高阶线性比例制导系统脱靶量幂级数解
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摘要
脱靶量是导弹制导系统设计和评估的重要指标,对于一阶环节线性比例制导系统,可以得到脱靶量的解析解,而对于更接近实际的高阶制导系统一般得不到解析解,通常由直接仿真或伴随仿真获得;研究了高阶线性比例制导系统脱靶量的幂级数解,为脱靶量的解算提供一种新的手段。首先,构造伴随系统,假设伴随系统的解为幂级数与指数函数乘积的形式;然后,利用幂级数法给出了脱靶量的幂级数解的系数递推关系;进一步严格证明了脱靶量幂级数解的收敛性;最后针对一阶环节和高阶二项式环节等特殊制导系统,通过选取适当的指数衰减参数,得到了幂级数解系数简化的递推关系,并且一阶环节制导系统的幂级数解和解析解是一致的。在计算脱靶量时,实际用到的是脱靶量幂级数部分和,而部分和项数的确定依赖于指数衰减参数。因此,还分析了指数衰减参数对幂级数解部分和的收敛速度的影响,并给出了指数衰减参数与部分和项数的选取方法,为幂级数解的应用奠定了基础。
Miss distance is a most important performance index for the design and evaluation of a missile guidance system.For a first-order linear propprtional guidance system,closed-form solutions miss distance can be obtained.However,such solutions do not exist for a generic higher-order guidance system matching reality better,in which case miss distance is usually achieved by direct simulation or adjoint technique.In this paper,power series solutions for miss distance of high-order linear proportional navigation are explored to provide a new method for calculation of miss distance.First,the adjoint system of a generic higher-order linear proportional navigation guidance system is constructed and normalized,and the solutions in form of the product of a power series and an exponential decay are assumed.Then a recursion relation for coefficients is derived by using the power series method.Moreover,it is proved that these power series solutions converge everywhere.For single-lag and higher-order binomial guidance systems,the recursion relation is simplified significantly by selecting the appropriate exponential decay constant.In practice,partial sums of power series are used to numerically calculate miss distance,and how many terms of partial sums are adequate is related to the exponential decay constant.Therefore,the influence of the exponential decay constant on the convergence rate of power series solutions is analyzed,and selection methods for the constant are proposed,laying the foundation for practical applications of power series solutions.
Miss distance is a most important performance index for the design and evaluation of a missile guidance system.For a first-order linear propprtional guidance system,closed-form solutions miss distance can be obtained.However,such solutions do not exist for a generic higher-order guidance system matching reality better,in which case miss distance is usually achieved by direct simulation or adjoint technique.In this paper,power series solutions for miss distance of high-order linear proportional navigation are explored to provide a new method for calculation of miss distance.First,the adjoint system of a generic higher-order linear proportional navigation guidance system is constructed and normalized,and the solutions in form of the product of a power series and an exponential decay are assumed.Then a recursion relation for coefficients is derived by using the power series method.Moreover,it is proved that these power series solutions converge everywhere.For single-lag and higher-order binomial guidance systems,the recursion relation is simplified significantly by selecting the appropriate exponential decay constant.In practice,partial sums of power series are used to numerically calculate miss distance,and how many terms of partial sums are adequate is related to the exponential decay constant.Therefore,the influence of the exponential decay constant on the convergence rate of power series solutions is analyzed,and selection methods for the constant are proposed,laying the foundation for practical applications of power series solutions.
引文
[1]ZARCHAN P.Tactical and strategic missile guidance[M].6th ed.Reston,VA:AIAA,2012:35-105.
[2]NESLINE F W,ZARCHAN P.A new look at classical vs modern homing missile guidance[J].Journal of Guidance,Control and Dynamics,1981,4(1):78-85.
[3]SU W S,YAO D N,LI K B,et al.A novel biased proportional navigation guidance law for close approach phase[J].Chinese Journal of Aeronautics,2016,29(1):228-237.
[4]YU W B,CHEN W C,YANG L,et al.Optimal terminal guidance for exoatmospheric interception[J].Chinese Journal of Aeronautics,2016,29(4):1052-1064.
