Performance Analysis for Distributed and Multiobjective Model Predictive Control — The role of Pareto fronts, multiobjective dissipativity and multiple equilibria

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Model predictive control 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 game theoretic or other generalized optimization settings are considered. The motivation for considering such generalized settings is primarily triggered by the growing interest in controlling large networks of systems in a distributed way, often by taking into account more than one optimization criterion. In such settings in general one cannot assume that a central optimal solution to the finite horizon optimal control problem can be found; instead, concepts like, e.g., Nash equilibria and Pareto optimality enter the picture. Smart grid applications form a particular class of such problems and while this proposal is not focused on a particular application area, smart grid control problems will serve as benchmark problems for evaluating our methods. Given such a situation, the central theme to be addressed in this project can be summarized in a single question: Given a distributed and/or multiobjective MPC scheme in which the solution to the finite horizon problem in each sampling instant satisfies some optimality property, can we conclude that the closed loop solution generated by the MPC scheme also enjoys a similar optimality property, at least in an approximate way? So, if for instance we can ensure that in each step an iterative algorithm (involving negotiations between a number of subsystems) can ensure that we reach a Nash equilibrium, under which conditions and for which optimality criteria can we ensure that the MPC closed loop is also (close to) a Nash equilibrium?In this project we will investigate both MPC schemes with and without stabilizing terminal constraints as well as economic MPC. In our analysis, dynamical properties of optimal finite horizon trajectories like turnpike properties are expected to play an important role. As these can be concluded from suitable dissipativity and controllability properties, the extension of such concepts to distributed and game theoretic contexts will have to be investigated. Moreover, numerical tools will be developed which allow to verify our theoretical results but also serve as a tool to build theoretical intuition and to identify reasonable assumptions on the systems under consideration.The project shall be carried out in close cooperation with the companion project "Fairness and Efficiency in Distributed Economic Model Predictive Control" proposed by Prof. Dr.-Ing. Frank Allgöwer, Universität Stuttgart. Both proposals address similar problem formulations, with this project concentrating on conceptual questions and their numerical verification and Prof. Allgöwer's proposal focusing on a constructive and algorithmic approach. As such, the projects ideally complement each other.