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Finite-Time Stability of Switched and Hybrid Systems with Unstable Modes

Kunal Garg, Dimitra Panagou

TL;DR

It is shown that even if the value of the generalized Lyapunov function increases between consecutive switches, finite- time stability can still be guaranteed if the finite-time convergent mode is active long enough.

Abstract

In this work, we study finite-time stability of switched and hybrid systems in the presence of unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of the system to be finite time stable. More specifically, we show that even if the value of the Lyapunov function increases in between two switches, i.e., if there are unstable modes in the system, finite-time stability can still be guaranteed if the finite time convergent mode is active long enough. In contrast to earlier work where the Lyapunov functions are required to be decreasing during the continuous flows and non-increasing at the discrete jumps, we allow the Lyapunov functions to increase \emph{both} during the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes. Then, we illustrate how the proposed finite-time stability conditions specialize for a class of switched systems, and present a method on the synthesis of a finite-time stabilizing switching signal for switched linear systems. As a case study, we design a finite-time stable output feedback controller for a linear switched system, in which only one of the modes is controllable and observable. Numerical example demonstrates the efficacy of the proposed methods.

Finite-Time Stability of Switched and Hybrid Systems with Unstable Modes

TL;DR

It is shown that even if the value of the generalized Lyapunov function increases between consecutive switches, finite- time stability can still be guaranteed if the finite-time convergent mode is active long enough.

Abstract

In this work, we study finite-time stability of switched and hybrid systems in the presence of unstable modes. We present sufficient conditions in terms of multiple Lyapunov functions for the origin of the system to be finite time stable. More specifically, we show that even if the value of the Lyapunov function increases in between two switches, i.e., if there are unstable modes in the system, finite-time stability can still be guaranteed if the finite time convergent mode is active long enough. In contrast to earlier work where the Lyapunov functions are required to be decreasing during the continuous flows and non-increasing at the discrete jumps, we allow the Lyapunov functions to increase \emph{both} during the continuous flows and the discrete jumps. As thus, the derived stability results are less conservative compared to the earlier results in the related literature, and in effect allow the hybrid system to have unstable modes. Then, we illustrate how the proposed finite-time stability conditions specialize for a class of switched systems, and present a method on the synthesis of a finite-time stabilizing switching signal for switched linear systems. As a case study, we design a finite-time stable output feedback controller for a linear switched system, in which only one of the modes is controllable and observable. Numerical example demonstrates the efficacy of the proposed methods.

Paper Structure

This paper contains 20 sections, 7 theorems, 40 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Stability of switched systems
  3. Stability of hybrid systems
  4. Related work on FTS
  5. Our Contributions
  6. Organization
  7. FTS of Hybrid Systems
  8. Preliminaries
  9. Main result
  10. Discussion on the main result
  11. FTS of Switched Systems
  12. FTS result
  13. Finite-Time Stabilizing Switching Signal
  14. FTS output-feedback for Switched Systems
  15. Simulations
  16. ...and 5 more sections

Key Result

Theorem 1

Suppose there exists a continuous function $V$: $D \rightarrow \mathbb{R}$ such that the following holds: (i) $V$ is positive definite (ii) There exist real numbers $c>0$ and $\alpha \in (0, 1)$ , and an open neighborhood $\mathcal{V}\subseteq D$ of the origin such that Then the origin is an FTS equilibrium for ex sys.

Figures (11)

  • Figure 1: The ball $B_\rho$, shown in dotted yellow, encloses $c$ sublevel sets of the Lyapunov functions $V_i$, whose boundaries are shown in solid lines.
  • Figure 2: Conditions (i), (ii) and (iii) of Theorem \ref{['FT Th 2']} regarding the allowable changes in the values of the Lyapunov functions. The increments shown by blue, red and black double-arrows pertain to condition (i), (ii) and (iii), respectively.
  • Figure 3: Switching signal $\sigma_f(t)$ for the considered hybrid system \ref{['hyb sys example']}.
  • Figure 4: The evolution of $x_1(t)$ and $x_2(t)$ for hybrid system \ref{['hyb sys example']}. The states can be seen switching signs during discrete jumps.
  • Figure 5: The evolution of $\|x(t)\|$ for \ref{['hyb sys example']}. The norm of the states reach a small neighborhood of the origin within a finite time.
  • ...and 6 more figures

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Theorem 1: bhat2000finite
  • Remark 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Definition 5
  • Lemma 1
  • Theorem 2
  • ...and 12 more