A Carleman Approximation-Based Approach to Address the Performance Criteria Issue of Economic Model Predictive Control Via Lyapunov Method

The last decade witnessed a widespread growth of Economic-oriented Model Predictive Control (EMPC) in the chemical, pharmaceutical and petroleum industries. Built upon MPC, which predicts future system behavior over a finite horizon and takes actions accordingly, EMPC aims at maximizing the economic profit of the system while satisfying performance and safety constraints ¹. One of the most significant challenges faced with EMPC is the difficulty in maintaining the system in a desired performance criteria.

Lyapunov-based approaches have been reported in many EMPC designs for their provable stability and performance properties ¹²³. Defined as Lyapunov-based EMPC (LEMPC), it has been applied with success to constrained nonlinear systems, switched systems and fault-tolerant control schemes ⁴⁵. Carleman approximation (also known as Carleman linearization)-based formulations have been applied to nonlinear MPC and Moving Horizon Estimation (MHE) ⁶⁷⁸⁹¹⁰. Nonlinear systems are approached with polynomial expressions and then represented with extended bilinear forms. This formulation enables analytical prediction of future states and analytically providing the sensitivity of the cost function to the manipulated inputs as the searching gradient. Consequently, (i) the optimization is relieved from equality constraints; (ii) the computation of optimal control policy is accelerated.

In this presentation, we propose a method to construct Lyapunov functions via Carleman approximation in the design of EMPC controllers. After expressing the dynamic constraints with polynomial forms, we represent the dynamic system in an extended bilinear expression. With the extended bilinear expression, we incorporate the dynamic constraints directly in the construction of Lyapunov function and in the objective function. Hence, we relieve the optimization from dynamic constraints. Meanwhile, it becomes easier for us to prove the boundedness of the economic performance criteria with the extended bilinear expression. A case study example uses a polymerization system to illustrate the proposed method. We design a controller that bounds the economic performance of the system in a desired set and prove the boundedness of the economic performance criteria.

Key words: Carleman Approximation, economic-oriented model predictive control, Lyapunov-based control

 

Reference:

1. Ellis M, Durand H, Christofides PD. A tutorial review of economic model predictive control methods. J Process Control. 2014;24(8):1156-1178. doi:10.1016/j.jprocont.2014.03.010.

2. Mhaskar P, El-Farra NH, Christofides PD. Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Syst Control Lett. 2006;55(8):650-659. doi:10.1016/j.sysconle.2005.09.014.

3. Heidarinejad M, Liu J, Christofides PD. Economic Model Predictive Control of Nonlinear Process Systems Using Lyapunov Techniques Mohsen. AIChE J. 2012;58(3):855-870. doi:10.1002/aic.

4. El-Farra NH, Mhaskar P, Christofides PD. Output feedback control of switched nonlinear systems using multiple Lyapunov functions. Syst Control Lett. 2005;54:1163-1182. doi:10.1016/j.sysconle.2005.04.005.

5. Munoz de la Pena D, Christofides PD. Lyapunov-Based Model Predictive Control of Nonlinear Systems Subject to Data Losses. IEEE Trans Automat Contr. 2008;53(9):2076-2089. doi:10.1109/TAC.2008.929401.

6. Armaou A, Ataei A. Piece-wise constant predictive feedback control of nonlinear systems. J Process Control. 2014;24(4):326-335. doi:10.1016/j.jprocont.2014.02.002.

7. Fang Y, Armaou A. Nonlinear Model Predictive Control using a bilinear Carleman linearization-based formulation for chemical processes. In: Proceedings of the American Control Conference.Vol 1.; 2015:5629-5634. doi:10.1109/ACC.2015.7172221.

8. Fang Y, Armaou A. Carleman approximation based Quasi-analytic Model Predictive Control for Nonlinear Systems. AIChE J. 2016. doi:10.1002/aic.15298.

9. Hashemian N, Armaou A. Fast Moving Horizon Estimation of Nonlinear Processes via Carleman Linearization. In: Proceedings of the American Control Conference.Vol 1.; 2015:3379-3385. doi: 10.1109/ACC.2015.7171854.

10. Hashemian N, Armaou A. Simulation, Model-Reduction, and State Estimation of a Two-Component Coagulation Process. AIChE J. 2016;62(5):1557-1567. doi: 10.1002/aic.15146.