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Output regulation for an unstable wave equation with output delay and one measurement only

Shen Wang, Zhong-Jie Han, Shuangxi Huang, Zhi-Xue Zhao

TL;DR

The paper tackles output regulation for a 1D unstable wave equation subjected to output delay and unknown disturbances. By translating the regulation problem into stabilization via regulator equations, it designs a backstepping-based feedforward regulator and partitions time into delay-free and delay-active intervals to stabilize the system. An adaptive, infinite-dimensional observer—built from a dynamic compensator and an adaptive internal model—tracks the unknown disturbance frequency, while a single-measurement error-based feedback controller ensures exponential convergence of the tracking error $e(t)$ to zero. The work provides a pioneering, first-of-its-kind solution for output regulation of distributed parameter systems with output delay, supported by numerical simulations.

Abstract

This paper addresses the output regulation problem for a one-dimensional unstable wave equation subject to output delay and all-channel disturbances with unknown frequencies and amplitudes. First, this problem is transformed into a stabilization problem for an unstable wave equation with output delay and disturbances by employing regulator equations. Subsequently, a backstepping-based feedforward regulator is proposed to exponentially stabilize this system. To track the states of the unstable wave equation, the time interval is partitioned into two segments. The observers and predictors are designed at these distinct intervals, respectively. Therein, the observers comprise two components: a state observer proposed via dynamic compensators and an adaptive observer designed by the adaptive internal model method. Finally, a novel error-based feedback controller is derived using a single measurement, ensuring exponential convergence of the tracking error to zero. This work establishes the pioneering solution to the output regulation problem for distributed parameter systems (DPS) with output delay. Numerical simulations are provided to illustrate the results.

Output regulation for an unstable wave equation with output delay and one measurement only

TL;DR

The paper tackles output regulation for a 1D unstable wave equation subjected to output delay and unknown disturbances. By translating the regulation problem into stabilization via regulator equations, it designs a backstepping-based feedforward regulator and partitions time into delay-free and delay-active intervals to stabilize the system. An adaptive, infinite-dimensional observer—built from a dynamic compensator and an adaptive internal model—tracks the unknown disturbance frequency, while a single-measurement error-based feedback controller ensures exponential convergence of the tracking error to zero. The work provides a pioneering, first-of-its-kind solution for output regulation of distributed parameter systems with output delay, supported by numerical simulations.

Abstract

This paper addresses the output regulation problem for a one-dimensional unstable wave equation subject to output delay and all-channel disturbances with unknown frequencies and amplitudes. First, this problem is transformed into a stabilization problem for an unstable wave equation with output delay and disturbances by employing regulator equations. Subsequently, a backstepping-based feedforward regulator is proposed to exponentially stabilize this system. To track the states of the unstable wave equation, the time interval is partitioned into two segments. The observers and predictors are designed at these distinct intervals, respectively. Therein, the observers comprise two components: a state observer proposed via dynamic compensators and an adaptive observer designed by the adaptive internal model method. Finally, a novel error-based feedback controller is derived using a single measurement, ensuring exponential convergence of the tracking error to zero. This work establishes the pioneering solution to the output regulation problem for distributed parameter systems (DPS) with output delay. Numerical simulations are provided to illustrate the results.

Paper Structure

This paper contains 20 sections, 10 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Contributions and novelties
  4. Organization
  5. Main results
  6. Problem statement
  7. Feedforward regulator design
  8. Error-based observers and predictors design
  9. Observers design
  10. Predictor design
  11. Error-based feedback controller
  12. Proof of main results
  13. Proof of Theorem \ref{['th1']}
  14. Proof of Theorem \ref{['th2']}
  15. Proof of Theorem \ref{['th3']}
  16. ...and 5 more sections

Key Result

Lemma 2.1

There exists a unique solution $\Pi \in C^2( (0,1);\mathbb{R}^{1 \times 2})$ to the initial value problem 6.

Figures (3)

  • Figure 1: Tracking performance, frequency estimate and evolution of $w(x,t)$ and $u(t)$ for $\tau=0.1$
  • Figure 2: Tracking performance, frequency estimate and evolution of $w(x,t)$ and $u(t)$ for $\tau=0.5$
  • Figure 3: Tracking performance, frequency estimate and evolution of $w(x,t)$ and $u(t)$ for $\tau=1$

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • ...and 1 more