(456h) Robust Model Based Control Via Closed-Loop Reference Trajectory Optimization

Model predictive control (MPC) relies on a dynamic model for prediction of future process behavior in order to calculate the control input. No model is perfect, and depending on the level of uncertainty, poor control performance can result if the uncertainty is not accounted for in the control calculation, particularly with regard to process constraint violations. This has led to the development of several robust MPC strategies over the past three decades, that can be broadly classified into open-loop and closed-loop strategies. In the former, the uncertainty propagation in the predicted response does not take future input action into account, whereas some form of feedback action is considered in the latter.

Lee and Yu (1997) present robust MPC algorithms that minimize the worst-case cost (min-max) based on either open-loop or and closed-loop prediction, the latter determined via a dynamic programming approach. Kothare et al. (1996) present a robust MPC algorithm that minimizes an upper bound on a worst-case performance objective, and utilizes a state feedback control law at each prediction step. Sakizlis et al. (2004) present a multi-parametric programming formulation of robust MPC, where they consider additive time-varying uncertainty. In a subsequent contribution, Pistikopoulos et al. (2009) consider uncertainty in the state-space matrices in a multi-parametric robust MPC formulation. Multi-scenario based robust MPC formulations have been proposed by several workers. Huang and Biegler (2009) present a robust nonlinear MPC formulation, in which the predicted response is propagated for a discrete number of uncertainty realizations in an open-loop manner. Lucia et al. (2013) present a multi-stage nonlinear MPC strategy in which the predicted future states and inputs evolve in response to uncertainty realizations through a scenario tree, and in Lucia et al. (2014), compare the multi-stage scenario-based approach to an open-loop prediction strategy and one that utilizes an affine control policy. A robust MPC strategy for supply chain operation proposed by Mastragostino et al. (2014) follows a two-stage scenario-based approach.

In previous work, the idea of rigorous closed-loop prediction has been applied in a dynamic real-time optimization (DRTO) formulation that computes set-point trajectories for an underlying MPC system based on the prediction of the future plant response under the action of constrained MPC (Jamaludin and Swartz, 2017a,b). This closed-loop DRTO formulation results in a multilevel optimization problem due to the MPC optimization subproblems along the DRTO prediction horizon, and is solved by reformulating the inner subproblems by algebraic constraints corresponding to their Karush-Kuhn-Tucker (KKT) optimality conditions, resulting in a single mathematical program with complementarity constraints (MPCC).

In the present study, a robust control strategy is proposed that follows a two-layer control structure in which set-point trajectories are computed at the upper layer and passed to an underlying plant MPC. This scheme is similar in structure to that of reference management or command governors (Bemporad et al., 1997; Bemporad and Mosca, 1998; Sugie and Yamamoto, 2001). However, in reference management, the primal control system is typically an unconstrained, linear control system, with the reference signal manipulated in order to handle constraints on the closed-loop response. The proposed scheme, on the other hand, utilizes constrained MPC as the core controller, with set-point trajectories calculated based on the predicted closed-loop response under uncertainty. The reference optimization layer considers multiple scenario realizations of the plant that covers the range of uncertain parameters. The predicted closed-loop response for each plant scenario is generated through a sequence of control calculations using the nominal MPC model, and a plant response calculation, with a separate closed-loop trajectory generated for each uncertain plant scenario. Only one reference trajectory is generated by an optimization problem and shared by all nominal MPC optimization subproblems along the closed-loop prediction horizon. Two case studies based on a linear transfer function with uncertain plant gain and a nonlinear evaporator process with an uncertain plant parameter are conducted, and demonstrate that the proposed control structure can avoid constraint violations and provide faster tracking capabilities despite different real plant parameters or sudden changes in such parameters along the simulation.

References

Bemporad, A., Casavola, A., Mosca, E. (1997). Nonlinear control of constrained linear systems via predictive reference management. IEEE Trans. Auto. Control 42(3), 340-349.

Bemporad, A. and Mosca, E. (1998). Fulfilling hard constraints in uncertain linear systems by reference managing. Automatica, 33(4), 451-461.

Huang, R. and Biegler, L.T. (2009). Robust nonlinear model predictive controller design based on multi-scenario formulation. Proc. American Control Conference, St. Louis, MO, 2341-2342.

Jamaludin, M.Z. and Swartz, C.L.E. (2017a). Dynamic Real-time Optimization with Closed-loop Prediction. AIChE J., 63, 3896 – 3911.

Jamaludin, M.Z. and Swartz, C.L.E. (2017b). Approximation of Closed-loop Prediction for Dynamic Real-time Optimization Calculations," Comput. Chem. Eng., 103, 23-38.

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

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

Lucia, S., Finkler, T., Engell, S. (2013). Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty. Journal of Process Control 23, 1306–1319.

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

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

Pistikopoulos, E.N., Faisca, N.P., Kouramas, K.I. and Panos, C. (2009). Explicit robust model predictive control. In Proceedings of the International Symposium on Advanced Control of Chemical Processes, 243-248.

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

Sugie, T. and Yamamoto, H. (2001). Reference management for closed loop systems with state and control constraints. Proc. ACC, 1426-1431.