Robust Nash equilibrium seeking based on semi-Markov switching topologies

TL;DR

This work tackles distributed Nash equilibrium seeking for N players with second-order dynamics in environments with disturbances and uncertain dynamics, under semi-Markov switching topologies. Two robust algorithms are developed: Algorithm 1 uses a supertwisting-based Integral Sliding-Mode Control with average consensus for finite-time disturbance rejection, while Algorithm 2 extends to semi-Markov switching and includes a sampled-data event-triggered mechanism to reduce communication. The authors establish exponential convergence of the reduced NE-seeking dynamics, mean-square consensus for action estimates under switching, and Zeno-free performance via Lyapunov-Krasovskii analysis and LMIs. A connectivity control game demonstrates practical effectiveness, showing robust convergence, accurate estimation, velocity stabilization, and substantial communication savings in realistic networks.

Abstract

This paper investigates a distributed robust Nash Equilibrium (NE) seeking problem in fluctuating environments. Specifically, the players, subject to the second-order dynamics, are considered to be influenced by external disturbances and uncertain dynamics while communicating via semi-Markov switching topologies. In such constantly changing network circumstances, the existence of disturbances and uncertain dynamics may directly affect the performance of most existing NE seeking algorithms. Moreover, the semi-Markov switching topologies may cause communication uncertainty, which are considered in NE seeking for the first time. To accommodate the above concerns, the following targets require to be reached simultaneously: (1) Disturbances and uncertain dynamics rejection in finite time; (2) Distributed estimation on unknown information required for players' cost functions; (3) A reasonable estimation consensus protocol under semi-Markov switching; (4) NE seeking for the second-order players. By combining supertwisting-based Integral Sliding-Mode Control (ISMC) with average consensus tracking, a novel robust NE seeking algorithm is constructed, incorporating an effective leader-follower consensus protocol. Furthermore, to lessen dispensable information transmission, a sampled-data-based event-triggered mechanism is introduced. Incorporating the advantages of both semi-Markov switching and event-triggered mechanism, another NE seeking algorithm is proposed. Through designing an appropriate Lyapunov-Krasovskii functional, it is shown that the leader-follower consensus can be achieved in the mean-square sense under event-triggered mechanism. Finally, a connectivity control game is formulated to illustrate the validity of the designed algorithms.

Figures (12)

• Figure 1: Communication topology of the vehicles.
• Figure 2: $x_{i1}(t)$, $x_{i2}(t)$ for the position trajectories produced by Algorithm 1.
• Figure 3: All the vehicles' trajectories in 2D plane.
• Figure 4: $v_{i1}(t)$, $v_{i2}(t)$ for the velocity trajectories produced by Algorithm 1.
• Figure 5: $x_{i1}(t)$, $x_{i2}(t)$ for the position trajectories produced by the algorithm in ye2023distributed.

Theorems & Definitions (29)

• Lemma II.1
• Lemma II.2
• Lemma II.3
• Lemma II.4
• Lemma II.5
• Definition II.1
• Definition II.2
• Remark II.1
• Remark II.2
• Remark III.1