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

TL;DR

This paper establishes less conservative conditions for finite-time stability in switched and hybrid systems with unstable modes, allowing Lyapunov functions to increase during flows and jumps, and provides methods for stabilizing switching signals and controllers.

Contribution

It introduces new sufficient conditions for finite-time stability that accommodate unstable modes by permitting Lyapunov function increases, expanding the applicability of stability analysis.

Findings

Finite-time stability can be guaranteed despite unstable modes.

Less conservative stability conditions compared to previous work.

Method for synthesizing stabilizing switching signals and controllers.

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.