Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs

The objective of the present paper is finite-dimensional observer-based control of the 1-D linear heat equation with constructive and feasible design conditions. We propose a method which is applicable to boundary or non-local sensing together with non-local actuation, or to Dirichlet actuation with non-local sensing. We use a modal decomposition approach. The dimension of the controller, N₀, is equal to the number of modes which decay slower than a given decay rate δ>0. The observer may have a larger dimension N≥N₀. The observer and controller gains are found separately by solving N₀×N₀-dimensional Lyapunov inequalities. We suggest a direct Lyapunov approach to the full-order closed-loop system and provide linear matrix inequalities (LMIs) for finding N and the exponential decay rate of the closed-loop system. We prove that the LMIs are always feasible for large enough N. The proposed method is different from existing qualitative methods that do not give easily verifiable and efficient bounds on the observer-based controller dimension and the resulting closed-loop performance. Numerical examples demonstrate that our LMI conditions lead to non-conservative bounds on N and the resulting decay rate.