Stabilization of an unstable reaction-diffusion PDE with input delay despite state and input quantization

TL;DR

This paper develops a predictor-feedback control method to stabilize an unstable reaction-diffusion PDE with input delay and quantization, combining backstepping and small-gain techniques to ensure global asymptotic stability.

Contribution

It introduces a switched predictor-feedback law that handles input delay and quantization in reaction-diffusion PDEs, extending stability results to quantized inputs.

Findings

Proven global asymptotic stability of the closed-loop system.

Designed a dynamic switching strategy for quantization range adjustment.

Extended stability results to systems with input quantization.

Abstract

We solve the global asymptotic stability problem of an unstable reaction-diffusion Partial Differential Equation (PDE) subject to input delay and state quantization developing a switched predictor-feedback law. To deal with the input delay, we reformulate the problem as an actuated transport PDE coupled with the original reaction-diffusion PDE. Then, we design a quantized predictor-based feedback mechanism that employs a dynamic switching strategy to adjust the quantization range and error over time. The stability of the closed-loop system is proven properly combining backstepping with a small-gain approach and input-to-state stability techniques, for deriving estimates on solutions, despite the quantization effect and the system's instability. We also extend this result to the input quantization case.