Brief paper: Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability

Authors:
Daniel Liberzon;Stephan Trenn
Affiliations:
Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA;Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany
Venue:
Automatica (Journal of IFAC)
Year:
2012

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Abstract

We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for linear switched DAEs to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov's direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively.