllable if for any w 1 ∈ B, w 2 ∈ B, there exists a T (≥ 0) ∈ T and a w ∈ B such that w(t) = w 1 (t) for t < 0, and w(t + T ) = w 2 (t) for t ≥ 0. Therefore, controllability refers to the fact that the past of any trajectory can be concatenated to the future of any other trajectory by a third trajectory if a transition period is allowed or equivalently it refers to the ability to switch from any one trajectory within the behavior to any other one, allowing some time delay. Controllabilty in Willems' sense is an intrinsic property of the dynamical system and it is independant of its representation. Now, denote the controllable elements of L q as L q cont , then it can be shown that, the behavior represented in (2.1.3), i.e., B = kerR d dt defines a controllable system if and only if R(s) has the same rank for all complex variable s ∈ C [reference]. In that case a very nice characterization of the controllable behavior B is that it admits an image representation, that is,

B ∈ L q cont ⇔ ∃M (ξ) ∈ R qו [ξ] such that B = im(M ( d dt )) (2.1.9)
that is, any manisfest variable w ∈ B is given by (2.1.8) for some latent variable . The opposite of a controllable system is, in a certain sense, an autonomous system. A behavior B is autonomous if the "past" of a trajectory, completely determines the "future" of that trajectory, i.e., if for all w 1 , w 2 ∈ B, w 1 (t) = w 2 (t) for t ≤ 0 implies w 1 (t) = w 2 (t) for all t. It can be shown that a behavior B is autonomous if and only if its generator R(ξ) ∈ R q×q [ξ] , i.e., R(ξ) is a square polynomial matrix, and detR(ξ) = 0, meaning that detR(ξ) is not identically equal to the zero polynomial or equivalently rankR(ξ) = q [PW97]. Note that an autonomous behavior B is characterized by an input cardinality m(B) = 0. Finally, define Σ to be stable if w ∈ B implies lim t→∞ w(t) = 0. Clearly, stability requires autonomy. .

FTC-based data-driven controller switching

As a researcher at ULB (Université Libre de Bruxelles) and member of the European project IFATIS from 2002 to 2005, I was in charge of the development of fault-tolerant control algorithms in the Figure 2.1: Projection-based FTC system structure framework of this project. The objective was, above all, that these FTC algorithms be implementable on real processes to demonstrate the feasibility of fault-tolerant control in practical situations. As argued in Chapter 1, model-based FDD for fault-tolerant control has inherent drawbacks, and the inevitable discrepancies between the actual operating plant and the model used for the FDD system design are the main sources of difficulties that result in false or missed alarms that significantly impair the performance of the FTC system. This is how, in order to circumvent this difficulties associated with plant modeling, I discovered the initial work of Michael Safonov [START_REF] Safonov | The Unfalsified Control Concept and Learning[END_REF] on the unfalsified control concept as a mean for a direct path from experiment to controller computation. Thanks to this concept, an approach to active fault-tolerant control (FTC) based on real-time closed-loop performance monitoring and experimental data without an explicit fault detection and isolation algorithm was derived [START_REF] Yamé | Performance-Based Switching for Fault Tolerant Control[END_REF]. The proposed approach sets forth the theory of unfalsified control to achieve a real-time control reconfiguration which is consistent with the performance objectives for a class of system faults. It was subsequently demonstrated to be effective on actuator faults and system components faults inducing abrupt changes in the plant dynamics [START_REF] Yamé | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF][START_REF] Yamé | Performance-based supervisory fault-tolerant control with application to the IFATIS two-tanks benchmark[END_REF]. The key feature of this real-time FTC procedure is that it makes no use of an online plant model and therefore it does not experience the drawbacks of fault-tolerant control systems made up of on-line model-based detection algorithm. A byproduct of the proposed model-free procedure is that the fault-tolerant controller is fast and reliable.The developed approach is in fact a mixed approach based on the model-based paradigm for the offline synthesis of a controller bank and the data-based paradigm for the real-time operation consisting in selecting a controller from the bank according to the real mode of the running plant. The supervisory level as depicted in the general figure 1.1 is thus reduced to a reconfiguration mechanism that selects the execution-level controller based on the real-time plant trajectories.

Structure of projection-based FTC systems

As stated above, the developped FTC approach uses both model-based and data-driven paradigms. It is assumed that from a preliminary analysis, e.g. via a FMCEA, that all faulty identified modes of interest that cover the possible abnormal modes of the plant as well as its healthy mode, prior knowledge has been acquired by modeling these plant modes. It is assumed further that a finite set of controllers C = {C 1 , C 2 , . . . , C N } has been constructed in such a way that in each faulty mode as well that the healthy mode of the plant, there is at least one stabilizing controller in that set which has the appropriate corrective action and is able to satisfy the control objectives. The above assumption is justified given the maturity of model-based control design methodologies for large classes of linear and nonlinear systems. It is worth to note that the family C of controllers can be viewed as generating a topology on the plant models (which describes all its faulty modes) via

Framing the problem in the behavioral paradigm

The autonomous FTC problem statement given by the triplet (O, P (α) , U (α) ) at the occurrence of faults f α (t) ∈ F requires for its solution to find the (new) set of constraints P (α) and the set of admissible controls U (α) . It is the role of the FDD module to acquire this knowledge of the constraints P (α) in real-time.

The question then arises as to whether the observation of only the external variables of the system allows the knowledge of the constraint P (α) ? Consider the feedback control loop of the execution level of the FTC system in figure 1.1. In the behavioral setting, the external variables of this feedback system which are observed are the input u and output y of the plant and the reference signal r. Define s = vec[r T , y T , u T ] as the vector of external variables (manifest variables) whose values lies in the signal space S of dimension s = r + y + u. The feedback loop of figure 1.1 translates into the behavioral framework as the interconnection of two systems where the interconnection is achieved through the terminals associated to the corresponding external variables as depicted in figure 2.2. With T ⊆ R, S ⊆ R s , the (generalized) plant and the controller are dynamical systems Σ P = (T, S, P), and Σ C = (T, S, C) whose behaviors are P ⊆ S T are C ⊆ S T . The interconnection of these two systems leads to the closed-loop dynamical system Σ K = (T, S, K) whose behavior K ⊆ S T is given by the set,

K = s ∈ S T |s ∈ P and s ∈ C ⇐⇒ K = P ∩ C (2.2.1)
that is, the closed-loop behavior K, called the implemented behavior or the controlled behavior, is simply the intersection the behavior P and C (see figure 2.3). The shift from the "classical system theory" paradigm to the behavioral paradigm is of paramount importance as revealed here by the above set-theoretic representation of feedback control systems. The objectives, or control

P ∩ C ⊆ D (2.2.2)
Checking for equation (2.2.2) obviously requires knowledge of the plant's behavior P, i.e., the set of all signals which can occur as outcomes of the dynamical system Σ P . However, only observed data or measurement from the actual system at run-time is available and this observed data gives a partial knowledge about the system. . Moreover, the evolving feature of dynamical systems makes the observation process explicitly dependent on time, as signals can be measured only till the present time. Let D be an observed (measured) data set, i.e., D ⊂ S T (see section 2.1.2) and denotes by D τ the observed data set on time intervals up to time τ . The behavior P of the plant is clearly a superset of this observed data set, i.e. D τ ⊆ O τ (P) where O τ : S T → S T is the time truncation operator defined by

(O τ (s)) (t) = s(t), t ≤ τ 0 t > τ
The following result provides a key data-based test for invalidation of the controllers which do not achieve the control objectives

Yamé & Kinnaert (2003)
Given the desired behavior D and the observed data D τ generated by the (unknown) plant P , the controller C achieves the control objective up to time τ if

O τ (O -1 τ (D τ ) ∩ C) ⊆ O τ (D). (2.2.3) where O -1 τ (D τ ) = s ∈ S T
s ∈ S T s = vec[ r T , y T m , u T m ] ∈ C (2.2.4)
The signals r ∈ (R r ) T are what are called the virtual references in [START_REF] Safonov | The Unfalsified Control Concept and Learning[END_REF].

Design of the supervisor in projection-based FTC structure

Since the set C of precomputed control laws makes the desired behavior D implementable for each faulty plant mode, the supervisor in the figure ?? aims at managing the switching of the controllers in the feedback loop according to the actual running plant mode. The supervisor should decide when to change the control law, and which controller should be placed in the closed-loop. A simple "conceptual solution" to the controller selection problem would be to evaluate experimentally each candidate controller's performance by applying it to the plant at run-time. Unfortunately, this "conceptual solution" is not realizable. It is also a fact that without further assumptions on the model of the plant in the real-time, it is logically impossible to verify that a controller from the set of controllers will implement the desired behavior. Now, assume that the desired behavior, i.e., the control objective, is defined in terms of a performance functional (a cost function)

D = s ∈ S T |J(s) ≤ γ (2.2.5)
where J : (R s ) T → R + , s → J(s) is a functional with γ ∈ R + a threshold below which the performance is considered satisfactory. If the set of signals (2.2.4) is available for each controller C i , (i = 1, ..., N ) of the precomputed set C, then the test (2.2.3) can be performed by checking that the signal s i corresponding to the controller C i in (2.2.4) belongs or does not belong to the desired behavior D by computing its associated cost J( s i ). We can thus compute the N performance indices {J( s i )} N 1 associated to the N controllers of the bank without having actually put them in the loop. It is shown in ( [START_REF] Yamé | Performance-Based Switching for Fault Tolerant Control[END_REF]) that under an admissibility condition on the set of control laws U , the signals r i (the virtual references) which are compatible with the behavior of the controller C i and that of the plant can be reconstructed uniquely, thereby yielding the manifest variable s i (i = 1, ..., N ) which belongs to the closed-loop behavior of the plant and the controller C i . For the admissible set U of stably causally-left-invertible control laws, the reconstructed virtual references are generated from linear stable filters driven by the measured data, i.e.,

r i = F C i u m y m , i = 1, ..., N (2.2.6)
where the filter F C i depends only on controller C i . The internal structure of the supervisor is shown in figure 2.4. The computation of the N performance indices {J( s i )} N 1 , with the aid of the performance index generator, allows for a ranking of the controllers and the one with the lowest cost is selected to be placed in the loop. The reconfiguration mechanism is enabled whenever the actual running controller is invalidated by the real-time data generated by the plant. Such control invalidation is a testimony of the occurence of unexpected changes in the feedback loop which requires a remedial action.

The controller selector block produces a piecewise constant signal σ : T → {1, 2, ..., N } (the switching signal) based on the performance indices {J(ŝ i )} N i=1 . In ([YK03]), the following controller selector logic is proposed

σ(t) = σ(t n ) for t n ≤ t < t n+1
(2.2.7)

with the updating rule

σ(t n+1 ) = σ(t n ), if O τ (P ∩ C) ⊆ O τ (D); argmin{J(ŝ i )} i =σ(tn) , if O τ (P ∩ C) O τ (D).
(2.2.8)

where the minimum lenght of time in which a controller is active in the loop, the dwell time τ D , can be fixed by collecting the measured data on time intervals [t n , t n + τ D ] with t n the instants of possible switchings. The above switching logic confers to the dwell time two properties: it lets the stable dynamics of the closed-loop switched system have enough time to decay before a next possible switching occurs and it bounds the detection delay, i.e. the time elapsed from the occurence of a fault to the controller invalidation. A short detection delay requirement will need a short dwell time τ D for the selector logic which clearly conflicts with the stability of the closed-loop switched system. The dwell time should result from a trade-off between the requirements on stability and the detection delay depending on the faults scenarios and their severity. The time for reconfiguration, i.e., the time needed after a controller invalidation to the selection of the next controller is however quasi-instantaneous needing only the computation time.

FTC-based data-driven controller redesign

In this section, I present a new approach to the on-line redesign of FTC controllers based solely on the data generated in real time by the unknown plant, which might be either in nominal mode or in failure mode with unidentified faults. This approach was developped as part of the doctoral thesis of Tushar JAIN [START_REF] Jain | Contribution à la synthèse de commandes tolérantes aux défauts par l'approche comportementale[END_REF]. Contrary to the "projection-based" approach which mixed two paradigms (model-based for offline controller design and data-based for real-time controller selection), the one presented here is genuinely a data-driven and model-free approach in which no online knowledge of the process is available. A key advantage, as well as in the previous approach, is that no explicit FDD module is required, which ipso facto avoids integration issues between the diagnosis module and the reconfiguration mechanism. The exposition is based on work published in [YS12; JYS12d; JYS12f; JYS12a; JYS11b; JYS11c; JYS10]. Before formalizing the fault-tolerant control problem without a prior i knowledge of the plant, we first present a general result giving the conditions under which a controller can achieve a closed-loop system that meets a specified performance objective. We formalize the fault-tolerant control problem as that of controlling an unknown system, possibly in nominal mode or in failure modes, to satisfy the control objective O at run-time.

Implementation of feedback control loops

For the sake of exposition, we limit ourselves to the conventional unity feedback structure. We use the "partial interconnection" paradigm of the behavioral approach in which control is viewed as restricting the behavior of a plant through some specific variables called the control variables. We are given a plant Σ P = (T, W × C, P) to be controlled which is viewed as a dynamical system with two types of signals, the first set of signals is the to-be-controlled variables w : T → W and the second set is the set of control or interconnection variables c : T → C where C is a c-dimensional space, say C = R c . The to-be-controlled variables are variables such as disturbances, reference signals, the tobe-controlled outputs or objective signals that enters in the performance specification of the control problem while the control variables are those variables available and attached to the controller. The controller is another dynamical system Σ C = (T, C, C) that imposes restrictions on the behavior of the control variables. When the plant and the controller are interconnected, they share the variables c. The interconnected system (the controlled system), denoted by Σ P ∧ c Σ C , obeys the laws of both the plant and the controller. We use the symbol ∧ c as the interconnection operation through the variables c. A control problem for a plant Σ P amounts to finding a controller Σ C in a given set of admissible controllers such that the controlled system Σ P ∧ c Σ C has desirable properties. These desirable properties can be expressed in terms of another dynamical system Σ D whose behavior D is called the specification.

A representation of a conventional feedback control structure as an interconnection in the behavioral setting is shown in figure (2.5) where the manisfest variable w are taken to be w = vec r T , y T , r and y being respectively the reference signal and the output of the plant, and the control variables are c = vec e T , u T with e the control the trackng error and u the control signal. Denote by s, the vector signal composed of the manisfest and control variables, s = vec w T , c T which takes its values in the space S having the dimension s = w + c with w = r + y and c = e + u. The full pant behavior is given by

P = s = vec w T , c T ∈ S T R 0 d dt w = M d dt c (2.3.1) with R 0 (ξ) ∈ R •×w [ξ] and M (ξ) ∈ R •×c [ξ].
The kernel representation of P is obtained by eliminating the control variables c from (2.3.1) and it is is formally given by the set

P w = w ∈ (R w ) T ∃c ∈ (R c ) T such that (w, c) ∈ P
The controller behavior is given by its kernel representation

C = c ∈ S T C d dt c = 0 (2.3.2) with C (ξ) ∈ R •×w [ξ].
The restrictions imposed on P by C through the control variables yields the full controlled behavior which is given by

K full = s = vec w T , c T |(w, c) ∈ P and c ∈ C (2.3.3)
By eliminating the variables c from (2.3.3), the manifest "w-behavior" of the controlled system is

K(C) = {w ∈ W T |∃c ∈ C such that (w, c) ∈ P}. (2.3.4)
Now, it is straightforward to see that whatever the controller C might be, the controlled behavior K(C) is not so arbitrary, it is in fact wedged in between two sets,

P 0 ⊆ K (C) ⊆ P w
where P 0 , called the hidden behavior, is the set consisting of all pairs (w, 0) ∈ (R w+c ) T satisfying to R 0 d dt w = 0 and included in the full plant behavior P. The control problem aims at assigning desirable properties expressed as a specification to be fullfilled by the controlled system K(C), this specification being given as an "w-behavior"

D = w ∈ S T |D d dt w = 0 (2.3.5)
A control problem reduces therefore in designing a controller C such that K (C) = D. If such a controller exists, the specification D is said to be achievable and we say that the controller C achieves D. Thanks to the set-theoretic characterization of dynamical systems, the solvability of the control problem becomes particularly clear in the behavioral framework. This is stated in "Willems implementability theorem" [START_REF] Willems | Synthesis of Dissipative Systems using Quadratic Differential Forms: Part I[END_REF].

Theorem 2.1 (Willems' Theorem). Let P be a behavior of the plant, and let D be a desired behavior.

Then the following statements are equivalent:

1. D is achievable with respect to the plant.

2. These exists a controller C that implements D.

3. P 0 ⊆ D ⊆ P w .

It follows that the control problem for a given plant, as seen from theorem 2.1, can be reformulated as finding an achievable specification D and then deriving the controller C which achieves it.

A key result, provided by A. J. van der Schaft [START_REF] Van Der Schaft | Achievable behavior of general systems[END_REF] and stated below in a general setting, is that such a controller that implements an achievable behavior can be explicitly constructed as a specific behavior.

Theorem 2.2. Let P be the behavior of the plant, and let D be the implementable desired behavior. Then the controller defined by the set

C = {c ∈ (R c ) R |∃ w such that ( w, c) ∈ P and w ∈ D}, (2.3.6)
implements the desired behavior D.

The controller (2.3.6) in Theorem 2.2 is termed as the canonical controller. An explicit representation of the canonical controller turns out to be easily derivable from the first inclusion in statement of in theorem 2.1. From the inclusion P 0 ⊆ D, it follows that there exists a polynomial matrix L (ξ) of appropriate dimension such that (see [START_REF] Blomberg | Algebraic Theory for Multivariable Linear Systems[END_REF])

D(ξ) = L (ξ) R 0 (ξ) (2.3.7)
Then, thanks to equation (2.3.1) of the full behavior of the plant P, we get the identity

L d dt M d dt c = 0 (2.3.8)
which yields a kernel representation of the canonical controller, i.e., C (ξ) = L (ξ) M (ξ) .

On achieving on-line data-driven fault-tolerant control

The problem of real-time control of a faulty plant might now be expressed as the problem of achieving the control objectives at any time, i.e. independently of the occurrence of a fault. From the behavioral perspective, the control objective O, both in nominal and failure modes, can be described by a desired behavior D defined in terms of the manifest variables w. It follows from the previous section that if this objective O is achievable for the unknown running plant, then there exists a controller that achieves this objective, and moreover this controller can be computed explicitly (see (2.3.8) ) with the knowledge of the behavioral equations of that plant through its polynomial matrices R 0 (ξ) and M (ξ) in (2.3.1). Note that, if the desired behavior is not achievable D, then no controller exists that can guarantee the fault tolerance. In that case, the objective should be modified in order to be achievable. In the following, we would have liked to exploit the concept of canonical controller directly, but unfortunately, as no a priori information on the plant's model or its behavioral equations is available in real-time, we can not do with the synthesis equation ((2.3.8).

A key result, obtained in [YS12; YS13], which does not use any knowledge of the behavioral equations of the plant, but only the real time data from the running plant is given next. Let the desired behavior kernel (polynomial) matrix D(ξ) = Q D (ξ) -P D (ξ) be partitioned comformably with the components of the manisfest variable w, and assume that Q D (ξ) = q D (ξ)I r , P D (ξ) = p D (ξ)I y , with q D (ξ), p D (ξ) ∈ R[ξ]. Let the controller kernel matrix in (2.3.2) be given as

C(ξ) = Q C (ξ) -P C (ξ) . Define the filtered plant signals (ū, ȳ) ū = Q D (ξ)u, ȳ = (Q D (ξ) -P D (ξ))y (2.3.9)
Then the following result holds:

Yamé & Sauter (2012,2013)
If the desired behavior D is implementable for the running plant P, subject to unknown faults, and controlled by a controller C in a unity feedback loop, then the following statements are equivalent:

1. The overall system is a real-time fault-tolerant control system.

2. The trajectories ȳ ū belongs to the controller behavior C, that is, the following differential equation holds

Q C (ξ)ȳ + P C (ξ)ū = 0. (2.3.10)
The proof of this result is given in [YS13; YS12] for SISO systems, and in (see also [START_REF] Jain | Active Fault-Tolerant Control Systems : A Behavioral System Theoretic Perspective[END_REF]) for multivariable systems. It provides a purely data-driven technique for the synthesis of a canonical controller that implements the desired closed-loop behavior without the knowledge the datagenerating plant. The computation of controller C, i.e., the solution of equation (2.3.10) with respect to Q C (ξ) and P C (ξ), is nothing more than a system identification problem that can be solved by usual continuous system identification algorithms [START_REF]Identification of Continuous-time Models from Sampled Data[END_REF]. Subsequent improvements and extensions to two-degree of freedom controllers and to the full interconnection paradigm have been derived and reported in [START_REF] Jain | Active Fault-Tolerant Control Systems : A Behavioral System Theoretic Perspective[END_REF].

Notes and references

The first part of this chapter on projection-based fault-tolerant control is based on the publications [START_REF] Yamé | Performance-Based Switching for Fault Tolerant Control[END_REF], [START_REF] Yamé | A fault accommodation strategy based on closed-loop performance monitoring[END_REF], [START_REF] Yamé | Performance-based supervisory fault-tolerant control with application to the IFATIS two-tanks benchmark[END_REF]. To the best of our knowledge, this was the first time the behavioral paradigm was used to formulate a (data-driven) fault tolerant control problem allowing an FTC scheme based on control performance monitoring and real-time plant data without explicit fault detection and isolation algorithms. The control reconfiguration in this scheme was fast because no convergence process needs to take place in the supervisor algorithm as in FDD-based scheme using observers or parameter estimation techniques. The effectiveness of the technique was demonstrated on the IFATIS two-tanks process benchmark and reported in [START_REF] Yamé | Performance-based supervisory fault-tolerant control with application to the IFATIS two-tanks benchmark[END_REF] and more works were performed in subsequent papers [JYS13b; JYS13a; JYS12c; JYS12e; JYS11a; JYS11c].. These encouraging results led us to further consider the behavioral paradigm as a reference mathematical framework for data-driven fault-tolerant control without recourse to any mathematical models of the controlled faulty plant. The main motivation were to be able to directly synthesize real time-FTC controllers based solely on the trajectories of the faulty plant to get rid of the issues related to model-based FTC delineated in section (1.1). The thesis of T. JAIN that I supervised was essentially devoted to this research in which the interconnection paradigm and the concept of canonical controller of the behavioral framework were highlighted to achieve a direct synthesis of FTC controllers based only on trajectories. The main related results appeared in the publications [START_REF] Jain | Case study on behavioral approach to fault-tolerant control: Application to an electric circuit[END_REF], [START_REF] Yamé | A Note on the Canonical Controller in the Behavioral System-theoretic Approach[END_REF], [START_REF] Jain | Trajectory-based real-time control of an electrical circuit against unknown faults[END_REF].

Introduction

As seen in the previous chapters, the FTC problem for plants subjects to failures can be dealt by two main approaches, i.e., the projection-based approach and the online controller redesign approach. In the former approach, assuming that a preliminary system analysis/modeling step have been performed and that a set of faulty modes has been retained, control engineers can design a set of fixed controllers, one for each faulty mode dynamics. When all these controllers, which are usually linear controllers, are packed in a digital signal processor, some way of switching between them as the plant evolves through all its nominal and faulty operating modes is required. This issue of transfering control authority between different controllers is not a trivial task. It is now well known that when transfering control authority between two controllers, some kind of coherence must be enforced between the trajectories of the output of the new online controller and that of the controller swapped out of the feedback loop in order to avoid undesirable harmful transients. A fault-tolerant control system aims normally at maintaining as much as possible the performance of the overall closed-loop system at all times, i.e., even after a fault occurrence. In the behavioral paradigm, closed-loop performance is easily characterized as a desired behavior. Therefore, it clearly appears that, from this paradigm, the performance specification of an FTC system should cover explicitly three time intervals over which the overall system operates under a fault occurrence:

(TI-1) fault-free period (TI-2) transient period during the reconfiguration, and (TI-3) steady-state period after the reconfiguration.

Note that usually, the offline design of the bank of ccontrollers and the control reconfiguration mechanism actually only guarantee that the desired behavior be satisfied during the first and third time intervals mentioned above.

Conditioned switching through smooth interconnection of behaviors

As demonstrated earlier, the system has been guaranteed to satisfy the desired behavior during the first and third courses of operation. In this section, we present a real-time algorithm to guarantee the system performance during the transient period of the controller reconfiguration mechanism, without using any a priori knowledge of the plant model.

