(17b) Robust Multi-Scenario MPC with Embedded Closed-Loop Prediction

Model Predictive Control (MPC) relies on an accurate model of the plant behavior to perform efficiently. In order to mitigate the impact of plant uncertainty on control system performance, robust MPC methods have been developed. Robust MPC has the potential to provide superior performance than its nominal counterpart in many applications, since models are never perfectly accurate and disturbances never fully eliminated. Different strategies have been developed with the goal of providing robustness to an MPC application, including worst-case (Kothare et. al., 1996; Lee and Yu, 1997), tube (Langson et al., 2004), multi-parametric (Sakizlis et. al., 2004), and multi-scenario (Lucia et al., 2013) approaches. These strategies can generally be classified into either closed-loop (CL) or open-loop (OL) methods. CL methods seek to account for future control action when making present control moves, whereas OL methods do not take this feedback mechanism into account.

Our focus in this paper is on multi-scenario approaches. Lucia et al. (2013) present a robust MPC scheme based on a multistage stochastic formulation in which the evolution of uncertainty in time is represented as a scenario tree. Scenario and bundle decomposition approaches are investigated for reducing the computation time which grows significantly with an increasing number of stages and uncertain parameters. This method is closed-loop in character due to the treatment of future control actions as recourse decisions. Lucia et. al. (2014) apply the multistage scenario tree approach to economic MPC (EMPC), and demonstrate its performance through application to a batch polymerization system. Mastragostino et al. (2014) consider robust MPC of supply chain systems that include binary decision variables that arise in production scheduling. They utilize a scenario-based approach, and present open-loop prediction and two-stage stochastic formulations, the latter which approximates closed-loop prediction through recourse input decisions. Holtorf at al. (2019), present a scenario-based robust nonlinear MPC method in which the scenario tree, based on worst-case uncertain parameter realizations, is adaptively generated at each time step based on constraint sensitivity analysis.

A key feature of the multi-scenario method is that it computes future control moves under a range of future possible plant conditions. However, for computational tractability branching is typically terminated at a certain point, referred to as the robust horizon, beyond which responses to uncertain plant behavior are not accounted for. Therefore, this paper seeks to extend the functional length of control actions under uncertain plant behavior by directly embedding future MPC optimization problems in the robust MPC formulation. This work draws on prior work in our group that utilizes the predicted closed-loop response of a plant under the action of constrained MPC. Jamaludin and Swartz (2017) employ a rigorous CL prediction approach where embedded MPC optimization problems are included in a dynamic real-time optimization (DRTO) formulation. Li and Swartz (2018) use this technique for coordination of distributed MPC systems using a single DRTO with multiple embedded MPC subproblems. This strategy explicitly solves for future control action and results in a multilevel optimization problem. The present study uses these embedded MPC subproblems to predict the MPC response to plant uncertainty realizations in a multi-scenario framework. By doing so, the robust horizon in which the future MPC behavior is considered can be extended without increasing the number of branches. The computational complexity therefore scales linearly with the robust horizon, rather than exponentially. The formulation of the proposed robust MPC approach will be presented, and its performance demonstrated through application to a linear SISO and nonlinear MIMO case study.

References

Holtorf, F., Mitsos, A., Biegler, L.T., 2019. Multistage NMPC with on-line generated scenario trees: Application to a semi-batch polymerization process, J. Proc. Control, 80, 167-179.

Jamaludin, M.Z. and Swartz, C.L.E., 2017. Dynamic real-time optimization with closed-loop prediction, AIChE J., 63 (9), 3896-3911.

Kothare, M.V., Balakrishnan, V. and Morari, M., 1996. Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10), 1361-1379.

Langson, W., Chryssochoos, I., Rakovic, S.V., Mayne, D.Q., 2004. Robust model predictive control using tubes, Automatica, 40, 125-133

Lee, J.H. and Yu, Z., 1997. Worst-case formulations of model predictive control for systems with bounded parameters. Automatica, 33(5), 763-781.

Li, H. and Swartz, C.L.E., 2018. Dynamic real-time optimization of distributed MPC systems using rigorous closed-looped prediction. Comp. Chem. Eng., 122, 356-371.

Lucia, S., Subramanian, S., Engell, S., 2013. Non-conservative robust model predictive control via scenario decomposition, 2013 IEEE Int. Conf. on Control Applications (CAA), 2013 IEEE Multi-Conference on Systems and Control, Hyderabad, India, Aug 28-30, 2013, 586-591.

Lucia, S., Andersson, J.A.E., Brandt, H., Diehl, M., Engell, S., 2014. Handling uncertainty in economic nonlinear model predictive control: A comparative case study, J. Proc. Control, 24, 1247-1259.

Mastragostino, R., Patel, S., Swartz, C.L.E., 2014. Robust decision making for hybrid process supply chain systems via model predictive control, Comp. Chem. Eng., 62, 37-55.

Sakizlis, V., Kakalis, N.M., Dua, V., Perkins, J.D. and Pistikopoulos, E.N., 2004. Design of robust model-based controllers via parametric programming. Automatica, 40(2), pp.189-201.