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

Recently a constructive method was introduced for finite-dimensional observer-based control of the 1D heat equation under Dirichlet actuation and non-local measurement. In this paper, we extend this method to Neumann actuation, where modal decomposition is applied to the original system (without dynamic extension) and L2 exponential stability is proved by a direct Lyapunov method. We provide reduced-order LMI conditions for finding the observer dimension N and resulting decay rate. The LMI dimension is defined by N0 +1 unstable modes and does not grow with N (which is larger than N0 +1). The obtained LMI is always feasible for large N, and feasibility for N implies feasibility for N +1. Differently from Dirichlet actuation, here we manage with delayed implementation of the controller in the presence of fast-varying (without constraints on the delay-derivative) input and output delays. We consider the case of interval input delay which is lower-bounded by r > 0. By employing Lyapunov functionals and Halanay's inequality, we derive LMIs for finding N, upper bounds on delays and the decay rate. A numerical example demonstrates the efficiency of our method.