Distributed proportional plus second-order spatial derivative control for distributed parameter systems subject to spatiotemporal uncertainties

In this paper, a robust distributed control design based on proportional plus second-order spatial derivative (P-sD ²) is proposed for exponential stabilization and minimization of spatial variation of a class of distributed parameter systems (DPSs) with spatiotemporal uncertainties, whose model is represented by parabolic partial differential equations with spatially varying coefficients. Based on the Lyapunov's direct method, a robust distributed P-sD ² controller is developed to not only exponentially stabilize the DPS for all admissible spatiotemporal uncertainties but also minimize the spatial variation of the process. The outcome of the robust distributed P-sD² control problem is formulated as a spatial differential bilinear matrix inequality (SDBMI) problem. A local optimization algorithm that the SDBMI is treated as a double spatial differential linear matrix inequality (SDLMI) is presented to solve this SDBMI problem. Furthermore, the SDLMI optimization problem can be approximately solved via the finite difference method and the existing convex optimization techniques. Finally, the proposed design method is successfully applied to feedback control problem of the FitzHugh-Nagumo equation.