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Lyapunov Characterization for ISS of Impulsive Switched Systems

Saeed Ahmed, Patrick Bachmann, Stephan Trenn

Abstract

In this study, we investigate the ISS of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is non-decreasing and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a technique for constructing a decreasing ISS-Lyapunov function from a non-decreasing one, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee the ISS of a particular class of impulsive switched systems when the switching signal is unknown.

Lyapunov Characterization for ISS of Impulsive Switched Systems

Abstract

In this study, we investigate the ISS of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is non-decreasing and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a technique for constructing a decreasing ISS-Lyapunov function from a non-decreasing one, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee the ISS of a particular class of impulsive switched systems when the switching signal is unknown.

Paper Structure

This paper contains 9 sections, 8 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Sufficient condition for ISS
  4. Sufficient and necessary condition for ISS
  5. Construction of ISS-Lyapunov functions
  6. Robustness with respect to the switching signal
  7. Conclusion
  8. Appendix
  9. Technical Lemmas

Key Result

Theorem 9

Consider the system $\Sigma$ for a given switching signal $\sigma$. If there exists a non-decreasing ISS-Lyapunov function as given in Definition def:nonDecrLF, then $\Sigma$ is ISS.

Figures (3)

  • Figure 1: Summary of our results
  • Figure 2: Construction of a decreasing ISS-Lyapunov function $W_\sigma(t,x(t))$ from a non-decreasing ISS-Lyapunov function $V_\sigma(t,x(t))$ and their relation.
  • Figure 3: Illustration of function $h_\sigma(p,t)$ for a fixed switching signal.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Remark 8
  • Theorem 9
  • proof
  • ...and 18 more