Dynamic Boundary Fuzzy Control Design of Semilinear Parabolic PDE Systems With Spatially Noncollocated Discrete Observation.

Abstract

The problem of dynamic boundary fuzzy control design is investigated in this paper for nonlinear parabolic partial differential equation (PDE) systems with spatially noncollocated discrete observation. Two cases of noncollocated discrete observation in space (i.e., pointwise observation in space and local piecewise uniform observation in space) are considered, respectively. The spatially noncollocated discrete observation makes the control design very difficult. Such design difficulty can be surmounted by an observer-based feedback control technique. It is assumed that the semilinear PDEs are accurately represented by a Takagi-Sugeno fuzzy PDE model. On the basis of the obtained fuzzy PDE model, a fuzzy Luenberger-type PDE state observer with above two cases of noncollocated discrete observation in space is first proposed for exponential estimation of the PDE system state. An observer-based dynamic fuzzy controller is then constructed such that the resulting closed-loop system is exponentially stable. Sufficient conditions on existence of such fuzzy controller are developed by Lyapunov technique with variations of vector-valued Poincaré-Wirtinger inequality, and presented in terms of linear matrix inequalities. Finally, extensive numerical simulation results of two examples are provided to support the proposed design method.