Où en sommes-nous (rappel des acquis) ? dans un bassin versant dans les habitations du quartier ou village ⇒ Un dispositif qui par l'édiction et l'adoption de règles limitées dans le temps et le champ social, permet d'échapper aux normes sociales et à certaines contraintes, puis de tester "autre chose" et se tester collectivement, pour enfin éventuellement stabiliser ces changements dans le réel Tous les jeux dits "sérieux" actuels, le sont-ils ainsi ?

  19/11/2020 Participation décisionnelle Qu'est-ce que la "participation décisionnelle" ? Quels sont les besoins des acteurs ? Quel est le "paysage" des acteurs et pratiques participatives ? Pourquoi et pour qui organiser ces webinaires ? Comment former ? quoi la "gestion participative de l'eau dans les territoires" ? Quelle participation pour SAGE, PGRE, PTGE, PAPI...? Quelle cohérence avec d'autres politiques ? Est-ce généralisable ? Quelles leçons en tirer ? 3/12/2020 Des cas exemplaires Des expériences contrastées : préparation participative pour le SAGE de la Drôme, conception participative de l'assainissement au Sénégal, Politique de l'Eau partagée en Nouvelle-Calédonie 10/12/2020 Discuter et évaluer des changements Quels changements attendus de la participation décisionnelle ? Pourquoi et comment en débattre ? Comment suivre et évaluer le processus et ses impacts ? Comment partager et utiliser ces informations ? 17/12/2020 Organiser un processus participatif Comment organiser un processus de participation décisionnelle ? Quelles étapes ? Quel cadre réglementaire ? Quels acteurs et quels rôles ? Comment se faire aider ? Quels méthodes et outils ? Quid du COVID ? 7/1/2021 Diagnostic et modélisation participative Qu'est-ce qu'un diagnostic ou un observatoire participatif ? Quels liens entre sciences citoyennes et participation décisionnelle ? Pourquoi et comment utiliser la modélisation participative ? Quels outils ? Comment concevoir et utiliser votre propre modèle pour porter un jeu de rôle ? Qu'est-ce que la Modélisation d'Accompagnement ? Quels outils supports : Wat-A-Game, EauEnJeu, Terristories, Jeu de Territoire, etc ? 21/1/2021 Prospective participative Qu'est-ce que la prospective participative ? Comment faire imaginer des futurs possibles et des chemins y menant ? Comment concilier rêve et réalité ? Quels outils ? 28/1/2021 Planification participative Comment faire créer et discuter des plans d'action intégrés, "multi-acteurs" ? Comment initier cela ? Quel format de propositions d'action ? Comment assembler et comparer des stratégies ? 4/2/2021 Participation numérique Quels usage, atouts et contraintes pour la participation numérique ? C'est quoi les "civic-tech" ? Comment soutenir un "vrai débat", un recueil d'avis, une carto participative ? Qu'apporte l'IA ? Qu'est-ce que CoOPILOT ? (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Au menu : de la Modélisation à la Simulation Participative (ID Webin. 6 → la modélisation participative permet d'associer tous les acteurs à la co-construction de modèles de leur situation socio-environnementale, pour…) 1. Les aider à structurer et partager un diagnostic 2. Préparer et s'approprier un support d'exploration des dynamiques par… → Simulation Participative (Jeu de rôle matériel, ou dispositif informatique), pour : • Apprendre à agir ensemble dans un environnement complexe • Inventer des actions nouvelles, et s'engager dans des chemins de changement (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Rien de neuf ? WALRUS "Water and LandUse Utilization Simulation" : Un jeu multi-joueurs à structure dynamique avec enjeux environnement et économie, et une base LEGO™ ! → auto-construit mais pas auto-modélisé Un jeu "sérieux" sur l'urbain (Feldt, 1972 -http://cluginfo.org ) by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 cf. Jeux de territoire (Lardon, Piveteau, 2005) Source: Chambon, Lardon, Caron, 2013 -LE JEU DE TERRITOIRE UN DISPOSITIF PARTICIPATIF FACILITANT L'INTERACTION ACTEURS -CHERCHEURS http://comeaulabo.ens-lyon.fr/IMG/pdf/JT_chercheurs-acteurs-ENS-mai2013-CHAMBON.pdf (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Défi : plusieurs niveaux de complexité dans les situations socio-environnementales... De multiples interactions… (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Défi : plusieurs niveaux de complexité dans les situations socio-environnementales... De multiples interactions et de multiples points de vues (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Jeux de rôle et Modélisation d'Accompagnement www.commod.org Autour d'un modèle co-conçu représentant la situation locale (espace, acteurs, ressources, actions, interactions), organiser des jeux de rôle pour permettre aux participants d'explorer ensemble l'impact d'actions individuelles et collectives, intégrant la réaction des "autres", des scénarii et la régulation par des institutions. SelfCormas (D'Aquino, Le Page, Bousquet, Bah, 2002) • Un travail précurseur qui engageait la coconception avec et pour les acteurs locaux (des agriculteurs sénégalais) d'un modèle et d'un jeu, le tout en 3 jours -Avec une simulation informatique "sur place" (Cormas) (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 (Rappel Webin 6) FAIRE MODÉLISER LES ACTEURS • Spécificités : ○ Amener un cadre, un "langage" aux acteur pour formuler et proposer ■ sur la complexité : incertitudes, ignorance ■ reconnaître la diversité des représentations ○ Faire produire des représentations explicites, discutables : "objets frontières", "intermédiaires" ■ négocier le partage des idées / visions ○ Un objet résultat "simulable", jouable, qui va représenter le monde et "résister" aux acteurs • Position vs. diagnostic ○ Diagnostic = des méthodes participatives pour un état des lieux ○ Modélisation participative = une méthode pouvant alimenter un diagnostic ou être utilisé avec d'autres motivations (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 *PUIS* FAIRE JOUER LES ACTEURS • Spécificité ○ Le jeu a été co-construit de manière transparente et appropriée ○ Les acteurs jouent leurs propres rôles, ou un autre ! ○ -> Le jeu est un modèle "légitime" dans le processus d'accompagnement (par opposition à un modèle valide) •Objectifs ○ Exploration du "modèle" partagé : comment il "fonctionne" ○ Test et mise en débat de scénarios ou de propositions d'actions ○ Apprentissages "sociaux": ■ Communicationnel (sur la manière de communiquer) ■ Expérientiel (par l'action) ■ Nominal (sur le fonctionnement de la situation de gestion et la place des actions individuelles) I m p o r t a n c e d u D é b r i e fi n g ! (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-dispose d'une grille d'évaluation avec plusieurs indicateurs : Pression phytosanitaire, Pression nitrate, Biodiversité, Satisfaction des autres usages et qualité du paysage. Après chaque tour, l'élu effectue un bilan et invite les acteurs à la concertation sur les pistes d'action possibles. Ce moment de dialogue permet d'expliciter les stratégies des acteurs, de donner lieu à des négociations, de nouer des alliances, etc. À la fin d'un jeu, on ouvre la discussion sur le lien avec la réalité… Un jeu qui met en lien acteurs agricoles et acteurs de l'eau (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Littosim (Becu & al, 2017) Un dispositif Commod "hybride" sur les submersions marines qui mobilise un support informatique de simulation pour accompagner le jeu de rôle. Sources : Littosim.hypotheses.org, 2021 (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 TerriStories (D'Aquino & al., 2017) Un dispositif simple de co-création de modèle et dialogue territorial + version informatique : gestion foncière et agricole au Sénégal (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 (cf. webin 3) Un jeu "expert" sur l'assainissement… pour les communautés Waste-WAG base de jeux de rôle pratiques sur les changements de pratiques en assainissement domestique et semi-collectif → aller vers une planification des pratiques puis un co-engagement Plus d'information sur http://eauenjeu.org Un kit de jeu pour explorer et débattre des enjeux de l'eau depuis l'habitation jusqu'au bassin versant (http://eauenjeu.org) • A partir de WAG, un kit pédagogique complet sur les usages et la gestion de l'eau • Du CE2 à la terminale, et plus • 2 supports de jeux de rôle de table :

