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Recently a constructive method was introduced for finite-dimensionalobserver-based control of 1D parabolic PDEs. In this paper we present animproved method in terms of the reduced-order LMIs (that significantly shortenthe computation time) and introduce predictors to manage with larger delays. Wetreat the case of a 1D heat equation under Neumann actuation and non-localmeasurement, that has not been studied yet. We apply modal decomposition andprove $L^2$ exponential stability by a direct Lyapunov method. We providereduced-order LMI conditions for finding the observer dimension $N$ andresulting decay rate. The LMI dimension does not grow with $N$. The LMI isalways feasible for large $N$, and feasibility for $N$ implies feasibility for$N+1$. For the first time we manage with delayed implementation of thecontroller in the presence of fast-varying (without any constraints on thedelay-derivative) input and output delays. To manage with larger delays, weconstruct classical observer-based predictors. For the known input delay, theLMIs dimension does not grow with $N$, whereas for unknown one the LMIsdimension grows, but it is ssentially smaller than in the existing results. Anumerical example demonstrates the efficiency of our method.

Delayed finite-dimensional observer-based control of 1D heat equation under Neumann actuation