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On formalism and stability of switched systems

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Abstract

In this paper, we formulate a uniform mathematical framework for studying switched systems with piecewise linear partitioned state space and state dependent switching. Based on known results from the theory of differential inclusions, we devise a Lyapunov stability theorem suitable for this class of switched systems. With this, we prove a Lyapunov stability theorem for piecewise linear switched systems by means of a concrete class of Lyapunov functions. Contrary to existing results on the subject, the stability theorems in this paper include Filippov (or relaxed) solutions and allow infinite switching in finite time. Finally, we show that for a class of piecewise linear switched systems, the inertia of the system is not sufficient to determine its stability. A number of examples are provided to illustrate the concepts discussed in this paper.

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Authors and Affiliations

  1. Department of Electronic Systems, Automation and Control, Aalborg University, Fredrik Bajers Vej 7 C, 9220, Aalborg East, Denmark

    John Leth & Rafael Wisniewski

Authors
  1. John Leth
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  2. Rafael Wisniewski
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Additional information

This research was supported by the Danish Council for Technology and Innovation.

John LETH received his M.S (2003) and Ph.D. (2007) degrees from the Department of Mathematical Sciences, Aalborg University, Denmark. Currently, he is employed as a postdoctoral researcher at the Department of Electronic Systems, Aalborg University. His research interests include mathematical control theory and hybrid systems.

Rafael WISNIEWSKI is a professor in the Section of Automation & Control, Department of Electronic Systems, Aalborg University. He receives his Ph.D. in Electrical Engineering in 1997, and Ph.D. in Mathematics in 2005. In 2007–2008, he was a control specialist at Danfoss A/S. His research interest is in system theory, in particular hybrid system.

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Leth, J., Wisniewski, R. On formalism and stability of switched systems. J. Control Theory Appl. 10, 176–183 (2012). https://doi.org/10.1007/s11768-012-0138-3

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  • DOI: https://doi.org/10.1007/s11768-012-0138-3

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