On switching transients and conditioned switching

When switching between two controllers, some kind of coherence must be enforced between the trajectories of the output of the new online controller and that of the controller swapped out of the feedback loop in order to avoid undesirable transients. The avoidance of these transients is known as the bumpless transfer problem. As noted in [YK04b; KDY09; ZT05], the concept of bumpless transfer was not precisely defined and it was some times misunderstood [START_REF] Peng | Anti-windup, bumpless, and conditioned transfer techniques for PID controllers[END_REF]. In the literature, it is usually understood either as a property of continuity (in the mathematical sense) of the plant input signal at switching times [AW96; Mor08], or as a plant input signal with minimized jumps at switching times. As reported in [PVH96; YK04b], this type of "input regularity" does not prevent the closed-loop system from exhibiting very poor transients with the new online switched controller. Furthermore, this kind of "input regularity" is not what is needed in most switched-mode systems, since switching between controllers is intended to achieve near-instantaneous or fast transitions between modes in order to maintain the stability and performance of the associated closed loops, such as in fault-tolerant control systems with multiple failures modes. It is noteworthy that in fault-tolerant control systems, controller switching is driven by sudden and drastic changes in the controlled plant dynamics and this requires responsive controls which may be contradictory with an almost continuous plant input signal. Hence jumps or abrupt changes in the input signal of the plant are actually inherent to FTC-based control reconfiguration. However, during the second timeinterval (TI-2) mentionned in section 3.1, the underlying control objective should not be violated as a result of these instantaneous events. It is therefore appropriate and meaningful to consider control switching as "bumpless" when transients are reduced to some extent or possibly eliminated from the closed-loop behavior despite the occurrence of jumps in the plant input signals and when the performance remains still good after mode switching. This notion of bumpless transfer, first considered by Hanus [Han79; HKH87] and coined "conditioned transfer" by Peng et al. [START_REF] Peng | Anti-windup, bumpless, and conditioned transfer techniques for PID controllers[END_REF], conveys the idea that switching should take place without perturbing the closed-loop system to depart from its desired performance. We put forward this concept and defined it in a simple and transparent way in the behavioral framework using the interconnection paradigm.

Consider the switched-mode FTC structure of figure 3.1, where the controller connected to the plant in the time-interval [0, t inter ) is controller C p and at time t inter , the second controller C f makes an interconnection with the unknown plant, where the subscript p and f denotes the past and the future with respect to the switching instant t inter . Generally, the above operation is performed so that the closed-loop with controller C f satisfies the desired behavior D. The overall system might be viewed as two individuals interconnected behaviors, defined as

K p = (r, y p , u p ) ∈ S T |(r, y p , u p ) ∈ P and (r, y p , u p ) ∈ C p } ∀t < t inter K f = {(r, y f , u f ) ∈ S T |(r, y f , u f ) ∈ P and (r, y f , u f ) ∈ C f } ∀t ≥ t inter (3.2.1)
where K p and K f are respectively the past behavior and future behaviors the interconnected system.

From the overall time axis perspective, the overall closed-loop system actually appears as a concatenation of the past behavior with the future behavior. The design methodologies for faulttolerant controllers are usually based on the synthesis of the closed-loop systems composed of the individual controllers such that the plant output trajectory y of the switched-mode system should be given by the ideal signal obtained by concatenating signals y p on (-∞, t inter ) with signal y f on [t inter , ∞), that is, y(t) = (y p tinter y f )(t) = y p (t) for t < t inter ) y f (t) for t ≥ t inter .

(3.2.2)

The above output trajectory would be achieved through the following ideal input signal to the plant

P u(t) = u id (t) = (u p tinter u f )(t) for t ∈ R (3.2.3)
where

w i = vec( u T i y T i
), i ∈ {p, f } , belongs to the behavior B of the unknown plant. A key feature of such ideal control u id is that its segment on [t inter , ∞), i.e., trajectory u f from t inter to ∞, is issued from the controller C f which has been constantly connected to the plant in closedloop [START_REF] Yamé | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF]. However, under the fault-tolerant control scheme, controller C f interconnects with the plant only after time t inter . Consequently its history, i.e. its state trajectory, on [0, t inter ) has no relation with the history of the plant in that time interval. It is well-known that the state variables paramerize the memory of a dynamical system [PW97, Chapter 4]. This lack of dynamical consistency between the state trajectories in the past and in the future of the closed-loop systems is the genuine cause of the switching transients [START_REF] Yamé | On bumps and reduction of switching transients in multicontroller systems[END_REF]. These transients lead the closed-looped system just after switching into an undesired behavior, that is, K f D. From the above, we formulate a precise and obvious definition underlying the concept of conditioned switching and express it through the behavioral interconnection paradigm as follows: Definition 3.1 (Real-time Smooth Interconnection). The interconnection of a controller C f and a running plant P at the switching instant t inter is said to be a smooth interconnection whenever K f ⊆ D.

Surprisingly, such a definition had never been proposed to capture the concept of bumpless transfer, perhaps because it is less obvious to formulate it in the classical framework. Observe that, in the above definition, no assumption is made on the time at which a control authority transfer takes place, as for example, assuming that a steady state has been reached before switching.

On achieving real-time smooth interconnection

The real-time interconnection is not "smooth" whenever controller switching leads to closed-loop signals which no longer belong to the desired behavior, such signals being usually termed as bumpy signals. The objective is therefore to implement C f in the running closed-loop system in a way such that the overall controlled behavior still satisfies the desired behavior despite the switching commands.

From now on, we will make reference to a time-dependent subset P t m of the behavior P. This subset is the set of signals experimentally measured on the time interval (-∞, t] where t is the current time. Let (u m , y m ) be the input/output (of the real plant) actually measured in an experimental setting, as e.g. in an actual closed-loop setting, then P t m is defined by

P t m = s = (r, u, y) ∈ S T s.t. u(τ ) -u m (τ ) y(τ ) -y m (τ ) = 0 for all τ ∈ (-∞, t] (3.2.4)
The next result [JY14; JY16] is of paramount importance and it is the basis for the derivation of the algorithm in achieving a smooth interconnection.

Jain & Yamé & (2014)

Let D be an implementable desired behavior and consider the controller switching scenario as depicted in Figure 3.1. Then, the interconnection between the controller C f and the unknown plant P is smooth if there exists a trajectory r such that s = (r, u m , y m ) ∈ P t m ∩ C f ⊆ D for all t < t inter . Indeed, let s f be the trajectory of the closed-loop switched-mode system after t inter , i.e., after switching C f in the loop. Then, under the stated condition, the achievability of a smooth interconnection of C f with P at time t inter is trivially equivalent to the fact that the concatenated signal s defined by

s = s tinter s f ⇐⇒ s(t) = s(t) t < t inter s f (t) t ≥ t inter (3.2.5)
should belong to the desired behavior, i.e.,

D s = s tinter s f (3.2.6)
which implies s f ∈ D, i.e., the interconnection is actually smooth. Remark 3.1. First, it is worth noticing that proposition 3.2.2 introduces a "virtual " behavior, i.e., a set of signals s = (r, u m , y m ) corresponding to a loop in which C f and P are fictively connected before time t inter (though in reality P is connected with C p for all t < t inter ). Second, the significance of the above result is that in order to achieve a smooth interconnection when C f will be effectively switched in the loop for all t ≥ t inter , this virtual loop should already satisfies to the desired behavior D.

The membership relationship (3.2.6) is a key requirement for achieving a smooth interconnection. Therefore, the following question arises: under what condition, the relationship (3.2.6) holds? From remark 3.1, the trajectory (3.2.5) might be viewed as that of the closed-loop system consisting of C f being constantly in the loop since ever. This trajectory is in fact splitted at t = t inter in two parts, that is, a past trajectory and a future trajectory. For a future trajectory in the behavior D to be a continuation of a past trajectory in D obviously requires some boundary conditions to be met at the splitting time. These boundary conditions can be expressed through state variables (see subsections 2.1.3 and 2.1.4). The manifest trajectories might therefore be explicitly parametrized with the state [START_REF] Polderman | Introduction to Mathematical Systems Theory: A Behavioral Approach[END_REF], so as to catch all the information about the past which are relevant to do a continuation of the trajectory in the future. Denote the state of the virtual closed-loop and future behavior of the system respectively by ˜ and f and parametrize the manifest trajectory of the virtual closed-loop and future manifest trajectory with respect to their states as s( ˜ ) and s f ( f ).

The following result which is a direct consequence of the property of the state answers the raised question above.

Jain & Yamé & (2014)

Let the manifest trajectories s( ˜ ) and s f ( f ) be elements of the behavior D, then their concatenation at time t = t inter also belongs to

D if ˜ (t - inter ) = f (t + inter ).
Partition the state of the controlled behavior P ∩ C explicitly as = vec (ζ P ) T (ζ C ) T where ζ P , ζ C are respectively the states of the plant and the controller. The state of the virtual closed-loop behavior P t m ∩ C f for t ∈ (-∞, t inter ) can be written as ˜ = vec( ( ζP ) T ( ζCf ) T ) and the state of the future behavior as f = vec( ). This means that the state of the to-be-switched controller C f should be initialized at the switching instant t = t inter with the value ζCf (t - inter ) of the state that controller C f would have achieved, had it been in the loop on the past time interval (-∞, t inter ).

(ζ P f ) T (ζ Cf f ) T )
The following algorithm summarizes the above results.

Algorithme 1 : Algorithm for smooth interconnection in real-time Data : P t m ⊆ P Result : Real-time smooth interconnection.

1: For t < t inter , observe the experimental plant behavior P t m and determine the online controller C f . 2: For t inter -ι ≤ t < t inter , compute the virtual manifest trajectory s, given in Proposition 3.2.2, and subsequently the corresponding virtual state ζ of the controller, with ι ≥ t settling where t settling is the settling time of the desired behavior D.

3: At t = t inter , initialize ζ Cf f as ζ Cf f (t + inter ) = ζCf (t - inter )
to achieve a smooth interconnection of P and C f . 4: Stop.

Remark 3.2. In step 2 of the above algorithm, the rationale behind considering a time interval [t inter -ι, t inter ) for computing the virtual signals comes from the fact that the computation of the reference trajectory r , and hence the construction of s should be done on all t < t inter to correctly calculate at the switching time t inter the value ζCf (t - inter ) of the state of the controller to be switched in the loop. However, from a practical standpoint related to the computation of a "very good" estimate for ζCf (t - inter ), it is quite sufficient to generate s over a finite-time window larger than the settling time of the desired behavior D.

An illustrative example

Consider an example where the desired behavior (i.e., the control objective) is to reconfigure the controller such that the y-trajectory follows the r-trajectory. The behaviors of the controllers are defined as where σ 2 + (.) is the operator resulting from iterating twice σ + (.) which is the shift-and-cut operator (see also [START_REF] Willems | The Behavioral Approach to Open and Interconnected Systems[END_REF]). Note that the above state representation does not make at the outset an inputoutput partition of the manifest variables. For example, if (r -y) is taken as the input and u as the output, the above equation can be written as a state equation Note that, as a by-product of representation (3.2.9), the virtual state of the controller C f and the reference trajectory r of the virtual closed-loop system are simultaneously generated. The closedloop signals are drawn in figure 3.2 that clearly shows the smooth interconnection between C f and P.

C j =    s ∈ S T n cj -n cj -d cj   r y u      = 0,

Conditioned switching through a common-dynamics realization of multiple controllers.

As seen in the previous section, the switching transients result from a mismatch between the joint "plant/controller" states just-before and just-after switching (see also [START_REF] Yamé | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF], [START_REF] Kinnaert | State Resetting for bumpless switching in supervisory control[END_REF]). Solutions for reducing switching transients would be to minimize or annihilate this mismatch at switching instants. This can be done in different ways, e.g. through controller state resetting at switching instants as done in the previous section (see also [START_REF] Kinnaert | State Resetting for bumpless switching in supervisory control[END_REF]) or through a virtual permanent connection of all idle controllers to the plant so as to ensure the consistency between the state of the plant and the states of all controllers at all times. An original approach to achieve the latter solution, as delineated first in [START_REF] Yamé | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF], is based on embedding the states of the set of switching controllers into a state-space of large dimension. This embedding allows the controllers' states to be "conditioned", in the sense of Hanus [START_REF] Hanus | Contribution à la Théorie des Régulateurs Conditionnés[END_REF], in a very natural way. We put forward this concept of conditioning the controllers' states through a generic structure, the so-called self-conditionable form, which allows seamlessly the realization of switching controllers that ensures smooth transition between different operating closed-loop modes driven by changing dynamics of the controlled plant. An important feature of the proposed technique is that no plant model is needed as may be the case for several bumpless transfer techniques based on model-based antiwindup circuitry attached to each switching controller [ZT05; ZT11].

On self-conditionable form of controllers

Consider a feedback control loop with a plant input nonlinearity N L which may be of actuator saturation type or input substitution type such as mode selection in switched systems. The general state space representation of a controller K in such feedback loop is given as [START_REF]Réalisations canoniques des fonctions de transfert (Transparents)[END_REF] where the dynamics of the system is driven by its output is said to be in self-conditionable form. This terminology comes from Hanus conditioning technique1 [START_REF] Hanus | Conditioning Technique, a General Antiwindup and Bumpless Transfer Method[END_REF] where he shows that with such a controller structure, there is no further need to build an auxiliary circuitry to be attached to the controller in order to realize the conditioning of the states of that controller. The conditioned controller is indeed directly obtained by replacing the controller output u by the plant input u * which is given by the output of the nonlinearity block N L, i.e.,

ς = F (ς, r, y) (3.3.1) u = G (ς, r, y) (3.3.2)
ς = F u (ς, y, u * ) u = G (ς, r, y) (3.3.5) u * = N L(u) (3.3.6)
This last controller structure (3.3.5) is said to be self-conditioned and the corresponding implementation is shown in figure 3.3.1 with the controller denoted by K sc where the subscript "sc" stands for "self-conditionable" form. The self-conditioned form of controllers have ipso facto natural antiwindup and bumpless transfer properties as pointed out in [START_REF] Hanus | Conditioning Technique, a General Antiwindup and Bumpless Transfer Method[END_REF]. This was also shown in [START_REF] Yamé | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF] that this is simply due to the fact that such scheme ensures that no mismatch of the joint "plant/controller" states occurs during input substitution or when the controller output u saturates. At this point, it is instructive to take a look-back on system representation in the behavioral paradigm. Denote the manifest variable of the plant by w * = vec (u * ) T , y T , then for linear controllers, equation (3.3.5) reads for for a zero-reference input (r = 0),

dς dt = F ς + Gw * (3.3.7)
for some matrices F and G comformable with the dimension of ς and w * . This equation shows that the self-conditioned form of the controller driving the plant can also be seen as a representation of the external behavior of the plant when it is in the feedback loop (see the dual pencil representation (2.1.7). In that case, the state ς of the self-conditioned controller clearly "supports" the w * trajectories of the plant and thus it may be considered also as a latent variable of plant itself. This interpretation sheds a new light to the self-conditioned form of controllers and could have some implications for a plant with an unknown model but controlled by a controller with known dynamical equations.

Nonminimal self-conditionable realization of multivariable controllers

The self-conditionable form as constructed through the "left inverse" operator in the preceding section may not be feasible if the "left inverse" operator does not exist, and even if an "inverse" operator exists, it could yield an unstable dynamics (3.3.3) albeit the original dynamics (3.3.1) is stable. In fault-tolerant control, the bank of controllers used to cover the operating modes of a plant subject to failures may consist of a number of heterogeneous controller structures, each control structure being adapted to one or different modes of the changing dynamics of the plant. In this case, a strategy for bumpless transfer among controllers would be attaching to each controller an antiwindup circuitry or a bumpless compensator [START_REF] Green | Linear Robust Control[END_REF], [START_REF] Zaccarian | The L2 (l2) bumpless transfer problem for linear plants: Its definition and solution[END_REF], however such technique would become quickly cumbersome as the number of controllers grows up. In what follows, I present an original construction of a self-conditionable form of a controller inspired from a model reference identifier suggested by G. Lüders and K. Narendra in [START_REF] Luders | A new canonical form for an adaptive observer[END_REF]. This self-conditionable form has a stable realization and it is easily extended to a set of linear controllers with heterogeneous structures. Consider N controllers given by their transfer matrices:

u i (s) = K i (s)e(s) = Bi (s) ãi (s) e(s) i = 1, • • • , N (3.3.8)
where e(s) stands for the control error r(s) -y(s), K i (s) is a (m × p) transfer matrix, Bi (s) is a (m × p) polynomial matrix and ãi (s) is a monic polynomial constructed as the smallest common multiple of all the denominators of the elements of K i (s). Set n = max i=1,N deg(ã i (s)) where deg(ã i (s)) denotes the degree of polynomial ãi (s). Then all transfer matrices K i (s) can be written in the form

K i (s) = B i (s) a i (s)
where the denominator polynomials a i (s), i = 1, • • • , N are monic polynomials of degree n, namely

a i (s) = s n + a i,1 s n-1 + • • • + a i,n-1 s + a i,n
and the numerator polynomial matrix can be written

B i (s) = B i,0 s n + B i,1 s n-1 + • • • + B i,n .
All additional roots of a i (s) with respect to the roots of ãi (s) should be placed in the open left-half plane. The following result provides a self-conditionable structure of any controller K i in the set of controllers (3.3.8).

Yamé & Kinnaert (2004)

A self-conditionable form for the ith controller in (3.3.8) can be written as

ζ1 (t) ζ2 (t) = A p 0 0 A m ζ 1 (t) ζ 2 (t) (3.3.9) + B p 0 -B m B i,0 B m e(t) u i (t) u i (t) = E i ζ 1 (t) + D i ζ 2 (t) + B i,0 e(t) (3.3.10) i = 1, • • • , N
In equation (3.3.9), matrices A m and B m are given by

A m =        0 I m 0 • • • • • • 0 0 0 I m 0 • • • 0 . . . . . . . . . . . . . . . 0 0 • • • • • • 0 I m -λ n I m -λ n-1 I m • • • • • • • • • -λ 1 I m        B m =      0 . . . 0 I m      with λ(s) = s n + λ 1 s n-1 + • • • + λ n ,
E i = B i,n B i,n-1 • • • B i,1 D i = d i,n I m d i,n-1 I m • • • d i,1 I m
where B i,j is defined by the equality

B i (s) -B i,0 a i (s) = B i,1 s n-1 + • • • + B i,n and d i,j = λ j -a i,j , j = 1, • • • , n.

Self-Conditioned Implementation of a Bank of Switching Controllers

Consider a switched-mode system composed of N controllers K i , (i = 1, ..., N ), designed for the different dynamical modes of the plant and denotes by σ the switching signal, i.e., a signal which maps the time axis R into the set of indices of the controllers, σ : R → I = {1, 2, ..., N }. The switchedmode control system operates as follows: at time t, the actual input u(t) to the plant is given by one of the outputs of the controllers, that is, u(t) = u i (t) if the switching signal σ(t) = i ∈ I, 1, 2, ..., N . Now, thanks to the conditionable form (3.3.9)-(3.3.10) of each individual controller K i , the set of all controllers {K i } N i=1 can be implemented as a single controller of which the input is e(t) and the output is the input signal u(t) to the plant. The state-space realization of this "single" controller is given next.

Yamé & Kinnaert (2004)-Common-dynamics self-conditioned switched-mode controllers

     • ζ (t) = Aζ (t) + B u u (t) + B e e(t) u i (t) = C i ζ (t) + B i,0 e (t) for i = 1, 2, ...N u (t) = u σ(t) (t) = u i (t) if σ (t) = i with i ∈ {1, 2, ...N } (3.3.11)
The state of all controllers packed in this single one is ζ = vec ζ T 1 , ζ T 2 and the matrices in equations (3.3.11) are the following block matrices

A = A p 0 0 A m , B u = 0 B m B e = B p -B m B σ(t),0 , C i = E i D i (3.3.12)
The state vector ζ is the common state to all controllers, whether they are actually connected in the loop or not. Let x P denotes the state of the unknown plant, then as a consequence of the controllers' common state, the joint "plant/controller" state, say χ = vec x T P , ζ T , is independent of the actual active controller K i in the loop, and in fact all idle controllers appears as being virtually connected to the plant. The following result is an immediate consequence of the common state realization of the set of switching controllers.

Yamé & Kinnaert (2004Kinnaert ( , 2007) ) Consider a switched-mode system composed of a set K of N controllers K i ,(i = 1, ..., N ), designed for a plant P. If each dynamical mode of the plant P is stabilizable by at least one controller K i ∈ K , then the realization (3.3.11) achieves bumpless switching.

Contrary to the widespread idea that a smooth transition is achieved by an "almost continuity" of the control signals at switching instants, it is worth observing here from (3.3.11), that the plant input can undergo jumps without causing switching transients. Indeed, the jump at the switching instant t s of the plant input given by ∆u(t s ) = u σ(t + s ) -u σ(t - s ) , with t - s < t s < t + s , might be different from zero. In fault-tolerant control systems, such jumps at the plant input might be necessary for a near-instantaneous control of the new faulty mode of the plant. On the other hand, it appears from the virtual connection of all the controllers to the plant that the joint "plant/controller" state χ(t) is in fact a continuous function at the switching instants. Actually, the state χ is a continuous function. An alternative version of the self-conditioned form (3.3.11) has also been developed for controllers with two degrees-of-freedom and this has been reported in [YQ10; YQK10].

Notes and references

This chapter has presented the problem of control authority transfer between several controllers in the behavioral paradigm and in a more classical paradigm. In the classical paradigm, we characterized the origin of switching transients in a very clear way and, based on this characterization, we suggested a new approach to reduce or annihilate these transients. The main idea underlying the proposed approach was inspired from the controller conditioned concept initially introduced by R. Hanus [START_REF] Hanus | Contribution à la Théorie des Régulateurs Conditionnés[END_REF]. We put forward this concept through a particular, albeit generic, controller parameterization which allows seamlessly a self-conditioning of several switching heteregeneous controllers. The resulting control structures achieve bumpless switching of these controllers, and the results were reported in several successive papers in the single-input/single-output and multivariable caases [START_REF] Yamé | Parameterization of Linear Controllers for Bumpless Switching in Multi-Controller Schemes[END_REF], [START_REF] Yamé | On bumps and reduction of switching transients in multicontroller systems[END_REF], [START_REF] Yamé | A Self-Conditioned Implementation of Switching Controllers for Smooth Transition in Multimode Systems[END_REF], [START_REF] Yamé | Conditioned switching between two-degree-of-freedom controllers for plants with changing dynamics[END_REF], [START_REF] Yamé | Parametrization of MIMO multi-controller schemes for bumpless switching[END_REF]. This work on bumpless switching has been refined from a behavioral point of view, a point of view that offers a very clear formulation of the notion of smooth switching between controllers. The key point was a definition that captures in a very simple way the concept of smooth interconnection between a controller and a plant by a membership characterization of closed-loop system trajectories to the control specification which is nothing but the desired behavior. The germ of this idea was in publication [START_REF] Kinnaert | State Resetting for bumpless switching in supervisory control[END_REF] and has been developed further in the papers [START_REF] Jain | On real-time smooth interconnection of online synthesized controllers in the behavioral framework[END_REF][YJ14] [START_REF] Jain | Synthesis and application of a real-time model-free behavioral controller with bumpless switching mechanism[END_REF]. The solutions to the smooth transfer of control authority problem presented in this chapter are both powerful and simple. They do not require any additional compensation circuit or optimization problem to be solved as is usually the case in many bumpless transfer schemes, nor they require a model of the controlled plant. Our work has been mainly driven by the need to address this problem for fault tolerant control systems, and the proposed above solutions have been for that purpose. Bumpless switching is still an active research area and many works are now addressing this issue in the specific context of fault-tolerant control systems. A nice recent review on dealing with some aspects of bumpless switching is provided in [START_REF] Atoui | Toward switching/interpolating LPV control: A review[END_REF] within the general framework of LPV control which might include somme classes of fault-tolerant control systems. In this setting, fault-tolerant control is achieved through online controller reconfiguration based on receding-horizon optimal control, generally referred to as model-based predictive control. The online synthesis of the controller is initiated from the fault information provided by a fault detection and diagnosis module which is built upon a model of the plant. It is then possible to reformulate a control problem which takes into account the new constraints induced by the fault. These new constraints constitute a model of the plant from which the control optimization problem can be explicitly formulated and its solution naturally provides a controller which is matched to the new actual constraints and satisfies the objective thereby achieving fault-tolerance. Our research in this area has been mainly motivated by the needs of the european project Energy-in-Time and we have been particularly interested in developing fault-tolerant controls meeting energy efficiency requirements in building operation. This requirements led us to the development of fault tolerant control based on economic model-predictive control where the cost to be optimized is no longer the usual trajectory-tracking cost aiming at minimizing the tracking error between the reference signals and the output signals.

