[Paper Review] Boundary Output Feedback Stabilization of Reaction-Diffusion PDEs with Delayed Boundary Measurement

Hugo Lhachemi, Christophe Prieur|arXiv (Cornell University)|Jun 25, 2021

Stability and Controllability of Differential Equations29 references4 citations

TL;DR

This paper proposes a boundary output feedback control strategy for 1-D reaction-diffusion PDEs with arbitrarily long delayed boundary measurements. It combines a finite-dimensional observer to estimate delayed system modes and a predictor to compensate for the delay, ensuring exponential stability in the $H^1$ norm for any given delay when the observer order is sufficiently large.

ABSTRACT

This paper addresses the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs with delayed boundary measurement. The output takes the form of a either Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component which is employed to obtain the control input to be applied at current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided the order of the observer is selected large enough. Taking advantage of this result, we discuss the extension of the control strategy to the case of simultaneous input and output delays.

Motivation & Objective

• Address the challenge of stabilizing 1-D reaction-diffusion PDEs when boundary measurements are delayed by an arbitrarily large time.
• Overcome the instability risks introduced by output delays in feedback control of distributed parameter systems.
• Develop a control strategy that ensures exponential stability in the $H^1$ norm despite delayed measurements.
• Extend the framework to handle simultaneous input and output delays by combining with existing input-delay compensation techniques.
• Provide a numerically verifiable LMI-based design procedure for controller and observer parameters.

Proposed method

• Design a finite-dimensional observer to estimate the first $N_0$ modes of the PDE based on delayed boundary measurements.
• Use a predictor component to reconstruct the current control input from the delayed observer output, compensating for measurement delay.
• Apply spectral-reduction and scaling-based methods to handle Dirichlet/Neumann boundary conditions directly without requiring time-derivative of control input.
• Formulate the closed-loop stability problem using a Lyapunov-Krasovskii functional to derive sufficient LMI conditions for exponential stability.
• Construct the state estimate $\tilde{X}(t) = \mathrm{col}(X(t), \zeta(t - h_o))$ to incorporate delayed measurements into the observer dynamics.
• Ensure feasibility of the LMI constraints by selecting a sufficiently large observer order $N_0$, guaranteeing stability for any given output delay $h_o > 0$.

Experimental results

Research questions

• RQ1 Can exponential stabilization be achieved for 1-D reaction-diffusion PDEs with arbitrarily long output delays using boundary output feedback?
• RQ2 How can a finite-dimensional observer be designed to estimate system modes from delayed boundary measurements without requiring derivative feedback?
• RQ3 What control architecture enables compensation for output delays while maintaining stability in the $H^1$ norm?
• RQ4 Can the proposed observer-predictor framework be extended to handle simultaneous input and output delays?
• RQ5 Under what conditions are the LMI-based stability constraints feasible for arbitrary output delays?

Key findings

• For any given output delay $h_o > 0$, exponential stability of the closed-loop system in the $H^1$ norm is guaranteed if the observer order $N_0$ is sufficiently large.
• The LMI conditions derived in Theorem 6.1 are always feasible for $N$ selected large enough, ensuring the existence of a stabilizing controller.
• The proposed control strategy achieves exponential decay of the $H^1$-norm of the PDE state and observer states at rate $2\delta > 0$, with $\delta$ related to the spectral gap of the PDE.
• The method is applicable to general 1-D reaction-diffusion PDEs with Dirichlet, Neumann, or Robin boundary conditions and measurements.
• The framework can be extended to handle simultaneous input and output delays by combining with the input-delay compensation method from Lhachemi & Prieur (2021).
• The stability result holds under mild regularity assumptions on the PDE coefficients and initial conditions in $H^2(0,1)$ with compatible boundary constraints.