• Multi-thématiques : -quantité, qualité, inondations, développement durable, bien-être et lien social, associations, gestion et politiques publiques, élections, économie, emploi… Diffusé dans l'Educ. Nat à 5000 unités

  retenir sur les Jeux de Rôle et Commod ! 1. Un dispositif exploratoire et décisionnel permettant de confronter les acteurs à la complexité de l'action locale et aux résistances des autres acteurs, et donc de co-évoluer vers une option d'action collective nouvelle : oser tester des options socialement difficiles dans un contexte facilitant et les rendre discutables 2. Dans la modélisation d'accompagnement, on préconise en amont des jeux de rôle de passer par la modélisation participative comme mode de structuration des connaissances, d'appropriation du support et d'apprentissage social préalable. Mais le jeu confronte "durement" les acteurs à leurs propres visions ! 3. Un rôle crucial pour le débriefing pour donner du sens à l'expérience (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Références Nicolas Becu, Marion Amalric, Brice Anselme, Elise Beck, Xavier Bertin, Etienne Delay, Nathalie Long, Nicolas Marilleau, Cécilia Pignon-Mussaud, Frédéric Rousseaux, 2017, Participatory simulation to foster social learning on coastal flooding prevention, Environmental Modelling & Software, Volume 98, Pages 1-11, ISSN 1364-8152, https://doi.org/10.1016/j.envsoft.2017.09.003 (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Un jeu c'est... « une action libre, sentie comme fictive et située en dehors de la vie courante, capable néanmoins d'absorber totalement le joueur [...] qui s'accomplit en un temps et un espace expressément circonscrits, se déroule avec un ordre selon des règles données, et suscite dans la vie des relations [...] accentuant par le déguisement leur étrangeté vis-à-vis du monde habituel » (J. Huizinga, 1951).

(cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 A

(cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020Une session de jeu

Rés'eaulution Diffuse

Un jeu de rôle pour la concertation sur l'enjeu des pollutions diffuses agricoles


Keywords: Rouchier, Barbet, 2020, Ed. Matériologiques

Quelle expertise pour, sur ou contre la participation décisionnelle ?

Cf. le cycle P<>P de la Participation Décisionnelle (Webin 1), 1. Qu'est-ce qu'un "expert" ? Et un "expert profane" ou "expert du quotidien" ? Quel lien avec la science ? 2. Quand et comment l'expertise intervient-elle dans la participation décisionnelle ? Quels rôles pour participant.e, expert.e, décideur ? 3. Dans ces processus participatifs, comment utiliser des connaissances et modèles issus de l'expertise ? Comment aborder les incertitudes ? 4. Comment organiser le dialogue avec des experts ?

Type Jury Citoyen :

Grp. tiré au sort + dialogue avec 140 experts + groupe d'appui de 14 experts → "co-construction" du plan citoyen

  COVID-19 & expertise vs. participation... décisionnelle ? Demander à des experts de choisir et organiser la participation des citoyens ? Ou demander aux citoyens de s'organiser eux-mêmes ? Aux experts médicaux de dire où on en est, avec les données disponibles Et si les citoyens participaient volontairement au suivi ? Et disaient ce qu'ils veulent savoir ? Que les modèles nous indiquent où l'on va ! Quid "Après" le COVID ? Question sociale vitale... MM. les experts, qui vacciner by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Expert… formel ? Expert profane ?... d'usage ? "L'expression d'une connaissance formulée en réponse à une demande de ceux qui ont une décision à prendre, en sachant que cette réponse est destinée à être intégrée à un processus de décision" (Académie des Sciences) → une désignation ! • Frontières scientifique -politique ? Expérience requise ? • Partialité / (sur-)cadrage de l'expertise • Construction collective et effets d'arbitrage dans l'expertise → "épistémique", groupes de consensus • Incertitude, précaution, doutes • Construction délibérative (Habermas) ⇒ "c'est précisément en tant que non-spécialistes que les citoyens « ordinaires » sont le plus souvent appelés à participer, leur savoir d'usage non formalisé ou leur bon sens étant censé compléter un savoir technique qui serait inadéquat s'il était isolé" (Sintomer, 2008) "savoir professionnel diffus" "participants aussi compétents que les responsables" « expertise par délégation » « contre-expertise » (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Quelle expertise dans la participation décisionnelle ? 1. (Expert.e en participation) Orienter / conseiller le processus 2. (Expert.e thématique, sur les sujets traités) a. Informer le processus décisionnel en amont, analyser la situation (diagnostic) b. Aider les participants à comprendre les enjeux / questions (débats, jurys citoyens) c. Proposer des solutions / options d. Evaluer / discuter des propositions de participant.e.s, avec ou sans modèle e. Conseiller un décideur principal f. Détailler des plans de mise en oeuvre, la faisabilité économique ou technique 3. (Experte.e SHS en analyse de processus) Suivre et évaluer by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020