Predictive control : a generic approach to model-based FTC

Consider a plant P and a set F = {f α (t)} α∈I of possible faults which may occur during the plant operation where f α (t) is a particular fault indexed by some α ∈ I , with I being an index set. Under the occurrence of fault f α (t), the plant P may be described or modeled by an associated dynamical constraint P (α) . Assume that the fault-free situation, f 0 (t) = 0 corresponding to index α = 0, is given by the nominal plant constraint P = P (0) . Then, the fault-tolerant control problem, as recalled from chapter 1, is characterized by a triplet (O (α) , P (α) , U (α) ) which needs to be solved autonomously at run-time of the feedback system. It turns out that model predictive control fits well to such a problem formulation defined by a control objective, a dynamic constraint described by an available mathematical model and the specification of admissible control laws. Let us frame the model predictive control problem within its classical setting using a state-space model of a linear dynamical plant having x(k) ∈ R n , u(k) ∈ R m and y(k) ∈ R p as state, input and output vectors, respectively, at time k. The standard (nominal) linear MPC problem for regulating the plant to the origin is formulated as the quadratic programming problem [RMD17]

O (0) : min U k J(U k , x(k)) = x T ( k + N | k)P x( k + N | k) + N -1 j=0 x T ( k + j| k)Qx( k + j| k) + u T (k + j)Ru(k + j) (4.1.1)
subject to:

P (0) : x( k + j + 1| k) = Ax( k + j| k) + Bu( k + j| k), j ≥ 0 x( k| k) = x(k) (4.1.2) x min ≤ x( k + j| k) ≤ x max j = 1, ..., N - 1
U (0) : u min ≤ u(k + j) ≤ u max j = 0, 1, ..., N -1 (4.1.3)
where

U k = {u(k + j)} N -1 j=0 , Q = Q T 0, R = R T 0, P 0
, N is the prediction horizon, and the subscripts min and max denote lower and upper bounds. The admissible control laws U (0) is the set of controls which satisfy the constraints 4.1.3 on the control signal. The problem (O (0) , P (0) , U (0) ) is solved repetitively, in a receding horizon fashion, at each time k for the current measurement x(k) and the vector of predicted state variables, x( k + 1| k), ...x( k + j| k) at time k + 1, ..., k + j, respectively, and the corresponding optimal control actions U * k = {u * (k), ..., u * (k + N -1} is obtained. The input that is applied to the plant is the first control action u(k) = u * (k). Thanks to the re-computation of the control signal at each step by solving an optimization problem, the nominal control problem can be extended to the fault-tolerant control problem by simply changing the problem formulation as the system evolves and goes through different operating modes P (α) , α ∈ I . Consequently, model-based predictive control offers the possibility of a generic approach to achieve fault tolerance by online reconfiguring the controller in response to a serious failure [START_REF] Marian | Predictive control: with constraints[END_REF]. Explicitly, this will be realized through a combination of predictive control based on detection and diagnosis of the occurring fault and on a high-fidelity model that represents the current mode of the operating plant. Specifically, if a fault affects the capabilities of the controlled system, then the dynamic and algebraic constraints and/or the control objectives as well as the admissible controls can be modified, and the online controller reconfiguration reduces simply to the search for a solution to the problem (O (α) , P (α) , U (α) ) with α the index of the identified fault in the set F .

Fault-tolerant economic model predictive control (FTeMPC)

I initiated this research topic through our participation in the European Energy-in-Time project for which a major task focused on solving building energy management problems during non-nominal operations resulting from heating, ventilation and air conditioning (HVAC) equipment failures. These non-nominal operations have an extremely negative impact on the energy performance of buildings, on the financial costs related to energy supply and on the thermal comfort of the occupants. The issue here is how to solve the trade-offs, under such HVAC failures, between the cost of energy supply and the control performance specified by the thermal comfort of building occupants. The classical approaches to solving such type of dilemma are generally made by optimal control synthesis techniques where the cost associated with the operation of the systems is optimized only according to static criteria which choose among the feasible operating points those which minimize this cost. The resulting overall scheme is a two-level hierarchical optimization, i.e., a static optimization at the top level for the computation of the operating points and then a dynamic optimization for the low-level control loop and such hierarchical structure invokes a time-scale separation between the two layers. The satic optimization is executed at a very slow rate of the order of several hours, while the feedback control layer computes input signals to the plant at a faster rate of the order of seconds or minutes. For the low-level layer, model predictive control has proven to be very useful because of its ability to control multivariable systems while taking into account the constraints and performance criteria of control. Though this two-level optimization solution is suitable for systems in nominal operation, it seems much less suitable for the problem of building energy management when failure modes occur. Indeed, the occurrence of these modes involves rapid or abrupt changes of dynamics with relatively short transient periods and the time scales required for static and dynamic optimization in such two-level hierarchical schemes should be almost of the same order. This obviously no longer allows these two optimizations to be performed sequentially one after the other.

Under the above considerations, the question which arises is how to directly optimize the economic objective (energy cost) within the dynamical low-level feedback layer? That is the central issue tackled in the thesis of T. Darure [Dar17] that I co-supervized with F. Hamelin. The main idea of this research has been to formulate the control optimization problem for building energy performance, in both nominal and non-nominal modes, as a single-level optimization problem with an economic objective under the constraint of a dynamic model and under occupant comfort constraints. The possibility of a generic approach to fault tolerance, as delineated in section 4.1, by reconfiguring the controller in response to a failure leads naturally to the receding-horizon optimal control techniques where the energy cost is explicitly formulated as a control objective O, in contrast to the quadratic functionals dealing with regulation or trajectory tracking in standard MPC. I have been interested in formulations of the eMPC problem in the context of two different applications, the first being applications to building control and the second to wind turbine control problems. These application-oriented formulations of eMPC have been presented in the literature where an appropriate cost function and constraints have been formulated after an in-depth knowledge of the application has been gained. Some additional elements added to the eMPC formulation are tailored for the particular application to allow for desirable performance properties. These properties are typically demonstrated and evaluated through simulations [DYH16b; DYH16a], [JY19; JY18a].

On FTeMPC of building HVAC systems

Our first eMPC-based FTC application addresses multi-zone building HVAC systems as found in non-residential buildings such as office buildings, hotels, commercial buildings, etc. A schematic of such multizone buildings with n zones (or rooms) is shown in figure 4.1 and the building is equipped with an HVAC system composed of a central air-handling unit (AHU) and variable-air-volume (VAV) boxes each appended to a zone. The AHU consists of an air mixer, a heating coil and a fan that blows heated air into the ducts. The system model is mandatory for MPC and it is derived as a linearized thermal dynamical system which describes the overall multizone building with its HVAC units. Let T i and ṁi denote respectively the temperature of room i (i ∈ n = {1, ..., n}, and the air mass flow rate delivered in the room by its VAV box, and denotes by T s the supply airflow temperature at the outlet of the AHU. Introduce the following vectors,

x = vec T 1 T 2 • • • T n , u = vec[ u 0 , u ], where u 0 = T s and u = vec[ ṁ1 , ṁ2 , • • • , ṁn ]
. Then, the discrete-time linearized thermal dynamical model of the multizone building reads

M P ≡ x(k + 1) = Ax(k) + B u u(k) + Ed(k) y(k) = x(k) (4.2.1)
where d(k) is the disturbance which accounts for the outside temperature and internal heat gains due to occupancy and electronic devices in the zones. The matrices A, B, E are compatible matrices with the associated signals. It is worth noticing that, for energy efficiency purposes in VAV-based HVAC systems, the supply airflow temperature u 0 = T s is normally kept at a constant value u

(0) 0 = T (0)
s in the nominal operating mode of the building. However, in order to have a fault-tolerance orientedmodel, this variable has been added here in the control vector to create a redundancy in actuation when a VAV damper lock-in-place failure occurs.

Our work has focused on VAV damper lock-in-place failures, which are serious failures that have some of the most negative impacts on building energy consumption and occupant comfort.

The problem is formulated as in (4.1.1) save that the cost function is expressed explicitly as an economic objective function that reflects the actual energy cost of operating the facility. The energy cost is due to the energy consumption of the AHU which is contributed by the heating energy demand at the heating coil and the electrical energy required by the fan to deliver the mass airflow to the variable-air-volume (VAV) boxes. If π hc (t) is the heat transfer rate (or power) of the heating coil and π f (t) the power of the fan, then the energy consumption per unit time of the AHU is π AHU (t) = π hc (t) + π f (t) and the total energy consumption on the time interval [t 0 , t N ] is given by W = ´tN t 0 π AHU (τ )dτ . This can be written equivalently in a discrete-time form as

W = N -1 k=0 ´tk+1 t k π AHU (τ )dτ if the time interval is viewed as [t 0 , t N ]=∪ N -1 k=0 [t k , t k+1 ), i.e.
, the union of N consecutive (disjoint) time intervals. The total energy cost, within a constant factor, is then given by

V N (u, x, d) = N -1 k=0 k (u (k) , x (k) , d (k)) (4.2.2)
where the stage cost

k (u (k) , x (k) , d(k)) = ´tk+1 t k
π AHU (τ )dτ is a nonlinear function of its arguments. Under the nominal and faulty modes, the system states and control variables are subject to constraints imposed by the thermal requirements and the VAVs damper limits Nominal mode

C (0) 1 x min i ≤ x i ≤ x max i , ∀i ∈ n C (0) 2 u min i ≤ u i ≤ u max i , ∀i ∈ n C (0) 3 u 0 = u (0) 0 (4.2.3)
for the nominal mode, and Faulty mode: α:th VAV damper stuck

C (α) 1 x min i ≤ x i ≤ x max i , ∀i ∈ n C (α) 2
1. for the α:thVAV

: u α = u stuck α 2. for all j ∈ n \ {α} : u min j ≤ u j ≤ u max j C (α) 3 u (0) 0 -εT η ≤ u 0 ≤ u (0) 0 + εT η (4.2.4)
for the faulty modes.

The constraints C 1 relates to the comfort bounds, C 2 to the VAV dampers limits and constraints C 3 to the supply air-temperature. A key point in the constraints formulation is that the supply air-temperature u 0 is viewed as a constant "parameter" in nominal mode, but in case of failure it becomes a decision variable used as a redundant control signal that can vary in a range centered on its nominal value, u

(0) 0 -εT η ≤ u 0 (t) ≤ u (0)
0 + εT η , where T η is a constant temperature and ε is a scalar variable which is the magnitude of the relaxation of the constraint. The constraints C (α) 2 with regards to the α:th VAV under the faulty mode is automatically set to the stuck value u stuck α estimated by the FDD algorithm. The constraints P (α) describing the plant and the related admissible control U (α) can be written as the following conjonction of constraints:

P (α) = M P ∧ C (α) 1 and U (α) = C (α) 2 ∧ C (α) 3
where α ∈ {0, 1, ..., n} with α = 0 corresponding to the nominal (fault-free) mode and the symbol ∧ stands for the conjonction. The FTeMPC is performed through solving repetitively, at each time k, the problem (O, P (α) , U (α) ) where α results from the identification of the stucked VAV damper by the FDD algorithm whose structure and synthesis process will not be detailed here. More explicitly, the fault-tolerant control is the solution of the constrained optimal control problem at time k formulated as

Darure, Yamé & Hamelin (2016) O : min U k ,ε V N (U k , x(k)) + ρε 2
subject to:

P (α) : x( k + j + 1| k) = Ax( k + j| k) + B u u (k + j) + E d ( k + j| k) , j ≥ 0 x( k| k) = x(k) (4.2.5) x min ≤ x ( k + j| k) ≤ x max j = 1, ..., N -1 u(k + j) ∈ U (α) : j = 0, 1, ..., N -1 (4.2.6)
where U k = {u (k) , ...u (k + N -1)} is the set of predicted control inputs at time k and x ( k + j| k), d ( k + j| k) the predicted state and predicted disturbance made at time k. The above FTC formulation leads to optimal performance with the following characteristics.

• The scalar ε, i.e. the magnitude of the relaxation of the supply air-temperature constraint

C (α)
3 , is treated as an optimization decision variable • When all VAVs are healthy, with no faults detected by the FDD algorithm, the weight ρ is set to zero in (4.2.5).

• Whenever a fault is detected and isolated, the optimization problem is adapted to the faulty mode by modifying accordingly the constraints in (4.2.5) with ρ > 0.

Problem (4.2.5) is solved repetitively at each time k for the current measurement x(k) and for the corresponding mode as indicated by the FDD algorithm along with the predicted state variables {x ( k + j| k)} N j=1 and the predicted disturbances

{ d ( k + j| k)} N j=1 .
Then, the corresponding optimal sequence U k = {u (k) , ..., u (k + N -1)} minimizing the economic objective criteria O while meeting all constraints is obtained, and its first element u (k) is applied to the system. This leads seamlessly to a fault-adaptive control algorithm capable of dealing with stuck damper failures.

When implemented on a simulated building HVAC-VAV system, it was observed that the supply airflow rates exhibit some kind of oscillatory behaviors which are fluctuations in the set-point trajectories computed by the eMPC controller. See figure 4.2(a) below corresponding to a building with 4 identical zones, where it is seen that the air mass flow at the VAV of zone 1 is very noisy, although the temperature comfort bounds given by constraint C

(u) = N -1 k=0 u(k) 2 R where u(k) = u(k) -u(k -1) and u(k) 2 R = ( u(k)) T R u(k)
for some positive definite matrix R. Such control smoothing could have been used here, but a close examination of the control signals shows that they are rapidly varying signals with superimposed fluctuations behaving like noisy signals. In order to avoid removing the genuine rapid variations in the original signal if a quadratic smoothing is used, we suggest a reconstruction of the desired set-point signals that removes much of the fluctuations, while still preserving the sharp edge or rapid variations in the set-point signals. This reconstruction has been achieved through 1 -regularization, or more precisely, through a total variation functional added to the original eMPC cost. This functional term is just

ψ (U k ) = DU k 1 = N -1 k=0 u(k + 1) -u(k) 1 where x 1 = |x 1 | + |x 2 | + ... + |x n | is the 1 -norm of vector x = [x 1 , x 2 , ..., x n ] T ,
and D is the bidiagonal block matrix of dimension compatible with the sequence U k = {u(k), u(k + 1), ..., u(k + N -1} viewed as a vector, with I the identity matrix with size compatible with the dimension of vector u. Note that, the vector u here does not contain the supply airflow temperature u 0 = T s , and it only relates to the mass airflows to the n VAV terminal boxes. By integrating this regularization term into the economic cost, we can immediately see the benefits of such strategy on the control system. Figure 4.2 is a simulation of a zone temperature and its VAV control in nominal situation witnin a four zones building. It can be seen from figure 4.2(a) that the eMPC policy allows to regulate the temperature of the zone in an optimal way, setting the temperature at its minimum comfort value to consume the least energy. On the other hand, we can see that the control signal, i.e., the set point trajectory sent to the local VAV controller is very fluctuating around some a time-varying mean. These fluctuations can lead to premature aging of the VAV actuators and considerably reduce their life span. The modifiaction of the economic cost by an 1 regularization term can be seen to smooth very efficiently the control signals without violating the thermal comfort bounds (see figure 4.2(b)).

D =        -I I 0 • • • 0 0 0 0 -I I • • • 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • -I I 0 0 0 0 • • • 0 -I I        (

On FTeMPC of wind turbines

The optimization performed to achieve FTeMPC for the building HVAC system in the previous section, though using a linear model of the building HVAC system, is a nonlinear optimization due to the energy cost function which is strongly nonlinear in terms of the decision variables. Such an optimization does not generally guarantee a global optimum. However, due to the slow thermal dynamical behavior of buildings and the way VAVs are locally controlled, the eMPC based faulttolerant control problem remains quite tractable. A serious drawback of using MPC is that the complexity of the control optimization problem can increase significantly depending on the size and structure of the system model and particularly for systems with fast dynamics. Indeed, dealing with fault tolerant control of wind turbines subject to pitch subsystem and converter subsystem failures, we have been confronted with computational challenges due to the non-convexity of the control optimization problem which renders the problem very difficult to solve. One approach to solve this problem, given the success of real-time convex optimization [START_REF] Boyd | Convex optimization[END_REF], has been to move towards a convexification of the eMPC-based fault-tolerant control problem for wind turbines (WT).

We summarize below the main steps in convexifying the economic optimization problem at hand and we refer, for the details, to the papers [JY18b; JY19]. The main objective of the control here is to achieve power maximization and fatigue reduction of the wind turbine under failures of the WT subsystems previously mentioned.

The wind turbine composed of the aerodynamic rotor subsystem, flexible drive-train shaft subsystem, generator, and flexible tower subsystem has a nonlinear model given by a state equation of the form

ẋ(t) = ψ(x, u, d) (4.2.7)
where the state x(t), control u(t), and disturbance d(t), are

x(t) = vec T g (t) ω r (t) ω g (t) θ dt (t) x t (t) ẋt (t) , u(t) = vec T g,ref (t) β(t) , d(t)
= v w (t), (4.2.8)
and ψ(.) is a non-linear vector-valued function of its arguments. The components of the state x are respectively, T g (t): the WT generator torque, ω r (t): the rotor angular velocity, ω g (t): the generator angular velocity, θ dt (t): the torsion angle of the drive-train, x t (t): the displacement of the nacelle from its equilibrium position and ẋt (t): the tower-top speed. The components of the control input u(t) are respectively, T g,ref (t): the reference command for the generator torque, and β(t): the blade pitch angle, while the disturbance d(t) is the wind speed v w (t). The electrical power, P g (t), extracted from the generator is expressed, in terms of T g (t) and the generator efficiency η g , as P g (t) = η g T g (t)ω g (t) and it is constrained by the inequality P min ≤ P g (t) ≤ P rated where P rated is the rated generator power. Other system variables are box-constrained and given by

    ω r,min ω g,max β min P min     ≤     ω r (t) ω g (t) β(t) P g (t)     ≤     ω r,max ω g,max β P max     (4.2.9)
The economic model predictive control problem for power maximization and fatigue reduction of the WT is formalized as

min u J = ˆt0 +T f t 0 (x(τ ), u(τ ), d(τ ))dτ (4.2.10a) subject to (4.2.7) (4.2.10b) x(t 0 ) = x 0 (4.2.10c) (4.2.9) ∀τ ∈ [t 0 , t 0 + T f ] (4.2.10d)
min ǔ J = ˆt0 +T f t 0 (x(τ ), ǔ(τ ), d(τ ))dτ (4.2.16a) s.t. (4.2.15) (4.2.16b) x(t 0 ) = x0 (4.2.16c) G ap (x(τ ), ǔ(τ ), d(τ )
) ≤ 0 ∀τ ∈ [t 0 , t 0 + T f ] (4.2.16d)
The conceptual picture 4.3 summarizes the main steps in the proposed convexification of the FTeMPC problem.

Towards optimal system decomposition for distributed MPC

My interest in distributed MPC dates back to the thesis work of Kamel Menighed [START_REF] Menighed | Commandes coopératives embarquées et tolérantes aux défauts[END_REF] on fault-tolerant embedded cooperative controls. Results have been obtained in the derivation of (non)cooperative distributed fault-tolerant ccontrollers with separable cost of the regulation/tracking type. The results obtained were based on ad hoc decompositions of large-scale systems by rewriting their models in a decentralized form with interactions between the decentralized submodels. The distributed MPC was in fact non-cooperative in the sense that, although sub-systems interactions were taken into account for the computation of each local control, the local controllers only optimized a local cost associated with the corresponding sub-system, possibly by exchanging information with other neighbor subsystems via a digital network [START_REF] Menighed | Fault tolerant cooperative control: A distributed model predictive control approach[END_REF]. Clearly, with different competing local objectives, no convergence were guaranteed in such non-cooperative distibted MPC. Later on, I have been interested in extending this work, which was of a generic nature, to the control of largescale buildings aiming at building energy efficiency operation. This research has been conducted within the context of the doctoral thesis of T. Darure which was supported by the european projet Energy-In-Time.

Let us recall the linearized state-space model of an overall large-scale building,

ż = Az + Bu + Ed (4.3.1)
where the state z ∈ R nz and the control signal u ∈ R nu , and d is the disturbance signal. The centralized model predictive control problem for this system can be formulated in a general nonlinear mathematical progmaming form as:

P c : min x f (x) (4.3.2) s.t. h(x) = 0 x min ≤ x ≤ x max
where, on a prediction horizon N , the optimization variables x ∈ R N (nz+nu) are specifically defined as

x = vec[u T ( 0| 0), z T ( 1| 0), u T ( 1| 0), z T ( 2| 0), ..., u T ( N -1| 0), z T ( N | 0)] T ] Function f (.) in (4.3.
2) is the objective function and h = [h 1 , ..., h n h ] T is the vector of n h equality constraints representing the dynamics of the system. Let denote by n the dimension of x, i.e., n = N (n z + n u ). In terms of the decision variable x, the system dynamics (4.3.1) is described by the equality constraint

h(x) = Ax -b = 0 (4.3.3)
where the (N n z × n) matrix A and the N n z -dimensional vector b are given by

A =        B -I nz 0 0 0 • • • 0 0 0 0 A B -I nz 0 • • • 0 0 0 . . . . . . . . . . . . . . . . . . A B -I nz 0 0 0 0 • • • • • • 0 0 A B -I nz        (4.3.4) b =      Az(0) + Ed (0 |0 ) Ed (1 |0 ) . . . Ed (N -1 |0 )      (4.3.5)
The forecast of disturbances d is assumed available and consequently vector b is a known vector. Throughout, the following technical hypothesis is assumed for problem P c : Assumption 1. The cost function f is convex and twice differentiable

The solution of the above centralized problem may be viewed as the reference optimal target and we aim at deriving a distributed model predictive control from the above centralized problem P c . The common approach for tackling this problem is to decompose the original problem into smaller ones by first decomposing, in some way, the large-scale system (4.3.1) into several subsystems. Such decomposition can be derived directly from the physical description of the system which might suggest a way to organize the system variables into groups. Alternativly, the system decomposition may rely on more systematic approaches, particularly when the only information at disposal is a mathematical model with no insigths into how the subsytems have to be chosen. For an overview of such system decomposition approaches, see [START_REF] Zecevic | Control of Complex Systems: Structural Constraints and Uncertainty[END_REF]. These decomposition techniques are usually decoupled from the control problem and results in solutions which may be far from the target solution, i.e., that of the centralized problem 4.3.2. Towards a better solution to the distributed MPC, we put forward the idea of integrating the system decomposition in the optimal control problem formulation. The procedure may be summarized into two steps: i)-decomposition of the decision variables x into p-subvectors xi , (i = 1, 2, .., p) that highlight p interconnected subsystems, this decompistion is based on the KKT matrix characterizing the optimality condition, and ii)-design of coordinated distributed p sub-controllers.

The KKT matrix of equality constrained optimization problems

The Lagrange function for problem P c (4.3.2) reads

L(x) = f (x) + λ T h(x) (4.3.6)
where λ = [λ 1 , ..., λ n h ] T is the vector of Lagrange multipliers with n h = N n z . As the bounds on the variables will not affect the decomposition, we will ignore the associated inequalities. The search for a KKT (Karush-Kuhn-Tucker) point to problem P c comes down to solving the system of n + n h equations in n + n h unknowns (x, λ)

∇L(x, λ) = ∇ x L(x, λ) ∇ λ L(x, λ) = ∇f (x) + J T h (x)λ h(x) = 0 (4.3.7)
where J h (x) is the Jacobian matrix of h(x) and ∇ x L(x, λ) and ∇ λ L(x, λ) are the gradients of L with respect to x and λ. The nonlinear equation (4.3.7) in (x, λ) can be solved iteratively using Newton's method ([nocedal], [START_REF] Boyd | Convex optimization[END_REF], [START_REF] Wright | The interior-point revolution in optimization: History, recent developments, and lasting consequences[END_REF]) from a starting guess (x 0 , λ 0 ) . The linearized system to be solved at iteration k for the Newton step

s k = x k+1 -x k , δ k = λ k+1 -λ k is K c (x k , λ k ) s k δ k = - ∇f (x k ) + J T h (x k )λ k h(x k ) (4.3.8)
where matrix

K c (x, λ) = ∇ 2 x L(x, λ) J T h (x) J h (x) 0 (4.3.9)
is the KKT matrix of the centralized constrained optimization problem (4.3.2) with ∇ 2 x L being the Hessian matrix of L with respect to variable x. A close look at the KKT matrix of the centralized constrained optimization problem reveals some key information about the problem and the system structure. Indeed, the left-upper block ∇ 2

x L reveals the separability of the cost function with respect to the decision variable x. If ∇ 2

x L shows up as a block-diagonal matrix, then the cost function has a separability property in the sense that it can be expressed as the sum of functions of sub-vectors of the decision vector x. As for block J h (x), it is really nothing else than the sensitivity matrix of the system dynamics with respect to vector x. Hence, the off-diagonal coefficients in this matrix represent the degree of coupling between the original components of vector x. It turns out that this matrix bears structural information about the system dynamics and in case this matrix is sparse, this makes it convenient to identify groups of variables in weak or strong interaction.

