Control of collective dynamics via mean field and and inverse mean field game theory

通过平均场和逆平均场博弈论控制集体动力学

• 批准号: RGPIN-2022-05402
• 负责人: Malhamé, Roland
• 金额: $ 3.35万
• 依托单位: École Polytechnique de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=754020
• 关键词: Control; collective; dynamics; via; mean

Mean field game theory has emerged in the first decade of the century and has quickly become one of the most effective ways of analyzing and controlling the aggregate behavior of large multi-agent systems. The idea can best be explained through the motion of fish schools whereby as the number of fish in a fish school increases indefinitely, the impact of one fish motion on the rest of the school becomes negligible, and each fish starts perceiving a "crowd effect" around it. At that point, we like to think of a generic fish as an agent associated with a cost function sensitive to how its own state is positioned with respect to the group of other agent states, hereby called the "mean field". Since the group behavior around the agent has gained inertia, given that behavior, the generic agent has to solve an optimal control problem to decide for its optimal moving strategy. By aggregating the optimal responses of agents, for consistency, one should be able to recover the group behavior assumed in the first place. This  is mathematically characterized as a fixed-point calculation. It is the basis of a powerful technique for constructing approximate Nash equilibria in large-scale games based on decentralized control laws.   The result is particularly attractive when dealing with the so-called linear quadratic (LQ) games situation because the aggregate behaves like a single appropriate dynamic agent. We propose a research program to extend as far as possible the application potential of the LQ games framework for the decentralized control of group dynamics through a prescriptive game approach where we design the cost functions with a specific desired aggregate goal in mind. We propose to test the limits of this design approach in terms of the objectives of classical control namely reference signal following  and disturbance rejection. In particular, we propose to develop the equivalent of a notion of "internal model principle", whereby  a system can follow a given reference signal only if it is intrinsically complex enough to generate that signal under the actions of internal initial conditions. In addition, we would like to investigate the possibilities of inverse-Nash design approaches, whereby one develops the differential equations that the cost coefficients must  satisfy when the aggregate is assumed to follow a target trajectory, and verify existence of solutions.    While classical mean field game theory has assumed instantaneous all to all agent influences, we would like to extend the theory over networks where communication and thus influences  propagate only, as in fish schools, from peer to peer. This results in delayed mutual influences of agents. Thus we would like to enhance the classical mean field game dynamic model with a communication layer, and study the mutual interactions of these two layers.  Applications are envisioned in smart grids, the channeling of crowd dynamics, and the decentralized control of micro-robotic swarms.

平均场博弈论出现于本世纪第一个十年，并迅速成为分析和控制大型多智能体系统总体行为的最有效方法之一，该想法可以通过鱼群的运动得到最好的解释。随着鱼群中鱼的数量无限增加，一条鱼的运动对鱼群中其他鱼的影响可以忽略不计，并且每条鱼开始感知到它周围的“群体效应”。通用鱼作为与相关的代理成本函数对其自身状态相对于其他智能体状态组的定位敏感，因此称为“平均场”，因为智能体周围的群体行为已经获得了惯性，给定该行为，通用智能体必须解决一个问题。最优控制问题，通过聚合代理的最优响应，为了一致性，应该能够恢复最初假设的群体行为，这在数学上被描述为定点计算。构建近似的强大技术的基础基于分散控制律的大规模博弈中的纳什均衡在处理所谓的线性二次（LQ）博弈情况时特别有吸引力，因为总体行为就像单个适当的动态代理。通过规定性博弈方法尽可能扩展 LQ 博弈框架在群体动态的去中心化控制方面的应用潜力，在该方法中我们设计了具有特定期望总体目标的成本函数，我们建议测试这种设计方法的局限性。按照经典控制的目标，即参考信号跟踪和干扰抑制，特别是，我们建议开发“内模型原理”的等效概念，因此，只有当系统本质上足够复杂以生成给定的参考信号时，系统才能跟踪给定的参考信号。此外，我们还想研究逆纳什设计方法的可能性，即当假设总体遵循目标轨迹时，成本系数必须满足的微分方程，并验证解的存在性。虽然经典平均场博弈论假设所有代理的影响都是瞬时的，但我们希望将该理论扩展到网络上，在网络中，通信和影响仅在鱼群中从对等体传播，这导致代理的相互影响延迟。因此，我们希望通过通信层增强经典的平均场博弈动态模型，并研究这两个层的相互交互在智能电网、人群动态的引导和分散控制中的应用。微型机器人群。