Approximate tracking control for nonlinear multi-player systems with deferred asymmetric time-varying full-state constraints

This paper proposes a set of Nash equilibrium tracking control strategies based on mixed zero-sum (MZS) game for the continuous-time nonlinear multi-player systems with deferred asymmetric time-varying (DATV) full-state constraints and unknown initial state. Firstly, an improved shift transformation is used to modify the original constrained system with an unknown initial state into a barrier transformable constrained system. Then, based on the barrier transformable constrained system and predefined reference trajectory, an unconstrained augmented system is formed through the application of the barrier function (BF) transformation. Furthermore, the MZS game Nash equilibrium tracking control strategies are derived by establishing the tracking error related quadratic cost functions and corresponding HJ functions for different players. On this basis, a critic-only structure is established to approximate the control strategy of every player online. By employing Lyapunov theory, it is proven that the neural network weights and tracking error are uniformly ultimately bounded (UUB) within DATV full-state constraints. Simulation experiments of a three-player nonlinear system demonstrate that our algorithm successfully handles deferred state constraints and unknown initial conditions, ensuring that the system states follow the desired reference trajectories. Simulation results further validate the uniform ultimate boundedness of neural network weights and tracking errors.