Systems involving quantisation arise in many areas of engineering, especially when digital implementations are involved. In this thesis we consider different aspects of quantisation in feedback control systems. We study two topics of interest: (a) quantisers that quadratically stabilise a given system and are efficient in the use of their quantisation levels and (b) the derivation of ultimate bounds for perturbed systems, especially when the perturbations arise from the use of quantisers. In the first part of the thesis we address problem (a) above. We consider quadratic stabilisation of discrete-time multiple-input systems by means of quantised static feedback and we measure the efficiency of a quantiser via the concept of quantisation density. Intuitively, the lower the density of a quantiser is, the more separated its quantisation levels are. We thus deal with the problem of optimising density over all quantisers that quadratically stabilise a given system with respect to a given control Lyapunov function. Most of the available results on this problem treat single-input systems, and the ones that deal with the multiple-input case consider only two-input systems. In this thesis, we derive several new results for multiple-input systems and also provide an alternative approach to deal with the single-input case. Our new results for multiple-input systems include the derivation of the structure of optimal quantisers and the explicit design of multivariable quantisers with finite density that are able to quadratically stabilise systems having an arbitrary number of inputs. For single-input systems, we provide an alternative approach to the analysis and design of optimal quantisers by establishing a link between the separation of the quantisation levels of a quantiser and the size of its quantisation regions. In the second part of the thesis we address problem (b) above. In the presence of perturbations, asymptotic stabilisation may not be possible. However, there may exist a bounded region that contains the equilibrium point and has the property that the system trajectories converge to this bounded region. When this bounded region exists, we say that the system trajectories are ultimately bounded, and that this bounded region is an ultimate bound for the system. The size of the ultimate bound quantifies the performance of the system in steady state. Hence, it is important to derive ultimate bounds that are as tight as possible. This part of the thesis addresses the problem of ultimate bound computation in settings involving several scalar quantisers, each having different features. We consider each quantised variable in the system to be a perturbed copy of the corresponding unquantised variable. This turns the original quantised system into a perturbed system, where the perturbation has a natural \emph{componentwise} bound. Moreover, according to the type of quantiser employed, the perturbation bound may depend on the system state. Typical methods to estimate ultimate bounds are based on the use of Lyapunov functions and usually require a bound on the norm of the perturbation. Applying these methods in the setting considered here may disregard important information on the structure of the perturbation bound. We therefore derive ultimate bounds on the system states that explicitly take account of the componentwise structure of the perturbation bound. The ultimate bounds derived also have a componentwise form, and can be systematically computed without having to, e.g. select a suitable Lyapunov function for the system. The results of this part of the thesis, though motivated by quantised systems, apply to more general perturbations, not necessarily arising from quantisation. / PhD Doctorate

Cette thèse aborde le problème de l'analyse de la stabilité et de la conception des lois de commande pour les systèmes non linéaires mis sous la forme de modèles flous continus de type Takagi-Sugeno. L'analyse de stabilité est généralement basée sur la méthode directe de Lyapunov. Plusieurs approches existent dans la littérature, basées sur des fonctions de Lyapunov quadratiques sont proposées pour résoudre ce problème, les résultats obtenus à l'aide des telles fonctions introduisent un conservatisme qui peut être très préjudiciable. Pour surmonter ce problème, différentes approches basées sur des fonctions de Lyapunov non quadratiques ont été proposées, néanmoins ces approches sont basées sur desconditions très restrictives. L'idée développée dans ce travail est d'utiliser des fonctions de Lyapunov non quadratiques et des contrôleurs non-PDC afin d'en tirer des conditions de stabilité et de stabilisation moins conservatives. Les propositions principales sont : l'utilisation des bornes locales des dérivées partielles au lieu des dérivés des fonctions d’appartenances, le découplage du gain du régulateur des variables de décision de la fonction Lyapunov, l’utilisation des fonctions de Lyapunov floues polynomiales dans l’environnement des polynômes et la proposition de la synthèse de contrôleur vérifiant certaines limites de dérivés respectées dans une région de la modélisation à la place de les vérifier a posteriori. Ces nouvelles approches permettent de proposer des conditions locales afin de stabiliser les modèles flous continus de type T-S, y compris ceux qui n'admettent pas une stabilisation quadratique et obtenir des domaines de stabilité plus grand. Plusieurs exemples de simulation sont choisis afin de vérifier les résultats présentésdans cette thèse. / This thesis deals with the problem of stability analysis and control design for nonlinear systems in the form of continuous-time Takagi-Sugeno models. The approach to stability analysis is usually based on the direct Lyapunov method. Several approaches in the literature, based on quadratic Lyapunov functions, are proposed to solve this problem ; the results obtained using such functions introduce a conservatism that can be very detrimental. To overcome this problem, various approaches based on non-quadratic Lyapunov functions have also been recently presented; however, these approaches are based on very conservative bounds or too restrictive conditions. The idea developed in this work is to use non-quadratic Lyapunov functions and non-PDC controller in order to derive less conservative stability and stabilization conditions. The main proposals are : using local bounds in partial derivatives instead of time derivatives of the memberships,decoupling the controller gain from the Lyapunov function decision variables, using fuzzy Lyapunov functions in polynomial settings and proposing the synthesis of controller ensuring a priori known time-derivative bounds are fulfilled in a modelling region instead of checking them a posteriori. These new approaches allow proposing local conditions to stabilize continuous T-S fuzzy systems including those that do not admit a quadratic stabilization. Several simulation examples are chosen to verify the results given in this dissertation.