Ce travail de thèse traite de l’analyse de la stabilité et la stabilisation des systèmes non-linéaires représentés par des modèles T-S. L’objectif est de réduire le conservatisme des conditions de stabilité, obtenue par la méthode directe de Lyapunov, et écrites, dans la mesure du possible, sous forme de LMIs. Dans ce cadre, deux contributions principales ont été apportées. Tout d’abord, nous avons proposé de nouvelles conditions de synthèse non-quadratique de lois de commande, strictement LMIs et sans restriction d’ordre, pour les modèles T-S via des FLICs. En effet, dans ce contexte, les résultats de la littérature ne sont valables que pour les modèles T-S d’ordre inférieur ou égal à 2. Afin de lever cette restriction, les conditions ont été obtenues grâce à la démonstration d’une propriété de dualité. Ensuite, peu de travaux traitant de la spécification des performances en boucle fermée, de nouvelles conditions LMIs (quadratiques et non-quadratiques) ont été proposées via le concept de D-stabilité. Dans un premier temps, la synthèse de lois de commande PDC et non-PDC D-stabilisantes a été proposée pour les modèles T-S nominaux. Ensuite, ces résultats ont été étendus au cas des modèles T-S incertains. De plus, nous avons mis en évidence, au travers d’un exemple de D-stabilisation en attitude d’un modèle de drone quadrirotor, que les modèles T-S incertains pouvaient être avantageusement considérés lorsque les non-linéarités d’un modèle non-linéaire dépendent à la fois de l’état et de l’entrée. / This work deals with the stability analysis and the stabilisation of nonlinear systems represented by T-S models.The goal is to reduce the conservatism of the stability conditions, obtained through the direct Lyapunov methodand written, when it is possible, as LMIs. In this framework, two main contributions has been proposed. First ofall, we have proposed some new conditions based on FLICs, strictly LMIs and without any order restrictions, forthe non-quadratic design of control laws devoted to stabilize T-S models. Indeed, in this non-quadratic context,the existing works are only available for 2nd order T-S models. In order to unlock this restriction, the proposed conditions have been obtained based on the proof of a dual property. Then, starting from the fact that few worksdeals with the closed-loop performances specification, some new LMI conditions (quadratic and non-quadratic)have been proposed via the D-stability concept. As a first step, D-stabilizing PDC and non-PDC controller designhas been considered for nominal T-S models. Then, these results have been extended to uncertain T-S models.Moreover, it has been highlighted, from an example of the attitude D-stabilization of a quadrotor model, that wecan make use of uncertain T-S models to cope with nonlinear models involving nonlinearities depending on bothstate and input variables.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23545号 / 情博第775号 / 新制||情||132(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 太田 快人, 教授 山下 信雄, 教授 大塚 敏之 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Ce travail de thèse traite de la problématique d’estimation des défauts et de l’étathybride pour une classe de systèmes linéaires à commutations. L’objectif est de développerune méthode afin de synthétiser un observateur et un estimateur dédiésrespectivement à l’estimation de l’état hybride et des défauts. Après la présentationd’un état de l’art sur les techniques d’estimation, de stabilité et de diagnosticpour les systèmes linéaires à commutations, la thèse est scindée en deux parties.La première partie propose une méthode d’estimation de l’état continu et desdéfauts dans le cas où l’état discret du système est connu. En se basant sur unetransformation de coordonnées qui découple un sous-ensemble de l’état du systèmedes défauts, nous avons synthétisé dans un premier temps un observateur hybridepour estimer l’état continu du système, et dans un second temps, un estimateurpermettant la reconstruction des défauts. L’estimateur de défauts proposé dépendde la dérivée de la sortie du système. Pour cette raison, un différenciateur robusteet exact basé sur des techniques des modes glissants est utilisé. Dans la secondepartie de ce mémoire, l’état discret du système est supposé inconnu. Une approchebasée sur des méthodes algébriques est proposée afin d’estimer les instants decommutation entre les différents sous-systèmes. Par la suite, l’estimation de l’étathybride (état continu et état discret) et des défauts est considérée dans le cas oùl’état discret du système est inconnu. Ce dernier est reconstruit en se basant surles instant de commutation estimé et sur une séquence de commutation connue.L’état continu du système est estimé en se basant sur une méthode de placementde pôles permettant d’améliorer les performances de la phase transitoire. Enfin, enexploitant des résultats trouvés dans la première partie, l’estimation des défautsest considérée en estimant la sortie du système avec un différenciateur algébrique.Ce différenciateur donne des résultats plus intéressants