微分对策的数值解法及鲁棒性问题研究
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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。
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.
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。
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.
引文
[1] Issacas R. Differential Games. New York: John Wiley and Sons, 1965.
[2] Friedman A. Differential Games. New York: John Wiley Interscience, 1971.
[3]谢政.对策论.长沙:国防科技大学出版社,2004.
[4]郑立辉,冯珊,张兢田等.微分对策在期权定价中的应用:数值分析.华中理工大学学报,1998,26(11):47-49.
[5] Eitan Alman, Odile Pourtallier. Advances in Dynamic Games and Applications. 8th International Symposium of Dynamic Games and Applications, 2001.
[6] Jerzy A. Filar, Koichi Mizukami, Vladimir Gaitsory. Advances in Dynamic Games and Applications. 7th International Symposium of Dynamic Games and Applications, 2000.
[7] GJ.Olsder. New Trends in Dynamic Games and Applications. 6th International Symposium of Dynamic Games and Applications, 1995.
[8] M.Bardi, T.Parthasarathy, T.E.S.Raghavan. Stochastic and Differential Games-Theory and Numerical Methods. Annals of The International Society of Dynamic Games-Volume4 Applications, 1998.
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[11]刘德铭,黄振高.对策论及其应用.长沙:国防科技技术大学出版社,1995.
[12]沙基昌.数理战术初探.长沙:国防科技大学讲义,1991.
[13]董志荣.微分对策的基本原理及其在军事对抗中的应用.北京:国防工业出版社,1983.
[14]年晓红,黄琳.微分对策理论及其应用研究的新进展.控制与决策,2004,19(2):128-133.
[15] Yong J. On the Isaacs equation of differential games of fixed duration. Optimal Theory, 1986, 50(2): 359-364.
[16] BerkovitzLD. Differential games of generalized pur-suit and evasion. SIAMJ Control. l986, 24(3): 361-372.
[17]刘三阳,张卓奎.微分对策研究进展.工程数学学报,2000年,17(增刊),41-82.
[18]罗飞,李登峰,陈庆华.微分对策界栅理论在舰艇作战能力评估中的应用.火力与指挥控制,2003,28(5):104-105.
[19] Ngo V L, Koji S. Some results on the Markov equilibrium of a class of homogenous Differential games. J Economic Behavior Organization. 1998, 33(3-4): 557-566.
[20] Basar T, Bernard P. H∞-Optimal Control and Related Minimax Design Problems. Boston, Massa-chusetts: Birkhauser, 1991.
[21]方绍琨,李登峰.微分对策及其在军事领域的研究进展.指挥控制与仿真.2008,30(1):114-117.
[22]左斌,杨长波,李静.基于微分对策的导弹攻击策略.海军航空工程学院学报,2005,20(5):524-526.
[23]李登峰,谭安胜,罗飞.兵力增援微分对策优化模型及解法.运筹管理,2002,11(4):16-20.
[24]李登峰,陈庆华.兵力增援微分对策优化模型及解法.火力与指挥控制,2004,29(1):23-25.
[25]汤善同.微分对策制导规律与改进的比例导引制导规律性能比较.宇航学报,2002,23(6):38-42,61.
[26]陈磊,王海丽,任萱.实时微分控制的算法研究.航天控制,2001,(4):48-52.
[27] Ian M. Mitchell, Alexandre M. Bayen, Claire J. Tomlin. A Time-Dependent Hamilton–Jacobi Formulation of Reachable Sets for Continuous Dynamic Games. Transactions on Automatic Control, 2005, 50(7): 947-957.
[28]周锐,李惠峰.神经网络理论在微分对策中的应用.北京航空航天大学学报,2000,26(6):666-668.
[29]周锐.基于神经网络的微分对策控制器设计.控制与决策,2003,18(1):123-125.
[30]李晓磊,邵之江,钱积新.一种基于动物自治体的寻优模式:鱼群算法.系统工程理论与实践,2002,2(211):32-38.
[31]王正初.基于人工鱼群算法的复杂系统可靠性优化.台州学院学报,2008,30(3):28-31.
[32]王锡淮,郑晓鸣,肖健梅.求解约束优化问题的人工鱼群算法.计算机工程与应用,2007,43(3):40-42.
[33] Xin Yao, Yong Liu, Guangming Lin. Evolutionary programming made faster. IEEE Trans Evol Compute, 1999; 3(2): 82-102.
