[Paper Review] PI regulation control of a 1-D semilinear wave equation

Hugo Lhachemi, Christophe Prieur+1 more|arXiv (Cornell University)|Jun 18, 2020

Stability and Controllability of Differential Equations44 references4 citations

TL;DR

This paper proposes a Proportional-Integral (PI) regulation control law for the left Neumann trace of a 1-D semilinear wave equation using boundary control via the right Neumann trace. By combining a preliminary velocity feedback to stabilize most modes and projecting trajectories onto a Riesz basis of the unstable modes, the authors design a finite-dimensional augmented model for PI control, proving local exponential stability and set-point regulation via Lyapunov analysis.

ABSTRACT

This paper is concerned with the Proportional Integral (PI) regulation control of the left Neu-mann trace of a one-dimensional semilinear wave equation. The control input is selected as the right Neumann trace. The control design goes as follows. First, a preliminary (classical) velocity feedback is applied in order to shift all but a finite number of the eivenvalues of the underlying unbounded operator into the open left half-plane. We then leverage on the projection of the system trajectories into an adequate Riesz basis to obtain a truncated model of the system capturing the remaining unstable modes. Local stability of the resulting closed-loop infinite-dimensional system composed of the semilinear wave equation, the preliminary velocity feedback, and the PI controller, is obtained through the study of an adequate Lyapunov function. Finally, an estimate assessing the set point tracking performance of the left Neumann trace is derived.

Motivation & Objective

• To address the challenge of regulating the left Neumann trace of a 1-D semilinear wave equation using boundary control.
• To extend PI control design to infinite-dimensional systems with unbounded control operators and nonlinearities.
• To ensure local exponential stability and set-point regulation of the left Neumann trace despite the presence of unstable modes.
• To develop a systematic control design combining velocity feedback, Riesz basis projection, and PI augmentation for infinite-dimensional systems.

Proposed method

• Apply a preliminary velocity feedback to shift all but finitely many eigenvalues of the unbounded operator into the open left half-plane.
• Project the system trajectories onto a Riesz basis formed by the generalized eigenstructures of the unbounded operator to derive a finite-dimensional truncated model of the unstable modes.
• Augment the truncated model with an integral component to form a PI controller for regulation control.
• Use a Lyapunov function to prove local exponential stability of the closed-loop infinite-dimensional system.
• Derive an estimate on the set-point tracking performance of the left Neumann trace based on the Lyapunov analysis.
• Leverage spectral analysis and asymptotic eigenvalue expansion to justify the Riesz basis structure and its perturbation from a reference basis.

Experimental results

Research questions

• RQ1 Can PI control be effectively applied to regulate the Neumann trace of a 1-D semilinear wave equation with unbounded control operator?
• RQ2 How can unstable modes in a semilinear wave equation be captured and stabilized using a finite-dimensional approximation derived from a Riesz basis?
• RQ3 What control structure ensures both local stability and set-point regulation in the presence of nonlinearity and boundary control?
• RQ4 How does the combination of velocity feedback and PI control affect the spectral properties and stability of the closed-loop system?

Key findings

• The proposed control design ensures local exponential stability of the closed-loop system composed of the semilinear wave equation, velocity feedback, and PI controller.
• The Riesz basis projection enables an accurate finite-dimensional approximation of the unstable modes, facilitating controller design.
• An estimate on the set-point tracking performance of the left Neumann trace is derived, quantifying regulation accuracy.
• The eigenvalues of the closed-loop system are shown to be geometrically simple and quadratically close to a reference Riesz basis, ensuring robustness of the spectral structure.
• The algebraic multiplicity of real eigenvalues is proven to be odd, supporting the stability analysis framework.