【作者】 尚志强；

【导师】 石鸿雁；

【作者基本信息】 沈阳工业大学 ， 应用数学， 2009， 硕士

【摘要】 微分对策是指在局中人之间进行对策活动时要用到微分方程来描述对策现象或规律的一种对策,是解决对抗与竞争问题的有力工具。近年来微分对策已广泛应用于经济、工程、生物等各个领域,而且这种应用还在不断向其它领域渗透,军事领域中的微分对策研究一直是微分对策理论发展的动力和热点,因此微分对策的研究具有非常重要的意义。虽然微分对策理论的研究及应用有了极大的发展,但在追逃微分对策模型的建立及求解、线性二次型微分对策的不确定性等方面的研究尚不充分,本文在这几方面做了一定的工作。本文对人工鱼群算法加以改进,建立一类空间追逃微分对策模型并利用改进后的人工鱼群算法进行求解,此外对线性二次型微分对策的鲁棒性展开深入研究,具体研究内容如下:1.深入研究了人工鱼群算法的原理、实施步骤以及特点和应用现状,利用逐步减小步长的方法对人工鱼群算法加以改进,通过对一些测试函数进行仿真实验,验证了该算法在寻优速度和计算精确度上均优于基本人工鱼群算法。2.针对一类三维空间中的追逃型微分对策模型,考虑动力学中的最小转弯半径因素,通过增加模型参数,建立了多参数比较符合实际问题的模型;将追逃微分对策模型的求解问题归结为约束函数优化问题,利用人工鱼群算法研究了空间追逃微分对策问题的数值解法,并应用改进后的人工鱼群算法对所建立的模型进行求解,避免了直接求解微分对策问题复杂的两点边值问题,仿真结果验证了该算法具有较强的鲁棒性,缩短了计算时间。3.针对一类带有不确定项的多人线性二次型微分对策模型,基于最优控制理论,将Hamilton-Jacobi方程转化为Riccati方程,通过求解Riccati方程设计出闭环系统的状态反馈控制器,给出了这类微分对策稳定的控制方案。更多还原

【Abstract】 Differential game refers to a countermeasure using differential equations to describe the phenomena or the law among the players when they begin the game. It can resolve the confrontation and the competition effectively. In recent years, differential game has been widely applied in the field of economic, engineering, biology and others. And such kind of application is penetrating to other industries. The research of differential game in the military field has been the hot topic and regarded as the impetus of the development of game theory. The model of military confrontation is mostly non-linear differential game model, such as: pursuit-evasion, interception, cooperation, etc. Therefore, the studying of differential game is very practical. Although the research and application of differential game theory has developed soundly, the model building and it’s solving of pursuit-evasion differential game and the uncertainty of linear quadratic differential game hasn’t been fully studied. Certain work will be done in this paper.Firstly, the artificial fish-swarm algorithm is improved in this paper and a model of pursuit-evasion differential game in space is established. Moreover, the paper applies the improved artificial fish-swarm algorithm to solving. Then linear quadratic differential games and it’s robustness is studied deeply. Specific studies are as follows:1. It studies the implementation steps, the characteristics and application status of the artificial fish-swarm algorithm theory deeply. By way of reducing the step size gradually, the artificial fish-swarm algorithm is improved. The simulating experiment of test function verifies that the improved algorithm is superior to the basic artificial fish-swarm algorithm in optimization speed and accuracy.2. Taking the factors of the smallest turning radius with dynamics into consideration and by way of increasing the model parameters, it establishes a multi-parameters mode that is more in line with practical problems for a class of three-dimensional space pursuit-evasion differential game model. The solving of the model of the pursuit-evasion differential game is taken as the optimization problems of constrained function. The numerical solution of pursuit-evasion differential game in space is studied by artificial fish-swarm algorithm. The model established is solved by improved artificial fish-swarm algorithm too. This method avoids solving the complex two point boundary value problem directly. The simulation result shows that this algorithm has stronger robustness and reduces computing time.3. Based on the theory of optimal control, the Hamilton-Jacobi equation is transformed into the Riccati equation for a class of multi-players linear quadratic differential game with uncertainties. After solving the Riccati equation, the controller of state feedback of the closed loop system is designed and a stable control scheme with this type of differential game is carried out.更多还原