vis-à-vis du bruit par rapportau différenciateur basé sur les techniques des modes glissants utilisé dans lapremière partie. / This work deals with the problem of estimation of fault and hybrid state for a classof switched linear systems. The objective is to develop a method to synthesize anobserver and an estimator dedicated respectively to the estimation of the hybridstate and the faults. After presenting a state of the art for estimation, stabilityand diagnostic techniques for switched linear systems, the report is divided intotwo parts. The first part proposes a method for estimating the continuous stateand the faults in the case where the discrete state of the system is known. Basedon a coordinate transformation which decouples a subset of the state of the systemof faults, we first synthesized a hybrid observer to estimate the continuous stateof the system and, in a second step, an estimator allowing the reconstructionof faults. The proposed fault estimator depends on the derivative of the systemoutput. For this reason, a robust and accurate differentiator based on sliding modetechniques is used. In the second part of this paper, the discrete state of the systemis assumed unknown. An algebraic approach is proposed to estimate the switchingtimes between the different subsystems. Thereafter, the estimation of the hybridstate (continuous and discrete state) and of the faults is considered in the casewhere the discrete state of the system is unknown. The latter is reconstructedfrom the estimated switching times and on a known switching sequence. Thecontinuous state of the system is estimated using a pole placement method allowingimprove the performances of the transient phase. Finally, by exploiting the resultsfound in the first part, the estimation of the faults is considered by estimatingthe output of the system with an algebraic differentiator. This differentiator givesmore interesting results at the noise compared to the differentiator based on thesliding mode techniques used in the first part.

Le travail de recherche concrétisé par ce mémoire de thèse se trouve à l’intersection de deux domaines importants, la commande robuste des systèmes linéaires (LTI, LPV, en commutation) à temps discret affectés par des perturbations permanentes bornées et des contraintes et les ensembles invariants ellipsoïdaux maximal ou minimal. La première partie de ce mémoire se focalise sur l’analyse de la stabilité entrée-état (en anglais ISS) du système par rapport à une perturbation bornée et le calcul des ensembles invariants ellipsoïdaux minimal ou maximal (ou sous forme d’ellipsoïdes tronqués) satisfaisant les contraintes. La deuxième partie envisage la synthèse d’une commande par retour d’état ISS stable et robuste vis-à-vis de perturbations bornées, garantissant l’ellipsoïde invariant maximal satisfaisant les contraintes ; puis la synthèse d’une loi decommande par retour d’état et observateur ISS stable vis-à-vis de perturbations bornées, garantissant une certaine performance ; enfin la synthèse d’un paramètre de Youla afin de garantir la projection maximale sur le sous-espace de l’état initial. La projection obtenue possède alors un volume plus grand que celui obtenu sans le paramètre de Youla d’où une amélioration en termes de robustesse. Une dernière étape vise à obtenirun compromis entre la robustesse et la performance en utilisant des critères basés sur le placement de pôles ou sur la vitesse de décroissance de la fonction de Lyapunov. Tous les résultats théoriques obtenus sont exprimés sous forme d’inégalités matricielles et sont validés en simulation et de façon expérimentale dans le cadre de la commande d’un convertisseur de puissance. / The research concretized in this memory is located at the intersection of two important fields, the robust control of discrete-time linear systems (LTI, LPV, switched) affected by bounded disturbances and constraints and the ellipsoidal invariant sets theory.The first part of this memory focuses on the analysis of input-to-state stability (ISS) over a bounded perturbation and the computation of the maximal or minimal invariant ellipsoidal (or truncated ellipsoidal) set satisfying the constraints. The second part is considering the synthesis of a control state feedback law ISS stable and robust over bounded disturbances, ensuring the maximal ellipsoidal invariant set satisfying the constraints, then the synthesis of an observer-based control law ISS stable over bounded disturbances,ensuring a certain performance, and finally the design of a Youla parameter guaranteeing the maximal ellipsoidal projection on the initial state subspace. The resulting projection has a volume greater than the one obtained without the Youla parameter resulting an improvement in terms of robustness. A final step is to obtain a compromise between robustness and performance using criteria based on poles placement or on theLyapunov function decreasing rate. The theoretical results are expressed as matrix inequalities and are validated in simulation and and experimentally on a Buck DC-DC converter.

