Predictive control of parabolic PDEs with state and control constraints

Abstract This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with state and control constraints. Initially, the PDE is written as an infinite‐dimensional system in an appropriate Hilbert space. Next, modal decomposition techniques are used to derive a finite‐dimensional system that captures the dominant dynamics of the infinite‐dimensional system, and express the infinite‐dimensional state constraints in terms of the finite‐dimensional system state constraints. A number of model predictive control (MPC) formulations, designed on the basis of different finite‐dimensional approximations, are then presented and compared. The closed‐loop stability properties of the infinite‐dimensional system under the low order MPC controller designs are analysed, and sufficient conditions that guarantee stabilization and state constraint satisfaction for the infinite‐dimensional system under the reduced order MPC formulations are derived. Other formulations are also presented which differ in the way the evolution of the fast eigenmodes is accounted for in the performance objective and state constraints. The impact of these differences on the ability of the predictive controller to enforce closed‐loop stability and state constraints satisfaction in the infinite‐dimensional system is analysed. Finally, the MPC formulations are applied through simulations to the problem of stabilizing the spatially‐uniform unstable steady‐state of a linear parabolic PDE subject to state and control constraints. Copyright © 2006 John Wiley & Sons, Ltd.