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

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 <inline-formula> <tex-math notation= LaTeX >$N+1$ </tex-math></inline-formula> modes in the observer. We introduce a chain of <inline-formula> <tex-math notation= LaTeX >$M$ </tex-math></inline-formula> sub-predictors that leads to a closed-loop ODE system coupled with infinite-dimensional tail. Given an input delay <inline-formula> <tex-math notation= LaTeX >$r$ </tex-math></inline-formula>, we present LMI stability conditions for finding <inline-formula> <tex-math notation= LaTeX >$M$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$N$ </tex-math></inline-formula> and the resulting exponential decay rate and prove that the LMIs are always feasible for any <inline-formula> <tex-math notation= LaTeX >$r$ </tex-math></inline-formula>. We also consider a classical observer-based predictor and show that the corresponding LMI stability conditions are feasible for any <inline-formula> <tex-math notation= LaTeX >$r$ </tex-math></inline-formula> provided <inline-formula> <tex-math notation= LaTeX >$N$ </tex-math></inline-formula> 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.