•"⇒

  Il nous fallait apporter aux citoyens des faits et non des opinions, les aiguiller vers des sources, en étant en permanence sur le fil du rasoir car l'expert totalement objectif et neutre n'existe pas" (B. Léguet) • "Certains experts ont eu beaucoup moins de temps de parole que d'autres, le groupe d'appui a eu un accès illimité aux citoyens et leur a même parfois suggéré la rédaction de certaines propositions" (D. Courant) • "C'est une construction entre les citoyens et les experts, faite de nombreux allers-retours, un réseau complexe avec des rapports de pouvoir, d'influence." (M. Gaborit) *** Source : Le Monde, 19/6/2020 *** Experts à disposition ou pilotes ("on tap" / "on top", Sprain & al, 2012) ? → 2 positions utiles (Stamenkovic, 2020) ⇒ Forte influence des experts, inégalité dans les apports d'expertise (quand, comment), très peu de débats entre experts, imposition parfois, & manque d'experts -formateurs sur la délibération ! (Giraudet & al, 2021) Enjeux de participation en santé environnementale • Il y a plusieurs types de pollutions (Faugère 2002) ○ Non perçues et non mesurées ○ Non perçues mais mesurées ○ Perçues et mesurées ○ Perçues mais non mesurées • Les ressources épistémiques sont concentrées sur certains phénomènes (Frickel et al. 2010) ○ Connaissances sectorielles approfondies ○ Incertitudes persistantes ○ Ignorances reproduites • Les groupes sociaux sont inégalement affectés • Ils requièrent une place dans la recherche scientifique ayant un impact sur leur vie M. Duperrex L'intérêts des savoirs profanes • Rectifier la tendance au réductionnisme des savoirs experts (Corburn 2005) • Considérer les particularismes locaux et à l'hétérogénéité • Intégrer des voix exclues par souci de justice • Identifier des solutions low cost « Lorsque les profanes tentent de faire avancer la science, ils élargissent le cercle de la participation, créent de la valeur en identifiant des considérations supplémentaires, remettent en question les méthodes professionnelles en tant que représentations inadéquates de la somme totale de la réalité, et modifient fondamentalement les règles existantes du 'domaine scientifique' » (Corburn 2005: 72).

▪

  construction de méthodes pour appréhender des problèmes environnementaux complexes • Nouvelle cartographie précise des pollutions avec identification de « points chauds » • Identification de lacunes et faiblesses de la surveillance règlementaire • Idées pour de nouvelles recherches participatives sur les sous-produits de chloration Gramaglia, C., & Mélard, F. (2019). Looking for the cosmopolitical fish: Monitoring marine pollution with anglers and congers in the Gulf of Fos, Southern France. Science, Technology, & Human Values, 44(5), 814-842. IECP Caractériser et modéliser la situation de référence Explorer des scénarios de diminution des quotas d'eau Evaluer les conséquences économiques de la diminution des quotas d'eau pour lNiveau des quotas ▪ Conditions climatiques ▪ Prix de vente ▪ Types de sols Construction et mise en oeuvre de la PEP en Nouvelle Calédonie (cc)by.nc.sa, Cooplaage@watagame.info, UMR G-EAU, 2020 Processus PEP Nouvelle-Calédonie + expertise MISE Forum de l'eau 2018 -------------> 709 propositions d

« Ça nous a donné l'impression que tout le travail de production des données avant le forum et la commande des actions n'a servi à rien puisque rien n'a été utilisé pendant le forum », Technicien des services de la DAVAR "Je pense que 90% de ce qu'on y retrouve [dans la PEP] est issu des groupes de travail des techniciens en amont du forum" Cadre de la DAVAR

○ Comment répondre aux injonctions faites aux scientifiques de "dire la vérité" sur des visions parfois opposée de la situation ? Sans même avoir à démonter des "fakes news"... ○ Incompréhensions des incertitudes : les doutes autour de ce qu'on ne peut pas qualifier localement faute de mesures, ce qu'on ne sait pas encore mesurer "précisément", la variabilité et le hasard, l'imprévisibilité liée à la complexité, les incertitudes sociales...

• Impact sur l'innovation : rôle / apport respectif des "experts formels" (vision large, connaissance du domaine, mais cadrage "normalisé" par la communauté scientifique) vs. profanes (naïfs, simplificateurs, manipulés, mais ouverts, créatifs), et comment organiser l'interaction créative en particulier quand les solutions sont complexes, systémiques, sur les "systèmes socio-techniques" ?


L Boutat-Baddas
email: latifa.boutat-baddas@iut-longwy.uhp-nancy.fr

D Boutat
email: driss.boutat@ensi-bourges.fr

J P Barbot
email: barbot@ensea.fr

Observability Analysis by Poincaré Normal Forms

Keywords: Observability, Poincaré's normal form, observability singularity

This paper deals with quadratic equivalence, normal forms of observability, characteristic matrices and normal quadratic numbers for nonlinear Single-Input Single-Output (SISO) systems. We investigated both cases; nonlinear systems linearly observable and nonlinear systems with one linear unobservable mode. Particularly, the e¤ect of the normal quadratic numbers on the observer design is pointed out. Finally, a faster observability analysis is proposed using characteristic matrices and normal quadratic numbers. Throughout the paper, academic examples as well as bio-reactor example highlight our purpose.

Introduction

The fact of being able to write down explicit solutions for linear di¤erential equations allows a complete analysis of their behaviors. For this reason, the linearization problem of nonlinear systems, at least in the neighborhood of a singular point, is one approach for studying nonlinear systems. A rigorous mathematical technique which allows substantial progress in this approach is the normal form method. This method was …rst addressed by Poincaré (see [P]). Poincaré's theorem, which applies to analytic systems, shows that when certain non-resonance conditions are satis…ed there is an analytic change of coordinates which transforms a nonlinear system into a linear one. For linearly controllable and uncontrollable systems, homogeneous approximation was …rst addressed by A. Krener in [K]. His idea is to apply Poincaré's technique by using not only a di¤eomorphism but also a state feedback. Lately, linearization and Poincaré's method have been used for observability problems. More precisely, consider the following system:

: x = f (x) y = h(x) (1)
where vector …elds f : IR n ! IR n and h : IR n ! IR m are assumed to be smooth where f (0) = 0 and h(0) = 0:

The observability problem is whether we can estimate the current state x(t) from past observations y(s); s t without measuring all state variables. It is well-known that, by appropriate signal processing, we are often able to obtain good estimates of all state variables from measured outputs. The algorithm that performs this signal processing is called an observer. In the case of observable linear systems: f (x) = Ax and h (x) = Cx where A 2 IR n n and C 2 IR m n coupling the internal model and the innovation correction leads to the observer equations: If the pair (A; C) is observable, then we can choose the observer poles of A KC such that : e(t) ! 0 as t ! +1:

Motivated by these considerations, the observer linearization problem was raised. Is it possible to …nd a state neighborhood U of 0 in IR n , and a change of state coordinates z = (x) such that dynamic (1) is linear driven by nonlinear output-injection:

: z = Az (y): (2)
where : IR m ! IR n is a smooth vector …eld. Note that the output-injection term is canceled in the observation error dynamic for system (2).

