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Converse Lyapunov Results for Stability of Switched Systems with Average Dwell-Time

Matteo Della Rossa, Aneel Tanwani

TL;DR

The paper tackles stability of switched nonlinear systems under average dwell-time constraints by establishing a converse Lyapunov framework based on multiple Lyapunov functions. It shows that dwell-time based conditions do not necessarily carry over to average dwell-time, using a counterexample to highlight the gap. A central contribution is a necessary-and-sufficient set of inequalities, independent of subsystem flows, that characterize $ au$-UGB stability via functions $W_i$ with specific decay, cross-mode, and jump properties. The linear-subsystem specialization yields a norm-based converse, offering practical criteria for linear switched systems and clarifying the relationship between average dwell-time stability and minimal dwell-time bounds.

Abstract

This article provides a characterization of stability for switched nonlinear systems under average dwell-time constraints, in terms of necessary and sufficient conditions involving multiple Lyapunov functions. Earlier converse results focus on switched systems with dwell-time constraints only, and the resulting inequalities depend on the flow of individual subsystems. With the help of a counterexample, we show that a lower bound that guarantees stability for dwell-time switching signals may not necessarily imply stability for switching signals with same lower bound on the average dwell-time. Based on these two observations, we provide a converse result for the average dwell-time constrained systems in terms of inequalities which do not depend on the flow of individual subsystems and are easier to check. The particular case of linear switched systems is studied as a corollary to our main result.

Converse Lyapunov Results for Stability of Switched Systems with Average Dwell-Time

TL;DR

The paper tackles stability of switched nonlinear systems under average dwell-time constraints by establishing a converse Lyapunov framework based on multiple Lyapunov functions. It shows that dwell-time based conditions do not necessarily carry over to average dwell-time, using a counterexample to highlight the gap. A central contribution is a necessary-and-sufficient set of inequalities, independent of subsystem flows, that characterize -UGB stability via functions with specific decay, cross-mode, and jump properties. The linear-subsystem specialization yields a norm-based converse, offering practical criteria for linear switched systems and clarifying the relationship between average dwell-time stability and minimal dwell-time bounds.

Abstract

This article provides a characterization of stability for switched nonlinear systems under average dwell-time constraints, in terms of necessary and sufficient conditions involving multiple Lyapunov functions. Earlier converse results focus on switched systems with dwell-time constraints only, and the resulting inequalities depend on the flow of individual subsystems. With the help of a counterexample, we show that a lower bound that guarantees stability for dwell-time switching signals may not necessarily imply stability for switching signals with same lower bound on the average dwell-time. Based on these two observations, we provide a converse result for the average dwell-time constrained systems in terms of inequalities which do not depend on the flow of individual subsystems and are easier to check. The particular case of linear switched systems is studied as a corollary to our main result.
Paper Structure (8 sections, 10 theorems, 98 equations, 3 figures)

This paper contains 8 sections, 10 theorems, 98 equations, 3 figures.

Table of Contents

  1. Systems Class and Switching Signals
  2. Stability over Classes of Switching Signals: Review and First Results
  3. $\text{UGES}_\rho$ w.r.t. $\mathcal{S}_{\textit{dw}}(\tau)$ does not imply $\text{UGES}_\rho$ w.r.t. $\mathcal{S}_{\textit{adw}}(\tau,2)$
  4. Lyapunov (Converse) Result For $\tau\text{-UGB}$
  5. Linear Subsystems Case
  6. Conclusions
  7. Concatenation of Switching Signals
  8. Proof of local Lipschitz property

Key Result

Proposition 1

Consider any $\tau>0$ and $\mathcal{F}=\{f_1,\dots, f_{\rm m}\}\subset \Sigma(\mathbb{R}^n)$. System eq:SwitchedSystem is UGAS w.r.t. $\mathcal{S}_{\textit{dw}}(\tau)$ if and only if there exist $V_1,\dots, V_{\rm m}\in \text{Lip}_0(\mathbb{R}^n,\mathbb{R})$ and $\alpha_1,\alpha_2\in \mathcal{K}_\in (Linear Case): Consider $\mathcal{A}=\{A_1,\dots, A_{\rm m}\}\subset \mathbb{R}^{n\times n}$; given

Figures (3)

  • Figure 1: Considering the switching signal $\sigma\in \mathcal{S}_{\textit{adw}}(\tau,2)$ in \ref{['eq:SignalDestabilizing']}, and the initial condition $x_0=[1,0]^\top$, we represent the corresponding solution $\Phi_\sigma(\cdot,x_0):\mathbb{R}_+\to \mathbb{R}^2$, highlighting the fact that it is unbounded, i.e., $\lim_{t\to +\infty}|\Phi_\sigma(t,x_0)|=+\infty$.
  • Figure 2: The graph of the minimum $\tau$ in Example 1 obtained by the techniques of HafTan23, parametrized by the value of $\mu=e^{\alpha\tau}$.
  • Figure 3: Schematic collection of the main results for the nonlinear case. The notation "LFs" stands for "Lyapunov functions". The linear-case scheme is completely equivalent, mutatis mutandis, replacing UGAS by $\text{UGES}_\rho$ and $\tau\text{-UGB}$ by $\tau\text{-UGEB}_\rho$.

Theorems & Definitions (31)

  • Definition 1: (Average) Dwell-Time Signals
  • Remark 1
  • Definition 2: Stability Notions for a Given Class
  • Proposition 1: Lyapunov characterization for $\mathcal{S}_{\textit{dw}}(\tau)$
  • Lemma 1
  • proof
  • Definition 3: Jump Dependent Boundedness Notions
  • Remark 2: Arbitrary switching stability
  • Lemma 2
  • proof
  • ...and 21 more