[34]黄华娟,周永权.改进型人工鱼群算法及复杂函数全局优化方法.广西师范大学学报,2008,26(1):195-197.
[35]周卿吉,徐诚,周文松等.微分对策制导律的研究现状及展望.系统工程与电子技术,1997,11(2):40-45.
[36]罗飞,李登峰,陈庆华.微分对策界栅理论在舰艇作战能力评估中的应用.火力与指挥控制,2003,28(5):104-105,112.
[37]李登峰.舰艇抗击定量微分对策模型.海军大连舰艇学院学报,1999,22(1):55-56.
[38]徐诚,沈如松,周文松等.基于模糊理论的微分对策制导律.飞行力学,1997,15(1):86-90.
[39]李建勋,佟明安,金德琨.协商微分对策理论及其在多机空战分析中的应用.系统工程理论与实践,1997,10(6):68-72.
[40]盛蔚,李建勋,佟明安.微分对策论在多机协同攻击中的应用研究.系统工程与电子技术,1998,16(3):7-11.
[41]周锐.基于神经网络的微分对策控制器设计.控制与决策,2003,18(1):123-125.
[42]陈迎春,齐欢.基于协同进化的平面追逃微分对策研究.控制与决策,2009,24(3):383-387.
[43]秦艳琳,吴晓平,杨广.基于梯度迭代法的一类追逃对抗模型研究.海军工程大学学报.2005,17(4):108-112.
[44] Shan X J, Jiang M Y, Li J P. The Routing Optimization Based on Improved Artificial Fish Swarm Algorithm. Proceedings of the 6th World Congress on Intelligent Control and Automation, 2006: 3658-3662.
[45]李国勇.智能控制及其MATLAB实现.北京:电子工业出版社,2006.
[46] Abou-Kandil H, Bertrand P. Analytic for a Class of Linear-Quadratic Open-Loop Nash Games. Int. J Control, 1986, 43(3): 997-1002.
[47] Manuel Jimenez, Alex Poznyak. Robust and Adaptive Strategies with Pre-Identification via Sliding Mode Technique in LQ Differential Games. Proceedings of the 2006 American Control Conference, 2006: 4771-4776.
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[49] Tal Shima, Oded M. Golan. Linear Quadratic Differential Games Guidance Law for Dual Controlled Missiles. 2006: 834-842.
[50] Michel Delfour, Olivier Dello Sbarba. Linear-Quadratic Differential Games: Open and Closed Loop Strategies with or without Singularities. Decision and Control. 2007, 6(4): 802-806.
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[52]张成科,王行愚.线性二次微分对策鞍点策略的小波分析法.控制与决策,2001,16(4):443-451.
[53]王德进.优化控制理论.哈尔滨:哈尔滨工业大学出版社,2001.
[54] Xie L. Output feedback H∞control of systems with parameter uncertainty. Int J Control, 1996, 63(4):741-750.
[2] Friedman A. Differential Games. New York: John Wiley Interscience, 1971.
[3]谢政.对策论.长沙:国防科技大学出版社,2004.
[4]郑立辉,冯珊,张兢田等.微分对策在期权定价中的应用:数值分析.华中理工大学学报,1998,26(11):47-49.
[5] Eitan Alman, Odile Pourtallier. Advances in Dynamic Games and Applications. 8th International Symposium of Dynamic Games and Applications, 2001.
[6] Jerzy A. Filar, Koichi Mizukami, Vladimir Gaitsory. Advances in Dynamic Games and Applications. 7th International Symposium of Dynamic Games and Applications, 2000.
[7] GJ.Olsder. New Trends in Dynamic Games and Applications. 6th International Symposium of Dynamic Games and Applications, 1995.
[8] M.Bardi, T.Parthasarathy, T.E.S.Raghavan. Stochastic and Differential Games-Theory and Numerical Methods. Annals of The International Society of Dynamic Games-Volume4 Applications, 1998.
[9]张嗣瀛.微分对策.北京:科技出版社,1987.
[10]李登峰.微分对策及其应用.北京:国防工业出版社,2000.
[11]刘德铭,黄振高.对策论及其应用.长沙:国防科技技术大学出版社,1995.
[12]沙基昌.数理战术初探.长沙:国防科技大学讲义,1991.
[13]董志荣.微分对策的基本原理及其在军事对抗中的应用.北京:国防工业出版社,1983.
[14]年晓红,黄琳.微分对策理论及其应用研究的新进展.控制与决策,2004,19(2):128-133.