The di¤eomorphism must satisfy the following …rst-order partial di¤erential equation: @ @x (x)f (x) = A (x) (h(x)):

(3) Krener and Isidori in [KI] showed that equation (3) has a solution if and only if the following two conditions are satis…ed: i) the codistribution span n dh; dL f h; :::

; dL n 1 f h o is of rank n at 0; ii) h ; ad k f i
= 0 for all k = 1; 3; :::; 2n 1 where is the vector …eld which ful…ls the following equations:

dL k f h( ) = 0 for 0 k n 2 dL n 1 f h( ) = 1
Moreover in [KR], Krener and Respondek relax the above conditions by yielding a di¤eomorphism on the output. Precisely, if

Condition 1 i) For n even, it is necessary that for k = 1 : n 1 [ 1 ; k ] = 0 for k = 1 : n 1 and [ 1 ; n ] = (y) 1 ii) For n odd, it is necessary that for k = 1 : n 1 [ 1 ; n ] = [ 1 ; k ] = [ 2 ; k ] = 0 and [ 2 ; n ] (y) 1 2 Span ( 1 )
these conditions enable us to …nd a local di¤ eomorphism on the output F (y). Then, we consider the following dynamic system:

: x = f (x) y = F (y) (4)
where the dynamic is the same as in (1) and the output replaced by F (y) :Therefore, we say that system (1) is linearizable in the sense of Krener-Respondek if and only if system (4) is linearizable in the sense of Krener-Isidori.

Remark 2 This paper deals with system (1) when the Condition 1 is not veri…ed and with system (4) when the Condition 1 is veri…ed. Consequently, the proposed approach considers systems (1) or (4) which is not exactly linearizable.

An analytical approach to solve equation ( 3) is used by Kazantzis and Kravaris [KK2]. They considered the restricted form of this problem, where the outputinjection is linear, (y) = By: They showed, using a particular form of the Lyapunov Auxiliary Theorem [L] that (3) has a unique solution under certain assumptions. Recently, under very general conditions and for (y) unspeci…ed, Krener and Xiao [KX] proved the existence and uniqueness of this solution. This allows us to design an observer for a larger class of nonlinear systems.

In this paper and in the preliminary version [START_REF] Boutat-Baddas | Quadratic Observability normal form[END_REF], by using quadratic transformations modulo input-output injection, we put linearly observable and linearly unobservable in one direction systems in particular normal forms. In particular, we are interested in the characterization of quadratic equivalence modulo input-output injection of systems for which we can design an observer. We show that every system lies in an equivalence class and is characterized by a list of relatively easy computable numbers. Rigorously speaking, our problem statement is as follows:

1) How can we characterize the fact that a system is quadratically linearizable modulo an input-output injection? If it is not the case, then what is its quadratic normal form?

The answer to both questions is given in Theorem 12 for nonlinear systems linearly observable and in Theorem 14 for nonlinear systems with one linear unobservable mode.

2) How can we compute normal quadratic numbers, i.e. quadratic coe¢ cients which are not equal to zero in the quadratic normal form?

The answer to this question is given in Proposition 21 for nonlinear systems linearly observable and in Proposition 27 for nonlinear systems with one linear unobservable mode. Our motivations to use this kind of observability normal forms are:

If the linear approximation is observable the proposed observability normal form highlights the structural obstruction to transform the system in well known injection form. Obviously the approximation is only valid locally, but thanks to this form, it is possible to analyse which type of obstructions we have. Nevertheless, the main interest of the proposed observability normal form is when the linear approximation is unobservable, in this case thanks to the normal form and more particularly due to the resonant terms it is possible to determine if or not the system is observable thanks to high order terms (see Remark 15). So from this analysis in the paper it is shown that it is possible to design an observer (obviously only locally valid). We think that such observability bifurcation (or more simply observability singularity) may be very well analysed by the observabilty normal form and this not only at the point x = 0 and u = 0 but around all observabilty singularity point thanks to a coordinate change and output set up. The study of such normal form around a manifold of observabilty singularity must be one of our future works. Nevertheless, the possibility to design an observer of the same dimension as that of the system, instead of searching for the information in the extra derivative (see the de…nition of weakly locally observability), is to keep the consistence with respect to a global observer design (we have just to switch o¤ such observer close to the singularity, but without dimension problem). The normal form may be also used to analyse if the system stays or not in observability singularity domain.

The paper is organized as follows. Notations and the de…nition of the quadratic equivalence modulo an input-output injection are presented in section 2. In section 3, we give homological equations and quadratic normal forms for nonlinear systems in both cases: nonlinear systems linearly observable and nonlinear systems with one linear unobservable mode. In the same section, we give two illustrative examples. In the last section, we give an algorithm to compute normal quadratic numbers directly and an example to check the e¢ ciency of this algorithm.

Notations and de…nitions

Throughout this paper, we consider in a state neighborhood U

IR n of the origin, a nonlinear SISO system in the following form:

_ = f ( ) + g( )u y = C (5)
where, vector …elds f; g : U IR n ! IR n are assumed to be real and analytic. We assume that 0 is an equilibrium point i.e. f (0) = 0: We set by de…nition A := @f @ (0) and B := g(0). Then, system (6) can be rewritten in the following form:

_ z = Az + Bu + f [2] (z) + g [1] (z)u + O 3 y = Cz (6)
where:

f [2] (z) = 0 B B B B @ f [2] 1 (z) f [2] 2 (z) . . . f [2] n (z) 1 C C C C A ; g [1] (z) = 0 B B B B @ g [1] 1 (z) g [1] 2 (z) . . . g [1] n (z) 1 C C C C A
and

O 3 = O 3 (z; u)
where for all 1 i n, f

[2]

i (z) and g

[1]

i (z) are respectively homogeneous polynomials of degree 2, respectively 1 in components of z.

De…nition 3

i) The component f [2] (z) + g [1] (z)u is the quadratic part of system (6).

ii) A quadratic transformation is a di¤ eomorphism of the form:

x = z [2] (z)
where [2] (z) =

[2] 1 (z) ; ::::::;

[2] n (z)
T and for all 1 i n;

[2] i (z) is a homogeneous polynomial of degree two with respect to z: Now consider the following vector spaces

E 1 = IR n [y] and E 2 = IR n y 2
:; d n ) T 2 IR n :
Finally, for a …xed input u we set E = E 2 + uE 1 : For solving the problem stated above, we need the following assumption:

Assumption 4 The output is always taken equal to the …rst state component. Consequently, the di¤ eomorphism

(x = z [2] (z)) is such that [2] 1 (z) = 0:
This assumption is legitimate as soon as for (1) does not ful…l Krener-Respondek conditions. However, for system which ful…ls Condition 1 we do the assumption for the new output y = F(y) De…nition 5

1) Consider a second system:

: x = Ax + Bu + f [2] (x) + g [1] (x)u + O 3 y = Cx : (7)
If here exists a quadratic transformation

x = z [2] (z)
such that, f [2] (x) + g [1] (x)u equals f [2] (z) + g [1] (z)u modulo E, then we say that ( 7) and ( 6) are quadratically equivalent modulo input-output injections.

