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Basic problems in Stability and Design



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s C 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 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 0 P i s a piecewise constant function of time called a switching signal In specific situations the value o f o 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 chattering 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 B Problem Statement and Motivation X A x 2 Many systems encountered in practice exhibit switching between several subsystems that is dependent on various environmental factors Some examples of such systems are discussed in 11 13 Another source of motivation for studying switched systems comes from the rapidly developing area of switching control Control techniques based on switching between different controllers have been applied extensively in recent years particularly in the adaptive context where they have been shown to achieve stability and improve transient response see among many references 4 L6 The



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