Disturbance-to-State Stabilization and Quantized Control for Linear Hyperbolic Systems

TL;DR

This paper develops robust boundary control strategies for linear hyperbolic PDEs under measurement disturbances and quantization, providing stability estimates and bounds on the system state.

Contribution

It introduces a disturbance-to-state estimate for hyperbolic PDEs with boundary control and quantized measurements, ensuring robustness and practical stability.

Findings

Derived a disturbance-to-state estimate for hyperbolic PDEs.

Established stability conditions using Lyapunov functions.

Provided bounds on state trajectories under quantized control.

Abstract

We consider a system of linear hyperbolic PDEs where the state at one of the boundary points is controlled using the measurements of another boundary point. Because of the disturbances in the measurement, the problem of designing dynamic controllers is considered so that the closed-loop system is robust with respect to measurement errors. Assuming that the disturbance is a locally essentially bounded measurable function of time, we derive a disturbance-to-state estimate which provides an upper bound on the maximum norm of the state (with respect to the spatial variable) at each time in terms of $\mathcal{L}^\infty$-norm of the disturbance up to that time. The analysis is based on constructing a Lyapunov function for the closed-loop system, which leads to controller synthesis and the conditions on system dynamics required for stability. As an application of this stability notion, the problem of quantized control for hyperbolic PDEs is considered where the measurements sent to the controller are communicated using a quantizer of finite length. The presence of quantizer yields practical stability only, and the ultimate bounds on the norm of the state trajectory are also derived.