(303c) Machine Learning-Based Predictive Control of Nonlinear Parabolic PDE Systems

Diffusion-reaction processes arise naturally in a variety of industrial applications including rapid thermal processing, crystal growth and various deposition systems, and these processes are usually governed by nonlinear parabolic PDE systems. The traditional approach to controlling systems governed by parabolic PDEs involves application of Galerkin’s method and its variants (like eigenfunction expansion techniques) thereby deriving reduced-order finite-dimensional models of a small number of first-order nonlinear ordinary differential equations in time that describe the dominant dynamics of the PDE system (e.g., [2], [3]), which are used to design low-order controllers for the parabolic PDE system (e.g., [4]). Model predictive control (MPC) strategies have gained popularity because of its capability to address closed-loop stability in tandem with satisfying input and state constraints. The most important requirement of MPC is an accurate process model for the system, which can be constructed by accounting for the physiochemical mechanisms that drive the process, or derived using a data driven approach by developing recurrent neural network models based on simulated/experimentally obtained data (e.g., [1]).

This work focuses on designing and analyzing closed loop stability of machine-learning-based predictive control systems for nonlinear parabolic PDE systems using only process state trajectories from measurements. First, the Karhunen-Loeve expansion is used to derive dominant spatial empirical eigenfunctions of the nonlinear parabolic system using the process state trajectories, then these eigenfunctions are used as basis functions within a Galerkin’s model reduction framework to derive the temporal evolution of a small number of temporal modes capturing the dominant dynamics of the PDE system. Recurrent neural networks (RNN) are then used to model the data-derived reduced-order dominant dynamics of the parabolic PDE system based on extensive and carefully-chosen open-loop simulations within a desired operating region. Finally, Lyapunov based MPC techniques [1] are used to design controllers based on the RNN models for a diffusion-reaction process, and simulations are performed to evaluate their closed-loop performance.

References:

[1] Wu, Z., A. Tran, D. Rincon and P. D. Christofides (2019), "Machine Learning-Based Predictive Control of Nonlinear Processes. Part I: Theory,'' AIChE J., 65, e16729.

[2] Baker, J. and P. D. Christofides (2000), ''Finite-Dimensional Approximation and Control of Nonlinear Parabolic PDE Systems,'' Int. J. Contr., 73, 439-456.

[3] Lao, L., M. Ellis and P. D. Christofides (2014), "Economic Model Predictive Control of Parabolic PDE Systems: Addressing State Estimation and Computational Efficiency," J. Proc. Contr., 24, 448-462.

[4] Armaou, A. and P. D. Christofides (2001), ''Finite-Dimensional Control of Nonlinear Parabolic PDEs with Time-Dependent Spatial Domains Using Empirical Eigenfunctions,'' Int. J. Appl. Math. & Comp. Sci., 11, 287-317.