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

模型预测控制是一种在有限时间范围内通过迭代求解最优控制问题来计算反馈律的控制方法。本文考虑了这些最优控制问题不以标准形式给出，但考虑了博弈论或其他广义优化设置的问题表述。考虑这种广义设置的动机主要是由于对以分布式方式控制大型系统网络的兴趣日益增长，通常要考虑多个优化标准。一般来说，在这种情况下，人们不能假设可以找到有限视界最优控制问题的中心最优解；相反，纳什均衡和帕累托最优等概念进入了人们的视野。智能电网的应用构成了此类问题的一个特殊类别，虽然本提案不关注特定的应用领域，但智能电网控制问题将作为评估我们方法的基准问题。在这种情况下，本项目要解决的中心主题可以概括为一个问题：给定一个分布式和/或多目标MPC方案，其中每个采样时刻的有限视界问题的解满足一些最优性，我们是否可以得出结论，由MPC方案生成的闭环解也具有类似的最优性，至少以一种近似的方式？因此，例如，如果我们可以确保在每个步骤中迭代算法（涉及多个子系统之间的协商）可以确保我们达到纳什均衡，那么在哪些条件下以及哪种最优性标准下，我们可以确保MPC闭环也是（接近）纳什均衡？在这个项目中，我们将研究有和没有稳定终端约束的MPC方案以及经济MPC。在我们的分析中，最优有限水平轨迹的动力学特性与收费公路的特性一样可望发挥重要作用。由于这些可以从适当的耗散性和可控性中得出结论，因此必须研究将这些概念扩展到分布式和博弈论背景。此外，将开发数值工具，以验证我们的理论结果，但也作为建立理论直觉和确定所考虑的系统的合理假设的工具。本项目将与dr . ing教授的姊妹项目“分布式经济模型预测控制中的公平与效率”密切合作。弗兰克Allgöwer， Universität斯图加特。两项建议都涉及类似的问题表述，本项目侧重于概念问题及其数值验证，Allgöwer教授的建议侧重于建设性和算法方法。因此，这些项目理想地相互补充。

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.