2) If f [2] (x)+ g [1] (x)u is de…ned modulo an element of E then we say that ( 6) and ( 7) are quadratically equivalent modulo an input-output injection.

3) If f [2] (x) + g [1] (x)u 2 E we say that system (6) is quadratically linearizable modulo an input-output injection.

Remark 6 If (A; C) is an observable pair, then we can transform system (6) into the following form [B]:

8 < : _ z = A obs z + B obs u + f [2] (z) + g [1] (z)u + O 3 +O 3 (z; u) y = C obs z (8)
where:

A obs = 0 B B B B B B @ a 1 1 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 a n 1 0 0 1 a n 0 0 0 1 C C C C C C A ; B obs = 0 B B B B B B @ b 1 b 2 . . . . . . b n 1 C C C C C C A
and C obs = 1 0 ::: 0

Remark 7 If (A; C) has one unobservable real mode, then we can transform system (6) into the following form [B]:

8 > > > < > > > : _ z = A obs z + B obs u + f [2] (z) + g[1] (z)u +O 3 (z; u) _ zn = nzn + P n 1 i=1 izi + bnu + f [2] n (z) +g [1] n (z)u + O 3 (z; u) y = C obs z (9)
where:

z = 0 B B B B B B @ z 1 z 2 . . . . . . z n 1 1 C C C C C C A ; z = z z n ; A obs = 0 B B B B B B @ a 1 1 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 a n 2 0 0 1 a n 1 0 0 0 1 C C C C C C A ; B obs = b 1 b 2 ::: b n 1 T and
C obs = 1 0 ::: 0 :

Throughout the paper, we deal with systems in forms ( 8) and ( 9).

The following gives a de…nition of what we will call quadratic normal forms throughout this paper.

De…nition 8

1) We say that system (8) is in its quadratic normal form if: for 1 j n 1:

( [2] f g j = P n i=2 k1;ixiu and [2] f gn = P n j i=2 hi;jxixj + P n 1 i=2 kn;ixiu (10)
where

[2] f gj = f [2] j (z) + g [1]
j (z)u: Coe¢ cients: h i;j for 2 i j n and k n;i for 2 i n 1 are called normal quadratic numbers.

2) Through an abuse of language1 , we say that system ( 9) is in its normal form if: for 1 j n 2;:

8 > > > > < > > > > : [2] f g j = P n i=2 k1;ixiu and [2] f g n 1 = P n j i=2 hi;jxixj + h1;nx1xn + P n i=2 kn 1;ixiu; [2] f gn = P n j i=2 li;jxixj + P n j=2 l1;jx1x + P n i=2 kn;ixiu (11)
Coe¢ cients: h i;j for 2 i j n and k n 1;i for 2 i n 1 are called normal quadratic numbers:

Throughout this work we will use matrix calculus to compute normal quadratic numbers, for this reason we adopt the following notations:

[2] i (z) = z T iz and f [2] i (z) = z T fiz :
where for all 1 i n; i and f i are symmetric matrices. We make the notational

[2] i (z) and f [2]
i (z) for all 1 i n of indicating quadratic functions and the notational i and f i of indicating the corresponding matrices. In the same way, for all 1 i n, the 1 n vectors g i = (g i;1 ; ::::; g i;n ) are associated with functions g

[1] i (z) i.e. g [1] i (z) = z T g i = g T i z:
With those notations, we have: = ( 1 ; :::::; n )

T ; f = (f 1 ; ::::; f n )

T and g = (g 1 ; :::::; g n ) T :

3 Quadratic Normal Forms

In this section we will state two theorems concerning the setting of systems ( 8) and ( 9) in their normal forms ( 10) and ( 11) respectively. Each theorem is followed by an illustrative example. The end of this section is devoted to the necessary and su¢ cient conditions to cancel the quadratic terms in the last dynamic of normal form (11). With these intentions, we will start by giving the necessary and su¢ cient algebraic conditions under which two systems in the form of (6) are quadratically equivalent modulo an input-output injection.

Proposition 9 System (6) is quadratically equivalent modulo an input-output injection to system (7), if and only if the following two homological equations are satis…ed:

i) A [2] (z) @ [2] @z Az = f [2] (z) f [2] (z) modulo E2 ii) @ [2]
@z B = g [1] (z) g [1] (z) modulo E1

where

@ [2] @z Az := @ [2] 1 (z) @z
Az; ::::::;

@ [2] n (z) @z Az T and for all 1 i n @ [2] i (z) @z
is the Jacobian matrix of i (z) :

Now we will express these homological equations in terms of matrices, this allows us to compute in section 4 normal quadratic numbers. For this, we set by de…nition where i and i 2 IR . With this notation we have:

1;i y 2 = z T i z 2 E 2 and 1;i y = i z 2 E 1 :
The next result gives the matrix version of Proposition 9.

Corollary 10 System (6) is quadratically equivalent modulo an output-input-injection to system (7), if and only if there exist and such that:

A T + A = f f + 2B T = g g +
where i) in the observable case = ( 2 ; :::::; n ; 0) ; ii) in the unobservable case = 2 ; :::::; n 1 ; 0; n with n = P n i=2 i i : Proof. Consider the homological equations:

i) f [2] (z) f [2] (z) + [2] (z1) = A [2] (z) @ [2]
@z Az ii) g [1] (z) g [1] (z) + [1] (z1) = @ [2] @z B

Using matrix notations:

[2] (z) = z T 1z; ::::; z T nz T and [2] (z) = A [2] (z):

i) As in the linearly observable case

A = 0 B B B B B B @ a 1 1 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 a n 1 0 0 1 a n 0 0 0 1 C C C C C C A ; then [2] (z) = [2]
2 (z); :::::;

[2] n (z); 0 because [2] 1 (z) = 0:
ii) For the unobservable case

A = 0 B B B B B B B B @ a 1 1 0 0 . . . 0 . . . . . . . . . . . . . . . . . . . . . 0 a n 2 0 0 1 a n 1 0 0 0 a 1 2 n 1 C C C C C C C C A ; we obtain (z) = [2]
2 (z) ; :::::;

[2] n 1 (z) ; 0; [2] n (z)
where

[2] n = n X i=2 i [2] i : Now, @ [2] (z) @z Az = @ [2] 1 (z) @z Az; :::; @ [2] n (z) @z Az ! T ;
as for all i 2 [1; n];

[2] i (z) = z T i z then @ [2] (z) @z Az = 0 B @ z T A T 1z + z T 1Az . . . z T A T nz + z T nAz 1 C A T
which by de…nition is:

