Multiobjective model predictive control for stabilizing cost criteria

Lars Grüne; Marleen Stieler

doi:10.3934/dcdsb.2018336

Article Contents

2019, Volume 24, Issue 8: 3905-3928. Doi: 10.3934/dcdsb.2018336

This issue Previous Article Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation Next Article Some regularity results for a double time-delayed 2D-Navier-Stokes model

Multiobjective model predictive control for stabilizing cost criteria

Lars Grüne^1, and
Marleen Stieler^1,

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

Received: April 2018

Revised: September 2018

Early access: January 2019

Published: August 2019

The authors are supported by DFG Grant Gr 1569/13-1

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 93B52, 93C10, 93C55, 91A12; Secondary: 90C29.

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Figure 1. Schematic illustration of a Pareto front for two objectives.

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Figure 2. Two bicriterion optimization problems with $ {\mathbb{R}}^2_{\geq 0} $-compact set of admissible values. The red parts indicate the nodominated values.

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Figure 3. Step (1) in Algorithm 2.

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Figure 4. 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).

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Figure 5. Trajectories of the six systems (phase plots).

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Figure 6. Performance without the constraints in step (1), Algorithm 2.

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Figure 7. Trajectories and accumulated performance without terminal constraints using Algorithm 3.

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Figure 8. Trajectories and accumulated performance without terminal constraints using Algorithm 4.

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