Transformation of the centralized KKT matrix

It is always possible to transform the centralized KKT matrix K c in (4.3.9), through some rows/columns permutation, into the following equivalent form

K c ∼ K d ∆ =      KKT 1 * • • • * * KKT 2 • • • * . . . . . . . . . . . . * * * KKT p      (4.3.10)
where the symbol ∼ stands for equivalent and the matrix K d is by definition the matrix displayed in (4.3.10). By the very transformation of K c into the form (4.3.10), p block-matrices appear on the diagonal of the transformed matrix K d and these block-matrices have the structure of KKT matrices. The block-matrices KKT i (i = 1, ..., p) may be viewed as those of some p subsystems (Σ 1 , Σ 2 , ..., Σ p ) composing the overall large-scale system so that the original decision vector x is fully equivalent to vector x = vec[x T 1 , ..., xT p ] where xj is a subvector which is the decision vector of subsystem Σ j . The ( * )-blocks in (4.3.10) are sparse block-matrices which bear the interaction between the subsystems. If these sparse block-matrices are identically zero matrices then matrix K c reduces to a block-diagonal matrix which may be regarded as the KKT matrix of a decentralized structure. Looking closely at (4.3.10), the equivalent centralized form may be viewed as that of a distributed structure in which p subsystems are clearly exhibited with interactions between these subsystems achieved through the * -blocks. It is thanks to this viewpoint that we denoted this matrix in (4.3.10) as K d where the subscript d refers to a distributed structure. The transformation of the centralized KKT matrix into an equivalent distributed KKT matrix as in (4.3.10) achieves the goal of decomposing problem (4.3.2) on the dynamics/constraints level as well as on the control level. In the literature, various methods describe the transformation of a symmetric matrix into a block-diagonal matrix form. The bandwidth of the centralized KKT matrix is further reduced by permuting its rows and columns such as to move all the nonzero elements of K c in a band as close as possible to the diagonal. This permutation can be achieved using the sparse reverse Cuthill-McKee (CM) algorithm [START_REF] Cuthill | Reducing the Bandwidth of Sparse Symmetric Matrices[END_REF]. The above system decomposition is summarized in the next algorithm: Note that for exhibiting the separable blocks, it is sufficient to set N = 2 (prediction horizon) in deriving the KKT matrix K c .

Distributed controllers synthesis procedure

Now, raises the issue of synthesizing the controllers for the identified p-subsystems in a coordinated way. The main idea and tool used to address this issue is an optimization procedure, the socalled optimality condition decomposition ( [START_REF] Conejo | Decomposition techniques in mathematical programming: engineering and science applications[END_REF]), which can be viewed as a modified Lagrangian relaxation. Thanks to the decomposition of x into p-subvectors, i.e., x ∼

x ∆ = vec[x The constraint h (x 1 , x2 , .., xp ) = 0 describes the overall system dynamics and is nothing more than the original constraint h(x 1 , ..., x n ) = 0 expressed as a function of the subvectors of the p subsystems of the decomposition. This constraint naturally decomposes into p vector-valued constraints describing the dynamics of the p subsystems Σ i (i = 1, ..., p) and how each subsystem Σ i interacts with other subsystems Σ j (j = i)(j = 1, ..., p). where the hi 's, i = 1, ..., p, are n h i -dimensional vector-valued constraints with p i=1 n h i = n h . These p vector-valued constraints hi (i = 1, 2, ..., p) are actually coupling constraints that if they were removed, the resulting optimization problem would be easier to solve. The iterative procedure underlying the optimality condition decomposition for solving problem (4.3.12) is to enforce "some separable approximation" for both f and hi (i = 1, 2, ..., p) by fixing some of the decision variables in these functions to their last computed values (indicated by overlining) to obtain the following p subproblems solved in parallel:

P i : min xi f ( x1 , ...,
i ≤ xi ≤ xmax i (i = 1, 2, ..., p) (4.3.13)
In (4.3.13), λj (j = 1, ..., p) denotes the dual variable (Lagrange multiplier) associated to the constraint hj and fixed at its last computed value in an iterative scheme. Note that the procedure looks like a modified Lagrangian relaxation, but it has the computational advantage that a single iteration for each subproblem P i can be performed before updating all the variables. Clearly, the benefits of this formulation rely in removing the coupling in the constraints from the original large-scale problem. Let x * = [x * 1 , ..., x * p ] T denotes the vector of optimal decision variables of the p subproblems (4.3.13), then the steps of this distributed control synthesis procedure can be summarized in the following algorithm:

Convergence analysis

Observe that, by the structure of problems P i in (4.3.13) built upon fixing all variables but the i th decision subvector xi , the original problem has been decomposed to yield a separable problem that approximates it. Indeed, problem P i can be rewritten as

P i : min xi φ i (x i ) subject to H i (x i ) = 0 xmin i ≤ xi ≤ xmax i (i =
φ(x) = p i=1 φ i (x i ) s.t. H(x) = [H T 1 (x 1 ), ..., H T p (x p )] T = 0 xmin ≤ x ≤ xmax (4.3.17)
Let K i be the KKT matrix of the optimization problem P i for i = 1, ..., p in (4.3.14), then the KKT matrix of the overall problem (4.3.17) is the diagonal matrix

Ǩ =       K 1 0 • • • 0 0 K 2 0 . . . . . . . . . 0 0 • • • 0 K p       (4.3.18)
The Newton steps in solving iteratively optimization problem (4.3.17) are given by the solution of the system of linear equations (see (4.3.8))

Ǩk π k = -γ k (4.3.19)
where π k = xk+1 -xk λk+1 -λk is the Newton step at iteration k, matrix Ǩk is the KKT matrix (4.3.18) evaluated at ξ k = vec[(x k ) T , ( λk ) T ], i.e., Ǩk = Ǩ(ξ k ) and

γ k = ∇φ(x k ) + J T H (x k ) λ H(x k ) (4.3.20)
Now, let x * denotes the optimal solution of the centralized problem (4.3.2) and let ξ * = vec[x * T , λ * T ] be the corresponding primal-dual point. For this optimal point, we make the following assumptions:

A simple illustrative exemple

To grasp the presented approach, we present below a very simple example which can be solved manually to illustrate the decomposition technique of a system and the synthesis of the optimal distributed decisions. Consider the following optimization problem

P c : min x f (x) (4.3.22) s.t. h(x) = 0 with f (x) = 2x 2 1 + x 2 2 -20x 1 -16x 2 (4.3.23) h(x) = x 2 1 + x 2 2 -12.5572 x 1 + x 2 -5 = 0 (4.3.24)
Note that the constraint h(x) is nonlinear here. The centralized solution of the problem is easily computed as

x * = 2.6692 2.3309 (4.3.25)

The system decomposition is trivial here but we will still illustrate it with algorithm 2. We first derive the KKT matrix K c from the Lagragian of the problem The explicit expression of KKT matrix K d is

K c =     4 + 2λ 1 0 0 2(1 + λ 1 ) 2x 1 1 2x 2 1 2x 1 2x 2 1 1 0 0 0 0     (
K d =     4 + 2λ 1 1 1 0 2x 1 0 0 1 2x 1 0 0 1 0 2x 2 2x 2 2(1 + λ 1 )     = KKT 1 * * KKT 2 (4.3.27)
Two blocks are trivially identified here with the new decision vector x consisting of subvectors (here scalars) x1 = x 1 and x2 = x 2 , that is, x equals to the original vector x = [x 1

x 2 ] T . The associated subsystems are h1 (x) = h 1 (x) for subvector x1 = x 1 and h2 (x) = h 2 (x) for subvector x2 = x 2 .

From this system decomposition, the synthesis of the coordinated distributed decision variables is performed by solving the two parallel subproblems (see (4.3.13)) min

x 1 (2x 2 1 + x2 2 -20x 1 -16x 2 ) + λ2 (x 1 + x2 -5) s.t
x

Notes and references

At the time we started this research within the Energy-in-Time project, very few works dealing with equipment failures in HVAC systems relied on economic model predictive control synthesis to address the fault-tolerant control problem. Most works on MPC-based FTC for buildings relied on quadratic objectives to ensure thermal comfort to the occupants. Our work shows that with appropriate adaptation of the objective and operating constraints to the failure characteristics, a simple and yet effective fault-tolerant control technique is obtained to ensure the energy efficiency of the overall building system. This constraints reconfiguration based-FTeMPC technique has been extended to other works that we have carried out on decentralized building control [START_REF] Subramaniam A Mona | Decentralized eMPC based fault tolerant control for energy efficiency in a multi-zone building[END_REF]. We introduced also a maintenance-aware MPC strategy thanks to the flexibility in formulating economic objectives. This is achieved through adding to the cost function related to the energy consumption by the HVAC-VAV equipments an 1 -regularization term aiming at minimizing the maintenance cost of HVAC-VAV terminal boxes. The rationale behind 1 -regularization term was to remove setpoint trajectories fluctuations without jeopardizing the true setpoint waveform which might have sharp edges. It is noteworthy that, not removing these persistent fluctuations would lead to increased wear of the VAVs which in turn might cause earlier equipment failures. This regularization technique directly increases equipment reliability while maintaining very economical operating costs in terms of maintenance costs. To the best of our knowledge, there were no work dealing with such economic MPC-based 1 or total variation regularization to tackle actuators wearing issues through judicious control actions.With regards to distributed control, the solution we provided to the problem is novel, by linking the system decomposition to the formulation of the optimization problem through the KKT matrix. The optimality of the decision variables of the derived distributed model-predictive control algorithm is achieved as that (i.e., global minimum) of the target reference centralized MPC.

When it came to extend the eMPC to the fault-tolerant control problem with respect to faults in wind turbines (WT), we have been confronted to the issue of solving the underlying optimization problem as it is highly non-linear and non-convex. The roots of this non-linearity and non-convexity of the optimization come from the non-linear dynamics of the WT in the decision variables and from the constraints and objective which are non-convex. To tackle this problem, we have explored the idea of convexification to make the problem tractable. This idea of convexifying the problem was found to be possible through a modification of the decision variables that makes the original nonlinear dynamics of the WT appear as a linear dynamics and in addition makes the control objective convex in these decision variables. Then, the eMPC-based FTC problem could be solved to obtain a global solution using efficient algorithms.

Introduction

Fault detection and diagnosis (FDD) is in its own right an important aspect for dependable systems and a key component in active fault-tolerant control systems as described in chapter 1. For fault-tolerant control systems, the precise estimation of the fault by the FDD module might be mandatory for achieving control reconfiguration which matches the faulty plant mode. The design methodologies of FDD algorithms are essentially based on mathematical models of the monitored plants. Consequently, the effectiveness of such model-based design techniques largely depends on the derivation of accurate and suitable models, including an uncertainty description and a characterization of the noises and disturbances acting on the plant. A powerful and classical approach to the design of FDD algorithms is based on linear observation techniques of dynamic systems linearized around an operating point, and this linearized-model based approach has proven to be very successful in many applications. Over the years, we have developed some techniques based on this approach, in particular for the estimation of rapidly varying faults [YMA11; MYA10] and more specifically for the detection and estimation of torque bias faults in Wind Turbines [START_REF] Jain | Fault-Tolerant Economic Model Predictive Control for Wind Turbines[END_REF] and for actuator faults in HVAC equipment of buildings [START_REF] Darure | Model-based fault-tolerant control of VAV damper lock-in place failure in a multizone building[END_REF][START_REF] Darure | Fault-adaptive control of VAV damper stuck in a multizone building[END_REF]. In this last case concerning the FDD of HVAC building systems, it turns out that model linearization methods are less appropriate because of the wide operating range of the building dynamics intrinsically linked to the outdoor climate and the building occupancy. This difficulty led us to deal directly with the thermal model of building-HVAC systems as derived from a physics-informed modeling. This physics-informed building model is non-linear, but it has a certain structure that makes it appear as a bilinear model. Although this research work was carried out within the framework of fault diagnosis of building HVAC systems with a mathematical structure of bilinear type for their modeling, the obtained results can be extended to other dynamic systems having a similar dynamic structure.

Bilinear-observer based FDD

Consider a building-HVAC system with n zones each equipped with a VAV terminal unit. We refer to figure 4.1 of chapter 4 for a pictorial view of the overall building system. The system subject to VAV actuator faults is described by the following physics-informed model which can be concisely written as (for the details, see [START_REF] Subramaniam | Bilinear model-based diagnosis of lock-in-place failures of variable-air-volume HVAC systems of multizone buildings[END_REF])

ẋ = Ax + D(u ⊗ x) + Bu + Ed + F (t)f a y = Cx (5.2.1)
where the state x = vec T T z T T w , with T z = vec T 1 T 2 • • • T n the vector of the n zonetemperatures, and T w the vector of the p walls' temperatures. Vector u is the vector of n air mass flow rates delivered to the VAVs, d is the disturbance which is the vector consisting of n internal heat gains produced by the occupancy in each zone and the outside temperature and

f a = f 1 • • • f n T
is the vector of damper faults in the n VAV-units. Matrix F (t) = F 1 (t) 0 p×n , with F 1 a nonsingular (n × n) diagonal matrix. The term D(u ⊗ x),where ⊗ is the kronecker product, represents the bilinear terms and reads

D(u ⊗ x) = n i=1 D i ṁi (t)x(t) with D = D 1 • • • D n
, where D i 's are square matrices of order (n + p). Vector y is the measured zone temperatures and thus matrix C reduces to C = I n 0 n×p .

The fault detection problem is solved through the generation of residuals based on the observer

ẋ = Ax + Bu + D u (t)x + Ed -K u (t)(y -ŷ) ŷ = C x (5.2.2)
where the bilinear term is written as D u (t)x with D u (t) = n i=1 D i ṁi (t) and the time-varying gain K u (t) is constructed as K u (t) = (C T LCAC T +C T LCD u (t)C T ) where L is a (n×n) diagonal matrix to be designed. Set C = -CAC T C, L = C T L and Ā = A -L C, then under the mild condition that (A, C) detectable, the solution to the observer design problem is provided by the following result:

build matrices

A(ε) =         A 0,1 (ε) 0 • • • 0 0 γ 0,2 (ε)c 0 A 0,2 (ε) 0 γ 0,3 (ε)c 0 . . . . . . . . . . . . . . . A 0,s-1 (ε) 0 0 γ 0,s (ε)c 0 A 0,s (ε)        
(5.2.8)

Γ (ε) =        γ 0,1 (ε) 0 . . . 0        , C = 0 0 • • • 0 1 (5.2.9)
For a noisy signal y 1 , the estimation of its derivative is given by the next result:

A. M. Subramanian, T. Jain and J.J. Yamé (2020)

Let s be a positive integer and consider the following 2s-dimensional state-space realization of a filter driven by a noisy signal y

1 ζ = A(ε)ζ + Γ (ε)y 1 z 1 = Cζ (5.2.10)
where matrices A(ε), Γ (ε) and C are given by (5.2.8) and (5.2.9). Then, the output z 1 of the filter is an estimate of the derivative of y 1 , i.e., z 1 = dy 1 dt .

Remark 5.1. One can see that the filter (5.2.10) is a cascade of s second-order filters (5.2.6). Thus, the filtering process consists in filtering successively the (noisy) driving signal y 1 to "extract" its underlying smooth signal, which obtained by constructing the "ξ 1 -states" of the successive high-gain observers. At the final s th stage, the state ξ 2 of this last stage is nothing but the sought derivative of y 1 . The computation of the derivative of each output y i , (i = 1, ..., n) proceeds in the same way with the filter (5.2.10).

Robust adaptive fault estimator

This section contains work being currently evaluated for an international journal. A preliminary, albeit not complete, version has been deposited on the IEEE TechRxiv preprints server. It is a continuation of our previous work on the diagnosis of VAV faults for a larger class of faults for these terminal units of building HVAC systems, in cases where not all disturbances d(t) are measurable and zone temperature measurements are subject to uniformly distributed additive noise η(t). In reality, the external climate and particularly the outdoor temperature T oa may significantly impact the building dynamics as an unknown disturbance input with very large variations. Clearly, the inherent uncertainty in the weather and hence in the outdoor air temperature is a major challenge that requires a proper treatment in the design of FDD algorithms for building HVAC systems.

The problem addressed here is how to design a bilinear observer for robust adaptive estimation of the magnitude of VAV damper faults under the effect of not only measured but also unmeasured disturbances and under the effect of sensor noises. i zone temperatur measurements. Under these conditions, the main non-trivial issues to be solved for this fault estimation problem are: how to detect and isolate the faults, how to remove the effect of unmeasured disturbance on fault estimation and how to attenuate the effect of sensor measurement noises on fault estimation?

The structure of the building-HVAC systems under consideration is that depicted in figure 4.1 and the same as considered in the previous section. Under faults which can be of different types, i.e., "lock-in-place dampers" faults, "constant bias damper position" faults and "loss of efficiency in VAV dampers" faults, the informed-physics dynamics of the overall faulty system reads

(Σ f ) : ẋ z = A B E 0 • F (t) C 0 • 0 • I • 0 • × [x T , u T , d T , η T , f T a ] T + D(u ⊗ x) 0 • (5.3.1)
where f a denotes the fault vector and F (t

)
     ẋ ŷ ḟa      =      A -LC B E m L F (t) C 0 • 0 • 0 • 0 • -θ(t) * GC 0 • 0 • θ(t) * G 0 •      × xT u T d T m z T f T a T + φ(u, z) T 0 • 0 • T (5.3.2)
where '*' represents the convolution operator, x(t) denotes the state estimate, ŷ(t) the output estimate. It is worth noticing that the unknown fault f a (t) is viewed as a time-varying parameter whose estimate fa (t) is obtained by the adjusting law given by ḟa (t) in (5.3.2), and the to-be-designed parameters are the constant matrices L and G, and ϑ(t) the impulse response of a causal filter. The main steps towards the design of these parameters proceed through a sufficient condition for the existence of matrix G and then, the design of L and the filter with impulse response ϑ(t). The key to this process is a transformation of the non-linear dynamics of the state estimation error ε(t) = x(t) -x(t) into a linear error-dynamics under an appropriate introduction of a new variable η(t) = vec[η T , η T u ], where η u = diag(u 1 (t), • • • , u n (t)) × η(t). This transformation followed by a similarity transformation ε = Γζ, with matrix Γ being the modal matrix of à = A -LC, i.e., Γ -1 ÃΓ = Ā is a diagonal matrix, yields finally the errors dynamics as

ζ ζ y = Ā B Ēum F (t) 0 • C 0 • 0 • 0 • I • × ζ T ηT d T um (f a -fa ) T η T T (5.3.3)
where

ζ y = z -ŷ, Ā = Γ -1 ÃΓ, B = Γ -1 B, Ēum = Γ -1 E um , F (t) = Γ -1 F (t) and C = CΓ.
The design of G relies on the next proposition: The proof of this proposition is constructive and relies on a judicious construction of the nonsingular (n × n)-matrix G as

G =              1 - h 1 h 2 0 • • • 0 0 0 1 - h 2 h 3 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 • • • 1 - h n-1 h n - h n h 1 0 0 • • • 0 1              (5.3.4)
which readily verifies the conditions of the proposition.

The main result achieving the design of the bilinear observer-based robust adaptive estimator (RAFE) is stated next and a sketch of the proof can be find in [SJY21]: Proposition 5.2. Let ϑ(t) be the causal filter given by

ϑ(t) = β 1 β 2 × vec δ(t) 1(t)
where matrices β 1 , β 2 ∈ R n×n , and δ(t), 1(t) ar respectively the Dirac delta function and the Heaviside function respectively. Let

Ȃ = Γ -1 0 • 0 • I • (A-LC)-Bβ 1 GC B -β 2 GC 0 • Γ 0 • 0 • I • Ȃ = Γ -1 0 • 0 • I • -H z (t)β 1 GC H z (t) 0 • 0 • Γ 0 • 0 • I •
where Γ is the modal matrix of A -LC and H z (t) = F (t) -B.

If there exist matrices β 1 , β 2 , and L such that (i) -Ȃ is Hurwitz (ii)-C F (t)(f a -fa ) ∈ ker (G), where fa = f a -fa denotes the fault estimation error (iii) -Ȃ(t) < λ min (N ) 2λmax(M ) , for some positive definite symmetric matrices N, M satisfying

ȂT M + M Ȃ = -N
Then the fault estimation error is independent of the unmeasured disturbance signal d um and the effect of measurement noise η on fa is attenuated.

Implementation

The implementation of the overall robust fault diagnosis methodology consisting of fault detection, isolation, and adaptive estimation of VAV damper faults. The design matrices L, G, β 1 , and β 2 of the RAFE are synthesized using Propositions 5.1 and 5.2. Prior to the estimation of a fault, the zone having a faulty VAV damper should be identified. Denote by RG the overall residual generator. This generator is composed of n residual generators RG i , i = 1, 2, . . . , n, where each RG i outputs a signal r i (t) which is a vector belonging to R n and computed using the formula r i (t) = Υ i Gζ y (t, fa = 0)

(5.3.5)

r(t) = vec r T 1 (t) r T 2 (t) • • • r T n (t) = vec           r 11 r 12 . . . r 1n      T      r 21 r 22 . . . r 2n      T • • •      r n1 r n2 . . . r nn      T     
and the square r(t) 2 of the Euclidean norm of r(t) is computed. The matrix Υ i in (5.3.5) is designed such that whenever there is a damper fault in the VAV terminal unit of the i-th zone, then only the i-th element of the vector r i (t) is zero and the i-th elements of the other residual vectors r k (t), ∀k = i, are greater than some threshold ρ, that is,

(r ii (t) = 0) ∧ (|r ki | ≥ ) holds = true, fault is in i-th zone false i-th zone is fault-free (5.3.6)
∧where r ki (t) denotes the i-th element of vector r k (t), | • | denotes the modulus and the symbol "∧" denotes the conjunction operation. Using (5.3.6), single and multiple VAV damper faults can be successfully isolated. Fault isolation is preceded by a fault detection stage carried out as in ((5.2.4))

The implementation of the complete robust fault detection and diagnosis methodology is shown in the block diagram of Fig. 5.1, where based on the measured signals, (z, u, d m ), a fault occurrence is detected using the detection logic (5.2.4) followed by an isolation of the faulty VAV dampers using the rule (5.3.6), and finally the adaptive fault estimator is triggered for estimating precisely the magnitude of the fault using (5.3.2) with parameter matrices computed from propositions 5.1 and 5.2. The various steps for implementing the proposed FDDA are summarized in the pseudocode given next. Compute Γ such that Ā = Γ -1 (A -LC)Γ is diagonal 5:

Compute B, C, Ēum , F (t) according to (5.3.3)

6:

Select Υ i , i = 1, 2, . . . , n according to (5.3.5) while r(t) ≥ do 7:

Identify fault actuator(s) according to 5.3.6

8:

Estimate the fault magnitude using 5.3.2 end 9: end procedure

Simulation

A simulated example is provided here to demonstrate the effectiveness of the robust adaptive fault estimator on a one-story office building with three spaces (zones) as shown in figure 5.2 with the occupancy profile and the outside air temperature. One fault scenario consisting of mutually exclusive VAV lock-in-place faults is presented here with the following designed matrices of the robust adaptive fault estimator : L = 0.0005

I 3 0 6×3 , β 1 = 0.002I 3 , β 2 = 0.0005I 3 .
For residuals generation, in the three-zones building, three matrices Υ i , i = 1, 2, 3 are selected as

Υ 1 = V -1   10 0 0 0 20 3 -20 √ 3 0 0 10   V, Υ 2 = V -1   10 0 0 0 10 0 20 3 0 -20 √ 3   V, Υ 3 = V -1    10 √ 3 10 0 20 3 -20 √ 3 0 0 0 10    V
where the columns of V are the generalized eigenvectors of matrix G.

