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Converse Lyapunov Results for Switched Systems with Lower and Upper Bounds on Switching Intervals

Matteo Della Rossa

TL;DR

This work addresses stability of continuous-time switched systems under both lower and upper bounds on switching intervals by developing converse Lyapunov results with multiple Lyapunov functions. It proves necessary and sufficient conditions for global uniform asymptotic stability on the switching-class $\mathcal{S}_{dw}(\tau_1,\tau_2)$ (and $\mathcal{S}_{dw}^{\star}(\tau_1,\tau_2)$), using Lyapunov families $V_i^-,V_i^+$ (and $W_i^-,W_i^+$) and a constructive decay via $U_\sigma$, with links to graph representations. In the linear case, the stability characterization reduces to the existence of Lyapunov multinorms, and practical SDP/LMIs are developed for quadratic Lyapunov candidates, enabling computational checks and robustness analysis against switching delays. Numerical examples demonstrate improved conservatism and alignment with existing results, illustrating the utility of the proposed conditions for event-triggered and constrained switching scenarios. The results unify nonlinear and linear cases under a common framework and suggest extensions to broader switching classes and hybrid systems.

Abstract

The topic of this manuscript is the stability analysis of continuous-time switched nonlinear systems with constraints on the admissible switching signals. Our particular focus lies in considering signals characterized by upper and lower bounds on the length of the switching intervals. We adapt and extend the existing theory of multiple Lyapunov functions, providing converse results and thus a complete characterization of uniform stability for this class of systems. We specify our results in the context of switched linear systems, providing the equivalence of exponential stability and the existence of multiple Lyapunov norms. By restricting the class of candidate Lyapunov functions to the set of quadratic functions, we are able to provide semidefinite-optimization-based numerical schemes to check the proposed conditions. We provide numerical examples to illustrate our approach and highlight its advantages over existing methods.

Converse Lyapunov Results for Switched Systems with Lower and Upper Bounds on Switching Intervals

TL;DR

This work addresses stability of continuous-time switched systems under both lower and upper bounds on switching intervals by developing converse Lyapunov results with multiple Lyapunov functions. It proves necessary and sufficient conditions for global uniform asymptotic stability on the switching-class (and ), using Lyapunov families (and ) and a constructive decay via , with links to graph representations. In the linear case, the stability characterization reduces to the existence of Lyapunov multinorms, and practical SDP/LMIs are developed for quadratic Lyapunov candidates, enabling computational checks and robustness analysis against switching delays. Numerical examples demonstrate improved conservatism and alignment with existing results, illustrating the utility of the proposed conditions for event-triggered and constrained switching scenarios. The results unify nonlinear and linear cases under a common framework and suggest extensions to broader switching classes and hybrid systems.

Abstract

The topic of this manuscript is the stability analysis of continuous-time switched nonlinear systems with constraints on the admissible switching signals. Our particular focus lies in considering signals characterized by upper and lower bounds on the length of the switching intervals. We adapt and extend the existing theory of multiple Lyapunov functions, providing converse results and thus a complete characterization of uniform stability for this class of systems. We specify our results in the context of switched linear systems, providing the equivalence of exponential stability and the existence of multiple Lyapunov norms. By restricting the class of candidate Lyapunov functions to the set of quadratic functions, we are able to provide semidefinite-optimization-based numerical schemes to check the proposed conditions. We provide numerical examples to illustrate our approach and highlight its advantages over existing methods.
Paper Structure (9 sections, 10 theorems, 71 equations, 2 figures)

This paper contains 9 sections, 10 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Signals with Upper-Lower Switching Bounds
  5. Converse Lyapunov Results
  6. Linear Case
  7. Numerical Examples
  8. Conclusion
  9. Technical Lemmas

Key Result

Lemma 1

Consider a set of vector fields $\mathcal{F}=\{f_1,\dots, f_M\}\subset \text{Lip}_{\ell}(\mathbb{R}^n,\mathbb{R}^n)$ satisfying Assumption assumt:Regularity and $\tau_2\geq \tau_1\geq 0$. System eq:SwitchedSystem is GUAS on $\mathcal{S}_{\text{dw}}(\tau_1,\tau_2)$ if and only if it is GUAS on $\math

Figures (2)

  • Figure 1: The graph representations of the partitions and Lyapunov constructions used in Theorem \ref{['Thm:ConverseResult1']} and Proposition \ref{['Prop:ConverseResult2']}. Any arrow stands for a required inequality: for instance, an edge from the node $U_a$ to the node $U_b$ labeled by a (set of) operator(s) $\Phi_s(\tau,\cdot)$ represents the inequality $U_b(\Phi_s(\tau,x))\leq e^\tau U_a(x)$, $\forall \,x\in \mathbb{R}^n$.
  • Figure 2: Example of diverging trajectory of \ref{['eq:LinearSwitchingExample']} (with $\varepsilon=0.1$), starting at $x_0=[0.5,0]^\top$, under the switching rule defined in \ref{['eq:DivergingSwitchingrule']}. Red color stands for the subsystem $\dot x=A_1x$, blue for $\dot x=A_2x$.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1: Limiting cases
  • Proposition 1
  • proof
  • Remark 2: Possible Generalizations and Numerical Verification of the Conditions
  • ...and 15 more