Switched nonlinear differential algebraic equations: Solution theory,Lyapunov functions,and stability |
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| Authors: | Daniel Liberzon Stephan Trenn |
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| Affiliation: | 1. Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL, USA;2. Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany;1. Department of Informatics, Bioengineering, Robotics, and Systems Engineering -DIBRIS, University of Genoa, Italy;2. Dipartimento di Ingegneria dell’Informazione -DINFO, University of Florence, Italy;3. ITM, Faculty of Mathematics and Natural Sciences, University of Groningen, The Netherlands;1. Intel Labs, Intel Tecnología de México, Av. del Bosque 1001, Colonia El Bajío, 45019 Zapopan, Jalisco, Mexico;2. Institute of Information Theory and Automation of the Czech Academy of Sciences, P.O. Box 18, 182 08 Prague, Czech Republic;3. CINVESTAV Unidad Guadalajara, Av. del Bosque 1145, Colonia El Bajío, 45019 Zapopan, Jalisco, Mexico;1. State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, PR China;2. College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China;1. School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran;2. The Control and Intelligent Processing Center of Excellence (CIPCE), School of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran;1. Bernoulli Institute for Mathematics, CS and AI, University of Groningen, The Netherlands;2. Univ. Lyon 1, Université Claude Bernard Lyon 1, CNRS, LAGEP UMR 5007, Villeurbanne, France;1. Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China;2. College of Automation, Harbin Engineering University, Harbin 150001, China |
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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. |
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