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Delayed finite-dimensional observer-based control of 2D linear parabolic PDEs

Pengfei Wang, Emilia Fridman

TL;DR

This work tackles delay-robust stabilization of a 2D linear heat equation using a finite-dimensional observer-based controller derived via modal decomposition. A vector Lyapunov functional combined with vector Halanay inequality is introduced to compensate fast-varying output delays in the infinite-dimensional tail, and LMIs are developed to determine the necessary observer dimension and admissible delays. Theoretical results establish exponential stability and asymptotic feasibility of the LMIs as the observer order grows, with numerical experiments showing the vector Halanay approach yields larger allowable delays than the classical scalar version for sufficiently large $N$. The framework supports both non-local actuation/measurement and boundary sensing/actuation, enabling delay-robust control of 2D diffusion processes with reduced sensor/actuator counts and networked control considerations.

Abstract

Recently, a constructive method was suggested for finite-dimensional observer-based control of 1D linear heat equation, which is robust to input/output delays. In this paper, we aim to extend this method to the 2D case with general time-varying input/output delays (known output delay and unknown input delay) or sawtooth delays (that correspond to network-based control). We use the modal decomposition approach and consider boundary or non-local sensing together with non-local actuation, or Neumann actuation with non-local sensing. To compensate the output delay that appears in the infinite-dimensional part of the closed-loop system, for the first time for delayed PDEs we suggest a vector Lyapunov functional combined with the recently introduced vector Halanay inequality. We provide linear matrix inequality (LMI) conditions for finding the observer dimension and upper bounds on delays that preserve the exponential stability. We prove that the LMIs are always feasible for large enough observer dimension and small enough upper bounds on delays. A numerical example demonstrates the efficiency of our method and shows that the employment of vector Halanay's inequality allows for larger delays than the classical scalar Halanay inequality for comparatively large observer dimension.

Delayed finite-dimensional observer-based control of 2D linear parabolic PDEs

TL;DR

This work tackles delay-robust stabilization of a 2D linear heat equation using a finite-dimensional observer-based controller derived via modal decomposition. A vector Lyapunov functional combined with vector Halanay inequality is introduced to compensate fast-varying output delays in the infinite-dimensional tail, and LMIs are developed to determine the necessary observer dimension and admissible delays. Theoretical results establish exponential stability and asymptotic feasibility of the LMIs as the observer order grows, with numerical experiments showing the vector Halanay approach yields larger allowable delays than the classical scalar version for sufficiently large . The framework supports both non-local actuation/measurement and boundary sensing/actuation, enabling delay-robust control of 2D diffusion processes with reduced sensor/actuator counts and networked control considerations.

Abstract

Recently, a constructive method was suggested for finite-dimensional observer-based control of 1D linear heat equation, which is robust to input/output delays. In this paper, we aim to extend this method to the 2D case with general time-varying input/output delays (known output delay and unknown input delay) or sawtooth delays (that correspond to network-based control). We use the modal decomposition approach and consider boundary or non-local sensing together with non-local actuation, or Neumann actuation with non-local sensing. To compensate the output delay that appears in the infinite-dimensional part of the closed-loop system, for the first time for delayed PDEs we suggest a vector Lyapunov functional combined with the recently introduced vector Halanay inequality. We provide linear matrix inequality (LMI) conditions for finding the observer dimension and upper bounds on delays that preserve the exponential stability. We prove that the LMIs are always feasible for large enough observer dimension and small enough upper bounds on delays. A numerical example demonstrates the efficiency of our method and shows that the employment of vector Halanay's inequality allows for larger delays than the classical scalar Halanay inequality for comparatively large observer dimension.
Paper Structure (8 sections, 6 theorems, 85 equations, 1 figure, 3 tables)

This paper contains 8 sections, 6 theorems, 85 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Non-local actuation and measurement
  3. System under consideration and controller design
  4. Stability analysis and main results
  5. Non-local actuation and boundary measurement
  6. Boundary actuation and non-local measurement
  7. Numerical examples
  8. Conclusions

Key Result

Lemma 1.1

(strauss2007partial) For eigenvalues eigenvalue2, we have $\lim_{N\rightarrow\infty}\frac{\lambda_{N}}{N}=\frac{4\pi}{|\Omega|}$, where $|\Omega|$ is the area of $\Omega$.

Figures (1)

  • Figure 1: Evolutions $\|z(\cdot,t)\|^{2}_{L^{2}}$ (Theorem \ref{['thm1']}), $\|\nabla z(\cdot,t)\|^{2}_{L^{2}}$ (Theorem \ref{['thm3']}), and $\|z(\cdot,t)\|^{2}_{L^{2}}$ (Theorem \ref{['thm2']}) VS $t$.

Theorems & Definitions (13)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 2.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.1
  • Remark 3.2
  • ...and 3 more