The scenario is as follows: the lock-in-place failure appears in VAV dampers of zone-3, zone-2, and zone-1 at the end of day 5, 3, and 1 respectively and disappears or is corrected in the respective zones by the end of the next day. The lock-in-places and their corresponding constant airflow rates are given in Table 5.1.

Under these faults, the zone temperatures are shown in figure 5.3(a). It should be noted that the large variations in the zone temperature measurements appear due to the departure and arrival of occupants. The norm of the residual signal is recorded in figure 5.3(b). Selecting the threshold values as ρ = 2 • C, and = 5 • C, the fault is successfully detected at the end of day 1, 3, and 5 according to (5.2.4). In addition, when the damper fault disappears, the residual signal norm falls below the threshold value. For fault isolation, the vector signals r i (t), i = 1, 2, 3, recorded in figure 5.3(c), are used to identify the faulty damper. The robust fault estimates (see Table 5.1 for the amplitude of the actual faults) are recorded in figure (5.4) showing clearly that the adaptive fault estimator successfully reconstructs the airflow rates at the input of the zones in the fault-free mode as well as in the faulty condition. This reconstruction is achieved under noisy sensor measurements and, without having any information of unknown large disturbance d um .

Notes and references

In this chapter, I have chosen to present only my very recent research work on model-based FDD in the context of building monitoring to ensure their energy efficiency in the event of HVAC equipment failure. The work focuses on a physics-informed model, which turns out to be a bilinear model, and which describes as closely as possible the thermal dynamics of the building-HVAC system in a wide range of climatic conditions. This is in contrast with most current approaches, based on linearized models of the building dynamics, which for this reason are not adapted to the monitoring of the building taking into account its wide operating. A novel bilinear observer-based FDD algorithm for lock-in-place failure of HVAC-VAV dampers in multi-zone buildings has been presented with the original feature that both an estimation of the zones temperatures and that of the walls are obtained, the latter being seldom measured by sensors. A particularly simple and novel approach using the estimates of the walls temperatures and the derivative of the zones temperature measurements, computed accurately in real-time using high gain observer techniques, allows to estimate the faults.

The related publications appeared in [START_REF] Subramaniam | Diagnosis of Actuator Faults in VAV-HVAC system using a Bilinear Observer[END_REF], [START_REF] Mona Subramaniam | Output Injected Nonlinear Observer for Diagnosing Faults in Multi-zone Building[END_REF], [START_REF] Subramaniam | Bilinear model-based diagnosis of lock-in-place failures of variable-air-volume HVAC systems of multizone buildings[END_REF]. The precise computation of the estimates of the magnitudes of the faults is essentially motivated by their subsequent use in MPC-based fault-tolerant control laws of the type described in the previous chapter. This research in model-based FDD is currently being continued to make the proposed solutions much more robust to uncertainties related to outdoor temperature and measurement noises. To this end, we develop a robust and adaptive version for fault estimation that is summarized in its major steps in the last section of this chapter. A paper written on this robust solution is currently under review, but some of the preliminary results can be found in a preprint posted on the open access IEEE TechRxiv server [START_REF] Subramaniam | Robust adaptive fault estimation for multizone building VAV terminal units[END_REF].

Epilogue

The writing of this habilitation thesis was an opportunity to review the topics on which I have been working with my doctoral students at the Centre de Recherche en Automatique de Nancy. These topics, however, do not cover all the questions that I have dealt with, but they allow to draw up a rather clear vision of the paradigmatic frameworks in which these works have been carried out.

Two "modeling" paradigms constitute the backdrop of these research activities and are declined for one, in the form of a classical model-based framework and for the other in the form of a trajectorybased framework. The foundation of the later framework is the behavioral approach to system theory introduced by J.C. Willems as a modeling paradigm in which a dynamical system is simply viewed as a set of trajectories, which defines its behavior. To move to trajectory-based modeling constitues a true paradigm shift where dynamic modeling aims at coming up with a specification of the behavior and control comes down to restricting the behavior. It was at this general set-theoretic level that I thought it appropriate to tackle fault-tolerant control so as not to be tied to the need for a mathematical model derived from various assumptions. This approach to FTC where the data directly generated by the controlled process specifies the "model" was clearly a break with the classical model-based framework. In view of the results obtained in this behavioral paradigm, we believe that this approach to fault-tolerant control that we have initiated is viable and that it needs to be further developped in the future. In this respect, the bubbling of ideas observed in the last three years, following the re-discovery of J.C. Willems' ideas by young researchers, show that the concept of behavior is becoming a central notion to control synthesis thanks to the data revolution that we are currently observing with powerful computing devices and the ubiquity of sensors in all technological sectors. To return to the issues of fault-tolerant control in the behavioral paradigm, many questions remain open, nevertheless I will not discuss these questions here in this brief conclusion since they form the last part of the document in the form of my research prospectives. The other paradigm of the backdrop of our research activities in fault-tolerant control is the more standard one of the model-based approach. In this framework, we were as much interested in the synthesis of control laws as in that of fault detection and diagnosis.The main control technique used in most of these works is that of receding horizon optimal control which lends itself well to the formulation of fault-tolerant control problems. Though not expounded in this memoir, this was the main control strategy developed within the doctoral thesis of K. MENIGHED on fault-tolerant control of embedded and networked systems for which a distributed non-cooperative MPC solution was derived for linear large-scale systems. Later on, with well-identified applicative targets, such as buildings and wind turbine systems, this research focused on practical aspects driven by energy efficiency considerations or economic issues. The major issues were to formulate these economic requirements through appropriate cost functionals for the underlying optimal control. Along with this, and to achieve fault-tolerant control, we have developed model-based FDD algorithms based on linearized models but also on physics-informed models which closely match the actual dynamics of the controlled plant. In particular, this research activity has focused on bilinear-observer based FDD geared towards monitoring and fault-tolerant control of building-HVAC systems. The results obtained on this use case (buildings) can be extended to systems with similar structure. This is an ongoing research topic with still unresolved issues, among which the robustness with respect to different types of uncertainties.

Part III

Perspectives de recherche

Chapitre 6

Projet scientifique

Contexte

Les principales directions de recherche dans lesquelles je souhaite mener mes travaux sur les quelques années à venir sont présentées ici en m'appuyant naturellement sur l'expertise acquise sur les sujets abordés jusqu'à présent. Mes travaux actuels de recherche au CRAN sont développés essentiellement autour du thème de la commande tolérante aux défauts (FTC -Fault-Tolerant Control) sur deux axes principaux, l'un dans le cadre comportemental de la théorie des systèmes et l'autre dans le cadre des techniques basées sur la commande optimale contrainte à coût économique et à horizon glissant. Le premier axe de recherche est fondamentalement de nature théorique et méthodologique via le développement d'outils de conception de commandes FTC dans un cadre mathématique nonconventionnel de la théorie des systèmes. Le deuxième axe de recherche est surtout motivé par diverses applications, notamment dans les domaines de la commande FTC des éoliennes et de la gestion efficiente de l'énergie dans les bâtiments non résidentiels mais aussi par la problématique de la commande tolérant aux défauts des systèmes contrôlés via un réseau de communication numérique. Dans ces applications, la commande optimale contrainte à horizon glissant permet d'intégrer la tolérance aux pannes de manière relativement simple en : (i) redéfinissant les contraintes pour représenter certaines fautes (généralement des fautes d'actionneurs), (ii) modifiant le modèle interne et (iii) modifiant les objectifs de contrôle pour refléter les limitations dues à un fonctionnement défaillant. Sur ces deux axes principaux de recherche en FTC, de nombreuses problématiques importantes sous-jacentes ont été traitées parmi lesquelles je citerai les problèmes de commutation entre contrôleurs FTC et leur conditionnement, le développement d'observateurs robustes et/ou adaptatifs pour la détection et l'estimation des défauts aussi bien pour des systèmes linéaires que bilinéaires, et l'optimisation distribuée des grands systèmes et des systèmes répartis. Ces travaux ont été majoritairement menés avec mes doctorants et aussi pour certains en collaborations internationales avec le Service d'Automatique et d'Analyse des Système de l'Université Libre de Bruxelles et l'Indian Institute of Technology, Mandi.

Bien que mes thématiques de recherche en FTC et les travaux associés soient très variés, je les ai cadré ici pour plus de lisibilité dans les deux axes principaux susmentionnés. Ces travaux ouvrent naturellement sur de nombreuses perspectives qui peuvent faire l'objet de recherches ponctuelles que je ne mentionnerai pas ici. Je souhaite, dans cette partie d'ouverture de mes travaux, exposer mon projet scientifique, à court et moyen terme, en le déclinant sous la forme de deux programmes de recherche. Je ne m'aventurerai donc pas ici sur le long terme, i.e., sur un horizon au delà de sept ans. Ces deux projets sont naturellement dans le prolongement des mes travaux actuels et se situent sur deux volets différents avec des intersections possibles, un volet à dominante applicative et un volet à dominante théorique. 1)-le volet à dominante applicative est un programme de recherche à court terme (i.e, sur les trois années à venir) qui porte sur la problématique de l'efficacité énergétique des bâtiments par le développement de techniques de commande centrée sur l'occupant. Nos travaux sur l'efficacité énergétique, lors du projet européen Energy-In-Time, ont été focalisés sur la commande et le diagnostic des systèmes et équipements HVAC, sans tenir compte des informations spécifiques aux occupants, autre que le nombre d'occupants d'une zone thermique. Or l'occupant et ses préférences en terme de confort sont au coeur de l'environnement du bâtiment et peuvent impacter considérablement l'efficience énergétique globale du bâtiment. Par exemple, dans le contexte de la gestion de la "réponse à la demande" des bâtiments intelligents où des stratégies visent à motiver les occupants à modifier leur consommation d'énergie afin d'atteindre certains objectifs de minimisation des charges de pointe ou de déplacement des charges vers des périodes où la production d'énergie renouvelable est importante, le comportement et la préférence des occupants sont un facteur clé pour le succès de ces stratégies.

2) -le volet à dominante théorique est un programme de recherche à court et moyen terme dans le prolongement de mes travaux en commande tolérante aux défauts dans le cadre comportemental. Ce projet est fortement motivé par l'émergence et l'importance de l'utilisation des données comme un moyen direct de modélisation, de prédiction et de commande des systèmes complexes dont les systèmes tolérants aux défauts font partie. Le cadre comportemental fournit un formalisme mathématique dont nous avons été les premiers à identifier la pertinence au regard de la synthèse basée données des systèmes tolérants aux défauts. Les résultats obtenus à l'aide ce formalisme pour certaines structures de commande FTC nous encourage à étendre nos travaux à des lois de type contrôle optimal basé uniquement sur des trajectoires ou séries temporelles générées par le système contrôlé. Cette extension de nos travaux a en ligne de mire la conception d'algorithmes de commande FTC basée sur les techniques de contrôle optimal piloté par les données, avec un accent particulier sur les commandes contraintes à horizon glissant. Récemment, une nouvelle initiative de l'Agence Internationale de l'énergie (AIE), l'Annexe 66 ([Yan+17]) a donné une impulsion à de nouveaux travaux de recherche sur la modélisation du comportement de l'occupant dans son interaction avec le bâtiment. L'objectif de ces travaux est de mettre en évidence l'impact réel du comportement des occupants sur les besoins énergétiques du bâtiment, de comprendre les mécanismes et moteurs comportementaux des occupants et de développer un modèle « universel ou standard » utilisable dans les logiciels de simulation thermique dynamique. Ces travaux, bien que très importants dans la compréhension du comportement humain comme moteur de la consommation énergétique, se limitent pour le moment au développement de modèles pour la simulation de performance des bâtiments en phase de conception.

Efficacité

En phase opérationnel d'exploitation des bâtiments, les approches de contrôle de l'efficacité énergétique active reposent toujours sur les facteurs techniques et physiques que sont l'enveloppe (bâti), les conditions climatiques et les équipements de chauffage, de ventilation et de conditionnement d'air (HVAC). Sur la base de ces facteurs, les systèmes technologiques actuels permettent uniquement de mesurer des grandeurs physiques (e.g., à l'aide de capteurs de température, de présence pour l'éclairage, d'émissions de CO2, etc.) et de réguler la consommation électrique des bâtiments afin de réaliser une réduction sous-optimale de la consommation énergétique. Comparativement aux techniques de contrôle prédictif fondées sur la dynamique de l'enveloppe et des conditions de confort intérieur, des prévisions météorologiques ou éventuellement des sources d'énergie à privilégier à un moment donné compte tenue de la demande, les travaux prenant en compte le facteur clé non -technique qu'est le comportement humain dans la problématique du contrôle actif de l'efficacité énergétique sont rares [START_REF] O'brien | Introducing IEA EBC Annex 79: Key challenges and opportunities in the field of occupant-centric building design and operation[END_REF], exceptés quelques travaux rapportés très récemment dans la littérature scientifique [START_REF] Yang | From occupants to occupants: A review of the occupant information understanding for building HVAC occupant-centric control[END_REF].

Au regard de cette lacune importante de la recherche sur le contrôle actif de l'efficacité énergétique axé sur le comportement humain, et dans la continuité de nos travaux FDI/FTC sur la cible applicative « bâtiments », les objectifs scientifiques de notre projet sont de : (a) -développer des « modèles » de représentation dynamique de bâtiments intégrant le comportement des occupants et orientés vers la commande, (b) -concevoir des stratégies de commande intelligente et tolérante aux défauts, centrées sur l'occupant, pour une réduction optimale de la consommation énergétique, (c) -mettre en oeuvre et valider les solutions développées sur des benchmarks (logiciels et expérimentaux).

Dans cette optique, comprendre le comportement des occupants d'un bâtiment et leurs habitudes d'occupation est nécessaire pour minimiser la consommation d'énergie tout en satisfaisant leur confort. Comme mentionné plus haut, les occupants influencent directement les équipements du bâtiment qui consomment de l'énergie, tels que les systèmes de chauffage, de ventilation et de climatisation, les fours, les réservoirs d'eau chaude, etc. Leurs comportements représentent en réalité une énorme source d'incertitude dans la modélisation classique de la dynamique thermique des bâtiments basée sur les modèles de connaissances classiques issues des lois de la physique et de la thermodynamique. De ce fait, l'occupant, lorsqu'il est pris en compte, est usuellement considéré comme une entrée exogène ou une perturbation au regard des problèmes de contrôle automatique dans les approches actuelles décrites précédemment. Le point de vue original et novateur que je souhaite adopter dans ce projet de recherche est de considérer plutôt le comportement des occupants comme une caractéristique endogène de la dynamique globale d'un bâtiment. Une question importante de recherche qui découle de ce point de vue est comment cette caractéristique endogène de la dynamique peut être correctement identifiée et modélisée pour un contrôle de l'efficacité énergétique ? Je propose de décliner cette question importante en deux problématiques centrales : (i) -Comment construire un « modèle » dynamique de bâtiment axé sur le comportement des occupants et adapté à l'objectif de contrôle optimal de l'efficacité énergétique ? (ii) -Quelles méthodes de commandes avancées permettraient de répondre aux exigences de l'efficience énergétique du bâtiment sous les contraintes du confort intérieur et des variations du comportement humain ? Comment concevoir à partir de ces méthodes des algorithmes de commande centrée sur l'occupant ? La question du contrôle de l'efficacité énergétique des bâtiments telle que formulée via les problématiques ci-dessus présente des difficultés non triviales et des verrous scientifiques à lever en raison des contraintes cumulatives qu'elles impliquent au regard de la modélisation thermique et de la prise en compte du comportement des occupants.

Une première analyse de l'état de l'art a permis l'identification de verrous/difficultés relatifs aux problématiques centrales de ce projet scientifique :

• Verrous conceptuels : Comment caractériser formellement l'information sur l'occupant (comportement, préférences, ...) ? Quel périmètre faut-il définir pour ce concept de comportement ? Quels dispositifs de mesures ou quels capteurs permettraient d'acquérir des signaux relatifs au comportement de l'occupant pour réaliser des boucles de rétroaction ?

• Verrous méthodologiques : Quels types de représentation (sous forme de données ou de modèles) pourraient décrire aux mieux la dynamique interactive humain/bâtiment en vue de la commande automatique ? Quelles techniques de synthèse du contrôle seraient appropriées au type de la représentation du système ?

• Verrous techniques et applicatifs : Comment intégrer des outils de simulation disponibles (e.g., MATLAB/Simulink, TRNSYS, EnergyPlus, ...) en une plateforme de simulation logicielle pour la dynamique humain/bâtiment afin de tester les solutions retenues Au regard de la complexité d'une prise en compte du comportement et des préférences de l'occupant dans une approche classique de l'automatique, une voie de recherche privilégiée que je projete d'explorer prioritairement au niveau méthodologique est celle de la modélisation et de la commande basées sur les données par des techniques d'apprentissage automatique.

• Une première raison pour cette prospective est que les données dans le secteur du bâtiment sont omniprésentes grâce aux technologies du numérique qui permettent aujourd'hui de déployer différents types de capteurs pour collecter des données pertinentes sur la façon dont l'énergie est consommée dans un bâtiment. L'analyse des données collectées peut permettre de comprendre la manière dont les occupants réagissent à leur environnement intérieur et par conséquent aider à révéler certains paramètres relatifs à l'information sur l'occupant. L'information pertinente pour la commande centrée sur l'occupant peut être par exemple la présence, le mouvement, l'utilisation des fenêtres, l'ombrage, le réglage du thermostat, l'éclairage, l'utilisation d'appareils, etc. La détermination de ces variables (e.g., la présence ou l'absence d'occupants, le nombre d'occupants, la localisation, les activités, les interactions avec des équipements,...), vus comme « l'état » du système à un instant donné, sera donc cruciale pour injecter de l'intelligence dans les systèmes de contrôle de l'efficacité énergétique.

• Une deuxième raison est que l'objectif de la commande axée sur l'occupant est de trouver une séquence d'entrées optimale qui conduit le système dynamique globale à minimiser sa consommation énergétique, compte tenues des informations et des données sur les occupants. Le défi majeur ici, mentionné plus haut dans les verrous méthodologiques, sera certainement la difficulté à établir un modèle à partir des lois et principes physiques compte tenue de la complexité de l'interaction humain/batîment.

Dans cette perspective d'une vision méthodolique basées sur les données, il serait possible de tirer parti de nos formulations du problème de commande économique à horizon glissant pour l'efficacité énergétique via l'établissement d'un modèle expliquant les données par des techniques d'apprentisssage supervisé. En effet si l'on dispose de données de séries temporelles {(x k , u k )} N -1 k=0 sur une fenêtre de longueur N où les données sont classées comme des commandes {u k } ou comme des états (ou variables latentes) {x k }, le problème d'optimisation suivant min Φ N -1 k=0

x k+1 -Φ(x k , u k ) 2 -→ Φ fournit formellement un modèle dynamique de prédiction x k+1 ≈ Φ(x k , u k ). Il est évident qu'une telle approche, avec une dynamique inconnue de l'interaction occupants/bâtiment, soulève de nombreuses questions :

• Si les grandeurs de commande sont facilement identifiables, le choix des données issues des capteurs fournissant des informations sur le « comportement » et pouvant être étiquettées comme des variables latentes est une question ouverte.

• Quelles sont parmi les variables latentes retenues celles qui contribuent de manière significative à la dynamique réelle du système ?

• Comment paramétriser la fonction non-linéaire Φ pour « capturer » la dynamique de l'interaction occupant/bâtiment ? Est-il possible que ce modèle non-linéaire soit parcimonieux de manière à avoir peu de termes possibles dans le modèle nécessaire pour expliquer les données ?

• Une paramétrisation de type linéaire Φ(x, u)= Ax+Bu où A et B sont des matrices de dimensions appropriées permet-elle de décrire de manière plus simple (ou localement) et approximativement cette dynamique en vue d'une prédiction pour le problème de contrôle optimale ? Pour assurer une robustesse de la commande optimale, est-il possible de déterminer des bornes d'incertitudes sur ce modèle linéaire vu comme un modèle nominal ? Ou faut-il réajuster ce modèle périodiquement pour une commande optimale de type contrôle adaptatif ?

Sans avoir été exhaustif sur les questions inhérentes à cette approche de modélisation d'une dynamique inconnue, elle présente l'avantage de pouvoir envisager la commande centrée occupant pour l'efficacité énergétique en deux étapes classiques, i.e., celle de la prédiction du modèle de la dynamique suivie de l'étape de synthèse de la commande optimale à horizon glissant.

La question de la modélisation du comportement des occupants reste donc une question centrale pour la commande axée sur l'occupant en vue de l'efficacité énergétique. Une alternative basée données très proche des techniques de commande optimale de l'automatique classique, mais issue des méthodes de l'apprentissage par machine, est l'apprentissage par renforcement qui peut être vu pratiquement comme une méthode de contrôle optimal adaptatif lorsque la dynamique est complètement inconnue [START_REF] Lewis | Reinforcement Learning and Feedback Control: Using Natural Decision Methods to Design Optimal Adaptive Controllers[END_REF]. La méthode utilise un principe similaire au principe de la programmation dynamique de Bellman pour synthétiser des politiques de commandes optimales sur la base uniquement du critère de performance sans nécessiter de modèle du système sous contrôle [START_REF] Meyn | Control Systems and Reinforcement Learning[END_REF]. Ceci conduit naturellement à faire l'hypothèse que l'apprentissage automatique par renforcement serait une stratégie de contrôle plausible pour la commande centrée sur l'occupant en vue de l'efficience énergétique des bâtiments. C'est une approche que je souhaite pleinement investiguer dans ce projet compte tenues de ses variantes possibles en complément de l'approche indirecte décrite précédemment.

Les problématiques de ce projet de recherche peuvent servir directement de trame de fond à des sujets de thèse. J'envisage donc pour ce projet un travail de recherche dans une logique collaborative avec un doctorant et des équipes de recherche en automatique ou effectuant des travaux sur l'efficacité énergétique active des bâtiments. A cet égard, un projet PHC-Toubkal 2023, dont je suis le porteur, vient d'être soumis. Il réunit des collègues (H. Jamouli et son équipe) de l'Ecole Nationale des Sciences Appliquées de l'Université d'Agadir (Maroc) et l'équipe du CRAN impliquée sur l'application « Bâtiment Intelligents ». La nouvelle plate-forme « Eco-Sûr » du CRAN dont la construction s'achève bientôt servira de benchmark expérimental pour les travaux menés dans le cadre de ce projet. Un aspect important de ce projet de recherche, au regard des retombées sur nos activités de formation dans le Master EEA -CEE « Parcours Contrôle de l'Efficacité Energétique », est qu'il pourra impliquer sur certains travaux de ce projet des étudiants du Master (orientation Recherche) et/ou éventuellement des étudiants dans le cadre du dispositif ORION.

Commande optimale tolérante aux défauts basée données par l'approche comportementale

Ce projet s'inscrit dans la continuité de mon activité de recherche sur la commande tolérante aux défauts (FTC) dans le cadre mathématique de la théorie comportementale des systèmes dynamiques (behavioral system theory) introduite par J.C. Willems à la fin des années 80. J'ai été le précurseur de cette approche comportementale en commande tolérante aux défauts dès l'année 2003 où il était encore trop tôt de prévoir dans quelle mesure les idées de J.C. Willems feraient partie du fondement général de la théorie des systèmes ou ceux-ci sont simplement vus comme un ensemble de trajectoires ou de séries temporelles. Cet ensemble de trajectoires contraintes par la dynamique du système qui les a généré constitue ce que J.C. Willems définit comme le comportement du système par opposition aux modèles classiques de l'automatique. L'une de mes principales motivations pour cette approche comportementale en FTC est partie de l'idée que les données de séries temporelles ou de trajectoires issues d'un système dynamique contiennent généralement des informations essentielles sur le système. Par conséquent, le problème de synthèse des contrôleurs FTC pourrait être uniquement basé sur les données générées en temps réel par le système dans un mode de défaillance inconnu plutôt que sur des hypothèses ou des informations a priori sur le système exprimées sous forme de modèles. Ces données générés par le système ne sont rien d'autres que le comportement du système au sens de J.C. Willems et la théorie comportementale des systèmes fournit un cadre mathématique élégant et rigoureux pour l'analyse et la synthèse basées données des systèmes dy-namiques.

