Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003
WeA05-5
Approximate Stabilisation of Uncertain Hybrid Systems with Controllable Transitions Yan Gao1 , John Lygeros2 , Marc Quincampoix3 and Nicolas Seube4 1 School of Management University of Shanghai for Science and Technology Jungong Road 516, Shanghai 200093, China Email:
[email protected] 2 Department of Electrical and Computer Engineering University of Patras Rio, Patras, GR 26500, Greece Email:
[email protected] 3 D´epartement de Mathematiques Universit´e de Bretagne Occidentale, 6 Avenue Le Gorgeu, B. P. 809, 29285 Brest Cedex, France Email:
[email protected] 4 ENSIETA, 2 Rue Francois Verny, 29806 Brest Cedex 9, France Email:
[email protected] Abstract−Stabilization of uncertain hybrid systems with controllable transitions is considered. Uncertainty enters in the form of a disturbance input that can affect both the continuous and the discrete dynamics. A method for designing piecewise constant feedback controllers is developed. The controllers achieve approximate exponential convergence of the runs of the closed loop system to the zero level set of a Lyapunov function. I. INTRODUCTION Stability conditions for hybrid systems have been the topic of intense research in recent years. Many of the proposed methods involve the use of Lyapunov function. These methods are especially useful for certain classes of systems (for example, switched linear systems) for which computationally efficient methods such as LMI’s can be used to construct Lyapunov functions automatically. An overview of the different issues and approaches in this area can be found in [3]. The topic of stabilization has been somewhat less extensively studied. Much of the work in this area deals with switched systems (usually linear and/or planar). The proposed stabilization schemes typically
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involve selecting appropriate times for switching between a set of given systems. In some cases this approach has been extended to robust stabilization schemes for systems that involve certain types of uncertainty [11, 6]. In this paper we concentrate on this last issue of robust stabilization. We consider a fairly wide class of hybrid systems, with non-linear continuous dynamics and nonlinear transition functions. Uncertainty enters both the continuous evolution and discrete transitions, in the form of an uncontrollable disturbance input. Earlier results in this direction [5] established conditions under which stabilization is possible for hybrid systems like these, were all transitions are forced (in the sense that they have to take place whenever the state reaches a predetermined part of the state space). In this paper we extend the results of [5] to also allow controllable transitions. In particular, in addition to states where transitions have to take place, our model also allows for states where transitions can take place if the control and/or the disturbance inputs want them to. We again use the result of [5] to establish conditions under which a controller that approximately stabilizes the system exponentially can be designed. The controller signal is piecewise constant and involves feedback whenever a discrete transition
takes place and at regular sampling times along continuous evolution. The advantage of such a controller from a practical point of view is that it is easy to implement, using an interrupt driven sampled data system and a zero order hold.
II. STABILIZATION PROBLEM The development of the stabilization scheme makes use of a few concepts from non-smooth analysis and viability theory. We briefly review the necessary material below; for a more thorough treatment the reader is referred to [1]. Let h·, ·i denote the standard inner product in
