A Neural-Network based Approach for Nash Equilibrium Seeking in Mixed-order Multi-player Games

Noticing that agents with different dynamics may work together, this paper considers Nash equilibrium computation for a class of games in which first-order integrator-type players and second-order integrator-type players interact in a distributed network. To deal with this situation, we firstly exploit a centralized method for full information games. In the considered scenario, the players can employ its own gradient information, though it may rely on all players' actions. Based on the proposed centralized algorithm, we further develop a distributed counterpart. Different from the centralized one, the players are assumed to have limited access into the other players' actions. In addition, noticing that unmodeled dynamics and disturbances are inevitable for practical engineering systems, the paper further considers games in which the players' dynamics are suffering from unmodeled dynamics and time-varying disturbances. In this situation, an adaptive neural network is utilized to approximate the unmodeled dynamics and disturbances, based on which a centralized Nash equilibrium seeking algorithm and a distributed Nash equilibrium seeking algorithm are established successively. Appropriate Lyapunov functions are constructed to investigate the effectiveness of the proposed methods analytically. It is shown that if the considered mixed-order game is free of unmodeled dynamics and disturbances, the proposed method would drive the players' actions to the Nash equilibrium exponentially. Moreover, if unmodeled dynamics and disturbances are considered, the players' actions would converge to arbitrarily small neighborhood of the Nash equilibrium. Lastly, the theoretical results are numerically verified by simulation examples