Nos contributions significatives sur le sujet ont été notamment la conception de contrôleurs FTC sans modules explicites de diagnostic de défauts à base de modèle grâce au concept de contrôleur canonique (se référer à la monographie [START_REF] Jain | Active Fault-Tolerant Control Systems : A Behavioral System Theoretic Perspective[END_REF]). La caractéristique singulière de ce contrôleur réside dans sa capacité à réaliser exactement le comportement spécifié en boucle fermée lorsqu'il est connecté au système et permet le développement de techniques de synthèse à la volée de contrôleurs FTC pour les modes de défaillances inconnues sur la base des données temps-réel générées par le système contrôlé. Les résultats ont permis de montrer comment un tel contrôleur qui réalise une spécification de performance en boucle fermée peut être conçu en temps réel uniquement à partir des données entrées/sorties du système commandé. Toutefois, une analyse mathématique approfondie du régulateur canonique dans une configuration à boucle unitaire a permis de révéler certaines de ses propriétés et de comprendre que le contrôleur canonique, bien que réalisant la spécification en boucle fermée, pouvait aussi la déstabiliser. A la lumière de cette analyse et de la caractérisation de la commande canonique comme une commande directement "pilotée par les données", des questions fondamentales émergent parmi lesquelles les problématiques suivantes constituent des points durs :

1. Comment garantir a priori la stabilité du système FTC puisqu'aucun modèle n'est utilisé en ligne pour la synthèse du contrôleur canonique ?

2. Pour quelle classe de systèmes dynamiques le contrôleur canonique FTC est-il déstabilisant ?

3. Quelles connaissances a priori doit-on avoir sur la dynamique des modes nominaux et défaillants du système contrôlé ? Ou dans quelle mesure des informations préalables sur le système sont nécessaires pour la conception en ligne du contrôleur canonique FTC ?

Il ressort des problématiques 2 et 3 ci-dessus qu'une difficulté particulière dans la synthèse des commandes FTC par la théorie comportementale est « l'identification » de la classe des systèmes dynamiques pour lesquels le régulateur canonique est déstabilisant. Une identification de cette classe de systèmes impliquerait clairement que la mise en oeuvre du régulateur canonique FTC exige une connaissance a priori du système commandé. On notera qu'en pratique l'on dispose rarement de cette connaissance en temps-réel des modes défaillants du système commandé. Ces problématiques peuvent se reformuler de manière équivalente comme suit :

(i) -Quel est le « degré » d'information préalable nécessaire à la conception du contrôleur canonique ?

(ii) -Quelles contraintes complémentaires doit-on imposer, au regard de la stabilité, sur la spécification (i.e., le comportement désiré en boucle fermée) autres que celles initialement imposées pour la réalisabilité du contrôleur canonique ? L'analyse mathématique approfondie du contrôleur canonique mentionée plus haut a révelé que la présence de zéros à déphasage non-minimal dans le mode actif (nominal ou défaillant) du système contrôlé est l'une des causes principales d'une déstabilisation du système FTC dans une configuration à boucle unitaire. Il est important de noter ici que les zéros à déphasage non-minimal sont susceptibles d'apparaître lorsque des défauts se produisent dans les actionneurs et/ou les capteurs, puisque l'on sait que les zéros des systèmes dynamiques dépendent de la manière dont les capteurs et les actionneurs sont couplés aux variables d'état du système.

Parmi les pistes de recherche possibles envisagées, une piste que je privilégie en particulier consistera à rechercher par construction une (ou des) structure(s) de bouclage relaxant la nécessité d'une connaissance a priori (ou à défaut la réduction maximale d'une telle connaissance) du système commandé afin d'assurer la synthèse en ligne, basée uniquement sur les données entrées/sorties, du régulateur canonique FTC stabilisant pour les dynamiques des modes défaillants inconnus. L'idée directrice sous-jacente à cette piste étant avant tout de pouvoir garantir par construction (i.e., une preuve par construction) la stabilité du système FTC via une structure de bouclage offrant plus de de degrés de liberté que le bouclage classique unitaire afin de dissocier le problème de stabilisation du système de celui de sa performance telle qu'exprimée par une spécification du comportement de la boucle fermée.

Notons que la conception du régulateur canonique conduit directement à une structure explicite sous forme d'un système dynamique réalisable dans l'une ou l'autre des représentations classiques (internes ou externes, i.e., sous forme d'équations d'état, d'équations aux différences entrées/sorties, etc...). En outre, cette conception nécessite de spécifier le comportement désiré du système bouclé en mode défaillant par des équations comportementales qui s'expriment en général par des équations différentielles ou par des trajectoires dans le noyau d'un opérateur polynomial. La technique s'avère donc rigide si l'on souhaite l'étendre vers d'autres types de formulation du problème de commande FTC. En particulier, dans le contexte de nos travaux sur la commande prédictive des équipements HVAC des bâtiments, la question s'est posée de pouvoir formuler des problèmes de commandes optimales dans le cadre comportemental, où la spécification du comportement désiré s'exprimerait sous forme d'une fonctionnelle des trajectoires (fonction de coût, critère de performance) en lieu et place d'une équation comportementale.

La puissance actuelle des dispositifs de calcul et la disponibilité d'outils mathématiques et algorithmiques de traitement de données de grande dimension suscitent un fort regain d'intérêt dans l'utilisation des données pour la commande de systèmes dynamiques complexes. Si la synthèse de contrôleurs par des approches indirectes, i.e., passant par une identification du modèle du système sur la base de données puis suivie d'une deuxième étape de conception du contrôleur à partir du modèle identifié n'est pas nouvelle, les méthodes directes où la synthèse du contrôleur est directement pilotée par les données émergent comme une alternative efficace pour le contrôle des dynamiques difficilement modélisables ou inconnues. L'approche comportementale, controversée à l'époque où nous démarrons nos travaux, est redécouverte aujourd'hui comme l'un des cadres mathématiques appropriés pour l'analyse et la synthèse directe pilotées par les données. En effet, une récente vague d'activités de recherche impulsées par un résultat de J.C. Willems En particulier, les solutions aux problèmes FTC utilisant des techniques classiques basées sur des représentations d'état ou externes (LQR, model matching, H ∞ , MPC, etc...) deviennent reformulables directement à l'aide de la matrice de données. Une perspective particulière sur laquelle je souhaite porter mon attention est celui de la commande FTC pilotée par les données via le contrôle prédictif formulé à l'aide de la représentation image. En effet, l'une des difficultés majeures de la commande tolérante aux défauts basée sur le contrôle prédictif à base de modèles (MPC) est la nécessité de disposer justement d'un modèle post-défaut en temps réel pour le calcul des prédictions sur lesquelles sont opérées le calcul des commandes optimales dans le(s) mode(s) défaillant(s). Très récemment, le problème classique de la régulation linéaire et de la poursuite à coût quadratique (LQR) à horizon fini a été reformulé à l'aide de la représentation image et sa solution implantée via le principe de l'horizon fuyant conduit au contrôle prédictif basée données dans le cadre comportemental [CLD19][CLD19 ; Ber+21] [Hua+22]. Malheureusement, les solutions proposées ne sont pas transposables aux commandes tolérantes aux défauts pour la raison suivante : les données de la matrice de Hankel sont des données préalablement enregistrées en boucle ouverte. Or un système défaillant passe par des modes dynamiques qui ne sont pas le mode nominal décrit par l'espace image de cette matrice de Hankel des données préalablement enregistrées hors ligne. Se pose alors la question de la mise à jour de cette matrice de Hankel lors d'une défaillance :

• comment construire les fenêtres de données qui forment les colonnes de la matrices de Hankel suite à une défaillance du système au regard de l'exigence d'une rapidité à l'accommodation aux défauts ?

• comment mettre à jour cette matrice de Hankel des données en temps réel ?

• comment satisfaire à l'exigence de la persistance de l'excitation du signal d'entrée en boucle fermée pour que l'image de la matrice de Hankel soit l'ensemble des trajectoires dans le mode défaillant ?.

• est-il possible de formuler le problème d'optimisation du contrôle prédictif en y intégrant directement cette exigence de persistance de l'excitation de l'entrée de commande, soit dans la fonction de coût, ou soit sous forme de contrainte ? Cette problématique du double rôle du signal d'entrée, i.e., excitation et commande conjointe, est un problème de commande duale qui rejoint l'une de mes premières thématiques décrites dans mes activités de recherche antérieures à ma mobilité vers le CRAN.

De nombreuses autres questions restent ouvertes dans cette approche contrôle prédictif basée données du problème FTC dans le cadre comportemental, entre autres, la formulation d'objectifs de performance économique, les garanties de stabilité, l'extension de la méthode à la commande FTC des systèmes distribués ou de grande dimension et aux systèmes faiblement non-linéaires tels que les systèmes bilinéaires.

Des collaborations intéressantes peuvent être envisagées aussi bien avec des doctorants qu'avec des équipes de recherche en automatique et/ou en apprentissage automatique (machine learning/IA). En particulier, et compte tenus de nos travaux communs antérieurs en FTC par l'approche comportementale, je poursuivrai ma collaboration internationale avec l'équipe de Tushar JAIN de l'Indian Institute of Technology, Mandi, Inde.

I•

  Notice d'activités 5 Model-Based Fault-Diagnosis 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Bilinear-observer based FDD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Robust adaptive fault estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III Perspectives de recherche 6 Projet scientifique 6.1 Contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Efficacité Energétique des Bâtiments : Commande axée sur l'occupant . . . . . . . . 6.3 Commande optimale tolérante aux défauts basée données par l'approche comportementale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Membre élu du Conseil de secteur MIAE de la Faculté des Sciences et Technologies (2014-2018) • Responsable de la spécialité ATIE (Automatique et Traitement de l'Information Embarquées) du Master Ingénierie des Systèmes Complexes -ISC (2009-2013) • Référent du parcours Automatique du Master IS-EEAPR (2005-2009) • Responsables de plusieurs unités d'enseignements en Licence (L1, L2 -SPI, L3 Génie Civil) et Master 1, Master 2 (EAA-CEE, ISC) • Membre nommé des Jurys des semestres S7 et S8 (M1-EEAPR 2005-2009) et des semestres S9 et S10 (M2-ISC 2009-2013) • Membre des comités de sélection (concours MCF) -poste MCF N°61-MCF-0040, 2012 (ESIAL) -poste MCF N°61-MCF-1323, 2011 (IUT Longwy)

•

  Eléments sur les machines à états finis • Systèmes hybrides, automates temporisés • Commandes pilotées par le temps et commandes pilotées par des événements • Systèmes contrôlés en réseau et systèmes distribués • Aspects temps-réel des systèmes cyber-physiques : systèmes multitâches, ordonnancement, contraintes temporelles. • Notion sur la conception conjointe commande/ordonnancement UE -Diagnostic des systèmes (16h CM, 14h TD, 15h TP) Cette UE est enseignée aux étudiants du Master 2 EEA-CEE et ISC parcours SMS-PHM (Sûreté, Maintenance, Soutien -Prognosis Health Management). Ce module présente les techniques modernes de surveillance des systèmes à base de modèles (redondance analytique) et est organisé autour des points suivants : • Introduction au diagnostic et à la surveillance des systèmes, monitoring classique • Validation (réconciliation) de données • Cartes de contrôle : Shewart, EWMA, Cusum • Concepts de redondance : redondance matérielle et redondance analytique • Propriétés de base des algorithmes de détection et d'isolation de défauts • Génération de résidus à base de modèles -Méthode de l'espace de parité : statique et dynamique -Techniques à base d'observateurs, Filtre de Kalman -Techniques à base d'estimation paramétrique Les connaissances acquises dans ce cours sont directement mises en oeuvre dans un projet collaboratif d'intégration système sur la plateforme TELMA (AIPL) UE -Modélisation des Systèmes -Master 1 -(8h CM, 8h TD et 8hTP) Ce module d'automatique fondamentale concerne les étudiants du Master 1 EEA-CEE et a pour but d'élargir leurs compétences initiales dans le domaine de l'automatique continue. Il est consacré à l'étude des systèmes dynamiques par l'approche d'état et s'articule autour de cinq leçons : • Introduction à l'automatique : concepts et objectifs • Eléments de modélisation des systèmes dynamiques, Linéarisation taylorienne, Exemples • Analyse des systèmes linéaires dans l'espace d'état, stabilité externe et interne, Théorie de Lyapunov • Modèles échantillonnés

  Méthodes d'analyse en SdF (analyse fonctionnelle, APR, analyse quantitative) • Mathématiques de la SdF • Fiabilité des systèmes • Utilisation des graphes de Markov en SdF • Simulation Monte-Carlo en SdF UE -Commande Avancée des Systèmes Embarqués -Master 2 -(16h CM) Ce module est enseigné aux étudiants du parcours automatique (ATIE) du Master ISC 2ème année et des étudiants de l'ENSEM et de Polytech Nancy inscrits au Master ISC -recherche

Figure 1 :

 1 Figure 1: Architecture distribuée de commandes FTC/FDI avec communication via un réseau numérique (source: [Men10])

  co-dirigé quatre (4) thèses soutenues : • Kamel MENIGHED (01/09/2006 -23/09/2010).

•• 1 conférence plénière 2

 2 17 articles en revues internationales (dont 1 accepté le 28/10/22 et dont 3 revues antérieures à mes travaux au CRAN) + 1 article soumis 2 • 1 monographie • 2 chapitres de livres (antérieurs à mes travaux au CRAN) • 79 articles en conférences internationales (dont 19 conférences antérieures à mes travaux au CRAN) Mona Subramanian A., T. Jain, J.J. Yamé -Bilinear observer-based robust fault estimation with application to air-conditioning VAV systems, submitted to Journal of Process Control, on November 15th, 2021 (2nd round of reviews) • étudiants master recherche encadrés • stagiaire ingénieur-CNAM encadré Projets labellisés (porteur et/ou membre participant) • projets européens • projet ANR • projet PHC (programme Hubert Curien) • projet BQR d'équipes labellisées • projet CPER • projet Région Wallone (Belgique)

[ 1 -

 1 New] -K. Menighed, J.J. Yamé, I. Chekakta. A Non-Cooperative Distributed Model Predictive Control using Laguerre Functions for Large-Scale Interconnected Systems, Journal Européen des Systèmes Automatisés (ISSN -1269-6935), Accepté le 28/10/22[START_REF]Modélisation des systèmes dynamiques continus (Transparents)[END_REF] -A. M. Subramaniam, T. Jain, J. J. Yamé. Bilinear model-based diagnosis of lock-in-place failures of variable-air volume HVAC systems of multizone buildings. Journal of Building Engineering, Elsevier, 2020, 28, pp.101023. doi :10.1016/j.jobe.2019.101023[2] -T. Jain, J. J. Yamé. Health-aware fault-tolerant receding horizon control of wind turbines. Control Engineering Practice, Elsevier, 2020, 95 :104236. doi :10.1016/j.conengprac.2019.104236.[3] -T. Jain and J. J. Yamé ; Fault-Tolerant Economic Model Predictive Control for Wind Turbines. IEEE Transactions on Sustainable Energy, vol. 10, no. 4, pp. 1696-1704, Oct. 2019. doi : 10.1109/TSTE.2018.2869480 [4] -Jain, T., Yamé, J. J. (2016) Synthesis and application of a real-time model-free behavioral controller with bumpless switching mechanism. International Journal of Adaptive Control Signal Processing,30(5) : 736-749, May 2016, doi :10.1002/acs.2636.[5] -T. Jain, Yamé J.J. , and Sauter D., A novel trajectory-based online controller design approach to fault accommodation in NREL's 5MW wind turbine systems. Control Theory and Technology, 12(2) :122-131, May 2014a. doi : 10.1007/s11768-014-2234-z.[6] -Jain T, Yamé. J.J., On real-time smooth interconnection of online synthesized controllers in the behavioral framework. Systems Science & Control Engineering, 2(1) :398-405, April 2014. doi : 10.1080/21642583.2014.912568. [7] -Jain T., Yamé,J.J., Sauter D., Trajectory-based real-time control of an electrical circuit against unknown faults. Journal of The Franklin Institute, 351(2) :986-1000, February 2014, doi : 10.1016/j.jfranklin.2013.10.005. [8] -Jain T., Yamé J.J. , Sauter D., A novel approach to real-time fault accommodation in NREL's 5-MW wind tu rbine systems. IEEE Transactions on Sustainable Energy, 4(4) :1082-1090, July on Advanced Intelligent Mechatronics, AIM 2019, Hong Kong, China, July 2019. doi :10.1109/AIM.2019.8868354. [30] Subramaniam A.M., Jain T., and Yamé, J.J.,-Decentralized economic model predictive control for energy efficiency in a multi-zone building. Proceedings of the 8th IEEE International Conference on Power Electronics, Drives and Energy Systems, PEDES 2018, Chennai, India, December 2018, doi : 10.1109/PEDES.2018.8707751 [31] Jain T, Yamé. J.J., -eMPC Based Fault-Adaptive Control for Wind Turbine. Proceedings of the 57th IEEE Conference on Decision and Control, CDC 2018, Miami, Florida, United States, December 2018, pp. 6353-6358 doi : 10.1109/CDC.2018.8619190 [32] Jain T., Yamé J.J. -eMPC based fault-tolerant control for wind turbine converters. Proceedings of the 10th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes, SAFEPROCESS 2018, Warsaw, Poland, August 2018. pp. 1343-1348, doi : 10.1016/j.ifacol.2018.09.560. [33] Darure T., Puig V., Yamé J.J., Hamelin F., and Wang Y., -Distributed model predictive control applied to a VAV based HVAC system based on sensitivity analysis. Proceedings of the 6th IFAC Conference on Nonlinear Model Predictive Control, NMPC 2018, Madison, Wisconsin, United States, August 2018. doi : 10.1016/j.ifacol.2018.11.023.[34] -Darure T., Yamé J.J., Hamelin F., -Total variation regularized economic model predictive control applied to a multizone building. Proceedings of the 14th International Conference on Control, Automation, Robotics and Vision, ICARCV 2016, Phuket, Thailand, November 2016. doi : 10.1109/ICARCV.2016.7838700 [35] -Jain T., Yamé J.J., Sauter D., -Data-driven fault-tolerant control for energy efficiency in a multi-zone building. Proceedings of the 14th International Conference on Control, Automation, Robotics and Vision, ICARCV 2016, Phuket, Thailand, November 2016. doi : 10.1109/ICARCV.2016.7838703 [36] -Darure T., Yamé J.J., Hamelin F., -Model-based fault-tolerant control of VAV damper lock-in place failure in a multizone building. Proceedings of the 14th International Conference on Control, Automation, Robotics and Vision, ICARCV 2016, Phuket, Thailand, November 2016. doi : 10.1109/ICARCV.2016.7838699 [37] -Darure T., Yamé J.J., Hamelin F., -Fault-adaptive control of VAV damper stuck in a multizone building. Proceedings of the 3rd Conference on Control and Fault-Tolerant Systems, SysTol, Barcelona, Spain, September 2016, Pages : 170-176, doi :10.1109/SYSTOL.2016.7739746 [38] -Yamé J., Jain T., Sauter D., An Online Controller Redesign Based Fault-Tolerant Strategy for Thermal Comfort in a Multi-Zone Building, Proceedings of IEEE Multiconference on Systems and Control, Sydney, Australia, September 2015, pp :1901-1906 [39] -Sauter D , Yamé J.J., Aubrun C., and Hamelin F., Design of fault isolation filter for control reconfiguration : application to energy efficiency control in buildings. Proceedings of the 23rd Mediterranean Conference on Control & Automation, MED 2015, Torremolinos, Spain, June 2015, pp : for smooth transition in multimode systems. Proc. of the Conference on Control and Fault-Tolerant Systems, SysTol'10, pp. 528-533, France (2010) [65] -Jain T., Yamé J.J., Sauter D. -A real-time router fault accommodation. Proc. of the Conference on Control and Fault-Tolerant Systems, SysTol'10, pp. 899-903, France (2010) [66] -Jain T., Yamé J.J., Sauter D. -A model-based 2-DOF fault tolerant control strategy. Proc. of the 18th Mediterranean Conference on Control and Automation, MED'2010, pp. 1073-1078, Morocco (2010) [67] -Menighed K., Yamé J. J., Aubrun C.-Modification of an adaptive observer to improve its performance in actuator fault detection and estimation. Proc. of the 18th Mediterranean Conference on Control and Automation, MED'2010, pp. 75-81, Morocco (2010) [68] -Menighed K., Aubrun C., Yamé J. J., Distributed state estimation and model predictive control : application to fault tolerant control, Proceedings of the IEEE International Conference on Control and Automation, Christchurch, New Zealand, 2009, pp. 936-941, doi : 10.1109/ICCA.2009.5410390 [69] -M. Kinnaert, T. Delwiche, J. Yamé -State resetting for bumpless switching in supervisory control, European Control Conference, ECC'09, Budapest, Hungary, August 23-26, 2009 [70] -D. Sauter, J. Yamé, Boukhobza T. -Design of Fault isolation filters under intermittent communication, 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, SAFEPROCESS 2009, pp. 804-809, Barcelona, Spain, June 30th-July 3rd, 2009 [71] -Boughanmi N., Cruz-Sanchez H., Song Ye-Qiong , Al-Salami I., Yamé J.J. . Case study of wireless networked control system : CONECS and COWNECS platform. 2nd Workshop Surveillance, S ûreté et S écurité des Grands Systèmes -3SGS'09, Nancy, France, June 2009. [72] -Yamé J.J.., D. Sauter -A Real-Time Model-Free reconfiguration Mechanism for Fault-Tolerance: Application to a Hydraulic Process, Proceedings of the 10th Intl. Conf. on Control, Automation, Robotics and Vision, Hanoi, Vietnam, December 17-20, 2008, pp. 91-96 [73] -Yamé J.J., D. Sauter, C. Aubrun -Design of Fault-Tolerant Controllers for Guaranteed H 2 -Performance over Digital Networks with Time-Varying Communication Delays, Proceedings of the 17th IFAC World Congress, Seoul, Korea, July 6-11, 2008, pp.1273-1278 [74] -D. Theilliol, J-C. Ponsart, M. Rodrigues, S. Aberkane, Yamé J.J. -Design of Sensor Fault Diagnosis Method for Nonlinear Systems Described by Linear Polynomial Matrices Formulation: Application to a Winding Machine, Proceedings of the 17th IFAC World Congress, Seoul, Korea, July 6-11, 2008, pp.1890-1895 [75] -C. Cuzzocrea, A. Dandache, J.-P. Georges, P. Jean, F. Monteiro, D. Theilliol , Yamé J.J.-Analysis of Wireless Transmissions QoS relatively to the Dependability of a Networked Control System, Proceedings of the 23rd IAR Workshop on Advanced Control and Diagnosis, Coventry, UK, November 27-28, 2008, pp. 210-215 Mémoires [118] -Yamé J.J. -Contribution to the Mathematical Theory of Sampled-Data Control Systems, Thèse de doctorat en sciences appliquées, Université Libre de Bruxelles, Ecole Polytechnique, 2001 [119] -Yamé, J.J. -Commande Duale Adaptive des Systèmes Stochastiques, Mémoire du grade complémentaire d'Ingénieur en Automatique (3ème cycle), Université Libre de Bruxelles, Ecole Polytechnique, Décembre 1986 [120] -Yamé, J.J. -Propagation des Ondes Electromagnétiques dans les Corps Biologiques, Mémoire de fin d'études du Grade d'Ingénieur Civil (Ir), Université Libre de Bruxelles, Ecole Polytechnique, Juin 1985 Preprints (IEEE -TechRxiv) [121] -Subramaniam, Mona; Jain, Tushar; Yamé, Joseph (2021): Robust adaptive fault estimation for multizone building VAV terminal units. TechRxiv. Preprint. https://doi.org/10.36227/techrxiv.14826705.v1 (Usage metrics on 24/06/2022 : 773 downloads and 1007 downloads) Séminaires (Invité) T. Jain, J.J. Yamé, Sauter D. -Synergy of canonical control and unfalsified control concept to achieve fault tolerance Séminaire "Journées S3", ENSAM Paris, 08 novembre 2010 Yamé J.J. -Plate-forme Expérimentale Pont Roulant : Du modèle à la Commande Multivariable dans un Environnement Communicant.

Figure 1 . 1 :

 11 Figure 1.1: Generic fault-tolerant control structure

  p×m and a permutation matrix Π ∈ R q×q such that B consists of all w : R → R q for which there exist u : R → R m , x : R → R n and y : R → R p such that ξx = Ax + Bu, y = Cx + Du (2.1.5) and Πw = u y .

