# American Institute of Mathematical Sciences

August  2019, 24(8): 3905-3928. doi: 10.3934/dcdsb.2018336

## Multiobjective model predictive control for stabilizing cost criteria

 Chair of Applied Mathematics, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

Received  April 2018 Revised  September 2018 Published  January 2019

Fund Project: The authors are supported by DFG Grant Gr 1569/13-1.

In this paper we demonstrate how multiobjective optimal control problems can be solved by means of model predictive control. For our analysis we restrict ourselves to finite-dimensional control systems in discrete time. We show that convergence of the MPC closed-loop trajectory as well as upper bounds on the closed-loop performance for all objectives can be established if the ‘right’ Pareto-optimal control sequence is chosen in the iterations. It turns out that approximating the whole Pareto front is not necessary for that choice. Moreover, we provide statements on the relation of the MPC performance to the values of Pareto-optimal solutions on the infinite horizon, i.e. we investigate on the inifinite-horizon optimality of our MPC controller.

Citation: Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336
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##### References:
Schematic illustration of a Pareto front for two objectives.
Two bicriterion optimization problems with ${\mathbb{R}}^2_{\geq 0}$-compact set of admissible values. The red parts indicate the nodominated values.
Step (1) in Algorithm 2.
Accumulated performance of the six objectives (blue) compared to the value of the Pareto optimal control sequence ${\bf{u}}^{\star, N}_{x_0}$ from step (0), Algorithm 2 (red).
Trajectories of the six systems (phase plots).
Performance without the constraints in step (1), Algorithm 2.
Trajectories and accumulated performance without terminal constraints using Algorithm 3.
Trajectories and accumulated performance without terminal constraints using Algorithm 4.
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