Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs

TL;DR

This paper develops finite-dimensional observer-based controllers with sub-predictors for the 1D heat equation to compensate for input delays, providing LMI conditions for stability and demonstrating practical benefits over classical predictors.

Contribution

It introduces a novel chain of sub-predictors combined with modal decomposition, offering a systematic way to handle delays in PDE control with proven stability conditions.

Findings

LMIs are feasible for any delay r

Classical predictor requires large N for stability

Sub-predictors enable lower-dimensional observer design

Abstract

We study constant input delay compensation by using finite-dimensional observer-based controllers in the case of the 1D heat equation. We consider Neumann actuation with nonlocal measurement and employ modal decomposition with $N+1$ modes in the observer. We introduce a chain of $M$ sub-predictors that leads to a closed-loop ODE system coupled with infinite-dimensional tail. Given an input delay $r$, we present LMI stability conditions for finding $M$ and $N$ and the resulting exponential decay rate and prove that the LMIs are always feasible for any $r$. We also consider a classical observer-based predictor and show that the corresponding LMI stability conditions are feasible for any $r$ provided $N$ is large enough. A numerical example demonstrates that the classical predictor leads to a lower-dimensional observer. However, it is known to be hard for implementation due to the distributed input signal.