Figure 2 . 2 :

 22 Figure 2.2: Feedback loop as an interconnection of two systems in the behavioral framework

Figure 2

 2 Figure 2.3: Closed-loop behavior K = P ∩ C as an intersection of the plant and controller behaviors

  |s ∈ P and O τ (s) ∈ D τ is the preimage of (D τ ). The test (2.2.3) allows a controller to be validated or invalidate on the sole basis of the data generated by the plant without any assumption on the plant. The signals in the set O -1 τ (D τ ) ∩ C are those signals s ∈ S T which are compatible with the measured plant data (u m , y m ) and the controller behavior C. If the controller C were in the feedback loop when the measured data (u m , y m ) was acquired, then the set of such compatible signals would be

Figure 2 . 4 :

 24 Figure 2.4: Internal structure of the supervisor

Figure 2 . 5 :

 25 Figure 2.5: Conventional feedback control with its representation in the partial interconnection paradigm

Figure 3 . 1 :

 31 Figure 3.1: Switching from controller C p to controller C f

Figure 3

 3 Figure 3.2: Example : Non-smooth versus smooth interconnection

Figure 3

 3 Figure 3.3: Self-conditioned implementation of a controller

  a Hurwitz monic polynomial such that λ(s) = a i (s), and with I m , the m × m identity matrix . The definitions of A p and B p are similar to those of A m and B m , with p substituted for m. In equation (3.3.10), matrices E i and D i are given by

Chapter 4 Model

 4 -Based Fault-Tolerant Control : The Predictive Control Paradigm Abstract This chapter presents my research activities in fault tolerant control in the model-based paradigm.

Figure 4 . 1 :

 41 Figure 4.1: Multizone building HVAC-VAV system

x min 1 = 22 . 5 • C and x max 1 =

 12251 [START_REF] Gabsi | ε -Regularized Economic Model Predictive Control for Thermal Comfort in Multizone Buildings[END_REF].5 • C, is well respected. Note that the economic cost (4.2.2) used in optimization problem does not impose any smoothing on the control signals. The classical approach for such control smoothing in standard MPC-based on quadratic costs is obtained via a quadratic term penalizing control moves, that is, a term of the form ψ quad

  Figure 4.2: Optimal zone temperature and supply air mass flow rate to the zone's VAV -(Source: Darure, Yamé, Hamelin [DYH16c])

Figure 4 . 3 :

 43 Figure 4.3: Steps in the convexification of the eMPC based fault-tolerant control problem

Algorithme 2 : 1 : 4 :

 214 KKT matrix based system decompositionData : Input data : h = [h 1 , ..., h n h ] T ; x = vec[x T 1 , ..., x T n ]; f (x) Result : xi , (i =1, ..., p) Derive the KKT matrix K c for problem P c 2: Transform K c into K d using sparse reverse Cuthill-McKee algorithm 3: Identify separable blocks from KKT matrix K d Determine the sub-vectors xi , (i = 1, ..., p) based on the identified separable blocks.

Algorithme 3 :T 1 : 4 :

 314 OCD-based distributed control synthesis Data : Input data : f ; h = [ h1 , ..., hp ] T ; x min , x max , Initial guess xi , (i = 1, ..., p) Result : Optimal decision variables x * = [x * 1 , ..., x * p ] Solve in parallel the optimization problems P i , (i = 1, ..., p) given in (4.3.13) and obtain primal and dual search directions (the Newton step) s i and δ i for the given iteration 2: Update the primal and dual variables : xi = xi + s i ; λi = λi + δ i 3: Go to step 6 if variables do not change significantly in two consecutive iterations, otherwise continue Broadcast the updated variables on the communication network 5: Update the coupling constraints in (4.3.13) and go to step 1 6: Implement x * ← [ x1 , ..., xp ] T .

Assumption 2 .

 2 The Jacobian of the constraints, ∇h, has full row rank at x * Assumption 3. Matrices Ǩk are non-singular for any ξ * and the sequence { Ǩk } converges to a non-singular matrix Ǩ * = lim ξ k →ξ * Ǩk Denote by K * d the evaluation of KKT matrix (4.3.10) at the optimal primal-dual point ξ * , then clearly K * d π * = -γ * = 0. The following is the main result which characterizes the optimal distributed solution [Yam+19]: Yamé et al.(2019)-Convergence of distibuted MPC control to the target centralized solution Under Assumptions 1,2 and 3, if at the optimal point ξ * of the centralized problem (4.3.2) it holds that I -Ǩ * -1 K * d < 1 (4.3.21) then the procedure used in Algorithm 3 converges globally to x *

4 .3. 26 )

 426 Using the sparsity pattern command of MATLAB, "spy", we visualize the structure of K c in figure4.4 (left image). This KKT matrix is then transformed into an equivalent KKT matrix K d with reduced bandwith as displayed in figure4.4 (right image) by the reverse Cuthill-McKee reordering algorithm using MATLAB command "symrcm". The resulting permutation vector is p = [1 4 3 2].

Figure 4 . 4 :

 44 Figure 4.4: Original KKT matrix K c and its equivalent reordering K d

Proposition 5 . 1 .

 51 If C Ēum is a vector with non-zero entries, i.e., C Ēum = h 1 , • • • , h n T with h i = 0, i = 1, 2, . . . , n, then there exists a gain matrix G such that (i) -the vector C Ēum d um ∈ ker (G), and (ii) -C F (t)(f a -fa ) ∈ ker (G).

Figure 5 . 1 :

 51 Figure 5.1: Robust fault detection and diagnosis methodology for multizone building VAV terminal units

Algorithme 4 :

 4 Robust Adaptive Fault Estimator1: procedure RAFE (A, B, C, D, E m ) Compute β 1 , β 2 , L according to Proposition 5.2 4:

  Figure 5.2: One-storey building , outside air temperature and internal heat gains

Figure 5 Figure 5 . 4 :

 554 Figure 5.3: Scenario with mutually exclusive faults

  et al. dans le cadre comportemental, connu sous le nom de « lemme fondamental », a émergé au cours des deux dernières années[START_REF] Maupong | Data-driven control: A behavioral approach[END_REF][Waa+20][START_REF] De Persis | Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness[END_REF]. Ce lemme fondamental est devenu le pivot de ces efforts récents de recherche sur la commande pilotée par les données. En termes simples, ce lemme stipule que, sous des hypothèses appropriées, toute trajectoire entrée-sortie d'un système linéaire invariant dans le temps peut être décrite comme une combinaison linéaire d'une trajectoire unique enregistrée précédemment[START_REF] Willems | A note on persistency of excitation[END_REF]. Sans entrer dans des détails mathématiques, soit w = vec u T y T désigne le vecteur des entrées u et de sorties y du système et supposons qu'on dispose de mesures issues du système sous la forme d'une série temporelle w d = {w d (0), w d (1), ..., w d (N -1)} sur une fenêtre suffisamment large N , où l'indice d signifie que ces signaux sont des données expérimentales. Construisons la matrice de Hankel des données à L blocs-lignes comme suit, où L < N ,H L (w d ) =    w d (0) w d (1) • • • w d (N -L) . . . . . . • • • . . . w d (L -1) w d (L) • • • w d (N -1)condition que le signal d'entrée u est suffisamment riche (persistance de l'excitation), le comportement du système sur des horizons de longueur L, i.e. l'ensemble de ces trajectoires w = {w(k)} L-1+j k=j pour tout j ∈ Z, est dans l'espace image de la matrice de Hankel, i.e. w = H L (w d )ζ (6.3.2) où ζ est un vecteur arbitraire de dimension appropriée qui paramétrise ces trajectoires. La représentation basée données d'un système via (6.3.2) permet donc de générer tous ces signaux à l'aide du paramètre ζ et ouvre un vaste champ de possibilités pour l'analyse et la synthèse directe des systèmes dynamiques sans utilisation d'un modèle classique. On note que la représentation image (6.3.2) est étroitement liée à la représentation noyau de mes travaux en FTC décrite par des équations comportementales [JYS18] où les trajectoires w sont contraintes par la dynamique du système sous la forme R(ξ)w = 0 (6.3.3) où R(ξ) = R 0 + R 1 ξ + ... + R l ξ l est une matrice polynomial en ξ et où l'indéterminée ξ peut être vue comme l'opérateur de décalage temporel, i.e., ξw(t) = w(t + 1). En regardant de près cette représentation, par l'introduction d'une matrice R = R 0 R 1 ... R l associée à la matrice polynomiale R(ξ), alors pour des données w d = {w d (0), w d (1), ..., w d (N -1)} générées par le système, la représentation noyau s'écrit de manière équivalente RH l (w d ) = 0 (6.3.4) Ceci montre que le noyau à gauche de la matrice de Hankel des données caractérise le comportement du système. Cette observation permet d'étendre nos travaux en FTC avec un nouveau regard donné par le lemme fondamental de Willems.

  IntroductionJe ne reviendrai pas ici sur mes activités pédagogiques avant mon recrutement comme Maître de Conférences à l'Université de Lorraine en 2005. Depuis mon recrutement en septembre 2005 à l'UL, tous mes enseignements sont essentiellement assurés dans ma composante de rattachement pour la partie enseignement (Faculté des Sciences et Technologies -FST). J'effectue mon service statutaire dans tous les cycles de l'offre de formation LMD principalement dans le profil « automatique » mais j'ai eu aussi des interventions à Nancy-Polytech et à Télécom-Nancy ainsi qu'en formation continue auprès d'auditeurs du Conservatoire National des Arts et Métiers (Grand Est). J'ai en outre dispensé des modules d'enseignements en automatique pour la licence Ingénierie Electrique à Distance

(LIE CNED) et pour la préparation au concours d'agrégation externe des sciences industrielles de l'ingénieur. J'ai pris des responsabilités pédagogiques dès ma prise de fonction en 2005 et ce jusqu'en 2009 comme enseignant référant du parcours automatique du Master IS-EEAPR puis comme responsable de la spécialité « automatique et traitement de l'information embarqués » du master Ingénierie des systèmes complexes de 2009 à 2013. En 2021, j'ai pris la responsabilité du portage du parcours « Contrôle de l'Efficacité Energétique (CEE) » du Master EEA pour l'accréditation 2024-2028.

Je suis responsable d'UEs de Licence SPI, du Master EEA-CEE et du Master ISC. Mes enseignements se répartissent entre l'automatique (40%), la sûreté de fonctionnement, la surveillance et l'analyse des données (40%) et l'informatique temps-réel (5%) et des UEs de formation générale (mécanique des fluides, analyse numérique, méthodologie, découverte des sciences pour l'ingénieur) (15%).

  Travaux au Centre de Recherche en Automatique de Nancy Suite à mon intégration au CRAN dans le groupe thématique SURFDIAG (Sûreté de fonctionnement et diagnostic) en septembre 2005, le point central de mes travaux s'est porté sur la commande tolérante aux défauts (Fault-Tolerant Control -FTC). C'est l'une des thématiques majeures du CRAN qui relève aujourd'hui du projet "Commande tolérante -Diagnostic de défauts et Sûreté (CDuSûr)" du département Contrôle, Identification et Diagnostic (CID). Les thèses de Kamel Menighed, Tushar Jain, Darure Tejaswinee et Mona Subramaniam A que j'ai codirigées s'inscrivent toutes dans cette thématique centrale mais sous différents aspects théoriques, méthodologiques et applicatifs. En particulier, ces travaux en FTC ont concerné :Les méthodes auxquelles je m'intéresse dans le cadre de ces travaux sont aussi bien des méthodes à base de modèles que des méthodes à base de données n'exigeant pas a priori une connaissance du modèle du processus sous contrôle. Ces travaux ont permis d'apporter des solutions originales et efficaces aux problématiques FTC/FDD traitées, notamment en faisant appel à des techniques mathématiques variées relevant de la théorie des systèmes et des techniques d'optimisation. Les publications issues de ces travaux sont mentionnées dans la liste exhaustive de documents de ma production scientifique dans la section I.Sur les systèmes contrôlés en réseaux et les commandes coopératives et embarquéesAu sein du groupe thématique SURFDIAG que j'ai intégré à mon arrivée au CRAN, j'ai demandé à participer à l'équipe-projet DESYR (Diagnostic Embarqué et Reconfiguration des Systèmes de commande distribués via Réseaux) dont le responsable était Dominique Sauter. J'ai donc initialement orienté mes activités de recherche en FTC au CRAN vers la commande tolérante aux défauts des systèmes caractérisés par des boucles de commande/régulation et de supervision se fermant via des réseaux de communications numériques. Ces systèmes sont usuellement sous le sigle NCS pour «Networked Control Systems». Contrairement aux systèmes de commandes classiques, le couplage entre la commande et les phénomènes de communication dans ces systèmes ne peuvent être négligés. Ce couplage induit des contraintes imposées par les restrictions sur la bande passante des canaux de communication, l'échantillonnage et la quantification des signaux, les temps morts dans la transmission des signaux et les pertes de données, et rendent complexes les problèmes de FTC qui non seulement ne sont plus localisés mais peuvent être géographiquement répartis ou distribués[START_REF] Lunze | Control Theory of Digitally Networked Dynamic Systems[END_REF].

	• les systèmes contrôlés via des réseaux de communications numériques
	• les commandes coopératives et embarquées
	• l'approche comportementale en FTC et le développement de commandes FTC sans module
	de diagnostic

• la problématique des commutations de commandes dans les structures FTC multi-contrôleurs • les commandes prédictives économiques à base de modèle • la détection et l'estimation de défauts (Fault detection and Diagnosis -FDD) à base d'observateurs • les applications de la FTC aux éoliennes et aux systèmes de chauffage, ventilation et conditionnement d'air (HVAC) des bâtiments Au moment où je démarrai ces travaux, la conception et la mise en oeuvre de systèmes de détection et d'isolation de défauts (FDI -Fault Detection and Isolation) et de commande tolérante aux défauts pour ces systèmes étaient encore peu étudiées et ces travaux font d'ailleurs encore aujourd'hui l'objet de nombreux efforts de recherche. En outre, le projet SafeNECS «Safe Networked Control Systems» dont Dominique Sauter était le coordonateur dans un consortium CRAN, LORIA, LAAS, GIPSA-LAG et INRIA Rhones-Alpes, venait d'être retenu pour un démarrage en décembre 2005 avec un financement de l'Agence Nationale de Recherche (ANR). J'ai donc été impliqué à 50% (de mes activités de recherche) au projet SafeNECS dès son démarrage. C'est dans ce cadre, et en vue d'un support applicatif à nos travaux, que j'ai introduit avec Dominique Sauter une demande de financement BQR-Région Lorraine pour la construction d'une plate-forme expérimentale «Pont Roulant» dont j'ai réalisé la conception et le dimensionnement avec Abishek Gupta (étudiant stagiaire) que j'ai encadré en 2006. La plate-forme matérielle est un « double pont roulant » consistant en un dispositif composé de deux chariots mobiles plus ou moins autonomes embarquant des contrôleurs, des capteurs et des systèmes de communications sans fil et d'un calculateur central de supervision muni également d'un système de communication sans fil. Initialement dédié au projet SafeNecs, le «double pont roulant» servira aussi de benchmark de référence pour le projet CPER MISN -SECTAF (Systèmes Embarqués Communicants Tolérants aux Fautes) et de plateforme d'intégration système dans les enseignements du Master ISC-ATIE (Automatique et Traitement Embarqués de l'Information) Sous l'angle méthodologique, je me suis intéressé à deux aspects importants de la FTC via un réseau de communication: un aspect temporel relatif aux retards de communication et un aspect «spatial» relatif à la distribution de contrôleurs locaux FTC pour des systèmes fortement couplés ou de grande dimension. Sous l'aspect temporel des NCS, je me suis particulièrement intéressé aux problématiques des retards induits par le réseau en boucle fermée où ces retards sont modélisés comme des grandeurs τ (k) variant en fonction du temps k, et résultant des délais de communication entre capteurs et contrôleurs. Sous l'hypothèse (réaliste) que ces retards de communication τ (k) sont bornés, avec des bornes inférieures et des bornes supérieures connues, nous avons montré que la performance H 2 de la commande FTC pour des défaillances «actionneurs» peut être guarantie même en présence d'incertitudes (bornées) des paramètres du modèle mathématique du systèmes. Ce résultat impor-

  -systèmes en interactions se fondent sur une étape de diagnostic local pour chaque sous-système. Le diagnostic réalisé à l'aide d'observateurs permet la détection et l'estimation les défauts pour une mise à jour des modèles des sous-systèmes pour les commandes locales FTC. Ces commandes FTC s'appuient sur des algorithmes de commandes prédictives avec une coordination des commandes réalisée à l'aide des prédictions des interactions et les échanges d'information via le réseau de communication. Les travaux effectués dans le cadre de cette thèse ont donné à différentes communications dans des conférences internationales[START_REF] Yamé | Robust adaptive estimation of arbitrary time-varying actuator faults[END_REF][START_REF] Menighed | Cooperative control for fault tolerance in distributed networked systems : A model predictive control approach[END_REF][START_REF] Menighed | Fault-tolerant cooperative control : a distributed model predictive control approach[END_REF][START_REF] Menighed | Aubrun C -A slightly improved robust adaptive observer for actuator fault detection[END_REF] 67, 68].Cette thématique vise à poursuivre les travaux originaux de recherche en FTC que j'ai initié avant ma mobilité vers le CRAN. L'idée fondamentale de ce projet, motivée par les exigences de sécurité critique et de l'inadaptation des modèles classiques en présences de fortes incertitudes résultant des défaillances, est d'utiliser une approche de modélisation basée sur le concept de "comportement" pour concevoir des stratégies de commande tolérante aux défauts. L'approche comportementale des systèmes (behavioral system approach) est un nouveau paradigme de modélisation des systèmes dynamiques développé principalement par J.

	requiert pas l'utilisation explicite de modules de diagnostic, la détection des dysfonctionnements et
	défauts étant implicitement réalisée sur la base des trajectoires et de la violation des spécifications
	exprimées sous la forme d'une fonctionnelle (coût) ou sous la forme d'une dynamique désirée en
	boucle fermée. Le caractère générique des techniques FTC issues de l'approche comportementale
	permet l'application des résultats obtenus à une large classe de systèmes dynamiques.
	Sur la commande tolérante aux défauts par l'approche comportementale
	des
	comportements désirés. Le concept d'interconnexion de J.C. Willems [Wil97] utilisé comme idée de
	base de la commande FTC permet de spécifier l'ensemble admissible des trajectoires du système
	commandé qui sont compatibles avec les spécifications de performance. Les questions de réalisabilité
	et d'implémentabilité d'un comportement désiré résultant des contraintes d'interconnexion 'contrô-
	leur/système défaillant' sont des points essentiels de ce travail avec pour résultats marquants la
	synthèse de contrôleurs qualifiés de « canoniques » qui réalisent le comportement spécifié unique-
	ment à partir des trajectoires issues en temps réel du système contrôlé. Les méthodes proposées ne

La problématique centrale est d'assurer, dans une architecture distribuée, l'autonomie des sous-systèmes en cas de défaillances sévères du système (actionneurs, capteurs, composants) ou du réseau de communication entre différentes unités (sous-systèmes) du système global.

Plus particulèrement, la stratégie de commande tolérante aux défauts du système global décom-posé en sousC.

Willems [Wil91b ;[START_REF] Willems | On interconnections, control, and feedback[END_REF][START_REF] Willems | The Behavioral Approach to Open and Interconnected Systems[END_REF]

. Le point de vue central de cette approche, en rupture avec les approches classiques basées sur des représentations de type externe (transfert) ou interne (modèle d'état) consiste à identifier un système dynamique à l'ensemble de ses trajectoires ("comportement", behavior) sans distinction initiale entre les entrées et les sorties du système et fournit un cadre unifié pour l'étude mathématique des systèmes. Contrairement aux problèmes d'analyse qui font l'objet d'un certain nombre d'études dans ce paradigme, le domaine important de la commande a été très peu abordé

[START_REF] Van Der Schaft | Achievable behavior of general systems[END_REF]

et a fortiori il n'existe pratiquement pas de travaux relatifs à la commande tolérante aux défauts à l'exception de la série de travaux parus dans mes communications internationales [72,

[START_REF] Yamé | Reconfigurable control via online closed-loop performance assessment[END_REF][START_REF] Yamé | Performance-based supervisory fault tolerant control with application to the IFATIS two-tanks benchmark[END_REF][START_REF] Yamé | A Fault accommodation strategy based on closed-loop performance monitoring[END_REF][START_REF] Yamé | A Behavioral approach to real-time fault-tolerant control[END_REF][START_REF] Yamé | Performance-Based Switching for Fault-Tolerant Control[END_REF]

.

C'est en 2009 que j'ai proposé un sujet de thèse avec une co-direction de Dominique Sauter et Tushar JAIN

[START_REF] Jain | Contribution à la synthèse de commandes tolérantes aux défauts par l'approche comportementale[END_REF]

comme doctorant recruté grâce à une allocation ministérielle. Cette thèse a été entièrement consacrée à cette thématique avec pour objectifs de développer des méthodes génériques de FTC basées sur les trajectoires réelles issues du système en fonctionnement soumis à des défauts plutôt que sur des modes définis par des modèles résultant d'hypothèses faites a priori.

Des résultats importants et significatifs ont été obtenus dans le cadre de ces travaux, notamment sur des techniques FTC classées comme des méthodes dites de « projection » et sur des algorithmes de reconfiguration en ligne dans la classe des méthodes de « re-conception ». Dans la classe des méthodes de projection, nous avons développé une structure FTC multi-contrôleurs où chaque contrôleur de la banque de commandes est associé à un mode (nominal ou défaillant) et la tolérance aux défaillances est réalisé grâce à un superviseur qui identifie en temps-réel le mode actif et le contrôleur associé uniquement sur la base des trajectoires mesurées. Dans la classe des méthodes dites de « reconception », l'idée force est d'exclure les trajectoires (comportements) du système défaillant, ces trajectoires ne répondant pas aux spécifications de performance elles-mêmes considérées comme

  Le projet EiT (date de début de projet : 01/10/2013 -date de fin de projet : 27/01/2017) est un projet collaboratif NMP du 7e Programme Cadre (FP7-NMP, EeB.NMP.2013-4) d'une durée de 48 mois d'ub consortium international constitué par 13 partenaires industriels et universitaires européens (Espagne, Portugal, Irlande, France, Roumanie, Royaume Uni, Italie, Finlande). obtenu un financement BQR à hauteur de 20 000 € pour la construction de la plateforme expérimentale « double-pont roulant » au CRAN. Cette plateforme a servi de support aux activités de recherche (dans le cadre du projet ANR-SafeNecs) et de support d'enseignement pour les projets d'ingénierie système des étudiants du master ISC-ATIE .

