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Unformatted text preview: 1 Uniform Stability of Switched Linear Systems: Extensions of LaSalle’s Invariance Principle Jo˜ao P. Hespanha, Senior Member, IEEE Abstract —This paper addresses the uniform stability of switched linear systems, where uniformity refers to the con vergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set. We provide a collection of results that can be viewed as extensions of LaSalle’s Invariance Principle to certain classes of switched linear systems. Using these results one can deduce asymptotic stability using multiple Lyapunov functions whose Lie derivatives are only negative semi definite. Depending on the regularity assumptions placed on the switching signals, one may be able to conclude just asymptotic stability or (uniform) exponential stability. We show by counter example that the results obtained are tight. Index Terms —Switched systems, hybrid systems, LaSalle’s Invariance Principle, Stability. I. I NTRODUCTION Switched systems are typically represented by equations of the form ˙ x = f σ ( x ) , x ∈ R n , (1) where σ : [0 , ∞ ) → P denotes a piecewise constant signal that effectively “switches” the righthandside of the differential equation by selecting different vector fields from a parame terized family { f p : p ∈ P} . The time instants at which σ is discontinuous are called switching times . The key distinction between the switched system (1) and the timevarying system ˙ x = g ( x, t ) , x ∈ R n , t ≥ , (2) with g defined by g ( x, t ) := f σ ( t ) ( x ) , ∀ x ∈ R n , t ≥ , is that one typically associates a family of admissible switching signals S to (1) and studies the properties of the solutions to (1) as σ ranges over S . Clearly, for a single switching signal σ , (1) and (2) represent exactly the same object. The set of solutions to an (unswitched) system like (2) is parameterized solely by a set of initial conditions. However, the set of solutions to a switched linear system like (1) is parameterized both by a set of initial conditions and by an admissible set of switching signals S on which σ is assumed to lie. This poses important questions with respect to the uniformity of properties such as stability, convergence, etc., as σ ranges over S . This paper addresses the uniform asymptotic stability of switched systems, where uniformity refers to the multiple solutions that one obtains as the switching signal ranges over a given set. We consider two notions of asymptotic Revision date 10/10/2004: This version of the paper differs slightly from the published version. Please see the Acknowledgments Section. This material is based upon work supported by the National Science Foundation under the Grant No. ECS0242798....
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This document was uploaded on 12/29/2011.
 Fall '09

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