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Department of Mathematics

Chair of Applied Mathematics Prof. Dr. L. Grüne / Prof. Dr. A. Schiela

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DFG project "Performance analysis for distributed and multiobjective model predictive control — The role of Pareto fronts, multiobjective dissipativity and multiple equilibria"

start of the project: 2019, end of the project: 2023

contract number: GR 1569/13-2

funding institution: DFG (Research Grants)

PROJECT MEMBERS

Principal investigator

Prof. Dr. Lars Grüne

Project member

M.Sc. Lisa Krügel

External project members

Professor Dr.-Ing. Frank Allgöwer (University of Stuttgart)

Prof. Dr. Gabriele Eichfelder (Technische Universität Ilmenau)

Prof. Dr.-Ing. Matthias A. Müller (University of Hanover)

AIMS OF THE PROJECT

Model predictive control (MPC) is a control method which computes a feedback law by iteratively solving optimal control problems on finite time horizons. This proposal considers problem formulations in which these optimal control problem are not given in standard form but in which multiple objectives and/or multiple optimal equilibria occur. It is a continuation of a project in which basic results on closed-loop performance and stability of multiobjective and game theoretic MPC formulations have been obtained.
The results from the first period revealed that knowledge of the Pareto front of the multiobjective optimal control problems to be solved in each sampling interval is needed for further insight into the closed loop behavior. This concerns both the development of the finite horizon Pareto front along the closed loop and the relation between the finite horizon and infinite horizon Pareto fronts. This topic will be studied in close cooperation with Prof. Gabriele Eichfelder, TU Ilmenau.
Besides the Pareto fronts, the key properties for the analysis of economic MPC schemes are strict dissipativity and the turnpike property. First promising results on understanding these properties and their relation were obtained in the first funding period. Yet, it is still a long way towards a full understanding. This shall be reached in the second funding period. This insight will provide the theoretical basis for the following two goals.
The first goal is the construction of a Lyapunov function for the closed loop in the multiobjective setting. Since the usual candidate for this function, the (rotated) finite horizon optimal value function, is vector valued in the multiobjective case, generalized definitions of Lyapunov functions need to be employed to this aim.
The second goal is the understanding of economic MPC schemes in the presence of multiple optimal equilibria. These are likely to occur in multiobjective, but also in discounted scalar MPC schemes. So far, no economic MPC theory can deal with this situation and this gap shall be closed in this project. The key tool for achieving is the development of local and regional dissipativity properties and corresponding local and regional turnpike properties.
Like in the first funding period, the project shall be carried out in close cooperation with the group of Prof. Dr.-Ing. Frank Allgöwer, Universität Stuttgart, with the companion project "Distributed dissipativity and graph theoretic properties in distributed economic MPC". Both proposals build upon generalized notions of (strict) dissipativity that need to be developed. Thus, similar mathematical structures will be useful in the two proposals such that we can exploit significant synergy effects and continue the highly successful collaboration from the first funding period.

For more information visit the webpage of the project.


responsible for the content: Lars Grüne

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