@ [2] (z) @z Az = z T (A T + A)z:
In the same way we have

@ [2] i (z) @z B = B T i z + z T i B = 2B T i z
because i is symmetric. Using previous results, in homological equations of Proposition 9, we obtain equations stated in Corollary 10:

f f + = A T + A g g + = 2B T

Nonlinear systems linearly observable case

In this subsection, we will compute quadratic normal form for linearly observable systems. It is clear from the structure of the quadratic normal form (10) that a system is quadratically linearizable if and only if all its normal quadratic numbers are equal to zero. Thus, a system in the form ( 8) is quadratically linearizable if and only if its quadratic part satis…es the homological equations of Corollary 10 where f = 0 and g = 0. Now, under the Assumption 4 and thanks to the structure of matrix A obs ; homological equations for a quadratically linearizable system which is linearly observable are:

8 < : (a.1) for i 2 [2; n] i = A T obs i 1 + i 1Aobs + f i 1 (a.2) 0 = A T
obs n + nAobs + fn and 8 < :

(b.1) g1 + 1 = 0 (b.2) for i 2 [2; n] ; 2B T i + g i = 0
where fj = f j + j and gj = g j + j : Equation (a:1) gives explicitly i for all 2 i n: Then all quadratic terms in the (n 1) …rst dynamics are deleted.

Remark 11 Moreover, equation (a:2) is equivalent to the following equation:

n 1 X s=0 0 @ s X j=0 C s j A T s j fn s A j 1 A = 0
and equations (b:1) and (b:2) are re-written as follows: 8 < :

(b.1) g1 + 1 = 0 (b.2) for i 2 [2; n] 2B T P i 2 k=0 P k j=0 C k j A T k j f i 1 k A j + g i = 0
Under Assumption 4, using Proposition 9, we obtain the following theorem which gives normal forms for nonlinear linearly observable systems.

Theorem 12 There exists a quadratic di¤ eomorphism which transforms the quadratic part of system (8) into the quadratic normal form (10) modulo E: Let us consider an example to illustrate the above theorem:

Example 13 Consider the following system: 8 > < > :

: z1 = a1z1 + z2 + k1z 2 2 + l1z 2 3 : z2 = a2z1 + z3 + k2z 2 2 + l2z 2 3 : z3 = a3z1 + k3z 2 2 + l3z 2 3 y = z1: (12)
Homological equations of Proposition 9 for system (12) are:

8 > > < > > : [2] 1 (z) = 0 [2] 2 (z) = k1z 2 2 l1z 2 3 + 1;1 z 2 1 [2] 3 (z) = 2 1;1 z1 (a1z1 + z2) 2k1z2 (a2z1 + z3) 2l1a3z1z3 k2z 2 2 l2z 2 3 + 1;2 z 2 1
then by choosing:

1;1 = l 1 a 3 a 1 + 2k 1 a 2 + l 2 a 3 and 1;2 = 3 1;1 a 1 + k 1 a 2 a 1 + k 2 a 2 + k 1 a 3 ;
the quadratic normal form of system ( 12) is:

8 > > > < > > > : : x1 = a1x1 + x2 + 1;1 x 2 1 + O 3 : x2 = a2x1 + x3 + 1;2 x 2 1 + O 3 : x3 = a3x1 + k3 2 1;1 + 2k1a2 x 2 2 + 1;3 x 2 1 + (2l1a3 + 2k2) x2x3 + (l3 + 2k1) x 2 3 + O 3 y = x1:

Nonlinear systems with one linear unobservable mode case

In the same way as in the above subsection, under Assumption 4 and Proposition 9 we obtain the following result.

Theorem 14 There is a quadratic transformation which transforms the quadratic part of system (9) in to the quadratic normal form (11).

Now we are ready to highlight the the usefulness of our normal form to analyze the observability.

Remark 15

1) We call the unobservability submanifold the subset S n 1 of U given by:

S n 1 = 8 < : x 2 U; such that n 1 P i=1 h i;n x i + 2h n;n x n + k n 1;n u = 0 9 = ;
Thus, for a …xed input u; when system evolves on S n 1 we lose the linear and quadratic observability.

2) If x 2 S n 1 and k n 1;n 6 = 0; then with an appropriate change of input u (universal input [GB]), we can modify S n 1 to locally restore the quadratic observability. Moreover if k n 1;n = 0 and if there exists i 2 [1; n] such that k i;n 6 = 0, then quadratic observability is restored.

3) If x 2 S n 1 such that u is not a function of x n and for all i 2 [1; n] we have k i;n = 0; then we use coe¢ cient n to study the detectability propriety. For this we distinguish three cases: a) if n < 0 then the state x n is detectable, b) if n > 0 then x n is unstable and consequently undetectable, c) if n = 0 we can use the center manifold theory in order to analyze stability or instability of x n and consequently its detectability or undetectability.

Remark 16 Let [2] (x) = 0; ::::;

[2] n (x) be the quadratic part of the di¤ eomorphism which gives the above theorem. The di¤ eomorphism

[2] 2 (x) is well determined from f [2] 1 (x) and for all 3 i n 1, [2] i (x) is determined from f [2]
i (x) and the derivatives of

[2] j (x) and f [2] j (x) for 2 j i 1: However, the choice of [2] n (x) is free. So it is interesting to use [2]
n (x) to cancel quadratic terms in the last _

x n dynamic. For this [2] n (x) must ful…l the following equation:

n [2] n (x) + n 1 X i=1 i [2] i (x) = @ [2] n @x A obs x + [2] fn (13) where [2] fj = f [2] j (x) + [2] j (x 1 ):
Unfortunately, the above equation is not ful…lled for arbitrary n and (a i ) 1 i n ; as we will show in the next example.

Example 17 Consider the following system:

8 > > > < > > > : : z1 = a1z1 + z2 + k1z 2 2 + l1z 2 3 : z2 = a2z1 + k2z 2 2 + l2z 2 3 : z3 = 1z1 + 2z2 + 3z3 + k3z 2 2 +l3z 2 3 y = z1 : (14)
The homological equations of Proposition 9 for system (14) are:

[2] 1 (z) = 0 [2] 2 (z) = k1z 2 2 l1z 2 3 + 1;1 z 2 1 rewriting [2]
3 (z) = z T 3 z where:

3 = 0 @ 1;1 1;2 1; 3
1;2 2;2 2;3 1;3 2;3 3;3 1 A then, if 3 6 = 0 and a 2 6 = 0 we obtain: 0 @ 1;3 2;3 3;3

1 A = 0 @ 2 l 3 + 1 l 1 3 l 3 + 1 l 1 3 a 2 ( a1 2 1) l 3 + 1 l 1 3 1 A : If 3 6 = 0, a 2 6 = 0 and ( 3 a 1 )( 2 3 2 3 a 1 4a 2 ) 6 = 0, then: 0 @ 1;1 1;2 2;2 1 A = 0 @ ( 3 2a1) 2a2 0 1 ( 3 a1) a2 0 2 3 1 A 1 0 @ 2 1 1;3 2 1;1 + 1;2 2 1;3 + 1 2;3 2 2 2;3 + 2k1 + k3 1 A and [2] 3 (z) cancels all quadratic terms in _ z 3 : Finally, choosing 1;1 = k 1 a 2 we obtain: 8 > > > < > > > : : x1 = a1x1 + x2 + 1;1 x 2 1 + O 3 : x2 = a2x1 + k2x 2 2 + (2l1 3 + l2) x 2 3 + 2l1 1x1x3 +2l1 2x2x3 + 2a1 1;1 + 1;2 x 2 1 + O 3 : x3 = 1x1 + 2x2 + 3x3 + O 3 y = x1
Thus, for 3 6 = 0, a 2 6 = 0 and

( 3 a 1 )( 2 3 2 3 a 1 4a 2 ) 6 = 0
there is

[2]
3 (x) which ful…ls (13).

