Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Output regulation for a reaction-diffusion system with input delay and unknown frequency

Shen Wang, Zhong-Jie Han, Kai Liu, Zhi-Xue Zhao

TL;DR

This work tackles output regulation for a reaction-diffusion PDE subject to input delay and disturbances with unknown frequency. It develops a novel adaptive architecture that combines modal decomposition with a dual observer and an adaptive internal model to estimate both states and disturbances in real time, achieving exponential convergence of the tracking error $y_e(t)$ to the reference. The regulator design uses regulator equations, LMIs, and Lyapunov-Krasovskii analysis to handle delay and unknown frequency while allowing an unrestricted reaction coefficient $a$. Numerical simulations validate the approach, showing effective frequency estimation and robust tracking across parameter regimes, with potential applicability to broader parabolic PDEs. The framework advances distributed-parameter control by enabling simultaneous delay compensation and unknown-disturbance rejection without full-state feedback.

Abstract

This study solves the output regulation problem for a reaction-diffusion system confronting concurrent input delay and fully unidentified disturbances (encompassing both unknown frequencies and amplitudes) across all channels. The principal innovation emerges from a novel adaptive control architecture that synergizes the modal decomposition technique with a dual-observer mechanism, enabling real-time concurrent estimation of unmeasurable system states and disturbances through a state observer and an adaptive disturbance estimator. Unlike existing approaches limited to either delay compensation or partial disturbance rejection, our methodology overcomes the technical barrier of coordinating these two requirements through a rigorously constructed tracking-error-based controller, achieving exponential convergence of system output to reference signals. Numerical simulations are presented to validate the effectiveness of the proposed output feedback control strategy.

Output regulation for a reaction-diffusion system with input delay and unknown frequency

TL;DR

This work tackles output regulation for a reaction-diffusion PDE subject to input delay and disturbances with unknown frequency. It develops a novel adaptive architecture that combines modal decomposition with a dual observer and an adaptive internal model to estimate both states and disturbances in real time, achieving exponential convergence of the tracking error to the reference. The regulator design uses regulator equations, LMIs, and Lyapunov-Krasovskii analysis to handle delay and unknown frequency while allowing an unrestricted reaction coefficient . Numerical simulations validate the approach, showing effective frequency estimation and robust tracking across parameter regimes, with potential applicability to broader parabolic PDEs. The framework advances distributed-parameter control by enabling simultaneous delay compensation and unknown-disturbance rejection without full-state feedback.

Abstract

This study solves the output regulation problem for a reaction-diffusion system confronting concurrent input delay and fully unidentified disturbances (encompassing both unknown frequencies and amplitudes) across all channels. The principal innovation emerges from a novel adaptive control architecture that synergizes the modal decomposition technique with a dual-observer mechanism, enabling real-time concurrent estimation of unmeasurable system states and disturbances through a state observer and an adaptive disturbance estimator. Unlike existing approaches limited to either delay compensation or partial disturbance rejection, our methodology overcomes the technical barrier of coordinating these two requirements through a rigorously constructed tracking-error-based controller, achieving exponential convergence of system output to reference signals. Numerical simulations are presented to validate the effectiveness of the proposed output feedback control strategy.

Paper Structure

This paper contains 19 sections, 11 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Novelties
  4. Paper outline
  5. Statement of main results
  6. Problem formulation
  7. Feedforward regulator design
  8. Error-based observer design
  9. Error-based control design
  10. Proof of main results
  11. Proof of Proposition \ref{['le1']}
  12. Proof of Theorem \ref{['le2']}
  13. Proof of Proposition \ref{['th2']}
  14. Proof of Lemma \ref{['le5']}
  15. Proof of Theorem \ref{['th1']}
  16. ...and 4 more sections

Key Result

Lemma 2.1

The initial value problem 7 has a unique solution $\Gamma(\cdot) \!\in\! H^2(\!(0,1);\!\mathbb{R}^{1 \times 2})$.

Figures (2)

  • Figure 1: Simulation results for $a=1.5$
  • Figure 2: Simulation results for $a=0.5$

Theorems & Definitions (13)

  • Lemma 2.1
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.2
  • Lemma 2.4
  • Lemma 2.5
  • ...and 3 more