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.