Distributed Nash equilibrium seeking for networked games of multiple high-order systems with disturbance rejection and communication delay

Abstract

This paper investigates distributed Nash equilibrium seeking problems for networked games played by multiple high-order systems with disturbance rejection and communication delay, where each player’ payoff function is determined by all players’ actions. To be specific, a fixed-time disturbance observer is used to enhance the capacity of disturbance rejection. In virtue of the observed values, the backstepping technique is performed to design a seeking strategy on the basis of a synthesis of a leader–follower consensus protocol and the gradient play technique, where a fixed-time differentiator is used to overcome the problem of ‘explosion of terms.’ Detailed stability analysis is conducted via the Lyapunov–Krasovskii method to show the convergence of the players’ actions to the Nash equilibrium under both constant and time-varying delays. Moreover, the relationship between the allowed upper bound of the communication delay and the controller parameters is also obtained analytically. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed seeking algorithms.