Nonlinear Zero-Sum Differential Game Analysis by Singular Perturbation Methods

摘要

A class of nonlinear, zero-sum differential games, exhibiting time-scale separation properties, can be analyzed by singular-perturbation techniques. The merits of such an analysis, leading to an approximate game solution, as well as the 'well-posedness' of the formulation, are discussed. This approach is shown to be attractive for investigating pursuit-evasion problems; the original multidimensional differential game is decomposed to a 'simple pursuit' (free-stream) game and two independent (boundary-layer) optimal-control problems. Using multiple time-scale boundary-layer models results in a pair of uniformly valid zero-order composite feedback strategies. The dependence of suboptimal strategies on relative geometry and own-state measurements is demonstrated by a three dimensional, constant-speed example. For game analysis with realistic vehicle dynamics, the technique of forced singular perturbations and a variable modeling approach is proposed. Accuracy of the analysis is evaluated by comparison with the numerical solution of a time-optimal, variable-speed 'game of two cars' in the horizontal plane.