Copyright 0 IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998 DEVELOPMENTS IN TIME-VARYING FEEDBACK STABILIZATION OF NONLINEAR SYSTEMS Pascal Morin, Jean-Baptiste Pomet, Claude Samson *,1 * INRIA 2004 Route des Lucioles 06902 SOPHIA-ANTIPOLIS, FRANCE E-mail: (Firstname.Lastname}@inria.lr Abstract: Over the last ten years, time-varying feedback has emerged as a new approach for the asymptotic stabilization of highly nonlinear control systems. This paper gives an overview of the developments made in this area. In particular, relations between theoretical developments and practical motivations are emphasized. Copyright © 1998 IFAC Keywords: Nonlinear control systems, stabilization, time-varying feedback 1. INTRODUCTION We consider control systems of the form x = l(x,1J.), x E Rn, E Rm, with 1(0,0) = 0. To simplify, it is assumed that 1 is analytic in a neighborhood of (x = 0,1J. = 0). The general problem under consideration is the asymptotic stabilization, in the sense of Lyapunov, of the origin x = ° via the use of a feedback control law. Let us just recall that asymptotic stabilization means the satisfaction of two properties: i) stabil- ity of the origin of the closed-loop system, and ii) convergence to this point of all the solutions of the closed-loop system. Within this large set of systems, and with respect to the problem 01 asymptotic leedback stabilization, one may try to distinguish several subsets of in- terest. The first one (obvious and important) is composed of systems the tangent approximation of which, at the equilibrium point (x = 0,11.= 0), is stabilizable, i.e. which can be locally asymptoti- cally stabilized by means of a linear feedback 11. = Kx. This set contains, in particular, all control- lable systems which can be exactly linearized via either static or dynamic feedback, since controlla- bility of the tangent approximation is a necessary condition for this (Charlet et al., 1991). Most of 1 The authors are with the "GDR/PRC d'Automatique" 565 the existing feedback control literature has been, and still is, devoted to this class of systems. Al- though the function 1 (., .) may be nonlinear, the system may be viewed as almost linear in the sense that any feedback control technique developed for stabilizable linear systems successfully applies in order to ensure i) local stability, ii) fast (expo- nential) convergence of the system's trajectories to zero, and iii) some amount of robustness with respect to modeling errors. Another subset of potential interest is composed of systems which are controllable at the origin -in a ''reasonable'' sense-, but whose tangent approximation at (x = 0,1J. = 0) is not sta- bilizable, so that none of the classical feedback techniques based on some type of linearization applies to them. In this respect these systems are truly nonlinear, by contrast with those belonging to the first category. For most of these systems, there exist open-loop controls which rapidly drive the system's state to zero, when the initial error is not too large and in the absence of modeling errors. However, this does not imply, contrary to the case of linear systems, that asymptotically stabilizing feedbacks exist. As a matter of fact, studying this class of systems has revealed that non-existence of asymptotically stabilizing contin- uous pure-state feedbacks 11.( x) is a rather common situation, as first pointed out in (Sussmann, 1979).