Preprint Article Version 1 This version is not peer-reviewed

Almost Global Stability of Nonlinear Switched Systems with Time-Dependent Switching

Version 1 : Received: 25 September 2018 / Approved: 26 September 2018 / Online: 26 September 2018 (14:06:32 CEST)

How to cite: Karabacak, O..; Kivilcim, A.; Wisniewski, R.. Almost Global Stability of Nonlinear Switched Systems with Time-Dependent Switching. Preprints 2018, 2018090514 (doi: 10.20944/preprints201809.0514.v1). Karabacak, O..; Kivilcim, A.; Wisniewski, R.. Almost Global Stability of Nonlinear Switched Systems with Time-Dependent Switching. Preprints 2018, 2018090514 (doi: 10.20944/preprints201809.0514.v1).

Abstract

For a dynamical system, it is known that the existence of a Lyapunov density implies almost global stability of an equilibrium. It is then natural to ask whether the existence of a common Lyapunov density for a nonlinear switched system implies almost global stability, in the same way as a common Lyapunov function implies global stability for nonlinear switched systems. In this work, the answer to this question is shown to be affirmative as long as switchings satisfy a dwell-time constraint with an arbitrarily small dwell time. As a straightforward extension of this result, we employ multiple Lyapunov densities in analogy with the role of multiple Lyapunov functions for the global stability of switched systems. This gives rise to a minimum dwell time estimate to ensure almost global stability of nonlinear switched systems, when a common Lyapunov density does not exist. The results obtained for continuous-time switched systems are based on some sufficient conditions for the almost global stability of discrete-time non-autonomous systems. These conditions are obtained using the duality between Frobenius-Perron operator and Koopman operator.

Subject Areas

Almost global stability, nonlinear switched systems, common Lyapunov density, multiple Lyapunov density

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