(644f) Robust Explicit Model Predictive Control Via Robust Optimization

Process control applications ubiquitously involve uncertainty due to the modeling mismatch between the real process and its digital twin, variations of process characteristics which are assumed constant, and the presence of unmeasured process disturbances. Hence, the implementation of an uncertainty-unaware control strategy could lead to significant deviations from the desired operational goal or unavoidably result in infeasibility. To address this challenge, robust model predictive control strategies have been proposed to handle these uncertainties and guarantee the satisfaction of constraints [1]. In the context of robust multiparametric/explicit model predictive control, contributions have focused on additive process disturbances and linear constraints [2], min-max approaches for additive [3] and multiplicative uncertainties [4] or employed dynamic programming and robust optimization techniques [5]. However, an open question remains the calculation of a robust explicit process control law for linear quadratic regulator problems with multiplicative uncertainty, that avoids the computational cost of dynamic programming [6].

In this work, we present an algorithm that solves the aforementioned challenge by calculating a feedback control law using multiparametric programming and robust optimization techniques. Firstly, considering box model uncertainty intervals, and a discrete time-invariant linear state space model, we construct an appropriate terminal set constraint [7], and we reformulate the original problem to its robust counterpart with a single step [8,9]. Subsequently, linear transformations are applied that preserve the linearity of the constraints and express the feasible space in terms of the initial states and the decision variables of the system. As the last step of the approach, we solve the multiparametric quadratic programming problem to derive the robust explicit solution that guarantees that satisfaction of constraints in the presence of the uncertainty [10]. We apply these findings to a process control problem and show that the approach can be extended to the case where binary decisions are part of the problem formulation.

References

[1] Kouvaritakis, B., & Cannon, M. (2016). Model predictive control. Switzerland: Springer International Publishing, 38.

[2] Olaru, S., & Ayerbe, P. R. (2006, December). Robustification of explicit predictive control laws. In Proceedings of the 45th IEEE Conference on Decision and Control (pp. 4556-4561). IEEE.

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

[4] Bemporad, A., Borrelli, F., & Morari, M. (2003). Min-max control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9), 1600-1606.

[5] Kouramas, K. I., Panos, C., Faísca, N. P., & Pistikopoulos, E. N. (2013). An algorithm for robust explicit/multi-parametric model predictive control. Automatica, 49(2), 381-389.

[6] Pappas, I., Kenefake, D., Burnak, B., Avraamidou, S., Ganesh, H. S., Katz, J., Diangelakis, N.A., & Pistikopoulos, E. N. (2021). Multiparametric Programming in Process Systems Engineering: Recent Developments and Path Forward. Frontiers in Chemical Engineering, 2.

[7] Blanchini, F. (1999). Set invariance in control. Automatica, 35(11), 1747-1767.

[8] Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical programming, 88(3), 411-424.

[9] Dantzig, G.B., Eaves, B.C. (1973). Fourier-Motzkin elimination and its dual. Journal of Combinatorial Theory Series A, 14(3), 288-297.

[10] Oberdieck, R., Diangelakis, N. A., & Pistikopoulos, E. N. (2017). Explicit model predictive control: A connected-graph approach. Automatica, 76, 103-112.