基于人工鱼群算法的Lanchester方程微分对策问题的研究
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摘要
微分对策理论起源于军事并广泛应用于经济、工程、生物等各个领域。随着高新技术在军事中的广泛应用,使得微分对策问题的研究显得尤为重要。由于在实际的军事对抗中战机转瞬即逝,因此,寻求一种快速有效的解法,并且将智能方法与微分对策进行结合是微分对策理论未来发展的方向。本文针对微分对策的计算问题展开理论研究;针对微分对策在作战指挥中的某些问题展开了应用研究。主要内容包括:
1、研究了微分对策数值解的求解方法。智能优化的发展为微分对策理论研究提供了一种有效的方法和技术途径。本文深入研究了混沌优化、人工鱼群算法的原理、实施步骤以及特点,针对人工鱼群算法运行速度慢和优化精度较差的缺点及混沌变量具有随机性、遍历性及规律性的特点,将混沌优化与人工鱼群相结合提出高效的混沌人工鱼群混合算法,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。同时在一定的假设条件下证明出混沌人工鱼群算法依概率收敛于全局最优值。
2、研究了微分对策在作战指挥中的应用。基于Lanchester方程,着眼于最优火力分配问题研究。由于战争的多样性和复杂性,本文主要探讨在直瞄武器交战下单兵种对多兵种作战这类比较典型的战争模型,运用微分对策最大值最小值原理进行分析,得出最优分配策略,即多兵种一方应全力攻击单兵种一方;单兵种一方对多兵种一方的攻击顺序取决于双方的交战强度,应按交战强度从大到小顺序攻击,从而给出了军事对抗中兵力不占优势一方如何获取胜利的一种方法。
3、结合特拉法尔海战历史战例,在军事想定的基础上建立适当的数学模型,利用本文提出的混沌人工鱼群算法,对模型进行求解,同时对得到的结果做出分析和研究,得出作战双方火力分配的最优决策。
Differential games theory has been applied to economics, engineering, biology and allother fields. Especially in the world today, high technology in military affairs iswidespread usage, making the research of differential games problem more important. Thispaper directs at theoretical researches on the numerical calculation methods of differentialgame; also made researches on some problem of its application in attack, command andcontrol. Main contents include:
1、Numerical calculation methods to differential game are researched. Development ofartificial intelligence provides an effective method and technological approach fordifferential game theoretical research. Principle, implementation steps and characteristicsof the chaos optimization and artificial fish-swarm algorithm are researched deeply. Due tolow speed and inadequate optimal efficiency of AFSA and chaotic variables with therandomness, regularity, an efficiency optimal hybrid algorithm (SCAFSA) whichcombines chaos optimization and artificial fish-swarm is proposed. The simulatingexperiment of test function verifies that this hybrid algorithm is stronger than AFSA inglobal optimization and the search efficiency is higher. Theoretically, this paper provesthat under certain circumstances the hybrid algorithm is convergent in probability to globaloptimum.
2、Applications of differential game on operational commanding are researched.Based on Lanchester equation, the optimal fire allocation problem is focused on. Becausethe diversity and complexity of war, the article mainly discusses at straight war weaponsthe typical differential game model of one to many. Using the maximum and minimumprinciple in differential game, optimum allocation strategy is obtained. So the side withmore arms should attack the only arm with full power. The attack policy of single arm sidedepends on the descending order of battle intensity. And then the question of bad situationarm winning in military confront is solved.
3、In consideration of Battle of Trafalgar historical case, on the basis of the MilitaryScenario base appropriate mathematical model is set up, using SCAFSA which is proposedin the article. Then the results are analysed and researched properly, getting both sides inmilitary affairs optimizing decision of power distribution.
1、研究了微分对策数值解的求解方法。智能优化的发展为微分对策理论研究提供了一种有效的方法和技术途径。本文深入研究了混沌优化、人工鱼群算法的原理、实施步骤以及特点,针对人工鱼群算法运行速度慢和优化精度较差的缺点及混沌变量具有随机性、遍历性及规律性的特点,将混沌优化与人工鱼群相结合提出高效的混沌人工鱼群混合算法,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。同时在一定的假设条件下证明出混沌人工鱼群算法依概率收敛于全局最优值。
2、研究了微分对策在作战指挥中的应用。基于Lanchester方程,着眼于最优火力分配问题研究。由于战争的多样性和复杂性,本文主要探讨在直瞄武器交战下单兵种对多兵种作战这类比较典型的战争模型,运用微分对策最大值最小值原理进行分析,得出最优分配策略,即多兵种一方应全力攻击单兵种一方;单兵种一方对多兵种一方的攻击顺序取决于双方的交战强度,应按交战强度从大到小顺序攻击,从而给出了军事对抗中兵力不占优势一方如何获取胜利的一种方法。
3、结合特拉法尔海战历史战例,在军事想定的基础上建立适当的数学模型,利用本文提出的混沌人工鱼群算法,对模型进行求解,同时对得到的结果做出分析和研究,得出作战双方火力分配的最优决策。
Differential games theory has been applied to economics, engineering, biology and allother fields. Especially in the world today, high technology in military affairs iswidespread usage, making the research of differential games problem more important. Thispaper directs at theoretical researches on the numerical calculation methods of differentialgame; also made researches on some problem of its application in attack, command andcontrol. Main contents include:
1、Numerical calculation methods to differential game are researched. Development ofartificial intelligence provides an effective method and technological approach fordifferential game theoretical research. Principle, implementation steps and characteristicsof the chaos optimization and artificial fish-swarm algorithm are researched deeply. Due tolow speed and inadequate optimal efficiency of AFSA and chaotic variables with therandomness, regularity, an efficiency optimal hybrid algorithm (SCAFSA) whichcombines chaos optimization and artificial fish-swarm is proposed. The simulatingexperiment of test function verifies that this hybrid algorithm is stronger than AFSA inglobal optimization and the search efficiency is higher. Theoretically, this paper provesthat under certain circumstances the hybrid algorithm is convergent in probability to globaloptimum.
