Singular Perturbation Trajectory Following Algorithms for Min-Max Differential Games

Abstract

This chapter examines trajectory following algorithms for differential games subject to simple bounds on player strategy variables. These algorithms are trajectory following in the sense that closed-loop player strategies are generated directly by the solutions to ordinary differential equations. Player strategy differential equations are based upon Lyapunov optimizing control techniques and represent a balance between the current penetration rate for an appropriate descent function and the current cost accumulation rate. This numerical strategy eliminates the need to solve 1) a min-max optimization problem at each point along the state trajectory and 2) nonlinear two-point boundary-value problems. Furthermore, we address “stiff” systems of differential equations that arise during the design process and seriously degrade algorithmic performance. We use standard singular perturbation methodology to produce a numerically tractable algorithm. This results in the Efficient Cost Descent (ECD) algorithm which possesses desirable characteristics unique to the trajectory following method. Equally important as a specification of a new trajectory following algorithm is the observation and resolution of several issues regarding the design and implementation of a trajectory following algorithm in a differential game setting.