Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stability criteria for singularly perturbed impulsive linear switched systems

Ihab Haidar, Yacine Chitour, Jamal Daafouz, Paolo Mason, Mario Sigalotti

TL;DR

This paper analyzes stability of a class of singularly perturbed impulsive linear switched systems with mode-dependent slow/fast dynamics by constructing auxiliary single-scale systems. It derives upper and lower bounds on the limit of the maximal Lyapunov exponent as $\varepsilon\to0$ via reduced-order models: a slow-dynamics system $\bar{\Sigma}$, a transient-model $\hat{\Sigma}$, and a transient-inclusive system $\tilde{\Sigma}$, establishing a sandwich of exponents: $\lambda(\bar{\Sigma}) \le \liminf_{\varepsilon\to0} \lambda(\Sigma^{\varepsilon})$ and $\limsup_{\varepsilon\to0} \lambda(\Sigma^{\varepsilon}) \le \lambda(\tilde{\Sigma})$. Under a dwell-time constraint, a complete characterization emerges: the exponents of the reduced and auxiliary systems coincide with the limiting exponent of the original, yielding necessary and sufficient conditions for exponential stability for all small $\varepsilon$. The results are complemented by a complementary-case analysis and a numerical example that illustrate the applicability of the auxiliary-system approach. The techniques hinge on block-diagonalization, Tikhonov-type reductions, transient-dynamics modeling, and a Lyapunov-exponent framework for impulsive switched systems. Overall, the work provides rigorous, practically applicable stability criteria for complex multi-time-scale switched hybrids encountered in engineering contexts.

Abstract

We study a class of singularly perturbed impulsive linear switched systems exhibiting switching between slow and fast dynamics. To analyze their behavior, we construct auxiliary switched systems evolving in a single time scale. We prove that the stability or instability of these auxiliary systems directly determines that of the original system in the regime of small singular perturbation parameters.

Stability criteria for singularly perturbed impulsive linear switched systems

TL;DR

This paper analyzes stability of a class of singularly perturbed impulsive linear switched systems with mode-dependent slow/fast dynamics by constructing auxiliary single-scale systems. It derives upper and lower bounds on the limit of the maximal Lyapunov exponent as via reduced-order models: a slow-dynamics system , a transient-model , and a transient-inclusive system , establishing a sandwich of exponents: and . Under a dwell-time constraint, a complete characterization emerges: the exponents of the reduced and auxiliary systems coincide with the limiting exponent of the original, yielding necessary and sufficient conditions for exponential stability for all small . The results are complemented by a complementary-case analysis and a numerical example that illustrate the applicability of the auxiliary-system approach. The techniques hinge on block-diagonalization, Tikhonov-type reductions, transient-dynamics modeling, and a Lyapunov-exponent framework for impulsive switched systems. Overall, the work provides rigorous, practically applicable stability criteria for complex multi-time-scale switched hybrids encountered in engineering contexts.

Abstract

We study a class of singularly perturbed impulsive linear switched systems exhibiting switching between slow and fast dynamics. To analyze their behavior, we construct auxiliary switched systems evolving in a single time scale. We prove that the stability or instability of these auxiliary systems directly determines that of the original system in the regime of small singular perturbation parameters.

Paper Structure

This paper contains 24 sections, 17 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem formulation and main assumption
  4. Singularly perturbed switched system
  5. Impulsive linear switched systems
  6. Problem statement and first stability result
  7. Auxiliary switched systems and statement of the main results
  8. Block diagonalization
  9. Slow dynamics by Tikhonov's approach
  10. Transient dynamics
  11. Main results
  12. Lyapunov exponents of singularly perturbed and auxiliary systems
  13. Proof of Theorem \ref{['main theorem-necessary']}
  14. Proof of item \ref{['itemii']} of Theorem \ref{['main theorem-necessary']}
  15. Proof of item \ref{['itemiii']} of Theorem \ref{['main theorem-necessary']}
  16. ...and 9 more sections

Key Result

Lemma 1

Let ${\cal Y}=\{Z_2\mid (Z_1,Z_2)\in {\cal Z}\}$. Then $\lambda(\Delta_{{\cal Z},\tau})=+\infty$ if and only if $\tau=0$ and $\Xi_{{\cal Y}}$ is unbounded. Moreover, if $\tau>0$ or system $\Xi_{{\cal Y}}$ is bounded, then the following properties hold:

Figures (1)

  • Figure 1: Time evolution of system $\Sigma^{\varepsilon}_{\cal K}$ of Section \ref{['sec:example']} with $r=0.3$, $\varepsilon=0.1$, $X(0)=(1,0.4)^T$ corresponding to a periodic piecewise-constant switching signal whose period $0.05$ (left) or $0.5$ (right) is divided into three equal intervals corresponding to modes 1, 1, and 2. Dotted lines denote jumps, dashed lines the images of the jump matrices, i.e., the lines of equation $y=\pm x/2$.

Theorems & Definitions (34)

  • Remark 1
  • Definition 1
  • Lemma 1: technical-spin-off
  • Proposition 1
  • proof
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Lemma 2
  • ...and 24 more