[5]陈峰,肖业伦,陈万春.基于零控脱靶量的大气层外超远程拦截制导[J].航空学报,2009,30(9):1583-1589.CHEN F,XIAO Y L,CHEN W C.Guidance based on zero effort miss for super-range exoatmospheric intercept[J].Acta Aeronautica et Astronautica Sinica,2009,30(9):1583-1589(in Chinese).
[6]陈峰,王育林,肖业伦,等.基于预测脱靶量的远程拦截速度增益导引[J].航空学报,2008,29(6):1665-1672.CHEN F,WANG Y L,XIAO Y L,et al.Velocity-to-begained guidance based on predicted miss distance for longrange intercept[J].Acta Aeronautica et Astronautica Sinica,2008,29(6):1665-1672(in Chinese).
[7]WEISS M.Adjoint method for missile performance analysis on state-space models[J].Journal of Guidance,Control,and Dynamics,2005,28(2):236-248.
[8]HE T L,CHEN W C.A new interpretation of adjoint method in linear time-varying system analysis[C]∥IEEEInternational Conference on CIS&RAM.Piscataway,NJ:IEEE Press,2017:58-63.
[9]赫泰龙,陈万春,刘芳.高超声速飞行器平稳滑翔扰动运动伴随分析[J/OL].北京航空航天大学学报,(2018-04-18)[2018-04-20].http:∥kns.cnki.net/kcms/detail/11.2625.V.20180418.1057.003.html HE T L,CHEN W C,LIU F.Adjoint analysis of steady glide with disturbance motion for hypersonic vehicle[J/OL].Journal of Beijing University of Aeronautics and Astronautics,(2018-04-18)[2018-04-20].http:∥kns.cnki.net/kcms/detail/11.2625.V.20180418.1057.003.html(in Chinese).
[10]王辉,林德福,祁载康,等.时变最优的增强型比例导引及其脱靶量解析解[J].红外与激光工程,2013,42(3):692-698.WANG H,LIN D F,QI Z K,et al.Time-varying optimal augmented proportional navigation and miss distance closed-form solutions[J].Infrared and Laser Engineering,2013,42(3):692-698(in Chinese).
[11]KABAMBA P T,GIRARD A R.Fundamentals of aerospace navigation and guidance[M].Cambridge:Cambridge University Press,2014:141-144.
[12]INCE E L.Ordinary differential equations[M].New York:Dover Publication,Inc.,1978:158-185.
[13]WASOW W.Asymptotic expansions for ordinary differential equations[M].New York:Dover Publication,Inc.,1987:9-48.
[14]CODDINGTON E A,CARLSON R.Linear ordinary differential equations[M].Philadelphia:Society for Industrial and Applied Mathematics,1997:163-220.
[15]BENDER C M,ORSZAG S A.Advanced mathematical methods for scientists and engineers:Asymptotic methods and perturbation theory[M].Berlin:Springer,1999:61-145.
[16]DUNKEL O.Regular singular points of a system of homogeneous linear differential equations of the first order[J].Proceedings of the American Academy of Arts and Sciences,1902,38(9):341-370.
[17]VAZQUEZ-LEAL H,SARMIENTO-REYES A.Power series extender method for the solution of nonlinear differential equations[J/OL].Mathematical Problems in Engineering,2015,15(7):1-7.
[18]HOLT G C.Linear proportional navigation:An exact solution for a 3rd-order missile system[J].Proceedings of the Institution of Electrical Engineers,1977,124(12):1230-1236.
[19]BOAS R P.Partial sums of infinite series,and how they grow[J].The American Mathematical Monthly,1977,84(4):237-258.
[20]ASCHER,U,GREIF C.A first course in numerical methods[M].Philadelphia:Society for Industrial and Applied Mathematics,2011:35-55.
[21]HIGHAM N.Accuracy and stability of numerical algorithms[M].2nd ed.Philadelphia:Society for Industrial and Applied Mathematics,2002:93-104.
[2]NESLINE F W,ZARCHAN P.A new look at classical vs modern homing missile guidance[J].Journal of Guidance,Control and Dynamics,1981,4(1):78-85.
[3]SU W S,YAO D N,LI K B,et al.A novel biased proportional navigation guidance law for close approach phase[J].Chinese Journal of Aeronautics,2016,29(1):228-237.
