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Small-gain conditions for exponential incremental stability in feedback interconnections

Mohamed Yassine Arkhis, Denis Efimov

TL;DR

The paper addresses when interconnecting two incrementally exponentially stable systems preserves stability. It develops a small-gain theorem based on exponential Finsler Lyapunov functions, guaranteeing IES on a compact forward-invariant set, and, under stronger assumptions, global WIES; a practical example demonstrates the results and that Jacobian negativity is not strictly necessary. The work also provides corollaries that extend the result to ISpS subsystems and to global properties via an auxiliary Lyapunov function. This contributes a constructive framework for stability analysis of nonlinear interconnected systems with small coupling gains.

Abstract

We prove that under a small-gain condition, an interconnection of two globally incrementally exponentially stable systems inherits this property on any compact connected forward invariant set. It is also demonstrated that the interconnection inherits a weaker version of incremental exponential stability globally. An example illustrating the theoretical findings is given. The example also shows that the uniform negativity of the Jacobian is not necessary for incremental exponential stability.

Small-gain conditions for exponential incremental stability in feedback interconnections

TL;DR

The paper addresses when interconnecting two incrementally exponentially stable systems preserves stability. It develops a small-gain theorem based on exponential Finsler Lyapunov functions, guaranteeing IES on a compact forward-invariant set, and, under stronger assumptions, global WIES; a practical example demonstrates the results and that Jacobian negativity is not strictly necessary. The work also provides corollaries that extend the result to ISpS subsystems and to global properties via an auxiliary Lyapunov function. This contributes a constructive framework for stability analysis of nonlinear interconnected systems with small coupling gains.

Abstract

We prove that under a small-gain condition, an interconnection of two globally incrementally exponentially stable systems inherits this property on any compact connected forward invariant set. It is also demonstrated that the interconnection inherits a weaker version of incremental exponential stability globally. An example illustrating the theoretical findings is given. The example also shows that the uniform negativity of the Jacobian is not necessary for incremental exponential stability.

Paper Structure

This paper contains 9 sections, 5 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incremental exponential stability
  4. Input-to-state practical stability
  5. Problem statement
  6. Main results
  7. Example
  8. Numerical simulation
  9. Conclusion

Key Result

Theorem 1

forni2013differential Let $\mathcal{D}\subset \mathbb{R}^{n+m}$ be a connected forward invariant set for System 1. If there exists an exponential Finsler Lyapunov function $V:\mathbb{R}^{n+m}\times \mathbb{R}^{n+m}\to \mathbb{R}_+$, $i.e.$, a $\mathcal{C}^1$ function satisfying $\exists \alpha, \und where $\dot{V}(z,\delta z):=\frac{\partial V(z,\delta z)}{\partial z}f(z,t)+\frac{\partial V(z,\del

Figures (3)

  • Figure 1: $c=1,b=0.1,\varepsilon=1,\rho_1=\rho_2=1$.
  • Figure 2: $c=1,b=0.1,\varepsilon=1,\rho_1=\rho_2=0.1$.
  • Figure 3: $c=1,b=1,\varepsilon=0.9,\rho_1=\rho_2=1$.

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Remark 2
  • Definition 3
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 2
  • ...and 9 more