<p>Flight Control Systems are often designed in linearization points over a flight envelope and it must be proven to clearance authorities that the system works for different parameter variations and failures all over this envelope. </p><p>In this thesis µ-analysis is tried as a complement for linear analysis in the frequency plane. Using this method stability can be guaranteed for all static parameter combinations modelled and linear criteria such as phase and gain margins and most unstable eigenvalue can be included in the analysis. A way of including bounds on the parameter variations using parameter dependent Lyapunov functions is also tried. </p><p>To perform µ-analysis the system must be described as a Linear Fractional Transformation (LFT). This is a way of reformulating a parameter dependent system description as an interconnection of a nominal linear time invariant system and a structured parameter block. </p><p>A linear and a rational approximation of the system are used to make LFTs. These methods are compared. Four algorithms for calculation of the upper and lower bounds of µ are evaluated. The methods are tried on VEGAS, a SAAB research aircraft model. </p><p>µ-analysis works quite well for linear clearance. The rational approximation LFT gives best results and can be cleared for the criteria mentioned above. A combination of the algorithms is used for best results. When the Lyapunov based method is used the size of the problem grows quite fast and, due to numerical problems, stability can only be guaranteed for a reduced model.</p>

In this study, various stability control systems are developed to remove the lateral instability of a conventional articulated steer vehicle (ASV) during the oscillatory yaw motion or “snaking mode”. First, to identify the nature of the instability, some analyses are performed using several simplified models. These investigations are mainly focused on analyzing the effects of forward speed and of two main subsystems of the vehicle, the steering system and tires, on the stability. The basic insights into the stability behavior of the vehicle obtained from the stability analyses of the simplified models are verified by conducting some simulations with a virtual prototype of the vehicle in ADAMS. To determine the most critical operating condition with regard to the lateral stability and to identify the effects of vehicle parameters on the stability, various studies are performed by introducing some modifications to the simplified models. Based on these studies, the disturbed straight-line on-highway motion with constant forward speed is recognized as the most critical driving condition. Also, the examinations show that when the vehicle is traveling with differentials locked, the vehicle is less prone to the instability. The examinations show that when the vehicle is carrying a rear-mounted load having interaction with ground, the instability may happen if the vehicle moves on a relatively good off-road surface. Again, the results gained from the analyses related to the effects of the vehicle parameters and operating conditions on the stability are verified using simulations in ADAMS by making some changes in the virtual prototype for any case. To stabilize the vehicle during its most critical driving condition, some studies are directed to indicate the shortcomings of passive methods. Alternative solutions, including design of different types of stability control systems, are proposed to generate a stabilizing yaw moment. The proposed solutions include an active steering system with a classical controller, an active torque vectoring device with a robust full state feedback controller, and a differential braking system with a robust variable structure controller. The robust controllers are designed by using simplified models, which are also used to evaluate the ability to deal with the uncertainties of the vehicle parameters and its variable operating conditions. These controllers are also incorporated into the virtual prototype, and their capabilities to stabilize the vehicle in different operating conditions and while traveling on different surfaces during the snaking mode are shown.

