Adaptive Dynamic Programming for a Nonlinear Two-Player Non-Zero-Sum Differential Game With State and Input Constraints

Abstract

This paper investigates a nonlinear two-player non-zero-sum differential game with state and input constraints. To solve this problem, this paper constructs a neural network (NN) framework to approximate the solution of the Hamilton-Jacobi-Isaacs (HJI) equation. The adaptive dynamic programming (ADP) method is utilized where each player only needs one critic NN. To solve the issue of state and input saturations, this paper develops a novel constrained system for the differential game, firstly to make the states within the predetermined constraint set. Then, the non-quadratic expression is used to substitute the traditional quadratic expression for the two-player non-zero-sum differential game, and both of the inputs of the two players are constrained. With these treatments, the control input and the system are more in line with real-world applications. Moreover, the stability of the system is also analyzed using the Lyapunov theorem. Two numerical examples are presented to illustrate that the critic NN weights estimation errors and the system are uniformly ultimately bounded (UUB), and the state and input constraints can be achieved.