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

TL;DR

The paper tackles stabilizing an unstable reaction-diffusion PDE subject to input delay and state/input quantization. It reformulates the delay as an actuated transport PDE, applies backstepping to obtain a stable target system, and implements a switched predictor-feedback law with a dynamic quantization zoom. Using a combination of backstepping, small-gain, and ISS techniques, it proves global asymptotic stability of the PDE state in $L^2$ and the actuator state in $L^{\infty}$, with extensions to input quantization. The results advance robust stabilization of infinite-dimensional systems under practical digital implementation constraints and quantify how quantization and delay interact under a switched control strategy.

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.

Theorems & Definitions (10)

• Theorem III.1
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem IV.1
• Lemma 3
• proof
• Lemma 4
• proof