Flight Control Systems are often designed in linearization points over a flight envelope and it must be proven to clearance authorities that the system works for different parameter variations and failures all over this envelope. In this thesis µ-analysis is tried as a complement for linear analysis in the frequency plane. Using this method stability can be guaranteed for all static parameter combinations modelled and linear criteria such as phase and gain margins and most unstable eigenvalue can be included in the analysis. A way of including bounds on the parameter variations using parameter dependent Lyapunov functions is also tried. To perform µ-analysis the system must be described as a Linear Fractional Transformation (LFT). This is a way of reformulating a parameter dependent system description as an interconnection of a nominal linear time invariant system and a structured parameter block. A linear and a rational approximation of the system are used to make LFTs. These methods are compared. Four algorithms for calculation of the upper and lower bounds of µ are evaluated. The methods are tried on VEGAS, a SAAB research aircraft model. µ-analysis works quite well for linear clearance. The rational approximation LFT gives best results and can be cleared for the criteria mentioned above. A combination of the algorithms is used for best results. When the Lyapunov based method is used the size of the problem grows quite fast and, due to numerical problems, stability can only be guaranteed for a reduced model.

In this study, various stability control systems are developed to remove the lateral instability of a conventional articulated steer vehicle (ASV) during the oscillatory yaw motion or “snaking mode”. First, to identify the nature of the instability, some analyses are performed using several simplified models. These investigations are mainly focused on analyzing the effects of forward speed and of two main subsystems of the vehicle, the steering system and tires, on the stability. The basic insights into the stability behavior of the vehicle obtained from the stability analyses of the simplified models are verified by conducting some simulations with a virtual prototype of the vehicle in ADAMS. To determine the most critical operating condition with regard to the lateral stability and to identify the effects of vehicle parameters on the stability, various studies are performed by introducing some modifications to the simplified models. Based on these studies, the disturbed straight-line on-highway motion with constant forward speed is recognized as the most critical driving condition. Also, the examinations show that when the vehicle is traveling with differentials locked, the vehicle is less prone to the instability. The examinations show that when the vehicle is carrying a rear-mounted load having interaction with ground, the instability may happen if the vehicle moves on a relatively good off-road surface. Again, the results gained from the analyses related to the effects of the vehicle parameters and operating conditions on the stability are verified using simulations in ADAMS by making some changes in the virtual prototype for any case. To stabilize the vehicle during its most critical driving condition, some studies are directed to indicate the shortcomings of passive methods. Alternative solutions, including design of different types of stability control systems, are proposed to generate a stabilizing yaw moment. The proposed solutions include an active steering system with a classical controller, an active torque vectoring device with a robust full state feedback controller, and a differential braking system with a robust variable structure controller. The robust controllers are designed by using simplified models, which are also used to evaluate the ability to deal with the uncertainties of the vehicle parameters and its variable operating conditions. These controllers are also incorporated into the virtual prototype, and their capabilities to stabilize the vehicle in different operating conditions and while traveling on different surfaces during the snaking mode are shown.

This thesis deals with high angle of attack behaviour of a generic delta wing model aircraft. A high angle of attack wind tunnel database has been generated for this aircraft and based upon the bifurcation analysis of the data and the results of extensive simulations, it has been shown in the thesis that the post stall behaviour of this aircraft is both unstable and unpredictable. Unpredictability of aircraft behaviour arises from the fact that the aircraft response is oscillatory and divergent; the aircraft state trajectories do not settle down to any stable limit set and very often exceed valid aerodynamic database limits. This unpredictability of behaviour raises a major difficulty in the design of a procedure to recover the aircraft to normal flight regime in case the aircraft stalls and departs accidentally. A new methodology has been presented in this thesis to recover such unstable aircraft. In this methodology, a nonlinear controller is first designed at high angles of attack. This controller is connected by the pilot after the departure of the aircraft and the controller drives the aircraft to a well-defined spin condition. Thus, the controller makes the post stall aircraft behaviour predictable. Then a set of automatic recovery inputs is designed to reduce aircraft rotations and to lower the angle of attack. The present aircraft model is unstable at low angle of attack flight conditions as well