随机切换系统的纳什均衡精确能控性及线性二次最优控制

发布时间：2018-09-19 16:53

【摘要】：在本论文中,我们研究领导者和多个追随者之间的非合作动态博弈。这是动态博弈一个新的研究方向,超越了传统控制理论和博弈理论的框架。假设在对称信息框架下,微分博弈问题的纳什均衡点存在,我们将领导者视为第三方或其他非营利组织,这样可以忽略领导者的收益功能。考虑领导者给定策略时,追随者们的非合作动态博弈。在此框架下,我们重点关注领导者的调控系统的能力,在某种意义上,它反映了领导者对非合作博弈系统的影响。首先,本文研究了在系统信息均衡的条件下追随者们的最大收益问题。这是一个随机最优控制问题。在实际中,优化控制问题越来越多的受人关注并且深入研究,比如金融市场、能源系统等。本文通过引入了正倒向随机微分方程(FBSDE)控制理论。考虑了对称信息下的线性二次非零和微分博弈问题,研究追随者们最大收益解的构造方法,并基于极大值原理给出了随机切换系统最优解的解析表达式。其次,本文讨论了系统状态和控制器都受到随机扰动的线性切换系统。在随机Nash均衡状态存在的条件下,对线性切换系统中的随机控制问题进行了研究。随机控制系统在证明过程中,引入黎卡提(Riccati)方程和正倒向随机微分方程(FBSDE)理论,证明了系统方程解的存在和唯一性。再次,本文研究了 Nash均衡存在条件下,领导者宏观调控的可行性问题。借鉴Peng提出的BSDE理论知识,综合运用FBSDE理论,对于含有切换参数的一类随机系统,给出了纳什均衡终端精确能控的充要条件。进一步,给出了线性随机切换系统的纳什均衡精确能控的充要条件。同时还给出了 Nash均衡精确可控性的代数判据。最后,为体现倒向随机控制系统的实际应用价值,本文给出了一个市场上最优投资组合调控的例子。领导者的决策是一个控制过程,跟随者的最优收益可以被调控,说明我们研究的问题是有实际意义的。同时,通过Matlab数值仿真验证了提出的控制器可以精确控制系统模型。
[Abstract]:In this thesis, we study the non-cooperative dynamic game between leaders and many followers. This is a new research direction of dynamic game, which surpasses the frame of traditional control theory and game theory. Assuming that the Nash equilibrium exists in the framework of symmetric information, we treat the leader as a third party or other non-profit organization, so that the leader's income function can be ignored. When considering the leader's given strategy, the followers of the non-cooperative dynamic game. In this framework, we focus on the ability of the leader's regulatory system, which in a sense reflects the leader's influence on the non-cooperative game system. Firstly, this paper studies the maximum profit of followers under the condition of system information equilibrium. This is a stochastic optimal control problem. In practice, optimization control problems have attracted more and more attention, such as financial markets, energy systems and so on. In this paper, the (FBSDE) control theory of forward backward stochastic differential equation is introduced. The problem of linear quadratic nonzero sum differential game with symmetric information is considered. The method of constructing the maximum return solution of followers is studied and the analytical expression of the optimal solution of stochastic switched system is given based on the maximum principle. Secondly, we discuss the linear switched systems where both the system state and the controller are stochastic perturbed. Under the condition of the existence of stochastic Nash equilibrium, the stochastic control problem in linear switched systems is studied. In the process of proving stochastic control system, the existence and uniqueness of the solution of the system equation are proved by introducing the Rikati (Riccati) equation and the (FBSDE) theory of forward backward stochastic differential equation. Thirdly, this paper studies the feasibility of macro-control under the condition of Nash equilibrium. Based on the BSDE theory proposed by Peng and FBSDE theory, a necessary and sufficient condition for the precise controllability of Nash equilibrium terminal is given for a class of stochastic systems with switching parameters. Furthermore, a necessary and sufficient condition for the exact controllability of Nash equilibrium for linear stochastic switched systems is given. The algebraic criterion for the exact controllability of Nash equalization is also given. Finally, in order to reflect the practical application value of the backward stochastic control system, this paper gives an example of optimal portfolio control in the market. The leader's decision is a controlling process, and the optimal return of the follower can be regulated, which shows that the problem we study is of practical significance. At the same time, the Matlab numerical simulation shows that the proposed controller can accurately control the system model.
【学位授予单位】：北京交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O232

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