Differential Games for pursuit evasion problems have been investigated for many years. Differential games, with linear state equations and quadratic cost functions, are called Linear Quadratic Differential Game (LQDG). In these games, one defines two players a pursuer and an evader, where the former aims to minimize and the latter aims to maximize the same cost function (zero-sum games). The main advantage in using the LQDG formulation is that one gets Proportional Navigation (PN) like solutions with continuous control functions. One approach which plays a main role in the LQDG literature is Disturbance Attenuation (DA), whereby target maneuvers and measurement error are considered as external disturbances. In this approach, a general representation of the input-output relationship between disturbances and output performance measure is the DA function (or ratio). This function is bounded by the control.
This work revisits and elaborates upon this approach. The work contains an introduction of a representative case study, a “Simple Boat Guidance Problem” (SBGP), with perfect and imperfect information patterns. By the derivation of the analytical solution for this game, and by running some numerical simulations, we developed the optimal solution based on the critical values of the DA ratio. Moreover, we will study a Missile Guidance Engagement (MGE) problem, with and without Trajectory Shaping (TS). The qualitative and quantitative properties of the MGE solution, based on the critical DA ratio, were studied by extensive numerical simulations, and are shown to be different than the fixed DA ratio solutions. The various factors that influence the choice of parameters for choosing the optimal Trajectory Shaping matrix will be introduced.
