(456c) Robust Model Predictive Control with Decomposed Disturbance Subsets for Less Conservative Control

Tae Hoon Oh and Jong Min Lee*

School of Chemical and Biological Engineering, Seoul National University,

1 Gwanak-ro, Gwanak-gu, Seoul, 08826, Korea

Robust Model Predictive Control (RMPC) has been developed over two decades in order to handle the uncertainty associated with model prediction. The uncertainties originate from model-plant mismatch or unknown disturbances. Most of the existing formulations of RMPC minimize the worst-case cost, and its resulting control actions are excessively conservative and significantly affected by the knowledge of the uncertainties. Although several approaches [1,2] are proposed to reduce the conservatism, it is difficult to apply these methods to real processes due to the lack of systematic consideration of the probability of having the worst-case scenario [3]. Conservative control actions, which are often far from optimal ones, are attributable to the worst-case scenario. The probability of having such a worst-case is negligible because it is a single point in a disturbance set. Therefore, it is much more reasonable to design a controller that represents the worst-case as a set so that the probability of the predicted scenario is nontrivial.

This study proposes a computationally tractable linear RMPC formulation with additive disturbances. The proposed formulation involves a parameter that trade-offs control performance against the probability of the worst-case scenario. In order to consider the predicted disturbance as a set rather than a point, the original disturbance set is decomposed into several finite convex subsets at each time step within the prediction horizon of N. Collecting N-1 decomposed subsets gives the sequence of disturbance subsets where each sequence is considered as the candidate for the worst-case scenario and the maximization is taken over those finite sequences. To ensure that the cardinality of the sequences is finite even for a large N, minimum response probability is proposed so that the probability of each sequence should be higher than this threshold value. Then the cost for each sequence is evaluated using the parameterized tube MPC with the sequence of decomposed disturbance subsets [4].

The minimum response probability serves as a tuning parameter to adjust the conservatism of control. If the minimum response probability is 1, the formulation is identical with the original PTMPC approach. On the other hand, if the probability approaches 0, the pessimistic control action is taken which is similar to the original closed-loop worst-case RMPC [5]. Besides the minimum response probability, there are additional possibilities to extend the problem formulation. First, the way of decomposing the disturbance set into given number of subsets is not unique. This is crucial because the control actions are affected by the shape of decomposed subsets. The ideal method would be partitioning the disturbance set in the order of increasing cost, but the resulting subsets are not convex. In addition, it is well known that partitioning an arbitrary compact set into several convex subsets is an NP-hard problem [6]. In this study, the finite identical set approximates partitioning of the arbitrary set, although more precise decomposition could be done depending on the shape of disturbance set. The distribution of the disturbance probability is not unique because the minimum response probability constraints the probability of the sequence of N subsets only. Since it is practical to give more weights on the risk of near future, one suggestion is decomposing the disturbance set into smaller subsets as the time step is closer to the current time. The recursive feasibility and stability are analyzed with the general form of problem formulation. The simulation results suggest that this method could be utilized to adjust the control performance.

References

[1] Stoorvogel, A.A., Weeren, A.J.T.M., “The discrete-time Riccati equation related to the control problem” IEEE Transactions on Automatic Control 39(3) (1994): 686-691.

[2] Gabrel, V., Murat, C., Thiele, A., “Recent advances in robust optimization: an overview” European journal of operational research 235 (3) (2014): 471-483.

[3] Saltik, M.B., Ozkan, L. Ludlage, J.H., Weiland, S., Van den Hof, P.M., “An outlook on robust model predictive control algorithms: Reflections on performance and computational aspects.” Journal of Process Control 61 (2018): 77-102.

[4] Rakovic, S. V., Kouvaritakis, B., Cannon, M., Panos, C., Findeisen R. “Parameterized tube model predictive control.” IEEE Transactions on Automatic Control 57(11) (2012): 2746-2761.

[5] Scokaert, P.O.M., Mayne D.Q. “Min-max feedback model predictive control for constrained linear systems.” IEEE Transactions on Automatic Control 43(8) (1998): 1136-1142.

[6] Bernard, C. “Convex partitions of polyhedral: a lower bound and worst-case optimal algorithm.” SIAM Journal on Computing 13(3) (1984): 488-507.