Locally optimal controllers and globally inverse optimal controllers

In this paper we consider the problem of global asymptotic stabilization with prescribed local behavior. We show that this problem can be formulated in terms of control Lyapunov functions. Moreover, we show that if the local control law has been synthesized employing a LQ approach, then the associated Lyapunov function can be seen as the value function of an optimal problem with some specific local properties. We illustrate these results on two specific classes of systems: backstepping and feedforward systems. Finally, we show how this framework can be employed when considering the orbital transfer problem

A Region-Dependent Gain Condition for Asymptotic Stability

A sufficient condition for the stability of a system resulting from the interconnection of dynamical systems is given by the small gain theorem. Roughly speaking, to apply this theorem, it is required that the gains composition is continuous, increasing and upper bounded by the identity function. In this work, an alternative sufficient condition is presented for the case in which this criterion fails due to either lack of continuity or the bound of the composed gain is larger than the identity function. More precisely, the local (resp. non-local) asymptotic stability of the origin (resp. global attractivity of a compact set) is ensured by a region-dependent small gain condition. Under an additional condition that implies convergence of solutions for almost all initial conditions in a suitable domain, the almost global asymptotic stability of the origin is ensured. Two examples illustrate and motivate this approach

Stabilisation sous contraintes locales et globales

This theses concerns hybrid systems and small gain theorems. More precisely, firstly we computed hybrid stabilizers for nonlinear control systems for which the backstepping techniques do not apply, and we succeeded to combine local feedback laws and global controllers stabilizing a set "close" to the origin. The second contribution is on small gains theorem, by dealing with systems for which small gains conditions are satisfied only regionally. We were able to combine such region-dependent small-gain conditions for the global asymptotic stability and for the almost global asymptotic stability.Cette thèse concerne des systèmes hybrides et le théorème des petits-gains. Plus précisément, d'une part, nous avons calculé les lois de commande hybrides pour les systèmes non-linéaires lesquels les techniques de synthèse par backstepping ne s'applique pas et nous avons réussi à combiner les lois de commande locales et globales pour la stabilisation d'un ensemble «proche» de l'origine. La seconde contribution est sur ​​le théorème des petits-gains, en traitant avec des systèmes pour lesquels les conditions des petits-gains sont satisfaites seulement au niveau régional. Nous avons réussi à combiner ces conditions régionales pour le petit-gain pour la stabilité asymptotique et pour la presque stabilité asymptotique globale

Relaxed and Hybridized Backstepping

International audienceIn the present work, we consider nonlinear control systems for which there exist structural obstacles to the design of classical continuous backstepping feedback laws. We conceive feedback laws such that the origin of the closed-loop system is not globally asymptotically stable but a suitable attractor (strictly containing the origin) is practically asymptotically stable. A design method is suggested to build a hybrid feedback law combining a backstepping controller with a locally stabilizing controller. A constructive approach is also suggested employing a differential inclusion representation of the nonlinear dynamics. The results are illustrated for a nonlinear system which, due to its structure, does not have a priori any globally stabilizing backstepping controller