Parabolic partial differential equations naturally arise as an adequate representation of a large class of spatially distributed systems, such as diffusion-reaction processes, where the interplay between diffusive and reaction forces introduces complexity in the characterization of the system, for the purpose of process parameter identification and subsequent control. In this work we introduce a model predictive control (MPC) framework for the control of input and state constrained parabolic partial differential equation (PDEs) systems. Model predictive control (MPC) is one of the most popular control formulations among chemical engineers, manly due to its ability to account for the actuator (input) constraints that inevitably exist due to finite actuator power and its ability to handle state constraints within an optimal control setting. In controller synthesis, the initially parabolic partial differential equation of the diffusion reaction type is transformed by the Galerkin method into a system of ordinary differential equations (ODEs) that capture the dominant dynamics of the PDE system. Systems obtained in such a way (ODEs) are used as the basis for the synthesis of the MPC controller that explicitly accounts for the input and state constraints. Namely, the modified MPC formulation includes a penalty term that is directly added to the objective function and through the appropriate structure of the controller state constraints accounts for the infinite dimensional nature of the state of the PDE system. The MPC controller design method is successively applied to control of the diffusion-reaction process described by linear parabolic PDE, by demonstrating stabilization of the non-dimensional temperature profile around a spatially uniform unstable steady-state under satisfaction of the input (actuator) constraints and allowable non-dimensional temperature (state) constraints.
A partial differential equation model predictive control strategy: application to autoclave composite processing