[4]YU W B,CHEN W C,YANG L,et al.Optimal terminal guidance for exoatmospheric interception[J].Chinese Journal of Aeronautics,2016,29(4):1052-1064.
[5]陈峰,肖业伦,陈万春.基于零控脱靶量的大气层外超远程拦截制导[J].航空学报,2009,30(9):1583-1589.CHEN F,XIAO Y L,CHEN W C.Guidance based on zero effort miss for super-range exoatmospheric intercept[J].Acta Aeronautica et Astronautica Sinica,2009,30(9):1583-1589(in Chinese).
[6]陈峰,王育林,肖业伦,等.基于预测脱靶量的远程拦截速度增益导引[J].航空学报,2008,29(6):1665-1672.CHEN F,WANG Y L,XIAO Y L,et al.Velocity-to-begained guidance based on predicted miss distance for longrange intercept[J].Acta Aeronautica et Astronautica Sinica,2008,29(6):1665-1672(in Chinese).
[7]WEISS M.Adjoint method for missile performance analysis on state-space models[J].Journal of Guidance,Control,and Dynamics,2005,28(2):236-248.
[8]HE T L,CHEN W C.A new interpretation of adjoint method in linear time-varying system analysis[C]∥IEEEInternational Conference on CIS&RAM.Piscataway,NJ:IEEE Press,2017:58-63.
[9]赫泰龙,陈万春,刘芳.高超声速飞行器平稳滑翔扰动运动伴随分析[J/OL].北京航空航天大学学报,(2018-04-18)[2018-04-20].http:∥kns.cnki.net/kcms/detail/11.2625.V.20180418.1057.003.html HE T L,CHEN W C,LIU F.Adjoint analysis of steady glide with disturbance motion for hypersonic vehicle[J/OL].Journal of Beijing University of Aeronautics and Astronautics,(2018-04-18)[2018-04-20].http:∥kns.cnki.net/kcms/detail/11.2625.V.20180418.1057.003.html(in Chinese).
[10]王辉,林德福,祁载康,等.时变最优的增强型比例导引及其脱靶量解析解[J].红外与激光工程,2013,42(3):692-698.WANG H,LIN D F,QI Z K,et al.Time-varying optimal augmented proportional navigation and miss distance closed-form solutions[J].Infrared and Laser Engineering,2013,42(3):692-698(in Chinese).
[11]KABAMBA P T,GIRARD A R.Fundamentals of aerospace navigation and guidance[M].Cambridge:Cambridge University Press,2014:141-144.
[12]INCE E L.Ordinary differential equations[M].New York:Dover Publication,Inc.,1978:158-185.
[13]WASOW W.Asymptotic expansions for ordinary differential equations[M].New York:Dover Publication,Inc.,1987:9-48.
[14]CODDINGTON E A,CARLSON R.Linear ordinary differential equations[M].Philadelphia:Society for Industrial and Applied Mathematics,1997:163-220.
[15]BENDER C M,ORSZAG S A.Advanced mathematical methods for scientists and engineers:Asymptotic methods and perturbation theory[M].Berlin:Springer,1999:61-145.
[16]DUNKEL O.Regular singular points of a system of homogeneous linear differential equations of the first order[J].Proceedings of the American Academy of Arts and Sciences,1902,38(9):341-370.
[17]VAZQUEZ-LEAL H,SARMIENTO-REYES A.Power series extender method for the solution of nonlinear differential equations[J/OL].Mathematical Problems in Engineering,2015,15(7):1-7.
[18]HOLT G C.Linear proportional navigation:An exact solution for a 3rd-order missile system[J].Proceedings of the Institution of Electrical Engineers,1977,124(12):1230-1236.
[19]BOAS R P.Partial sums of infinite series,and how they grow[J].The American Mathematical Monthly,1977,84(4):237-258.
[20]ASCHER,U,GREIF C.A first course in numerical methods[M].Philadelphia:Society for Industrial and Applied Mathematics,2011:35-55.
[21]HIGHAM N.Accuracy and stability of numerical algorithms[M].2nd ed.Philadelphia:Society for Industrial and Applied Mathematics,2002:93-104.
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