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Unknown input observer design for a class of coupled wave PDE systems

Najmeh Ghaderi, Birgit Jacob

TL;DR

The paper addresses unknown-input state estimation for a class of coupled semilinear one-dimensional wave PDEs by developing a state observer and an $H_{ abla\infty}$ observer. It derives Lyapunov-based matrix-inequality conditions that guarantee asymptotic stability of the estimation error when $d(t)=0$ and establish an $L^2$-gain robustness bound $\| e \|_{L^2} \le \mu \| d \|_{L^2}$ for nonzero disturbances, with the results expressed as LMIs involving Lipschitz constant $\gamma$ and design variables. The H$_{\infty}$ design hinges on a block inequality $\Theta \prec 0$ and $\Gamma - I \prec 0$, ensuring stability and disturbance attenuation; the approach is demonstrated numerically on a pair of coupled waves, showing effective state reconstruction and compliance with the $H_{\infty}$ bound. Overall, the work advances robust estimation for distributed parameter systems with unknown inputs, though the feasibility conditions may be restrictive in practice.

Abstract

This paper deals with the problem of designing unknown input observers for a class of coupled semilinear wave partial differential equations (PDE) systems. A state observer is designed to estimate the uncertain coupled wave PDE systems. Then, the analysis of the asymptotic stability and $H_{\infty}$ performance for the observer design of coupled wave PDE systems is investigated. Some sufficient conditions of asymptotic stability for the observer error system with disturbance attenuation level are derived via matrix inequalities based on the Lyapunov stability theory. Finally, a numerical simulation is presented to demonstrate the effectiveness of the obtained result.

Unknown input observer design for a class of coupled wave PDE systems

TL;DR

The paper addresses unknown-input state estimation for a class of coupled semilinear one-dimensional wave PDEs by developing a state observer and an observer. It derives Lyapunov-based matrix-inequality conditions that guarantee asymptotic stability of the estimation error when and establish an -gain robustness bound for nonzero disturbances, with the results expressed as LMIs involving Lipschitz constant and design variables. The H design hinges on a block inequality and , ensuring stability and disturbance attenuation; the approach is demonstrated numerically on a pair of coupled waves, showing effective state reconstruction and compliance with the bound. Overall, the work advances robust estimation for distributed parameter systems with unknown inputs, though the feasibility conditions may be restrictive in practice.

Abstract

This paper deals with the problem of designing unknown input observers for a class of coupled semilinear wave partial differential equations (PDE) systems. A state observer is designed to estimate the uncertain coupled wave PDE systems. Then, the analysis of the asymptotic stability and performance for the observer design of coupled wave PDE systems is investigated. Some sufficient conditions of asymptotic stability for the observer error system with disturbance attenuation level are derived via matrix inequalities based on the Lyapunov stability theory. Finally, a numerical simulation is presented to demonstrate the effectiveness of the obtained result.

Paper Structure

This paper contains 7 sections, 3 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Observer design
  4. Stability analysis
  5. $H_{\infty}$ design
  6. Numerical simulation
  7. Conclusion

Key Result

Lemma 2.1

(cao) Let the set of $n$ by $n$ positive definite symmetric matrices be denoted by $\mathcal{S}_{n}^{+}$. Then for $x, y \in \mathbb R^{n}$ and $S \in \mathcal{S}_{n}^{+}$, we have

Figures (2)

  • Figure 1: Response of the original open-loop plant \ref{['main']}, the estimated system \ref{['observer']}, the error $e(x,t)=\widehat{w}(x,t)-w(x,t)$ with no disturbance ($d(t)=0$).
  • Figure 2: Response of the original open-loop plant \ref{['main']}, the estimated system \ref{['observer']}, the error system $e(x,t)=\widehat{w}(x,t)-w(x,t)$ in the presence of disturbance ($d(t) \neq 0$).

Theorems & Definitions (5)

  • Lemma 2.1
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof