(340i) Multi-Objective and Hierarchical Explicit Model Predictive Control

Optimal control problems can involve multiple, often conflicting objectives, including economic, tracking, safety, or environmental criteria. As these objectives result in different input trajectories and consequently to different operational behaviors in closed-loop, different control strategies have been proposed to incorporate these multiple objectives. Various approaches have been proposed in the open literature for the development of multi-objective model predictive controllers (MOMPC) including the ε-constraint approach [1] and the weighted-sum method [2]. Additionally, in [3] the full Pareto front with linear objectives and up to one quadratic objective was derived using multiparametric programming and the weighted-sum method. Apart from that, multiple objectives can also be ranked in a hierarchy, resulting in hierarchical model predictive control (HMPC) structures [4, 5], with the resulting formulation typically corresponding to a multi-level programming problem. Even though connections between multi-objective and hierarchical optimization have been introduced [6,7], this has not been the case for model predictive control frameworks.

This work proposes solution methods for both MOMPC and HMPC using multiparametric programming. Assuming that there exist two or (possibly) more conflicting control objectives described by linear or multiple convex quadratic functions, we develop multiparametric-based approaches for the derivation of the explicit Pareto front of MOMPC, and the explicit solution of HMPC. The MOMPC problem is reformulated into a multiparametric programming problem, which can then be exactly solved using state-of-the-art algorithms [8,9], while the HMPC problem is reformulated into a multiparametric multi-level programming problem, which can be exactly solved using the algorithm proposed in [10]. A case study on a chemostat with two competing objectives, an economic and a tracking objective, is used to illustrate the developed control strategies and compare them against each other, but also against pure tracking and pure economic model predictive controllers. The results of this study clearly indicate the effect of the different control strategies on the operation of the chemostat.

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[4] Avraamidou, S.; Pistikopoulos, E. N. A multi-parametric bi-level optimization strategy for hierarchical model predictive control. 27th European Symposium on Computer-Aided Process Engineering (ESCAPE-27); Elsevier, 2017; pp 1591-1596.

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[8] Oberdieck, R.; Pistikopoulos, E. N. Multi-objective optimization with convex quadratic cost functions: A multi-parametric programming approach. Computers & Chemical Engineering 2016, 85, 36-39.

[9] Diangelakis, N. A.; Pappas, I. S.; Pistikopoulos, E. N. On multiparametric/explicit NMPC for Quadratically Constrained Problems. 6th IFAC Conference on Nonlinear Model Predictive Control; Elsevier, 2018; pp 400-405.

[10] Faísca, N. P., Saraiva, P. M., Rustem, B., & Pistikopoulos, E. N. (2009). A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems. Computational management science, 6(4), 377-397.