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 presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach. Keywords. Piecewise linear systems, Lyapunov stability, linear matrix inequalities. 1. Introduction Construction of Lyapunov functions is one of the most fundamental problems in systems theory. The most direct application is stability analysis, but analogous problems appear more or less implicitly also in performance analysis, controller synthesis and system identification. Consequently, methods for constructing Lyapunov functions for general nonlinear systems is of great theoretical and practical interest. The objective of this paper is to develop a uniform and compu...

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 pap e we prove in a constructive way, the ee ale b e we e pie- a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot be e asilydely from those of the comp tline subsyste Inste we propose practical nume te base onmixe te line programming. Keywords--- Hybrid syste controllability,obse ability, pie line syste pie a#ne syste mixe teline programming I. Introducti In recent yearsb oth control and computer science haveb een attractedb y hybridsystem [1], [2], [23], [25], [26],b ecause they provide a unified framework fordescribgARB( cesses evolving accordingto continuous dynamics, discrete dynamics, and logic rules. The interest is mainly motivatedb y the large variety of practical situations, for instance real-time systems, where physical processes interact with digital controllers. Several modelingformalisms h...

Abstract — Hybrid systems which are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of a system (e.g., continuous-time dynamics, discrete-time dynamics, jump phenomena, switching and logic commands, and the like) are of great current interest. In the present paper we first formulate a model for hybrid dynamical systems which covers a very large class of systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of an invariant set (e.g., equilibrium) for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability). To demonstrate the applicability of the developed theory, we present specific examples of hybrid dynamical systems and we conduct a stability analysis of some of these examples (a class of sampleddata feedback control systems with a nonlinear (continuous-time) plant and a linear (discrete-time) controller, and a class of systems with impulse effects). Index Terms — Asymptotic stability, boundedness, dynamical system, equilibrium, exponential stability, hybrid, hybrid dynamical

this paper is to give a brief tutorial review of four different classes of hybrid systems of this type - each consists of a continuous-time process to be controlled, a parameterized family of candidate controllers, and an event driven switching logic. Three of the logics, called prerouted switching, hysteresis switching and dwelltime switching respectively, are simple strategies capable of determining in real time which candidate controller should be put in feedback with a process in order to achieve desired closed-loop performance. The fourth, called cyclic switching, has been devised to solve the long-standing stabilizability problem which arises in the synthesis of identifier-based adaptive controllers because of the existence of points in parameter space where the estimated model upon which certainty equivalence synthesis is based, loses stabilizability

Stability and robustness issues for hybrid systems are considered in this paper. General stability results that are extensions of classical Lyapunov theory have recently been formulated. However, these results are in general not straightforward to apply due to the following reasons. First, a search for multiple Lyapunov functions must be performed. However, existing theory does not unveil how to find such functions. Secondly, if the most general stability result is applied, knowledge about the continuous trajectory is required, at least at some time instants. Because of these drawbacks stronger conditions for stability are suggested, in which case it is shown that the search for Lyapunov functions can be formulated as a linear matrix inequality (LMI) problem for hybrid systems consisting of linear subsystems. Additionally, it is shown how robustness properties can be achieved when the Lyapunov functions are given. Specifically, it is described how to determine permitted switch regions ...

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 ...

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.