Journal
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
Volume 53, Issue 3, Pages 734-748Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TAC.2008.919257
Keywords
asymptotic stability; hybrid systems; robustness; smooth Lyapunov functions
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It is shown that (pre)asymptotic stability, which generalizes asymptotic stability, of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth Lyapunov function. This result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small, state-dependent perturbations. As a special case, we state a converse Lyapunov theorem for systems with logic variables and use this result to establish input-to-state stabilization using hybrid feedback control. The converse Lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying, weakly jumping parameters, to temporal regularization, to the insertion of jumps according to an average dwell-time rule, and to the insertion of flow according to a reverse average dwell-time rule.
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