C s Daniel Liberzon and A. Stephen Morse y a switched system, we mean a hybrid dynamical system consisting of afamily 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 sequcnces, stability for certain useful classes of switching sequences, and construction of sta- bilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in con- nection with various questions of interest in control theory and applications. Problem Statement and Motivation Many systems encountered in practice exhibit switching be- tween several subsystems that is dependent on various environ- mental factors. Some examples of such systems are discussed in [ 11-13]. Another source of motivation for studying switched sys- tems comes from the rapidly developing area of switching con- trol. Control techniques based on switching between different controllers have been applied extensively in recent years, partic- ularly in the adaptive context, where they have been shown to achieve stability and improve transient response (see, among many references, [4]-L6]). The importance of such control meth- ods also stems in part from the existence of systems that cannot be asymptotically stabilized by a single continuous feedback control law 171. Switched systems have numerous applications in control of mechanical systems, the automotive industry, aircraft and air traffic control, switching power converters, and many other fields. The book 18 I contains reports on various developments in some of these weas. In the last few years, every major control conference has had several regular and invited sessions on switching systems and control. Moreover, workshops and sym- posia devoted specifically to these topics are regularly taking place. Almost evcry major technical control journal has had or is planning to have a special issue on switched and hybrid systems. These sources can be consulted for further references. Mathematically, a swilched .systenz can be described by a dif- fcrenlial equation of the form B : [0, -) + Pis a piecewise constant function of time, called a switching signal. In specific situations, the value ofo at a given timet might just depend ont orx(t), or both, or may be generated using more sophisticated techniques such as hybrid feedback with memory in the loop. We assnme that the state of (1) does not jump at the switchinginstants, i.e., the solutionx(,) is cvcrywhere continuous. Note that the case of infinitely fast switching (chat- tering), which calls for a concept of generalized solution, is not considered in this article. The sct P is typically a compact (often finite) subset of a finite-dimcnsional linear vector space. In the particular case where all the individual subsystems are linear, we obtain ii switched linear system X: = A,x. (2) This class of systems is the one most commonly treated in the lit- erature. In this article, whenever possible, problems will be for- mulated and discussed in the more general context of the switched system (I). The first basic problem that we will consider can he formu- latcd as follows. Problem A. Find conditions that guarantee that the switched system (I) is asymptotically .stahlefor uny switching signal. One situation in which Problem A is of great importancc is when a given plant is being controlled by means of switching among a family of stabilizing controllers, each of which is de- signed for a specific task. The prototypical architecture for such a multicontroller switched system is shown in Fig. 1. A high- level decision maker (supervisor) determines which controller is to be connected in closed loop with the plant at each instant of time. Stability of the switched system can usually hc ensured by keeping each controller in the loop for a long enough time, to al- low the transient effects to dissipate. However, modern com- puter-controlled systems are capable of‘ very fast switching rates, which creates the need to be ablc to test stability of the switched system for arbitrarily last switching signals. We are assuming here that the individual subsystems have the origin as a common eq~iilibriuni point: f,](O) = 0, p t ‘P. Clearly, a necessary condition for (asymptotic) stability under arbitrary where {f;, : p E P) is a family of sufficiently regular functions from B” to R” that is parametrized by some index set P, and Liberzon ([email protected]. yule.edu) undMor,se ure with the De- partment of Electricrrl Engineering, Yak University, New Haven, CT 04520-8267. This research w(1.s .supported by ARO DAAH04- 95-1-0114, NSF ECS 9634146, andAFOSR F49620-97-I-OIOh‘. October 1999 59t- ll I I/ Fig. 1. Multicontroller architecture. switching is that all of the individual subsystems are (asymptoti- cally) stable. Indeed, if the 11th system is unstable, the switched system will he unstableil we reto(t) = p. To see that stability of all the individual subsystems is not sufficicnt, consider two scc- ond-order asymptotically stable systems whose trajectories are skctched in the top row of Fig. 2. Depending on a particular switching signal, the trajectories of thc switched system might look as shown in the bottom left corner (asymptotically stable) or as shown in the bottom right corner (unstable). The above example shows that Problem A is not trivial in the sense that it is possible to get instability by switching between as- ymptotically stable systems. (However, there are certain limita- tions as to what types of instability are possible in this case. For cxample, it is easy to see that the trajectorics of such a switched system cannot escape to infinity in finite time.) If this happens, one may ask whether the switched system will be asymptotically stable for certain useful classes of switching signals. This leads to the following problem. Problem B. IdenriJj, those cla.s.ses of .switching sijinals for which the switched system (I) is aymproticnlly stablc. Fig. 2. Possible trajectories ofa switched system. 60 Since it is often unreasonable to exclude constant switching signals of the formo(t) E p, Problem B will he considered under the assumption that all the individual subsystems arc asymptoti- cally stable. Basically, wc will find that stability is ensured if