(555b) Handling State Constraints and Economics in Feedback Control of Transport-Reaction Processes

Economic model predictive control (EMPC) is a practical optimal control-based technique that has recently gained widespread popularity within the process control community and beyond because of its unique quality of effectively integrating process economics and feedback control (e.g., [1]-[2]; see, also, [3] for an overview of recent results and references). However, most of EMPC systems have been designed for lumped parameter processes described by linear/nonlinear ordinary differential equation (ODE) systems. In our previous work [4]-[5], EMPC systems with a general economic cost function for parabolic PDE systems were proposed which operate the closed-loop system in a dynamically optimal fashion. Specifically, the EMPC schemes were developed with a reduced-order model (ROM) (i.e., low-order nonlinear ODE model) derived through Galerkin's method using analytical eigenfunctions [4] and empirical eigenfunctions derived by proper orthogonal decomposition (POD) [5], respectively. To achieve high accuracy of the ROM derived from the empirical eigenfunctions of the original PDE system, the POD method usually needs a large ensemble of solution data (snapshots) to contain as much local and global process dynamics as possible. Constructing such a large ensemble of snapshots becomes a significant challenge from a practical point of view; since currently, there is no general way to realize a representative ensemble. Based on this consideration, an adaptive proper orthogonal decomposition (APOD) methodology was proposed to recursively update the ensemble of snapshots and compute on-line the new empirical eigenfunctions in the on-line closed-loop operation of PDE systems [6]-[7]. The ROM accuracy is indeed limited by the number of the empirical eigenfunctions adopted for the ROM; in practice, when a process faces state constraints, the accuracy of the ROM based on a limited number of eigenfunctions may not be able to allow the controller to avoid a state constraint violation.

For hyperbolic PDE systems (e.g., convection-reaction processes where the convective phenomena dominate over diffusive ones), it is common that the eigenvalues of the spatial differential operator cluster along vertical or nearly vertical asymptotes in the complex plane, and thus, many modes are required to construct finite-dimensional models of desired accuracy. Considering this, the Galerkin's method would be computationally expensive for this case compared to finite difference method utilizing a sufficient large number of spatial discretization points. However, no work on applying EMPC to hyperbolic PDE systems as been completed.

Motivated by the above considerations, we address two open issues in the context of applying EMPC to transport-reaction processes. First, we focus on a system of nonlinear parabolic PDEs and propose a novel EMPC design integrating adaptive proper orthogonal decomposition (APOD) method with a high-order finite-difference method to handle state constraints. The computational efficiency and constraint handling properties of this scheme are evaluated using a tubular reactor example modeled by two nonlinear parabolic PDEs. Second, we formulate an EMPC system that accounts for both manipulated input and state constraints for a system of first-order hyperbolic PDEs. Various closed-loop simulations scenarios are presented to demonstrate the overall effectiveness of this EMPC scheme using a plug flow reactor example.

[1] Angeli D, Amrit R, Rawlings JB. On average performance and stability of economic model predictive control. IEEE Transactions on Automatic Control. 2012;57:1615-1626.
[2] Amrit R, Rawlings JB, Angeli D. Economic optimization using model predictive control with a terminal cost. Annual Reviews in Control. 2011;35:178-186.
[3] Ellis M, Durand H, Christofides PD. A tutorial review of economic model predictive control methods. Journal of Process Control, in press.
[4] Lao L, Ellis M, Christofides PD. Economic model predictive control of transport-reaction processes. Industrial \& Engineering Chemistry Research, in press.
[5] Lao L, Ellis M, Christofides PD. Economic model predictive control of parabolic PDE systems: Addressing state estimation and computational efficiency. Journal of Process Control. 2014;24:448-462.
[6] Ravindran SS. Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM Journal on Scientific Computing. 2002;23:1924-1942.
[7] Varshney A, Pitchaiah S, and Armaou A. Feedback control of dissipative PDE systems using adaptive model reduction. AIChE Journal.