Abstract: 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.

Key words: power series solution, proportional navigation, miss distance, linear differential equation, adjoint method, regular singular point