高阶线性比例制导系统脱靶量幂级数解

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。

Power series solution for miss distance of higher-order linear proportional navigation guidance systems

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.