Computation algorithm of normal quadratic numbers

In this section, we give an easily implementable algorithm in order to compute normal quadratic numbers of quadratic normal forms (10) and ( 11): This allows us to de…ne the observability quadratic equivalence class of a system without formally solving homological equations 9. For this, we introduce what we call characteristic matrices, of which some coef-…cients are the normal quadratic numbers. We consider again Example 13 to show the power of this algorithm. Moreover, for the linearly unobservable case we point out normal quadratic numbers which allow us to recover, at least locally, the quadratic observability, and so to be able to design an observer. At the end of this section, we highlight our purpose by the bio-reactor example.

Nonlinear systems linearly observable case

The next theorem explicitly gives the matrix family ( i ) 1 i n and ( i ) 1 i n which transforms the system into its quadratic normal form.

Theorem 18

i) The family ( i ) 1 i n viewed as real numbers, is the solution of the following algebraic linear system:

D 0 B B B B B B @ 1;1 1;2 . . . . . . 1;n 1 C C C C C C A = (CF ) T (15)
where

D = 0 B B B B B B @ d 1;1 d 1;n 1 1 . . . : : 1 0 . . . : : 0 0 d n 1;1 1 0 0 0 1 0 0 0 0 1 C C C C C C A ; F : = n 1 X s=0 0 @ s X j=0 C s j A T s j f n s A j 1
A and C = (1; 0:::::; 0) ; and for all 1 k n we have 0 ii) For 2 i n, matrices i which ful…lled equation (a:1) are:

B B B B B B B B B B @ d 1;k d 2;k . . . 1 0 . . . 0 1 C C C C C C C C C C A = 0 B B B B B B B B B B @ d 1;k d 2;k . . . d n (k 1) ; k 0 . . . 0 1 C C C C C C C C C C A = n k X j=0 C n k j A T j × A n k j C T
i = i 2 X k=0 k X j=0 C k j A T k j f i 1 k A j (16)
Remark 19 If matrix family ( i ) 1 i n and ( i ) 1 i n which are given in Theorem 18 below ful…l equations (a:2), (b:1) and (b:2) then system (8) is fully quadratically linearizable. However, in general these equations do not admit solutions. We use their left members to give the following de…nition:

De…nition 20 We call characteristic matrices associated with system (8) the following matrices:

M = P n 1 s=0 P s j=0 C s j A T s j f n s A j N1 = g1 + 1 ; and for i 2 [2; n] Ni = 2B T P i 2 k=0 P k j=0 C k j A T k j f i 1 k A j + g i (17)
We call characteristic numbers; coe¢ cients of characteristic matrices which are not generically equal to zero. Now we can state the link between normal quadratic numbers and elements of characteristic matrices.

We know from Remark 19 that M = 0 and N l = 0 for 1 l n are the conditions of the full quadratic linearization, thus we have:

Proposition 21 Quadratic normal numbers of normal form (10) are given by: 8i; j 2 [2; n] and l 2 [2; n] 8 > < > :

hi;j = 2Mi;j for i 6 = j hi;i = Mi;i k1;j = N1;j k l;j = N l;j where M i;j is the i th row and the j th column element of characteristic matrix M; and N l;j is the j th element of the row vector N l :

Remark 22

1) The fact that M 1;j = 0 for all 1 j n is equivalent to the algebraic linear system (15).

2) In the normal form (10) we have at most n(n 1) 2 normal quadratic numbers x i x j and n (n 1) normal quadratic numbers x i u. In fact, we have i) n n(n+1) 2 constraints given by the n symmetric matrices

(f i ) 1 i n ,
ii) n 2 constraints given by the n matrices (g i ) 1 i n ,

iii) (n 1) n(n+1) 2 symmetric matrices i 2 i n which are completely given by the (n 1) …rst dynamics and iv) n degrees of freedom given by output injections ( i ) 1 i n ; v) n degrees of freedom given by output injections ( i ) 1 i n :

Thus, the number of normal quadratic numbers is:

n n(n + 1) 2 (n 1) n(n + 1) 2 2n + n 2 = 3 2 n(n 1):
Example 23 Let us consider again (12) to highlight the power of the characteristic matrices to compute quadratic normal forms.

( :

z1 = a1z1 + z2 + k1z 2 2 + l1z 2 3 : z2 = a2z1 + z3 + k2z 2 2 + l2z 2 3 : z3 = a3z1 + k3z 2 2 + l3z 2 3 :
(18)

From Theorem 18, we obtain: 8 > < > :

1;1

= l1a3a1 + 2k1a2 + l2a3 1;2 = 3 1;1 a1 + k1a2a1 + k2a2 + k1a3 1;3 = 3a 2 1 1;1 1;1 a2 2a1 1;2 a 2 1 + a2 1;1 + 2a 2 2 k1 + 2a 2 3 l1 then 8 > > <
> > :

M 1;j = 0 for all 1 j 3 M 2;2 = 2a 2 k 1 k 3 + 2 1;1 M 2;3 = M 3;2 = a 3 l 1 k 2 M 3;3 = l 3 2k 1 :
By Proposition 21 the normal quadratic numbers are: 8 > < > :

h1;1 = h1;2 = h1;3 = 0 h2;2 = 2a2k1 + k3 2 1;1 h2;3 = 2a3l1 + 2k2 h3;3 = l3 + 2k1
So, the quadratic normal form associated to ( 18) is: 8 > < > :

: x1 = a1x1 + x2 + 1;1 x 2 1 + O 3 : x2 = a2x1 + x3 + 1;2 x 2 1 + O 3 : x3 = a3x1 + k3 2 1;1 + 2k1a2 x 2 2 + (l3 + 2k1) x 2 3 + (2l1a3 + 2k2) x2x3 + O 3 :
Conditions of the full quadratic linearization are M i;j = 0 for all 2 i j 3:

Nonlinear systems with one linear unobservable mode case

In this subsection, as previously for the linear observable case, …rstly we introduce and compute characteristic matrices of (11), we deduce normal quadratic numbers and …nally we discuss the quadratic observability and stability properties. For this, setting f = 0 and g = 0 in homological equations of Corollary 10, we obtain:

i = A T i 1 + i 1A + f i 1 | {z } and for i2[2;n 2] 0 = A T n 1 + n 1A + f n 1 = P n 2 s=0 P s j=0 C s j A T s j f n s A j n n = P n 1 i=1 i i + A T n + nA + fn
The following is corresponding result to Theorem 18 for linearly unobservable case.