2、Applications of differential game on operational commanding are researched.Based on Lanchester equation, the optimal fire allocation problem is focused on. Becausethe diversity and complexity of war, the article mainly discusses at straight war weaponsthe typical differential game model of one to many. Using the maximum and minimumprinciple in differential game, optimum allocation strategy is obtained. So the side withmore arms should attack the only arm with full power. The attack policy of single arm sidedepends on the descending order of battle intensity. And then the question of bad situationarm winning in military confront is solved.
3、In consideration of Battle of Trafalgar historical case, on the basis of the MilitaryScenario base appropriate mathematical model is set up, using SCAFSA which is proposedin the article. Then the results are analysed and researched properly, getting both sides inmilitary affairs optimizing decision of power distribution.
引文
[1]刘德铭，黄振高．对策论及其应用．长沙：国防科学技术大学出版社，1995．
[2] Isaacs R. Differential Games. New York: John Wiley & Sons, 1965
[3] Friedman A. Differential Games. New York: Wiley Inter.science, 1971
[4]刘德铭．对策论及其应用（上、下册）．长沙：国防科学技术大学研究生讲义，1985．
[5]张嗣赢．微分对策．北京：科学出版社，1987．
[6]董志荣．微分对策的基本原理及其在军事对抗中的应用．《国外舰船技术》火控技术类编辑室，1983
[7]沙基昌．数理战术学初探．长沙：国防科学技术大学出版社，1991．
[8]李登峰．微分对策及其应用．北京：国防工业出版社，2000．
[9] M.Bardi, T.Parthasarathy, T.E.S.Raghavan. Stochastic and Differential Games-Theory andNumerical Methods. Annals of The International Society of Dynamic Games-Volume4Applications, 1998.
[10] Eitan Alman, Odile Pourtallier. Advances in Dynamic Games and Applications. 8th InternationalSymposium of Dynamic Games and Applications, 2001.
[11] Jerzy A. Filar, Koichi Mizukami, Vladimir Gaitsory. Advances in Dynamic Games andApplications. 7th International Symposium of Dynamic Games and Applications, 2000.
[12] GJ.Olsder. New Trends in Dynamic Games and Applications. 6th International Symposium ofDynamic Games and Applications, 1995.
[13] J.G. Taylor. Lanchester Models of Warfare, Military Applications Section, Operation ResearchSociety of America, 1983.
[14] Cruz Jr JB, Chen C I. Series Nash solution of two person nonzero-sum linear-quadratic games. JOptim Theory Appl. 1971,7(4):240~257
[15] Xu H, Mizukami K. Linear- quadratic zero-sum differential games for generalized state spacesystems. IEEE Transon Automatic control, 1994, 39(1): 143~147
[16] Yong J. On the Isaacs equation of differential games of fixed duration. J Optim Theory Appl，1986, 50(2): 359~364
[17] Lukoyanov N Yu. A Hamilton-Jacobi type equation in control problems with hereditaryinformation. J Math Anal Appl. 2000, 64(4): 243~253
[18]徐自祥．微分对策理论及在作战指挥控制中的应用研究．西北工业大学，2006，1~12．
[19]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．
[20]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．
[21]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．
[22]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．
[23]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．
[24]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．
[25]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，2001，26(6)：666~668．
[26]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．
[27]许诚．基于神经网络和微分对策理论的制导律．系统工程与电子技术，1998，20(1)：l~3，28．
[28]周锐，张鹏．基于神经网络的鲁棒制导律设计．航空学报，2002，23(3)：262~264．
[29]许诚．基于模糊理论的微分对策制导律．飞行力学，1997，15(1)：86~90．
[30]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．
[31]沙昌基．数理战术学．北京：科学出版社，2003．
[32]于庆坤，张明泽，陈向勇．基于Lanchester方程的一类作战系统谋略分析．辽宁科技大学学报，2009，32(6)：581~584．
[33]郝柏林．从抛物线谈起——混沌动力学引论．上海：上海科技教育出版社，1993．