and therefore to stabilize the aircraft to a low angle of attack level flight, another controller is designed. The high angle of attack controller is disconnected and the low angle of attack controller is connected automatically during the recovery process. The entire methodology is tested using extensive non-linear six degree-of-freedom simulations and the efficacy of the technique is established. The nonlinear controller that stabilizes the aircraft to a spin condition is designed using feedback linearization. The stability of a closed loop system obtained using feedback linearization is determined by the stability of the zero dynamics of the open loop plant. It has been shown in literature that the eigenvalues of the linearized zero dynamics are the same as the transmission zeros of the linearized plant at the equilibrium point. It is also well known that the location of transmission zeros of a linear system can be changed by the choice of outputs. In this thesis it is shown that if it is possible to reassign the outputs, then the feedback linearization based design for a linear system becomes very similar to a controller design for eigenvalue assignment. This thesis presents a new two-step procedure to obtain a locally stable and optimally robust closed loop system using feedback linearization. In the first step of this procedure optimal locations of the transmission zeros are found and in the second step, optimal outputs are constructed to place the system transmission zeros at these locations. The same outputs can then be used to construct nonlinear feedback for the nonlinear system and the resultant closed loop system is guaranteed to be locally robustly stable. The high angle of attack controller is designed using this procedure and its performance is presented in the thesis. The stabilized spin equilibrium point of the closed loop system is also shown to have a large domain of attraction. Having designed a locally robust stabilizing controller, the thesis addresses the problem of the evaluation of robustness of the stability of the equilibrium point in a nonlinear framework. The thesis presents a general method to construct bounds on the additive perturbations of the system vector field over a large region in the domain of attraction of a stable equilibrium point using Lyapunov functions. If the system perturbations lie within these bounds, the system is guaranteed to be stable. The thesis first proposes a method to numerically construct a Lyapunov function over a large region in the domain of attraction. In this method a sequence of Lyapunov functions are constructed such that each function in the sequence gives a larger estimate of the domain of attraction than the previous one. The seminal idea for this method is obtained from the existing literature and this idea is considerably generalized. Using this method, it is possible to numerically obtain a Lyapunov function value at each point in the domain of attraction, but the Lyapunov function does not have an analytical form. Hence, it is proposed to represent this function using neural networks. The thesis then discusses a new method to construct perturbation bounds. It is shown that the perturbation bounds obtained over a large region in the domain of attraction using a single Lyapunov function is too conservative. Using the concept of sequence of Lyapunov functions, the thesis proposes three methods to obtain the least conservative bounds for an initial local Lyapunov function. These general ideas are then applied to the aircraft example and the bounds on the perturbation of the aerodynamic database are presented.

<p>This thesis consists of four independent papers concerningthe control of mobile robots in the context of obstacleavoidance and formation keeping.</p><p>The first paper describes a new theoreticallyv erifiableapproach to obstacle avoidance. It merges the ideas of twoprevious methods, with complementaryprop erties, byusing acombined control Lyapunov function (CLF) and model predictivecontrol (MPC) framework.</p><p>The second paper investigates the problem of moving a fixedformation of vehicles through a partiallykno wn environmentwith obstacles. Using an input to state (ISS) formulation theconcept of configuration space obstacles is generalized toleader follower formations. This generalization then makes itpossible to convert the problem into a standard single vehicleobstacle avoidance problem, such as the one considered in thefirst paper. The properties of goal convergence and safetyth uscarries over to the formation obstacle avoidance case.</p><p>In the third paper, coordination along trajectories of anonhomogenuos set of vehicles is considered. Byusing a controlLyapunov function approach, properties such as boundedformation error and finite completion time is shown.</p><p>Finally, the fourth paper applies a generalized version ofthe control in the third paper to translate,rotate and expanda formation. It is furthermore shown how a partial decouplingof formation keeping and formation mission can be achieved. Theapproach is then applied to a scenario of underwater vehiclesclimbing gradients in search for specific thermal/biologicalregions of interest. The sensor data fusion problem fordifferent formation configurations is investigated and anoptimal formation geometryis proposed.</p><p><b>Keywords:</b>Mobile Robots, Robot Control, ObstacleAvoidance, Multirobot System, Formation Control, NavigationFunction, Lyapunov Function, Model Predictive Control, RecedingHorizon Control, Gradient Climbing, Gradient Estimation.</p>