	Diffusion et rayonnement Activités éditoriales
	Rayonnement scientifique • Editeur associé (AssociateEditor) des revues internationales
	1. Mathematical Problems in Engineering (depuis 2015) 1. Participation à des jurys de thèse (hors établissement) Outre les jurys de thèse des quatre doctorants : A. Mona Subramanian (2021), T. Darure 2. Entropy (a MDPI Journal) depuis 2021
	(2017), T. Jain (2012), K. Menighed(2010) que j'ai co-dirigé, j'ai participé à deux jurys de
	thèse hors établissement en 2017 et en 2021. J'ai été rapporteur externe des thèses étrangères • Reviewer pour les Mathematical Reviews (MR) -American Mathematical Society
	(PhD) suivantes : (AMS) (2013)
	(a) DEVIKA K.B. (2017) • Activité régulière d'évaluation d'articles de revues (referee)
	Titre de la thèse : Sliding Mode Controller Design for Free-Flight Autopilot System -Je suis très fréquemment sollicité pour la relecture (reviewer) d'articles de journaux et de
	Thèse soutenue le 28/07/2017 [Supervisor (directeur de thèse) : Pr. Susy Thomas] conférences internationales parmi les plus importants :
	National Institute of Technology, Calicut, India
	1. Revues (automatique) : IEEE Transactions on Automatic Control, Automatica (IFAC),
	International Journal of Robust and Nonlinear Control, IEEE Transactions on Automa-(b) RAJASHREE RAGHAVAN (2021) tion and System Engineering, International Journal of Control (IJC) , Control Enginee-Titre de la thèse : Model Predictive Controller Designs for Multivariable Systems -[Su-ring Practice, IET Control Theory and Applications, International Journal of Adaptive pervisor (directeur de thèse) : Pr. Susy Thomas] Control and Signal Processing, Systems and Control Letters, Simulation Modelling Prac-National Institute of Technology, Calicut, India tice and Theory, Mathematical and Computer Modelling of Dynamical Systems
	2. Revues (énergie et bâtiments) : Science and Technology for the Built Environment (Taylor
	& Francis), Journal of Building Engineering (Elsevier), Energy & Buildings (Elsevier), 2. Prix scientifique Building and Environment (Elsevier) The Prize for the Best Theoretical Paper awarded to the paper : " Trajectory-based 3. Conférences internationales : IEEE Conference on Decision and Control (CDC), Ameri-fault accomodation in wind turbine systems " authored by T. Jain, J. Yamé and D. Sauter. can control conference (ACC), European Control Conference (ECC), IFAC World Congress,
	IFAC SafeProcess, IEEE Conference on Control Technology and Applications (CCTA),
	Ce prix a été décerné à cet article suite à mon exposé au 10th International Conference on IEEE-Med
	Control (UKACC) à Loughborough, UK, 9-11th, July, 2014
	Organisation de colloques
	3. Invitation dans des universités étrangères • Membre du comité d'organisation de la conférence IEEE Multi-Conference on Systems and
	11/2014 -Chercheur invité à l'Université des Sciences et Techniques de Pékin, Chine Control (IEEE, MSC'14, Antibes, France, 08-10 Octobre 2014)
	J'ai effectué un séjour de recherche sur invitation du Prof. Xiaoli Li, Department of Control
	Science and Engineering, University of Science and Technology Beijing
	• Autre encadrement scientifique
	Durant l'année académique 2008-2009, J'ai acceuilli et encadré (avec un taux d'encadrement 4. Conférence plénière (Plenary speaker)
	de 100%) au CRAN un stagiaire du Conservatoire National des Arts et Métiers (CNAM) pour J'ai donné une conférence plénière [16] sur mes travaux par l'approche comportementale en la préparation de son mémoire d'Ingénieur : commande tolérante aux défauts lors du 13th European Workshop on Advanced Control and
	Diagnosis (ACD 2016), Lille, France, 17-18
	-Christian CUZZOCREA
	-Titre : Modélisation et commande multivariable dans un environnement communiquant 5. Titulaire de la prime d'encadrement doctorale et de recherche 2015-2018 (avis
	d'un double pont roulant. CNU : A)

J'ai été le « Chair » du « Student Activities and Developing Nations Program » du IEEE pour cette conférence. J'ai géré ce programme doté d'un montant de US$ 10,000 qui a permis de sélectionner 14 étudiants (PhD students) parmi 17 candidatures en provenance de 10 pays différents (Spain, Germany, China, US, India, UK, Tunisia, The Netherlands, Russia, Italy, Canada). L'aide financière octroyée par le programme a couvert les frais d'inscription à la conférence et les frais de séjours (4 nuitées) dans deux hotels réservés à cet effet. Une attestation de participation et de présentation orale de l'article à la conférence a été délivrée à chaque doctorant après son exposé oral. • Organisateur de la session invitée : System Modeling, Control and Diagnosis for Energy Efficiency in Buildings. The 14th International Conference on Control, Automation, Robotics and Vision, November 13-15th, 2016, Phuket, Thailand • Co-président (Co-Chair) et member du comité d'organisation de la conference : International Symposium on Security and Safety of Complex Systems (2SCS), Agadir, Morocco, 2012 • Membre -Technical Program Committee des conférences internationales suivantes : -The 14th International Conference on Control, Automation, Vision, and Robotics, ICARCV, 2016, Phuket, Thailand -The 13th International Conference on Control, Automation, Vision, and Robotics, ICARCV, 2014, Singapore -International Conference on Engineering, Applied Sciences and System Modeling, ICEASSM'2017, 2017, Accra, Ghana -The 3rd World Conference on Complex Systems (WCSS'15), Marrakech, Morocco,2015 -The 2nd World Conference on Complex Systems (WCSS'14), Agadir, Morocco, 2014 • Autres services à la communauté : Présidence (modérateur) de sessions au congrès internationaux (sur les 12 dernières années) -Chair, session ThS2T1 : Energy Management. 4th Conference on Control and Fault-Tolerant Systems, September 18-20, 2019, Casablanca, Morocco -Chair, session WeB03 : Distributed Control II. 2019 American Control Conference, July 10-12, 2019, Philadelphia, PA, USA -Chair, session FrDT3 : FDI/FTC of Railways and Power Plants. Safeprocess 18, 29-31 August 2018, Warsaw, Poland -Co-chair, session « Fault-tolerant systems ». 2015 IEEE Multi-Conference on Systems and Control (MSC'15), Sydney, Australia 2015 -Co-chair, session « Energy Production ». 2014 IEEE Multi-Conference on Systems and Control (MSC'13), Antibes/Nice, France 2014 -Chair, session FrB06 : Linear Control methods II. 2013 IEEE Multi-Conference on Systems and Control (MSC'13), India, 2013 -Co-Chairman, session WeA06 : Control of Wind Energy Systems . 2013 IEEE Multi-Conference on Systems and Control (MSC'13), India, 2013 -Chairman, session Th42 : Intelligent systems and Automation, 12th International Conference on Control, Automation, Robotics & Vision, ICARCV 2012, China, 2012 -Chairman, session TuB03 : Fault Detection IV -50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011, Orlando, Florida, USA 2011 -Chairman, session WM3 : Cooperative control -11th International Conference on Control, Automation, Robotics & Vision, ICARCV 2010, Singapore, 2010 -Co-Chairman, session ThB2 : Nonlinear control and Fault diagnosis -Conference on Control and Fault-Tolerant Systems, SysTol'10, France, 2010 Responsabilités scientifiques de projets de recherche (Contrats de recherche évalués suite à appel à projet) 1. PROJET EUROPÉEN « ENERGY-IN-TIME » -(EiT) -Montage, négociation et administration scientifique J'ai été co-porteur (avec Dominique SAUTER), pour l'UL-CRAN, du projet Européen «Energy-In-Time (EiT) -Simulation Based Control for Energy Efficiency Building Operation and Maintenance ». Site web UE -Research Projects : http ://cordis.europa.eu/projects/rcn/108953_en.html J'ai participé activement au montage, à la négociation et à la rédaction du 'proposal' avec les 12 autres partenaires pour la soumission à l'appel d'offres. Le consortium a remporté l'appel d'offres de ce projet compétitif et d'envergure avec une enveloppe globale de financement de l'UE de € 7.7 Millions (la part UL-CRAN est de 380 000 €). J'ai été le responsable et contact scientifique du projet pour la partie UL-CRAN et en charge de la gestion scientifique et de la coordination des tâches/work-packages et de l'animation de l'équipe impliquée dans le projet (3 doctorants et 6 enseignants-chercheurs). Les tâches principales (work packages (WP)) du CRAN dans le cadre du projet ont porté sur la recherche et le développement de méthodes de modélisation structurelle et d'analyse de propagation de défauts, d'algorithmes de détection/diagnostic de défauts et la conception de modules de commande adaptative et tolérante aux fautes aussi bien au niveau composant qu'au niveau système. En outre, le CRAN a eu en charge un WP sur la maintenance prévisionnelle pour la maîtrise des performances énergétiques des bâtiments. Dans le cadre de ce projet, le financement obtenu a permis de financer trois thèses de doctorat : Darure Tejaswinee (co-directeurs : J. Yamé et F. Hamelin), Krishnan Srinivasarengan (co-directeurs : C. Aubrun et D. Maquin), Collince Christian Nzukam (co-directeurs : E. Levrat , A. Voisin, D. Sauter). 2. J'ai été porteur des projets suivants : (a) Projet TOURNESOL Franco-Belge (2009-2010) : Régulation Robuste des Systèmes Hybrides Incertains, Applications aux Dispositifs de Télé-opération et aux Eoliennes J'ai eu la responsabilité administrative et scientifique de ce projet que j'ai monté avec le Professeur Michel Kinnaert de l'Université Libre de Bruxelles (ULB) pour un partenariat de recherche fort entre le CRAN et le Service d'Automatique et d'Analyse des Systèmes (ULB) dans le cadre du programme Hubert CURIEN. (b) Projet BQR d'équipes labellisées (2006) : Système de Contrôle-Commande Coopératifs Embarqués (projet monté avec Dominique Sauter) J'ai 3. J'ai été fortement impliqué, en tant que membre d'équipe-projet, dans les projets de recherche suivants (rédaction de livrables, rapports d'avancement périodiques et présentations/exposés aux évaluateurs) : (a) Projet CPER MISN -SECTAF (2007-2009) : Systèmes Embarqués Communicants Tolérants aux Fautes (b) Projet ANR SafeNECS (2006-2009) : Systèmes Contrôlés en Réseaux Sûrs de Fonctionnement (c) Projet de Veille Technologique pour le Centre National des Etudes Spatiales -CNES (2008) (d) Projet européen IFATIS -Intelligent Fault Tolerant Control in Integrated Systems (2002-2005) (e) Projet CELOFA (2000-2001) -Centre d'Etudes de la Logique Floue et de ses Applications (f) En 2004, j'ai défendu avec succès, en réunion du Collège d'experts de « Research in Brussels », le projet « Advanced Supervision Systems for Reducing Environmental and Human Risks in Industrial Operations » et j'ai obtenu un financement pour un séjour de 6 mois du Professeur Daniele Romano du Department of Mechanical Engineering, University of Cagliari, Italie comme chercheur visiteur au Service d'Automatique et d'Analyse des Systèmes de l'Université de Bruxelles.

  BCM95] that despite a possible discontinuity of the manifest variables (attached to P) at the switching instant t inter , the state of the plant is continuous at t = t inter , i.e., ζ P

	p (t -inter ) = ζ P f (t + inter ) which implies that ζP (t -inter ) = ζ P f (t + inter ). From the above, it turns out that the boundary condition ˜ (t -inter ) = f (t + inter ) is achieved if and only if ζCf (t -inter ) = ζ Cf f (t + inter

for all t > t inter . Introduce the (past) state p = vec( (ζ P p ) T (ζ Cp p ) T ) of the actual loop P t m ∩ C p evolving on the time axis (-∞, t inter ) , then ζP = ζ P p . It is a well-known fact [

  where j ∈ {p, f} (3.2.7)If the past controller C p no longer satisfies the desired behavior, the supervisor switches the (future) controller C f in the closed-loop. The polynomials defining the kernel behavior (3.2.7) are respectively:n cp = (ξ+1)(53.33ξ+160), d cp = ξ(ξ+44), n cf = (ξ+1)(13.33ξ+40), d cf = ξ(ξ+14).Assume that the controller C f makes an interconnection with the unknown plant at time t inter = 8.4sec, then the resulting trajectories of the system as shown in Fig.3.2 clearly exhibits large switching transients. From the theory developed above, the fulfillment of the condition given in Lemma 3.2.2 is actually achieved through ζ Cf f (t + inter ) = ζCf (t - inter ). The differential equation describing the manifest behavior of the controller is given by d 2 dt 2 u + 14 d dt u = 13.33 d 2 dt An efficient algorithm, known as the shift-and-cut operation [RW97, Algorithm 1-3], is udes to compute the state trajectory as

	ζCf = χ	d dt	s =	σ + (C(ξ)) σ 2 + (C(ξ))	s
	ζCf = χ	d dt	s =	ξ + 14 13.33ξ + 53.33 -(13.33ξ + 53.33) 1 13.33 -13.33	 u r  y 	(3.2.8)

2 (r -y) + 53.33 d dt (r -y) + 40(r -y).

  xi-1 , xi , xi+1 , ..., xp ) + p j=1,j =i λj hj ( x1 , ..., xi-1 , xi , xi+1 , ..., xp ) subject to hi ( x1 , ..., xi-1 , xi , xi+1 , ..., xp ) = 0 xmin

  the time-varying fault distribution matrix as in the previous secion. . Now, partition the disturbance signal, d(t), into measured and unmeasured disturbances denoted by d m (t) and d um (t) respectively, i.e., d = vec d T m d T um and the associated matrix E as E = E m E um . The structure of the bilinear observer-based robust adaptive estimator is devised as

Table 5 .

 5

		1: Fault Scenario
	Fault Type	Damper-1	Damper-2	Damper-3
	Lock-in-place	0.67	0.68	0.7
	(Airflow rate) (0.061 kg/s) (0.064 kg/s) (0.071 kg/s)

  Energétique des Bâtiments : Commande axée sur l'occupantPour contribuer à la performance énergétique des bâtiments, deux leviers peuvent être activés : l'un relatif au bâti (enveloppe), l'autre relatif à l'utilisation juste nécessaire par une gestion active des systèmes et équipements techniques du bâtiment. On parle alors d'efficacité énergétique passive et d'efficacité énergétique active. Notons que l'efficacité énergétique passive relève de l'ingénierie des constructions et de l'architecture et se situe essentiellement dans la phase de conception des bâtiments.Le projet de recherche que nous proposons s'inscrit dans le cadre de l'efficacité énergétique active et a pour objectif de développer des commandes « intelligentes » pour atteindre l'efficience énergétique des bâtiments en phase opérationnelle d'exploitation. De nombreux travaux ont été réalisés ces dernières années sur cette approche active et plusieurs solutions novatrices pour réaliser des objectifs d'efficience énergétique en présence de diverses contraintes ont été proposées pour l'essentiel dans le cadre des dispositifs de gestion de l'énergie et de contrôle du confort intérieur [WKL17 ; TC13 ; KEM17]. En effet, les besoins énergétiques des bâtiments et le confort thermique intérieur sont des exigences contradictoires qui nécessitent des techniques de contrôle optimale pour atteindre des compromis entre ces deux objectifs. Les techniques avancées de commandes automatiques et en particulier le contrôle prédictif basé sur des techniques d'optimisation sous contraintes ont permis de faire des progrès notables dans la gestion énergétique des bâtiments. Le contrôle prédictif est en effet capable d'exploiter à la fois les prévisions des perturbations futures (e.g., les gains internes, les conditions météorologiques, le nombre d'occupants, etc.) et des exigences imposées (e.g., les plages de confort), pour anticiper les besoins énergétiques du bâtiment et optimiser son comportement thermique sur la base des objectifs de contrôle définis[START_REF] Serale | Model Predictive Control for Enhancing Building and HVAC System Energy Efficiency: Problem Formulation, Applications and Opportunities[END_REF]. La technique a été utilisée aussi bien pour la gestion efficace de l'énergie en fonctionnement nominal qu'en situation de défaillances des équipements pour s'accommoder aux défauts et réduire la consommation d'énergie lors d'apparition de modes de défaillances[START_REF] Darure | Fault-adaptive control of VAV damper stuck in a multizone building[END_REF][START_REF] Darure | Model-based fault-tolerant control of VAV damper lock-in place failure in a multizone building[END_REF]. Ces travaux sur le contrôle prédictif sont toujours d'actualité avec une prise en compte de différents aspects économiques, de gestion de la demande et de l'approvisionnement en énergie par des sources d'énergies renouvelables (solaire, éolien, . . .) pour des objectifs de haut niveau de confort thermique, de qualité de l'air de l'environnement intérieur et de haute efficacité énergétique[START_REF] Gabsi | ε -Regularized Economic Model Predictive Control for Thermal Comfort in Multizone Buildings[END_REF].Nos travaux antérieurs, dans le cadre du projet européen Enegy-In-Time avec la thèse de Darure Tejaswinee[START_REF] Darure | Contribution à l'optimisation de la performance énergétique des bâtiments de grande dimension : une approche intégrée diagnostic/commande économique et coopérative à horizon glissant[END_REF], se sont essentiellement portés sur des aspects économiques par le contrôle prédictif tolérant aux défauts des équipements HVAC. Certains types de « dysfonctionnements » causés par les occupants, et classés comme des défauts opérationnels par oppostion aux défauts systèmes et aux et la fermeture des fenêtres, la montée et la descente des stores et les déplacements entre les espaces, etc. . ., ont un impact significatif sur la consommation réelle d'énergie et le confort intérieur. Le retour d'expérience a effectivement confirmé que les performances énergétiques réelles en phase opérationnelle des bâtiments sont très différentes de celles prévues lors des calculs de conception via des logiciels de simulation de performance énergétique des bâtiments. En particulier, les résultats de l'Annexe 53 ([YHN17]), un projet de l'Agence Internationale de l'Énergie (AIE) mené entre 2008 et 2013, indiquent que l'une des causes majeures d'incertitudes dans les logiciels de simulation est effectivement la prise en compte insuffisante des occupants et de leurs comportements.

défauts équipements, n'ont pu faire l'objet d'une attention particulière. Or ce type de dysfonctionnement causé par l'humain, comme par exemple l'ouverture de la fenêtre d'une pièce/salle alors que la pièce/salle est en mode chauffage, a des conséquences néfastes sur la consommation énergétique du bâtiment. Les recherches montrent qu'environ 90% des gens passent leur temps à l'intérieur des bâtiments et 65% du temps dans des bureaux. Pour améliorer significativement l'efficacité énergétique des bâtiments dans lesquels vivent et/ou travaillent des individus, il apparait évident qu'il ne suffira pas de modifier les techniques de construction du bâti ou d'utiliser des technologies de contrôlecommande plus efficaces sur le plan énergétique [HFS20 ; Pan+21]. Parmi les nombreux facteurs dont dépend la consommation énergétique d'un bâtiment (e.g., l'enveloppe, l'orientation et les caractéristiques ambiantes telles que le chauffage, le refroidissement, l'éclairage, etc. . .), il s'avère donc que l'occupant et son comportement sont des facteurs clés extrêmement importants qui façonnent et influencent très fortement l'utilisation de l'énergie de manière complexe et parfois de manière contre-intuitive

[START_REF] Delzendeh | The impact of occupants behaviours on building energy analysis: A research review[END_REF]

. En effet, les actions de l'occupant, telles que le réglage d'un thermostat pour le confort, l'allumage et l'extinction des lumières, l'utilisation d'appareils électroménagers, l'ouverture

Les appels à citation par numéro, e.g.[START_REF]Modélisation des systèmes dynamiques continus (Transparents)[END_REF], se réfèrent aux documents de ma liste classée de publications de la section I, par contre les appels à citation par auteur, e.g.[START_REF] Lunze | Control Theory of Digitally Networked Dynamic Systems[END_REF], se réfèrent aux documents de la bibliographie.

in J. Baillieul, J.C. Willems (Eds) -Mathematical Control Theory, Springer-Verlag, New York, Inc., 1999

Note that the nominal control problem can be embedded in the general FTC problem by letting the index α = 0 correspond to the fault-free situation, i.e., f0(t) = 0

In a private discussion I had with Prof. R. Hanus in

2009, he explained that by lack of a more judicious term, he used the french expression "mettre en condition" which means "to prepare someone or something to take service as soon as possible" as an appropriate term for the task devoted to an antiwindup circuitry.

Remerciements

• Responsable scientifique du projet Energy-in-Time (2013-2017) (2009-2010)

• Participation à la mise en place de la nouvelle plateforme Eco-Sûr (

Conférences nationales avec comité de lecture et actes

[100] -Clanché F., Sauter D., Yamé J.J., Pont-roulant plate-forme d'intégration pour l'étude des systèmes coopératifs dans le contexte d'une commande multi-variable pour l'enseignement et la recherche. Actes 8 ème Colloque sur l'Enseignement des Technologies et des Sciences de l'Information et des Systèmes, CETSIS 2010, Grenoble, France (2010)

Workshops -Communications orales et résumés écrits

[101] -Darure T., Yamé, J.J., Hamelin F., -Fault diagnosis and adaptive control of VAV dampers failures, European Research Conference: Buildings. World Sustainable Energy days, 2017, (WESD -17), Wels, Austria, March 1-3, 2017

[102] -Darure T., Yamé, J.J., Hamelin F., Fault-adaptive control of HVAC VAV Terminal Units, Sustainable Places Conference, June 29-July 1, 2016, Anglet, France with the instantaneous economic cost given by

where F t is the thrust force on the rotor , Ḟt its time-derivative, and

is the kinetic energy of the rotor, and α i , i ∈ {1, 2, 3, 4, 5, 6} are positive constants. The economic interpretation of this cost is given in [JY18b; JY19] and the resulting eMPC problem (4.2.10) is a highly nonlinear and non-convex optimization problem. Indeed, though the functional in (4.2.10a) is a convex function of the quantities -P r , P g , Ḟt , ẋt and K r , it is unfortunately not a convex function of the decision variables u, x in (4.2.8) and moreover, the constraint set is not a convex set in terms of the decision variables.

Convexification of optimization problem

A key question towards convexifying the problem, from the observation of (4.2.11), would be whether it might be possible for the quantities (-P r , P g , Ḟt , ẋt , K r ) to be directly the decision variables of the problem? An insight into the dynamics of the wind turbine [START_REF] Hovgaard | Model predictive control for wind power gradients[END_REF] suggests that power and energy variables might be used to derive an equivalent model to (4.2.7)-(4.2.8) for the wind turbine. It turns out that the derivative of the instantaneous total energy K(t) stored in the rotational motion of the drive train, given by

allows for such an equivalent representation of the WT dynamics under the following transformation of variables:

The transformation (4.2.14) intrinsically leads to a linear dynamical model for the wind turbine in the new variables (x(t), ǔ(t)), i.e.,

for some matrices A W T , B W T of compatible dimensions and depending only on the wind turbine characteristics. Through this variables transformation, it appears that the instantaneous cost (x, ǔ, v w ) as given by (4.2.11) is now a convex function of the decision variables x, ǔ and the system dynamics is linear. However, at this point, the constraint set

, where the vector-valued function G(.) is the new function constraint expressed in terms of the new variables, is a non-convex set. This makes the optimization in the decision variables (x, ǔ) still a non-convex optimization problem. We have shown in [START_REF] Jain | Fault-Tolerant Economic Model Predictive Control for Wind Turbines[END_REF] that, to circumvent this issue, the non-convex components in G(.) can be closely approximated by convex functions yielding a convex set {(x(τ ), ǔ(τ ), d(τ )) : G ap (x(τ ), ǔ(τ ), d(τ )) ≤ 0} where G ap (.) is a tight convex approximation of G(.). Finally, by substituting G ap (.) for G(.), the original optimization problem is completely convexified and leads to a global optimal solution via readily available efficient solvers.

A. M. Subramanian, T. Jain and J.J. Yamé (2020)

Let N and M be positive definite matrices solutions of ĀT M +M Ā = -N . Then the observer (5.2.2) is an asymptotic observer for yhe fault-free system (5.2.1), i.e., lim t→∞ (x(t) -x(t)) = 0, if and only if there is a stabilizing matrix L such that

where λ min (.) and λ max (.) denote the minimum and maximum eigenvalues of a matrix respectively.

Fault detection is achieved through the residual vector r(t) computed by r(t) = Q(y(t) -ŷ(t)) for some scaling diagonal (n × n)-matrix Q with the logic

where is a given threshold value with units in • C and . is the Euclidean norm. An estimate of the actual airflow rate corresponding to the faulty VAV damper is required to achieve the diagnosis. The derivative of the output signal makes the fault signal explicitly appear in the equation in the following form ẏ(t) = Φ(t) + F 1 (t)f a (t) where Φ(t) can be estimated with part of the estimated state x provided by the bilinear observer. This part is the estimated wall temperatures. A key observation here is the non-singularity of F 1 (t) which allows to reconstruct the fault as

with Φ(t) the estimate of Φ(t). An innovative approach for computing in real-time the derivative of the output, which may be corrupted with noise, has been derived. It uses a cascade of high-gain linear observers in series as causal approximators of the derivatives of the output. The idea is based on the second-order linear filter ξ = A 0 ξ + γ 0 y 1 (5.2.6) with state vector ξ = ξ1 ξ2 T , and

where α 1 and α 2 are positive constants and ε > 0. This filter is driven by the scalar signal y 1 and when ε → 0 and y 1 is noise-free, the linear filter (5.2.6) is a high-gain observer whose state ξ2 reconstructs accurately the derivative dy 1 dt of signal y 1 . Now, perform the steps below:

1. choose an integer s greater than 1 and choose s couples (α k,1 , α k,2 ), k = 1, 2, ...s of positive real constants 2. construct matrices A 0,k (ε) and γ 0,k (ε), k = 1, 2, ...s, respectively, as matrices A 0 and γ 0 in (5.2.7) with constants α 1 and α 2 renamed as α k,1 and α k,2

Références bibliographiques

Part IV

Publications

Annexe A Sélection de 5 publications significatives