Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the exponential convergence of input-output signals of nonlinear feedback systems

Lanlan Su, Di Zhao, Sei Zhen Khong

TL;DR

This work tackles robust exponential convergence of endogenous input-output signals in discrete-time nonlinear feedbacks that may contain unbounded LTI and nonlinear blocks. It develops an IQC-based framework with exponential weighting and a directed-gap homotopy to certify a decay rate $\rho\in(0,1)$ that holds uniformly across an uncertainty set. A central contribution is Theorem \ref{th:fix rate}, which provides verifiable conditions, including a frequency-domain inequality with a self-adjoint multiplier $\Pi$, and an LMI-based mechanism to compute the rate bound when $\Pi\in\mathcal{RL}_{\infty}$. The framework is specialized to Lurye systems with static nonlinearities, yielding practical LMIs (e.g., LMI_sector) that deliver computable exponential-stability certificates, as demonstrated by illustrative examples with rates around $0.79$ to $0.88$. These results offer tractable, quantitative guarantees for exponential tail behavior in uncertain nonlinear feedback loops, with potential impact on robust control and optimization algorithms that rely on fast convergence.

Abstract

This note studies the exponential convergence of input-output signals of discrete-time nonlinear systems composed of a feedback interconnection of a linear time-invariant system and a nonlinear uncertainty. Both the open-loop subsystems are allowed to be unbounded. Integral-quadratic-constraint-based conditions are proposed for these uncertain feedback systems, including the Lurye type, to exhibit the property that the endogenous input-output signals enjoy an exponential convergence rate for all initial conditions of the linear time-invariant subsystem. The conditions are established via a combination of tools, including integral quadratic constraints, directed gap, and exponential weightings.

On the exponential convergence of input-output signals of nonlinear feedback systems

TL;DR

This work tackles robust exponential convergence of endogenous input-output signals in discrete-time nonlinear feedbacks that may contain unbounded LTI and nonlinear blocks. It develops an IQC-based framework with exponential weighting and a directed-gap homotopy to certify a decay rate that holds uniformly across an uncertainty set. A central contribution is Theorem \ref{th:fix rate}, which provides verifiable conditions, including a frequency-domain inequality with a self-adjoint multiplier , and an LMI-based mechanism to compute the rate bound when . The framework is specialized to Lurye systems with static nonlinearities, yielding practical LMIs (e.g., LMI_sector) that deliver computable exponential-stability certificates, as demonstrated by illustrative examples with rates around to . These results offer tractable, quantitative guarantees for exponential tail behavior in uncertain nonlinear feedback loops, with potential impact on robust control and optimization algorithms that rely on fast convergence.

Abstract

This note studies the exponential convergence of input-output signals of discrete-time nonlinear systems composed of a feedback interconnection of a linear time-invariant system and a nonlinear uncertainty. Both the open-loop subsystems are allowed to be unbounded. Integral-quadratic-constraint-based conditions are proposed for these uncertain feedback systems, including the Lurye type, to exhibit the property that the endogenous input-output signals enjoy an exponential convergence rate for all initial conditions of the linear time-invariant subsystem. The conditions are established via a combination of tools, including integral quadratic constraints, directed gap, and exponential weightings.
Paper Structure (10 sections, 10 theorems, 38 equations, 3 figures)

This paper contains 10 sections, 10 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Open-loop systems
  4. Closed-loop systems
  5. Main results
  6. Robust exponential stability
  7. An LMI characterisation
  8. Illustrative examples
  9. Lurye systems with static nonlinearity
  10. Conclusion

Key Result

Lemma 1

For locally Lipschitz continuous $G,\Delta:\ell_{2e}^{0+}\rightarrow\ell_{2e}^{0+}$, $[G,\Delta]$ is well-posed if $\gamma(G)\gamma(\Delta)<1$.

Figures (3)

  • Figure 1: Feedback system
  • Figure 2: $\rho$-scaled feedback system
  • Figure 3: The trajectories of $0.8825^{-k}y_k$ under various initial conditions

Theorems & Definitions (24)

  • Definition 1
  • Lemma 1: Willems1971nonlinear
  • Definition 2
  • Definition 3
  • Remark 1
  • Proposition 1
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 14 more