Delayed finite-dimensional observer-based control of 1D parabolic PDEs via reduced-order LMIs

Recently a constructive method was introduced for finite-dimensional observer-based control of 1D parabolic PDEs. In this paper we present an improved method in terms of the reduced-order LMIs (that significantly reduce the computation time) and introduce predictors to manage with larger delays. We treat the case of a 1D heat equation under Neumann actuation and non-local measurement, that has not been studied yet. We apply modal decomposition and prove L² exponential stability by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension N and resulting decay rate. The LMI dimension does not grow with N. The LMI is always feasible for large N, and feasibility for N implies feasibility for N+1. For the first time we manage with delayed implementation of the controller in the presence of fast-varying (without any constraints on the delay-derivative) input and output delays. To manage with larger delays, we construct classical observer-based predictors. For the known input delay, the LMIs’ dimension does not grow with N, whereas for unknown one the LMIs dimension grows, but it is essentially smaller than in the existing results. A numerical example demonstrates the efficiency of our method.