Mean Field, Distributed and Hybrid Control Systems

拟议的研究计划的总体目标是推进控制系统的基本分析和设计（i）分布式大规模系统（即没有集中控制或信息的高状态基数的系统）和（ii）具有连续和离散状态和动态的系统。这样的系统在当代网络化信息和基于计算的环境中是普遍存在的，因此它们的增强的控制和优化将带来巨大的好处。 平均场博弈（Mean Field Game，MFG）理论是一种新的分布式大规模系统分析与设计方法，它起源于NSERC，由提出者及其学生Minyi Huang和共同导师罗兰·马尔哈姆（Roland Malhame）在2003 - 2007年期间，以及J. M. Lasry和P. - L.狮子在2006 - 2007年。从那时起，该领域经历了蓬勃发展，由于这些和其他研究人员的持续贡献世界各地;此外，潜在的应用MFG方法和新的，具有挑战性的问题不断被发现在广泛的领域。 MFG理论建立了仅由随机动力系统代理在涉及大量代理的游戏中的局部反馈产生的近似纳什均衡的存在性。有限人口情况下的平衡和反馈律是通过基本无限人口MFG Hamilton-Jacobi-Bellman和（McKean-Vlasov）-Fokker-Planck-Kolmogorov偏微分方程的更简单的解决方案获得的，这些偏微分方程由通用代理的状态分布联系在一起，也称为系统的平均场。当代理人的数量趋于无穷大时，近似误差趋于零。 该提案映射了一个为期五年的研究计划，以下主题：（一）估计MF系统：MF系统理论的一个中心问题是，部分观察到的系统与主要代理（即非渐近消失人口规模）有随机平均场，必须递归估计沿着与主要代理的部分观察状态。状态估计理论的非线性部分可观察的MFG系统的主要和次要的代理，因此随机平均场，将具有重要意义，并将利用在（B）空间动态MF系统的发展，（c）联盟形成MF系统，（d）局部无限和大规模拓扑MF系统，（e）MF市场和拍卖模型。在大范围的系统和控制工程领域，混合控制系统是先进控制方法的同义词。最近的进展产生了复杂的混合最小值原理（HMP）理论和计算方法，以及混合动态规划（HDP）的并行理论。在提议者的工作的基础上，计划产生（i）统一的HMP-HDP理论，（ii）用于HDP问题的基本类的理论和相关算法，其中系统受到状态切换约束，以及（iii）集成的HDP -模型预测控制方法。整个计划的成功将产生8个博士学位和4个工程硕士学位，加上4个PDF，并将有助于新技术的出现，使分布式自适应代理网络能够控制随机环境中演变的大规模系统;主要例子（已在加拿大启动）包括集成智能电表、电动汽车充电系统和本地化发电，其中车辆和发电机本身使用混合系统控制算法进行优化。

The overall objective of the proposed research program is to advance the fundamental analysis and design of control systems for (i) distributed large scale systems (i.e. systems without centralized control or information which are of high state cardinality) and (ii) systems with both continuous and discrete states and dynamics. Such systems are ubiquitous in the contemporary networked information and computation based environment and so their enhanced control and optimization would bring great benefits. Mean Field Game (MFG) theory constitutes a fundamental new methodology for the analysis and design of distributed large scale systems which originated in NSERC supported work by the proposer together with his student Minyi Huang and the co-supervisor Roland Malhame' during 2003 – 2007, and, independently, by J.-M. Lasry and P.- L. Lions during 2006 – 2007. Since then the field has undergone vigorous development due to the continuing contributions of these and other researchers world wide; moreover, potential applications of MFG methodology and new, challenging problems are continually being discovered in a vast range of domains. MFG theory establishes the existence of approximate Nash equilibria generated solely by local feedback for stochastic dynamical system agents in games involving large numbers of agents. The equilibria and the feedback laws applied in the finite population case are obtained via the far simpler solutions to the fundamental infinite population MFG Hamilton-Jacobi-Bellman and (McKean-Vlasov)-Fokker-Planck-Kolmogorov PDEs which are linked by the state distribution of a generic agent, otherwise known as the system's mean field. As the number of agents goes to infinity the approximation error goes to zero. This proposal maps a five year research program on the following themes: (a) Estimation in MF Systems: a central problem for MF system theory is that partially observed systems with major agents (i.e. non-asymptotically vanishing with population size) have stochastic mean fields which must be recursively estimated along with the major agents' partially observed states. A state estimation theory for non-linear partially observed MFG systems with major and minor agents, and hence stochastic mean fields, will be of great significance and will be exploited in the development of (b) Spatially Dynamic MF Systems, (c) Coalition Formation in MF Systems, (d) Locally Infinite and Large Scale Topology MF Systems, and (e) MF Market and Auction Models. In large areas of systems and control engineering hybrid control systems are synonymous with advanced control methodology. Recent advances have generated both a sophisticated Hybrid Minimum Principle (HMP) theory and computational methods, and a parallel theory of Hybrid Dynamic Programming (HDP). Building on the proposer's work, it is planned to generate (i) a unified HMP - HDP theory, (ii) a theory and associated algorithms for the fundamental class of HDP problems where the systems are subject to state switching constraints, and (iii) an integrated HDP - Model Predictive Control methodology. Success in the overall program will yield 8 PhDs and 4 MEng degrees, plus 4 PDFs, and will contribute to the emergence of new techniques enabling networks of distributed adaptive agents to control large scale systems evolving in random environments; prime examples (already initiated in Canada) include the integration of smart meters, electric vehicle charging systems and localized generation, where the vehicles and generators are themselves optimized using hybrid system control algorithms.