Distributed Nash equilibrium seeking for networked games of multiple high-order systems with disturbance rejection and c
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ORIGINAL PAPER
Distributed Nash equilibrium seeking for networked games of multiple high-order systems with disturbance rejection and communication delay Xiaolin Ai
Received: 19 March 2020 / Accepted: 10 June 2020 © Springer Nature B.V. 2020
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. Keywords Nash equilibrium seeking · Communication delay · High-order systems · Leader–follower consensus · Disturbance rejection
X. Ai (B) College of Engineering, Peking University, Beijing 100871, China e-mail: [email protected]; [email protected]
1 Introduction Over the past several decades, noncooperative networked games of multiagent systems (MASs) have received tremendous attention from multidisciplinary researchers. This is mainly due to its wide potential applications in natural/social science and engineering, such as power allocation [1,2], cloud computation [3] and mobile sensor networks [4]. A fundamental and important issue in networked games is to seek the Nash equilibrium, which has emerged to be of both theoretical significance and practical relevance recently. For this problem, seeking strategy is developed for each player to minimize (or maximize) its own payoff function, which involves all the players’ actions, until the Nash equilibrium is achieved. To date, numerous distributed methodologies have been reported to seek the Nash equilibrium for networked games. In [5], an asynchronous gossip-based algorithm was designed based on estimates of local neighboring players’ actions to seek the Nash equilibrium for a networked game. Subsequently, this result was extended to the case that the interference graph is not complete in [6]. The authors of [7] studied the Nash equilibrium seeking problem for constrained networked games with nonlinear aggregates. In [8], a distributed gradient-based projection strategy was proposed to address the Nash equilibrium seeking problem with both private and coupled constraints. Moreover, stochastic extremum and Shahshahani g
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