Sub-Predictors for Finite-Dimensional Observer-Based Control of Stochastic Semilinear Parabolic PDEs

We study output-feedback control of 1D stochastic semilinear heat equation with constant input delay and nonlinear multiplicative noise where the nonlinearities satisfy globally Lipschitz condition. We consider the Neumann actuation and nonlocal measurement. To compensate delay r, we construct a chain of M+1 sub-predictors in the form of ODEs that correspond to the delay fraction r/M. Differently from the deterministic case, we add an additional sub-predictor to the chain that leads to the closed-loop system with the stochastic infinite-dimensional tail and the finite-dimensional part that consists of non-delayed stochastic equations and delayed deterministic ones. The latter essentially simplifies the Lyapunov-based mean-square L2 exponential stability analysis of the full-order closed-loop system. We employ corresponding Itô's formulas for stochastic ODEs and PDEs, respectively. Our stability analysis leads to LMIs which are shown to be feasible for any input delay provided M and the observer dimension are large enough and Lipschitz constants are small enough. A numerical example demonstrates the efficiency of the proposed approach.