Theorem 24

i) The family ( i ) 1 i n 1 viewed as real numbers, is the solution of the following algebraic linear system:

D 0 B B B B B B @ 1;1 1;2 . . . . . . 1;n 1 1 C C C C C C A = (CF ) T where D = 0 B B B B B B @ d 1;1 d 1;n 1 1 . . . : : 1 0 . . . : : 0 0 d n 1;1 1 0 0 0 1 0 0 0 0 1 C C C C C C A ; F : = n 2 X s=0 0 @ s X j=0 C s j A T s j f n 1 s A j
d 1;k d 2;k . . . 1 0 0 1 C C C C C C C A = 0 B B B B B B B @ d 1;k d 2;k . . . d n 1 (k 1) ; k 0 0 1 C C C C C C C A = n 1 k X j=0 C n 1 k j A T j × A n 1 k j C T
= i 2 X k=0 k X j=0 C k j A T k j f i 1 k A j Remark 25 If n 1 satis…es the equation (a:2) i.e. n 2 X s=1 s 1 X j=0 C s 1 j A T s j fn s A j+1 + fn 1 = 0
and n satis…es the following equation

n n = n 1 X i=1 i i + A T n + n A + fn and if (b.1) g1 + 1 = 0 (b.2) and for i 2 [2; n] 2B T P i 2 k=0 P k j=0 C k j A T k j f i 1 k A j + g i = 0
then system (9) is fully quadratically linearizable.

De…nition 26 Characteristic matrices of the observable part of ( 9) are given by:

M = P n 2 s=0 P s j=0 C s j A T s j f n 1 s A j ; N1 = g1 + 1 and for i 2 [2; n 1] Ni = 2B T P i 2 k=0 P k j=0 C k j A T k j f i 1 k A j + g i
Now, from the previous de…nition we present relations between normal quadratic numbers of (11) and elements of its characteristic matrices.

Proposition 27 Quadratic normal numbers associated with the observable part of normal form (11) are: 8i; j 2 [2; n] and l 2 [2; n 1] 8 > > > > < > > > > :

h 1;n = 2M 1;n h i;j = 2M i;j for i 6 = j 2 [2; n] h i;i = M i;i k 1;j = g 1 1 k l;j = N l;j

Remark 28 In the same way as for Remark 22; in the (n 1) …rst dynamics of normal form (11) we have at most n(n 1) 2 + 1 normal quadratic numbers x i x j and (n 1) 2 normal quadratic numbers x i u. Thus, the number of normal quadratic numbers is: n(3n 5) 2 + 2 = (n 1) n(n + 1) 2 (n 2) n(n + 1) 2 (n 1) + n (n 1) (n 1) :

Now, in order to highlight the usefulness of normal form in the observability analysis and the observer design, we chose to deal with the following bio-reactor dynamics.

Example 29 Consider the following bio-reactor dynamics [R]:

8 < :

: = 1 1 k ( 2 ) 1 y = 1 (19)
guaranteed for x 1 (t) = 0 for all t, then in order to take into account, we set E s = 1 if x 1 6 = 0 else E s = 0. This example shows that the observability singularity can be overcome thanks to quadratic normal form and its normal quadratic numbers which may contain information for the observer design. Here, observation information is given by rx 1 x 2 (the term h 1;n x 1 x n of the normal form).

Conclusion

In this paper, for nonlinear systems linearly observable and nonlinear systems with one linear unobservable mode, a particular quadratic normal form is given for observer design and detectability analysis around an equilibrium point. Using Poincaré's normal forms, we point out normal quadratic numbers that ensure the local observability. Moreover, in order to analyze the observability without transforming the systems in their normal forms, we have introduced what we have called characteristic matrices. Some of them are linked to the quadratic part of the drift and they characterize the quadratic observability without input. Others are linked to the so-called universal input and de…ne the choice of this input to preserve the observability. These matrices inform us about the quadratic observability before we begin the design of an observer. There are many areas where this study may be implemented; the synchronization of chaotic systems [NM], sensorless control of induction motors [CYBMM].

Nevertheless, some of these cases need an extra development of our method. For the synchronization of chaotic systems [START_REF] Boutat-Baddas | Observability bifurcation versus observing bifurcations[END_REF][START_REF] Boutat-Baddas | Sliding mode observers and observability singularity in chaotic synchronization[END_REF] some modi…cations to the output-injection will be considered and for the sensorless control of induction motors a generalization of the approach to multi-output systems will be necessary.

  estimate state and b y is the expected observation, K is the observer gain matrix. The estimate error e := x b x is driven by the following equation: : e = (A KC)e:

  whose elements are respectively (d 1 ; :::; d n )T y and (e 1 ; :::; e n ) T y 2where (e 1 ; :::; e n ) T and (d 1 ; ::

  matrices i which satisfy equation (a:1) are:

  i

The linear unobservable part has not yet been reduced.

and

are chosen in order to ensure at least the stability. However using the Remark 15, the submanifold observability singularity is S 1 = fx 2 U; such that h 1;2 x 1 = 0g. So, the asymptotic stability is not

where state = ( 1; 2 ) T represent respectively the concentration of biomass ( 1 ) and of substrate ( 2 ), and the ( 2 ) = r 2 (1 2 c ) is the growing function. In this example, the interesting equilibrium point with respect to the observability singularity is not at = (0; 0) but at 0 = (0; c) T .

After changing coordinate z = 0 , we obtain: 8 < :

This system is not linearly observable. Using the above notations

then the characteristic matrix of the …rst dynamic is

with 1;1 = 0 and normal quadratic numbers are:

h 1;2 = r and h 2;2 = 0 :

For the last dynamic we obtain:

and l 2;2 = 0 :

Then, observable normal form associated to the bioreactor is:

In [R] a solution was proposed, here our purpose is just to highlight the e¢ ciency of the observable normal form, for this we propose the following sliding mode observer:

, ŷ = x1 and if jx 1 j < " then E 1 = 0 else E 1 = 1, in the same way if e 1 = x 1 x1 is not on the constrained manifold (e 1 = 0) then E 2 = 0 else E 2 = 1. Moreover, ,