Adaptive optimal safety tracking control for multiplayer mixed zero-sum games of continuous-time systems

When the equipment is working, it is very important to avoid the occurrence of malignant accidents by providing a highly reliable safety protection means. In this paper, for multiplayer mixed zero-sum games, an optimal safety tracking control scheme based on adaptive dynamic programming (ADP) is proposed, and a control barrier function (CBF) is introduced into the value function of the system to ensure that the system operates within its safe region. Firstly, through system transformation, the original tracking problem is transformed into a state tracking error problem. Secondly, an augmented Hamilton-Jacobi-Bellman (HJB) equation is derived from the improved augmented error system and the value function. Different from traditional methods, this paper uses a single critic neural network (NN) instead of the actor-critic NN to approximate the Nash equilibrium solution of the system, and introduces a concurrent learning technique that can relax the traditional continuous excitation condition into a simplified condition of recording data. Then, according to the Lyapunov theory, the stability of the system is analyzed in detail. Finally, two simulation examples are used to verify the effectiveness of the proposed scheme.