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Stabilization of linear systems with multiple unknown time-varying input delays by linear time-varying feedback

Bin Zhou, Kai Zhang

TL;DR

This paper tackles stabilizing linear systems with multiple time-varying input delays when both the exact delays and their bounds are unknown. It develops a delay-ignorant, linear time-varying feedback by employing the solution to a parametric Lyapunov equation (PLE) and a decreasing time-varying gain $ heta(t)$, validated through a time-varying Lyapunov-Krasovskii-like functional that guarantees asymptotic stability. The approach yields a tractable, offline-designed controller $u(t)=-B^T P( heta(t)) x(t)$ and extends to an observer-based output feedback by truncating the observer, with rigorous Lyapunov-based proofs ensuring asymptotic stability of the closed-loop. Numerical simulations corroborate the theory, illustrating parameter effects on convergence and demonstrating practical viability for systems with multiple unknown time-varying delays.

Abstract

This paper addresses the stabilization of linear systems with multiple time-varying input delays. In scenarios where neither the exact delays information nor their bound is known, we propose a class of linear time-varying state feedback controllers by using the solution to a parametric Lyapunov equation (PLE). By leveraging the properties of the solution to the PLE and constructing a time-varying Lyapunov-Krasovskii-like functional, we prove that (the zero solution of) the closed-loop system is asymptotically stable. Furthermore, this result is extended to the observer-based output feedback case. The notable characteristic of these controllers is their utilization of linear time-varying gains. Furthermore, they are designed entirely independent of any knowledge of the time delays, resulting in controllers that are exceedingly easy to implement. Finally, a numerical example demonstrates the effectiveness of the proposed approaches.

Stabilization of linear systems with multiple unknown time-varying input delays by linear time-varying feedback

TL;DR

This paper tackles stabilizing linear systems with multiple time-varying input delays when both the exact delays and their bounds are unknown. It develops a delay-ignorant, linear time-varying feedback by employing the solution to a parametric Lyapunov equation (PLE) and a decreasing time-varying gain , validated through a time-varying Lyapunov-Krasovskii-like functional that guarantees asymptotic stability. The approach yields a tractable, offline-designed controller and extends to an observer-based output feedback by truncating the observer, with rigorous Lyapunov-based proofs ensuring asymptotic stability of the closed-loop. Numerical simulations corroborate the theory, illustrating parameter effects on convergence and demonstrating practical viability for systems with multiple unknown time-varying delays.

Abstract

This paper addresses the stabilization of linear systems with multiple time-varying input delays. In scenarios where neither the exact delays information nor their bound is known, we propose a class of linear time-varying state feedback controllers by using the solution to a parametric Lyapunov equation (PLE). By leveraging the properties of the solution to the PLE and constructing a time-varying Lyapunov-Krasovskii-like functional, we prove that (the zero solution of) the closed-loop system is asymptotically stable. Furthermore, this result is extended to the observer-based output feedback case. The notable characteristic of these controllers is their utilization of linear time-varying gains. Furthermore, they are designed entirely independent of any knowledge of the time delays, resulting in controllers that are exceedingly easy to implement. Finally, a numerical example demonstrates the effectiveness of the proposed approaches.
Paper Structure (7 sections, 5 theorems, 122 equations, 6 figures)

This paper contains 7 sections, 5 theorems, 122 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Problem Formulation
  4. Properties of Parametric Lyapunov Equations
  5. State Feedback Stabilization
  6. Observer-Based Output Feedback Stabilization
  7. Simulation

Key Result

Lemma 1

Assume that $(A,B)$ satisfies Assumption ass1 and the delays $\tau_{i}(t),\ i\in \mathbf{I}_{1}^{q}$ satisfy Assumption ass2. Let $P(\gamma)$ be the unique positive definite solution to the PLE zhou14book Then system (sys) can be stabilized by the following linear state feedback where $\gamma^{\ast}\in(0, 1/((1+\sqrt{3})n\sqrt{n}\bar{\tau})) >0$ is a constant depending on $\bar{\tau}$.

Figures (6)

  • Figure 1: Norm of $x(t)$ with different $\gamma_{0}$ ($\omega=1$ and $\mu=1/2$).
  • Figure 2: Norm of $x(t)$ with different $\mu$ ($\gamma_{0}=2$ and $\omega=1$).
  • Figure 3: Norm of $x(t)$ with different $\omega$ ($\gamma_{0}=2$ and $\mu=1/2$).
  • Figure 4: Controlled trajectories with $\gamma_{0}=2$, $\mu=1/2$ and $\omega=1.2$.
  • Figure 5: Control inputs with $\gamma_{0}=2$, $\mu=1/2$, and $\omega=1.2$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Example 1
  • Lemma 3
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2