(247a) On the Robust Explicit Model Predictive Control of Hybrid Discrete-Time Linear Systems

In robust model predictive control, the aim is to control a system in the face of model uncertainties [1-3]. For the case of discrete-time linear systems, one of the most challenging issues is encountered when the state-space matrices are uncertain, a problem which is termed multiplicative/parametric uncertainty. In the open literature, this type of problem has been tackled using the concepts of open-loop and closed-loop control [4]. In the open-loop approach, the aim is to identify a set of control actions which will guarantee constraint satisfaction over the entire horizon. However, this approach is quite conservative, as it does not consider the fact that future control actions will have future measurements at their disposal. Thus, the idea of closed-loop robust MPC has been formulated, which aims at finding a control action to apply â??nowâ?? such that there will exist a â??futureâ??, feasible control action. While such robust approaches have been applied extensively to the case of discrete-time continuous systems, only few approaches have been proposed regarding hybrid systems, i.e. systems featuring both continuous and discrete components [5].

In this work, we consider the development of novel closed-loop robust explicit MPC strategies for hybrid discrete-time linear systems. It extends our results for continuous systems [6], which is based on the following components: (i) a robust counterpart formulation for the special case of box-constrained uncertainty, guaranteeing constraint satisfaction, (ii) a multi-parametric linear programming (mp-LP) problem for each stage (starting from the final stage in a dynamic programming setting), and (iii) the recursive solution of the mp-LP problem yielding the initial set of states for which robust constraint satisfaction can be guaranteed. For the case of hybrid systems, we show that (i) a similar type of robust counterpart formulation can be derived under mild assumptions, (ii) this robust counterpart results in a multi-parametric mixed-integer linear programming (mp-MILP) problem which can be formulated for each stage, and (iii) using state-of-the-art methods the mp-MILP problem can be solved recursively. This then enables the formulation of the robust explicit hybrid MPC problem as a multi-parametric mixed-integer quadratic programming problem, for which the authors recently proposed the first exact solution algorithm [7]. Using a series of example problems, the applicability and scalability of this novel approach will be highlighted.

References

[1] Mayne, D. Q.; Rawlings, J. B.; Rao, C. V.; Scokaert, P. O. M. (2000) Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789 â?? 814.

[2] Kerrigan, E. C.; Maciejowski, J. M. (2004) Feedback min-max model predictive control using a single linear program: robust stability and the explicit solution. International Journal of Robust and Nonlinear Control, 14(4), 395 â?? 413.

[3] Wan, Z.; Kothare, M. V. (2003) An efficient off-line formulation of robust model predictive control using linear matrix inequalities. Automatica, 39(5), 837 â?? 846.

[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 â?? 1606.

[5] Nascu, I.; Oberdieck, R.; Pistikopoulos, E. N. (2015) Offset-free Explicit Hybrid Model Predictive Control of Intravenous Anaesthesia. In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics (SMC), 2475 â?? 2480.

[6] Oberdieck, R.; Misener, R.; Pistikopoulos, E. N. (2016) Robust explicit/multi-parametric control. AIChE Journal, in revision.

[7] Oberdieck, R.; Pistikopoulos, E. N. (2015) Explicit hybrid model-predictive control: The exact solution. Automatica, 58, 152 â?? 159.