Input-to-state stability of impulsive and switching hybrid systems with time-delay

  • Authors:

  • Jun Liu;Xinzhi Liu;Wei-Chau Xie

  • Affiliations:

  • Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2011

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Abstract

This paper investigates input-to-state stability (ISS) and integral input-to-state stability (iISS) of impulsive and switching hybrid systems with time-delay, using the method of multiple Lyapunov-Krasovskii functionals. It is shown that, even if all the subsystems governing the continuous dynamics, in the absence of impulses, are not ISS/iISS, impulses can successfully stabilize the system in the ISS/iISS sense, provided that there are no overly long intervals between impulses, i.e., the impulsive and switching signal satisfies a dwell-time upper bound condition. Moreover, these impulsive ISS/iISS stabilization results can be applied to systems with arbitrarily large time-delays. Conversely, in the case when all the subsystems governing the continuous dynamics are ISS/iISS in the absence of impulses, the ISS/iISS properties can be retained if the impulses and switching do not occur too frequently, i.e., the impulsive and switching signal satisfies a dwell-time lower bound condition. Several illustrative examples are presented, with their numerical simulations, to demonstrate the main results.