[34]李兵，蒋慰孙．混沌优化方法及应用．控制理论与控制工程，1997．14(4)：613~615．
[35]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(11)：32~38．
[36]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．
[37]盛骤，谢式千，潘承毅。概率论与数理统计。北京：高等教育出版社，2002．
[38]李宏，王宇平，焦永昌.解非线性两层规划问题的新的遗传算法及全局收敛性．系统工程理论与实践，2005，26(3)：62~71．
[39]刘勇，陆军，徐裕生等．多点收缩混沌优化方法及全局收敛性证明．纯粹数学与应用数学，2009，25(3)：491~496．
[40]曲良东，何登旭．一种混沌人工鱼群优化算法．计算机工程与应用，2010，46(6)：40~42．
[41]陈向勇．基于Lanchester平方律方程的一类海战实例的决策分析．东北大学，2008．
[42]陈向勇，井元伟，李春吉等．基于Lanchester平方律方程的一类海战实例的决策分析．东北大学学报，2009，30(4)：535~538．
[2] Isaacs R. Differential Games. New York: John Wiley & Sons, 1965
[3] Friedman A. Differential Games. New York: Wiley Inter.science, 1971
[4]刘德铭．对策论及其应用（上、下册）．长沙：国防科学技术大学研究生讲义，1985．
[5]张嗣赢．微分对策．北京：科学出版社，1987．
[6]董志荣．微分对策的基本原理及其在军事对抗中的应用．《国外舰船技术》火控技术类编辑室，1983
[7]沙基昌．数理战术学初探．长沙：国防科学技术大学出版社，1991．
[8]李登峰．微分对策及其应用．北京：国防工业出版社，2000．
[9] M.Bardi, T.Parthasarathy, T.E.S.Raghavan. Stochastic and Differential Games-Theory andNumerical Methods. Annals of The International Society of Dynamic Games-Volume4Applications, 1998.
[10] Eitan Alman, Odile Pourtallier. Advances in Dynamic Games and Applications. 8th InternationalSymposium of Dynamic Games and Applications, 2001.
[11] Jerzy A. Filar, Koichi Mizukami, Vladimir Gaitsory. Advances in Dynamic Games andApplications. 7th International Symposium of Dynamic Games and Applications, 2000.
[12] GJ.Olsder. New Trends in Dynamic Games and Applications. 6th International Symposium ofDynamic Games and Applications, 1995.
[13] J.G. Taylor. Lanchester Models of Warfare, Military Applications Section, Operation ResearchSociety of America, 1983.
[14] Cruz Jr JB, Chen C I. Series Nash solution of two person nonzero-sum linear-quadratic games. JOptim Theory Appl. 1971,7(4):240~257
[15] Xu H, Mizukami K. Linear- quadratic zero-sum differential games for generalized state spacesystems. IEEE Transon Automatic control, 1994, 39(1): 143~147
[16] Yong J. On the Isaacs equation of differential games of fixed duration. J Optim Theory Appl，1986, 50(2): 359~364
[17] Lukoyanov N Yu. A Hamilton-Jacobi type equation in control problems with hereditaryinformation. J Math Anal Appl. 2000, 64(4): 243~253
[18]徐自祥．微分对策理论及在作战指挥控制中的应用研究．西北工业大学，2006，1~12．
[19]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．
[20]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．
[21]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．
[22]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．
[23]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．
[24]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．
[25]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，2001，26(6)：666~668．
[26]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．
[27]许诚．基于神经网络和微分对策理论的制导律．系统工程与电子技术，1998，20(1)：l~3，28．
[28]周锐，张鹏．基于神经网络的鲁棒制导律设计．航空学报，2002，23(3)：262~264．
[29]许诚．基于模糊理论的微分对策制导律．飞行力学，1997，15(1)：86~90．
[30]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．
[31]沙昌基．数理战术学．北京：科学出版社，2003．
[32]于庆坤，张明泽，陈向勇．基于Lanchester方程的一类作战系统谋略分析．辽宁科技大学学报，2009，32(6)：581~584．
[33]郝柏林．从抛物线谈起——混沌动力学引论．上海：上海科技教育出版社，1993．
[34]李兵，蒋慰孙．混沌优化方法及应用．控制理论与控制工程，1997．14(4)：613~615．
[35]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(11)：32~38．
[36]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．
[37]盛骤，谢式千，潘承毅。概率论与数理统计。北京：高等教育出版社，2002．
[38]李宏，王宇平，焦永昌.解非线性两层规划问题的新的遗传算法及全局收敛性．系统工程理论与实践，2005，26(3)：62~71．
[39]刘勇，陆军，徐裕生等．多点收缩混沌优化方法及全局收敛性证明．纯粹数学与应用数学，2009，25(3)：491~496．
[40]曲良东，何登旭．一种混沌人工鱼群优化算法．计算机工程与应用，2010，46(6)：40~42．
[41]陈向勇．基于Lanchester平方律方程的一类海战实例的决策分析．东北大学，2008．
[42]陈向勇，井元伟，李春吉等．基于Lanchester平方律方程的一类海战实例的决策分析．东北大学学报，2009，30(4)：535~538．