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.

平均场博弈论兴起于本世纪头十年，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。这个想法可以通过鱼群的运动来最好地解释，当鱼群中的鱼数量无限增加时，一条鱼的运动对其他鱼群的影响变得可以忽略不计，每条鱼开始感知周围的“群体效应”。在这一点上，我们喜欢把普通的鱼想象成一个与成本函数相关的代理，它对自己的状态如何相对于其他代理状态组定位敏感，这里称之为“平均域”。由于智能体周围的群体行为具有惯性，在给定该行为的情况下，通用智能体必须解决一个最优控制问题，以确定其最优移动策略。为了一致性，通过聚合代理的最佳反应，人们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。它是在基于分散控制律的大规模博弈中构建近似纳什均衡的强大技术的基础。当处理所谓的线性二次博弈（LQ）情况时，结果特别吸引人，因为集合的行为就像一个适当的动态代理。我们提出了一个研究计划，通过一种规范的博弈方法，尽可能地扩展LQ博弈框架在群体动力学分散控制方面的应用潜力，在这种方法中，我们设计了带有特定期望总体目标的成本函数。我们建议根据经典控制的目标即参考信号跟踪和干扰抑制来测试这种设计方法的局限性。特别是，我们建议开发“内部模型原理”概念的等效概念，即系统只有在内部初始条件的作用下，其本质上足够复杂才能产生该信号时，才能遵循给定的参考信号。此外，我们希望研究反纳什设计方法的可能性，即当假定总体遵循目标轨迹时，开发成本系数必须满足的微分方程，并验证解的存在性。虽然经典的平均场博弈论假设了瞬时的所有对所有代理的影响，但我们希望将该理论扩展到网络中，在网络中，通信和影响仅在点对点之间传播，就像在鱼群中一样。这导致代理之间的相互影响延迟。因此，我们希望在经典的平均场博弈动力学模型上增加一个通信层，并研究这两层之间的相互作用。在智能电网、人群动态的引导和微型机器人群的分散控制中，应用被设想。