Optimal guidance cut-off strategy based on closed-form PNG solution for single lag system

A systematic way to determine both the guidance cut-off time and the terminal acceleration command is newly proposed for missiles guided by the proportional navigation (PN) toward a non-maneuvering target. In order to analyze the diverging behavior of the guidance loop in terminal phase, the actual PN guidance loop with first-order dynamic lag is modeled as the confluent hyper-geometric differential equation of relative kinematics and its closed-form solution is first derived. Based on this analytic solution, the optimal guidance cut-off problem is formulated as the minimization of a quadratic cost in terms of miss-distance and the constant acceleration command issuing after guidance cut-off time. By solving this optimization problem, it is shown that there always exists the guidance cut-off time nullifying zero-effort miss. Through the typical homing guidance scenarios for the stationary target, it is demonstrated that the proposed method guarantees the perfect interception even in the presence of the dynamic lag in the PN guidance loop.