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.
Similar content being viewed by others
References
Pang, J., Scutari, G., Facchinei, F., Wang, C.: Distributed power allocation with rate constraints in Gaussian parallel interference channels. IEEE Trans. Inf. Theory 54(8), 3471–3489 (2008)
Deligiannis, A., Panoui, A., Lambotharan, S., Chambers, J.: Game-theoretic power allocation and the Nash equilibrium analysis for a multistatic MIMO radar network. IEEE Trans. Signal Process. 65(24), 6397–6408 (2017)
Ardagna, D., Panicucci, B., Passacantando, M.: Generalized Nash equilibira for the service provisioning problem in cloud systems. IEEE Trans. Serv. Comput. 6(4), 429–442 (2013)
Stankovic, M., Johansson, K.: Distributed seeking of Nash equilibria with applications to mobile sensor networks. IEEE Trans. Autom. Control 57(4), 904–919 (2012)
Salehisadaghiani, F., Pavel, L.: Distributed Nash equilibrium seeking: a gossip-based algorithm. Automatica 72, 209–216 (2016)
Salehisadaghiani, F., Pavel, L.: Distributed Nash equilibrium seeking in aggregative graphical games. Automatica 87, 17–24 (2018)
Liang, S., Yu, P., Hong, Y.: Distributed Nash equilibrium seeking for aggregative games with coupled constraints. Automatica 85, 179–185 (2017)
Lu, K., Jing, G., Wang, L.: Distributed algorithms for searching generalized Nash equilibrium of noncooperative games. IEEE Trans. Cybern. 49(6), 2362–2371 (2019)
Liu, S., Krstic, M.: Stochastic Nash equilibrium seeking for games with general nonlinear payoffs. SIAM J. Control Optim. 49(4), 1659–1679 (2011)
Poveda, J., Quijiano, N.: Shahshahani gradient-like extremum seeking. Automatica 58, 51–59 (2015)
Koshal, J., Nedic, A., Shanbhag, U.: Distributed algorithms for aggregative games on graphs. Oper. Res. 63(3), 680–704 (2016)
Yi, P., Pavel, L.: Asynchronous distributed algorithms for seeking generalized Nash equilibria under full and partial-decision information. IEEE Trans. Cybern. 50(6), 2514–2526 (2020)
Ye, M., Hu, G.: Distributed Nash equilibrium seeking by a consensus based approach. IEEE Trans. Autom. Control 62(9), 4811–4818 (2017)
Ye, M., Hu, G.: Distributed Nash equilibrium seeking in multiagent games under switching communication topologies. IEEE Trans. Cybern. 48(11), 3208–3217 (2018)
Deng, Z., Liang, S.: Distributed algorithms for aggregative games of multiple heterogeneous Euler–Lagrange systems. Automatica 99, 246–252 (2019)
Ai, X., Yu, J., Jia, Z., Yang, D., Xu, X., Shen, Y.: Disturbance observer-based consensus tracking for nonlinear multiagent systems with switching topologies. Int. J. Robust Nonlinear Control 28(6), 2144–2160 (2018)
Deng, Z., Nian, X.: Distributed algorithm design for aggregative games of disturbed multiagent systems over weight-balanced digraphs. Int. J. Robust Nonlinear Control 28, 5344–5357 (2018)
Zhang, Y., Liang, S., Wang, X., Ji, H.: Distributed Nash equilibrium seeking for aggregative games with nonlinear dynamics under external disturbances. IEEE Trans. Cybern. (2019). https://doi.org/10.1109/TCYB.2019.2929394
Ai, X., Yu, J., Jia, Z., Shen, Y., Ma, P., Yang, D.: Adaptive robust consensus tracking for nonlinear second-order multi-agent systems with heterogeneous uncertainties. Int. J. Robust Nonlinear Control 27(18), 5082–5096 (2017)
Ai, X.: Adaptive robust bipartite consensus of high-order uncertain multi-agent systems over cooperation–competition networks. J. Frankl. Inst. 357(3), 1813–1831 (2020)
Yu, J., Wang, L.: Group consensus in multi-agent systems with switching topologies and communication delays. Syst. Control Lett. 59(6), 340–348 (2010)
Liu, K., Xie, G., Wang, L.: Consensus for multi-agent systems under double integrator dynamics with time-varying communication delays. Int. J. Robust Nonlinear Control 22(17), 1881–1898 (2012)
Ren, W., Beard, R.: Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. Autom. Control 50(5), 655–661 (2005)
Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
Ai, X., Yu, J.: Flatness-based finite-time leader-follower formation control of multiple quadrotors with external disturbances. Aerosp. Sci. Technol. 92, 20–33 (2019)
Ai, X., Yu, J.: Fixed-time trajectory tracking for a quadrotor with external disturbances: a flatness-based sliding mode control approach. Aerosp. Sci. Technol. 89, 58–76 (2019)
Ding, Z.: Adaptive consensus output regulation of a class of nonlinear systems with unknown high-frequency gain. Automatica 51, 3338–355 (2015)
Cai, H., Lewis, F., Hu, G., Huang, J.: The adaptive distributed observer approach to the cooperative output regulation of linear multi-agent systems. Automatica 75, 299–305 (2017)
Abdessameud, A., Tayebi, A.: Distributed output regulation of heterogeneous linear multi-agent systems with communication constraints. Automatica 91, 152–158 (2018)
Scutari, G., Facchinei, F., Pang, J.S., Palomar, D.P.: Real and complex monotone communication games. IEEE Trans. Inf. Theory 60(7), 4197–4231 (2014)
Basin, M., Yu, P., Shtessel, Y.: Finite- and fixed-time differentiators utilizing HOSM techniques. IET Control Theory Appl. 11(8), 1144–1152 (2017)
Swaroop, D., Hedrick, J.K., Yip, P.P., Gerdes, J.C.: Dynamic surface control for a class of nonlinear systems. IEEE Trans. Autom. Control 45(10), 1893–1899 (2000)
He, S., Wang, W., Wang, J.: Adaptive backstepping impact angle control with autopilot dynamics and acceleration saturation consideration. Int. J. Robust Nonlinear Control 27, 3777–3793 (2017)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Acknowledgements
The authors would like to thank the editor and all anonymous reviewers for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
All authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ai, X. Distributed Nash equilibrium seeking for networked games of multiple high-order systems with disturbance rejection and communication delay. Nonlinear Dyn 101, 961–976 (2020). https://doi.org/10.1007/s11071-020-05758-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05758-5