[15] Yong J. On the Isaacs equation of differential games of fixed duration. Optimal Theory, 1986, 50(2): 359-364.
[16] BerkovitzLD. Differential games of generalized pur-suit and evasion. SIAMJ Control. l986, 24(3): 361-372.
[17]刘三阳,张卓奎.微分对策研究进展.工程数学学报,2000年,17(增刊),41-82.
[18]罗飞,李登峰,陈庆华.微分对策界栅理论在舰艇作战能力评估中的应用.火力与指挥控制,2003,28(5):104-105.
[19] Ngo V L, Koji S. Some results on the Markov equilibrium of a class of homogenous Differential games. J Economic Behavior Organization. 1998, 33(3-4): 557-566.
[20] Basar T, Bernard P. H∞-Optimal Control and Related Minimax Design Problems. Boston, Massa-chusetts: Birkhauser, 1991.
[21]方绍琨,李登峰.微分对策及其在军事领域的研究进展.指挥控制与仿真.2008,30(1):114-117.
[22]左斌,杨长波,李静.基于微分对策的导弹攻击策略.海军航空工程学院学报,2005,20(5):524-526.
[23]李登峰,谭安胜,罗飞.兵力增援微分对策优化模型及解法.运筹管理,2002,11(4):16-20.
[24]李登峰,陈庆华.兵力增援微分对策优化模型及解法.火力与指挥控制,2004,29(1):23-25.
[25]汤善同.微分对策制导规律与改进的比例导引制导规律性能比较.宇航学报,2002,23(6):38-42,61.
[26]陈磊,王海丽,任萱.实时微分控制的算法研究.航天控制,2001,(4):48-52.
[27] Ian M. Mitchell, Alexandre M. Bayen, Claire J. Tomlin. A Time-Dependent Hamilton–Jacobi Formulation of Reachable Sets for Continuous Dynamic Games. Transactions on Automatic Control, 2005, 50(7): 947-957.
[28]周锐,李惠峰.神经网络理论在微分对策中的应用.北京航空航天大学学报,2000,26(6):666-668.
[29]周锐.基于神经网络的微分对策控制器设计.控制与决策,2003,18(1):123-125.
[30]李晓磊,邵之江,钱积新.一种基于动物自治体的寻优模式:鱼群算法.系统工程理论与实践,2002,2(211):32-38.
[31]王正初.基于人工鱼群算法的复杂系统可靠性优化.台州学院学报,2008,30(3):28-31.
[32]王锡淮,郑晓鸣,肖健梅.求解约束优化问题的人工鱼群算法.计算机工程与应用,2007,43(3):40-42.
[33] Xin Yao, Yong Liu, Guangming Lin. Evolutionary programming made faster. IEEE Trans Evol Compute, 1999; 3(2): 82-102.
[34]黄华娟,周永权.改进型人工鱼群算法及复杂函数全局优化方法.广西师范大学学报,2008,26(1):195-197.
[35]周卿吉,徐诚,周文松等.微分对策制导律的研究现状及展望.系统工程与电子技术,1997,11(2):40-45.
[36]罗飞,李登峰,陈庆华.微分对策界栅理论在舰艇作战能力评估中的应用.火力与指挥控制,2003,28(5):104-105,112.
[37]李登峰.舰艇抗击定量微分对策模型.海军大连舰艇学院学报,1999,22(1):55-56.
[38]徐诚,沈如松,周文松等.基于模糊理论的微分对策制导律.飞行力学,1997,15(1):86-90.
[39]李建勋,佟明安,金德琨.协商微分对策理论及其在多机空战分析中的应用.系统工程理论与实践,1997,10(6):68-72.
[40]盛蔚,李建勋,佟明安.微分对策论在多机协同攻击中的应用研究.系统工程与电子技术,1998,16(3):7-11.
[41]周锐.基于神经网络的微分对策控制器设计.控制与决策,2003,18(1):123-125.
[42]陈迎春,齐欢.基于协同进化的平面追逃微分对策研究.控制与决策,2009,24(3):383-387.
[43]秦艳琳,吴晓平,杨广.基于梯度迭代法的一类追逃对抗模型研究.海军工程大学学报.2005,17(4):108-112.
[44] Shan X J, Jiang M Y, Li J P. The Routing Optimization Based on Improved Artificial Fish Swarm Algorithm. Proceedings of the 6th World Congress on Intelligent Control and Automation, 2006: 3658-3662.
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