(188m) A Multi-Parametric Bi-Level Optimization Strategy for Hierarchical Model Predictive Control

Hierarchical control structures consist of a hierarchy of control levels, where every level is controlling a subset of the overall control variables, by manipulating a subset of the overall manipulated variables [1, 2]. In the case of hierarchical model predictive control (MPC) structures, each control level involves an optimization problem, with the resulting formulation corresponding to a multi-level optimization problem. The solution of this type of problems is very challenging, even for the case of linear MPC models at all levels, typically requiring the use of global optimization techniques and intense computational power [3, 4, 5].

In this work, we propose the use of a novel algorithm [6, 7] capable of providing the exact, global and parametric solution of bi-level programming problems for the solution of linear or quadratic hierarchical control problems. The derivation of hierarchical explicit/multi-parametric MPC (mp-MPC) controllers through the proposed algorithm, allows the controller to only do simple function evaluations at every control step, instead of solving the full bi-level optimization problem. We are illustrating the proposed methodology through a simple example of a two-level hierarchical control of a continuous stirred tank reactor (CSTR) system, with an economic objective function in the first control level, and a set-point tracking objective function in the second control level [8, 9]. The main idea of our approach is to treat the lower control level (set-point tracking) as a multi-parametric programming problem in which the input flowrate (optimization variable of the upper level control problem) along with system disturbances and states are considered as parameters. The resulting parametric solutions are then substituted into the upper level economic control problem, which can be solved as a set of single-level parametric programming problems. The obtained hierarchical controller is then able to effectively reject disturbances and maintain the system at the given set-points (driven by the second control level) in a more economical way (driven by the first control level) than a classical MPC controller.

References:

[1] Mesarović, M., Macko, D, Takahara, Y. Theory of hierarchical, multilevel systems (1970) Mathematics in science and engineering, 68, New York, London: Academic P.

[2] Scattolini, R. Architectures for distributed and hierarchical Model Predictive Control - A review (2009) Journal of Process Control, 19 (5), pp. 723-731.

[3] Mitsos, A. Global solution of nonlinear mixed-integer bilevel programs (2010) Journal of Global Optimization, 47 (4), pp. 557-582.

[4] Saharidis, G.K., Ierapetritou, M.G. Resolution method for mixed integer bi-level linear problems based on decomposition technique (2009) Journal of Global Optimization, 44 (1), pp. 29-51.

[5] Gümüş, Z.H., Floudas, C.A. Global optimization of mixed-integer bilevel programming problems (2005) Computational Management Science, 2 (3), pp. 181-212.

[6] Faísca, N.P., Saraiva, P.M., Rustem, B., Pistikopoulos, E.N. A multi-parametric programming approach for multilevel hierarchical and decentralized optimisation problems (2009) Computational Management Science, 6 (4), pp. 377-397.

[7] Avraamidou, S., Diangelakis, N. A., Pistikopoulos, E. N. Mixed Integer Bilevel Optimization through Multi-parametric Programming (2017) Foundations of Computer Aided Process Operations / Chemical Process Control; In Press. (http://folk.ntnu.no/skoge/prost/proceedings/focapo-cpc-2017/FOCAPO-CPC%202017%20Contributed%20Papers/73_FOCAPO_Contributed.pdf)

[8] Christofides, P.D., El-Farra, N.H. Special Issue: Economic nonlinear model predictive control (2014) Journal of Process Control, 24 (8), p. 1155.

[9] Ellis, M., Durand, H., Christofides, P. A tutorial review of economic model predictive control methods (2014) Journal of Process Control, 24 (8), pp. 1156–1178.