Efficient atmospheric and extra-atmospheric interception through optimally biased proportional navigation

Starting from a definition of the biased proportional navigation (PN) law and expression of the trajectory parameters, especially the lateral acceleration, in terms of a nondimensional bias parameter, an analytical minimization is made of the integrated (total) control effort with respect to the bias parameter. It is shown that optimum biasing may lead to significantly more control-efficient PN guidance in a wide variety of engagement situations, especially those involving higher target maneuvers. The performance of the optimally biased PN is compared with the standard (unbiased) PN law for the general case of a maneuvering target. The optimum bias is expressed through a simple algebraic equation readily solvable in real time even on small onboard processors. For the special case of the effective navigation being equal to three, the equation reduces to a quadratic, lending to an explicit expression for the optimum bias. Specific examples are provided for interception both inside and outside the atmosphere. It is shown that control effort savings up to a factor of three can be achieved through optimum biasing under realistic engagement conditions