  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.

Optimization and Control