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On Incremental Stability of Interconnected Switched Systems

Bhabani Shankar Dey, Indra Narayan Kar, Pushpak Jagtap

TL;DR

This work addresses the problem of ensuring incremental stability for interconnected switched nonlinear systems with state-dependent switching. It leverages contraction theory and matrix-measure techniques to derive sufficient $\delta$-ISS conditions for bimodal subsystems and analyzes their interconnection via feedback, cascade, and generalized network structures. A small-gain framework is developed to guarantee $\delta$-ISS of the overall system, with results validated through numerical examples that demonstrate exponential convergence of trajectory differences. The methods provide scalable, principled criteria for preserving incremental stability in complex, interconnected switched systems and outline directions for future data-driven extensions and time-varying switching scenarios.

Abstract

In this paper, the incremental stability of interconnected switched nonlinear systems is discussed. The nature of switching considered is state-dependent. The incremental stability of the switched interconnected system is a stronger property compared to the conventional notion of stability. Even if individual systems in the interconnected setting are stable, guaranteeing stability for the overall system is challenging. However, one of the important features of incremental stability is that the notion is preserved over interconnection. Here, leveraging the contraction-theoretic tools, we derive a set of sufficient conditions for the overall interconnection consisting of bimodal switched systems. To showcase the wider usability of our proposed results, we have also included the effect of external input, which leads to the study of incremental input-to-state stability ($δ$-ISS). For the case of feedback interconnection, the small gain characterisation is presented for the overall system's $δ$-ISS. Further, for a special case of feedback, i.e., cascade interconnection, the results are derived. The derived conditions are based on the matrix measure, making it computationally tractable and general. To make the results more general, a generalised interconnection of bimodal switched systems is studied and corresponding sufficient condition for $δ$-ISS are presented. Two numerical examples are demonstrated and supported with simulation results to verify the theoretical claims.

On Incremental Stability of Interconnected Switched Systems

TL;DR

This work addresses the problem of ensuring incremental stability for interconnected switched nonlinear systems with state-dependent switching. It leverages contraction theory and matrix-measure techniques to derive sufficient -ISS conditions for bimodal subsystems and analyzes their interconnection via feedback, cascade, and generalized network structures. A small-gain framework is developed to guarantee -ISS of the overall system, with results validated through numerical examples that demonstrate exponential convergence of trajectory differences. The methods provide scalable, principled criteria for preserving incremental stability in complex, interconnected switched systems and outline directions for future data-driven extensions and time-varying switching scenarios.

Abstract

In this paper, the incremental stability of interconnected switched nonlinear systems is discussed. The nature of switching considered is state-dependent. The incremental stability of the switched interconnected system is a stronger property compared to the conventional notion of stability. Even if individual systems in the interconnected setting are stable, guaranteeing stability for the overall system is challenging. However, one of the important features of incremental stability is that the notion is preserved over interconnection. Here, leveraging the contraction-theoretic tools, we derive a set of sufficient conditions for the overall interconnection consisting of bimodal switched systems. To showcase the wider usability of our proposed results, we have also included the effect of external input, which leads to the study of incremental input-to-state stability (-ISS). For the case of feedback interconnection, the small gain characterisation is presented for the overall system's -ISS. Further, for a special case of feedback, i.e., cascade interconnection, the results are derived. The derived conditions are based on the matrix measure, making it computationally tractable and general. To make the results more general, a generalised interconnection of bimodal switched systems is studied and corresponding sufficient condition for -ISS are presented. Two numerical examples are demonstrated and supported with simulation results to verify the theoretical claims.
Paper Structure (13 sections, 6 theorems, 57 equations, 8 figures)

This paper contains 13 sections, 6 theorems, 57 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Piece-wise Smooth (PWS) Systems
  4. Incremental Input-to-State Stability
  5. System Description and Problem Formulation
  6. Interconnected System
  7. Problem Statement
  8. Main Result
  9. Generalised Interconnection of Bimodal Switched Systems
  10. Results and Discussion
  11. Feedback Connection
  12. Cascade Connection
  13. CONCLUSION

Key Result

Lemma 2.6

fiore2016contraction Let $C\subseteq \mathbb{R}^n$ be some forward invariant set. If there exist some positive constants $c_1$, $c_2>0$, and certain norm defined in $C$, and corresponding induced matrix measure denoted as $\mu$ (as defined in Definition matrix_measure), then a bimodal switched syste

Figures (8)

  • Figure 1: Feedback interconnection of two switched nonlinear systems
  • Figure 2: Cascade connection of two switched nonlinear systems
  • Figure 3: General interconnection of switched nonlinear systems
  • Figure 4: The time evolution of variational state $\delta \xi_1$ for feedback interconnection(Solid Curve).
  • Figure 5: The time evolution of variational state $\delta \xi_2$ for feedback interconnection(Solid Curve).
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 4.1
  • Proof 4.2
  • Remark 4.3
  • ...and 5 more