Sub-Predictors and Classical Predictors for Finite-Dimensional Observer-Based Control of Parabolic PDEs

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.

I. Introduction

Finite-dimensional observer-based controllers for parabolic systems were designed by the modal decomposition approach in [1], [2], [3], [4], [5]. Recently, the first constructive LMI-based method for finite-dimensional observer-based controller was suggested in [6] for the 1D heat equation under nonlocal or Dirichlet actuation and nonlocal measurement. The observer dimension and the resulting exponential decay rate were found from simple LMI conditions. Finite-dimensional observer-based control of the Kuramoto-Sivashinsky equation with boundary actuation and point measurement was studied in [7].