By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. This article surveys recent developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.

This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavor, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper. Keywords---Hybrid systems, linear matrix inequalities, stability, stabilizability, switched systems. I.

In this paper we present two strategies to design a hybrid controller for a system described by several nonlinear vector fields. Besides the overall goal to find a controller that stabilizes the closed-loop hybrid system, the selection will also be made in such a way that an exponentially stable closed-loop system is obtained. The design strategies are based on stated stability and exponential stability theorems for hybrid systems. The first approach results in regions where it is possible to change vector fields guaranteeing (exponential) stability of the closed-loop hybrid system. The second design strategy utilizes the fact that a system is (exponentially) stable if it is always possible to choose a vector field that points in a direction such that the trajectory approaches the equilibrium point. These conditions can be verified by solving a linear matrix inequality (LMI) problem. The presented methods are illustrated by examples. Keywords: Hybrid systems, Hybrid controller, Stabili...

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving these problems in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability, and also represent problems in which significant progress has been made. We also comment on the inherent difficulty of determining stability of switched systems in general which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell time requirements and state dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems is reviewed. We briefly comment on the classical Lur’e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area.

In this paper, the problem of asymptotically stabilizing switched systems consisting of second-order LTI subsystems is studied and solved. In particular, necessary and suÆcient stabilizability conditions are derived and a design procedure to construct stabilizing switching laws is introduced. Switching is needed for the stabilization of a switched system if none of its subsystems is stable. Switched systems consisting of subsystems with unstable foci are studied rst and stabilizing conic switching control laws for such systems are introduced. In particular, necessary and suÆcient conditions for asymptotic stabilizability are derived for such systems. This result is then extended to switched systems with unstable nodes and saddle points. If a switched system is asymptotically stabilizable, then using the conic switching approach introduced earlier, asymptotically stabilizing switching control laws can be obtained. Furthermore, the conic switching laws derived in the paper are shown to ...

In this paper a brief overview of hybrid control systems is given and an introduction to several approaches in hybrid systems research with an extended list of references is presented. Emphasis is put on Petri net approaches to hybrid control.

In this paper, we investigate the disturbance attenuation properties of timecontrolled switched systems consisting of several linear time-invariant subsystems by using a dwell time approach incorporated with piecewise Lyapunov functions. First, we show that when all subsystems are Hurwitz stable and achieve a disturbance attenuation level smaller than # 0 , then the switched system can achieve any disturbance attenuation level larger than # 0 if the dwell time is chosen sufficiently large. This result is extended to the case where not all subsystems are Hurwitz stable, by showing that if the dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then a desirable disturbance attenuation level is guaranteed. Finally, a discussion is made on the case for which nonlinear norm-bounded perturbations exist in the subsystems.

We propose a notion of passivity for hybrid systems. Our work is motivated by problems in haptics and teleoperation where several computer controlled mechanical systems are connected through a communication channel. To account for time delays and to better react to user actions it is desirable to design controllers that can switch between different operating modes. Each of the interacting systems can be therefore naturally modeled as a hybrid system. A traditional passivity definition requires that a storage function exists that is common to all operating modes. We show that stability of the system can be guaranteed even if different storage function is found for each of the modes, provided appropriate conditions are satisfied when the system switches.

In recent work we proposed a general model for hybrid dynamical systems whose states are dened on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but dened on real time R + = [0; 1). The motions of this class of systems are in general discontinuous. This class of systems may be nite or innite dimensional. For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specic classes of hybrid dynamical systems. In the present paper we continue the...

We present a framework for designing stable control schemes for systems with changing dynamics (SCD). Such systems form a subset of hybrid systems; their stabilization is therefore a problem in hybrid control. It is often difficult or even impossible to design a single controller that would stabilize a SCD. An appealing alternative are switching control schemes, where a different controller is employed in each dynamic regime and the stability of the overall system is ensured through an appropriate switching scheme. We formulate a set of sufficient conditions for the stability of a switching control scheme. We show that by imposing a hierarchy among the controllers, sufficient conditions can be formulated in a form suitable for the controller design. The hierarchy is formally defined through a partial order. Our methodology is applied to stabilization of a two-wheel mobile robot of the Hilare type, where the wheels are allowed to slip.