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Nash equilibrium seeking in coalition games for multiple Euler-Lagrange systems: Analysis and application to USV swarm confrontation

Cheng Yuwen, Jialing Zhou, Meng Luan, Guanghui Wen, Tingwen Huang

Abstract

This paper addresses a class of Nash equilibrium (NE) seeking problems in coalition games involving both local and coupling constraints for multiple Euler-Lagrange (EL) systems subject to disturbances of unknown bounds. Within each coalition, agents cooperatively minimize a shared cost function while competing against other coalitions. A distributed strategy is proposed to seek the NE under informational constraints, where each agent has access only to its own action, cost function, and constraint parameters. In the proposed distributed NE seeking strategy, adaptive techniques are combined with sign functions to handle model uncertainties and disturbances with unknown bounds in the EL systems. To deal with the Lagrange multipliers associated with local and coupling constraints, primal-dual techniques are integrated with consensus protocols. Additionally, a dynamic average consensus algorithm is employed to estimate the gradient of the coalition cost function, while a leader-following protocol is utilized to estimate the actions of other agents. Under standard convexity and graph-connectivity assumptions, global convergence of the closed-loop EL system to the NE is established. As an illustrative application, a swarm confrontation of unmanned surface vehicles involving formation, encirclement, and interception tasks is modeled within the coalition game framework, and numerical simulations are conducted under this model to validate the theoretical results.

Nash equilibrium seeking in coalition games for multiple Euler-Lagrange systems: Analysis and application to USV swarm confrontation

Abstract

This paper addresses a class of Nash equilibrium (NE) seeking problems in coalition games involving both local and coupling constraints for multiple Euler-Lagrange (EL) systems subject to disturbances of unknown bounds. Within each coalition, agents cooperatively minimize a shared cost function while competing against other coalitions. A distributed strategy is proposed to seek the NE under informational constraints, where each agent has access only to its own action, cost function, and constraint parameters. In the proposed distributed NE seeking strategy, adaptive techniques are combined with sign functions to handle model uncertainties and disturbances with unknown bounds in the EL systems. To deal with the Lagrange multipliers associated with local and coupling constraints, primal-dual techniques are integrated with consensus protocols. Additionally, a dynamic average consensus algorithm is employed to estimate the gradient of the coalition cost function, while a leader-following protocol is utilized to estimate the actions of other agents. Under standard convexity and graph-connectivity assumptions, global convergence of the closed-loop EL system to the NE is established. As an illustrative application, a swarm confrontation of unmanned surface vehicles involving formation, encirclement, and interception tasks is modeled within the coalition game framework, and numerical simulations are conducted under this model to validate the theoretical results.

Paper Structure

This paper contains 14 sections, 8 theorems, 83 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph Theory
  4. Projection Operator
  5. Problem Statements
  6. Distributed Robust NE Seeking Strategy
  7. Algorithm Design
  8. Convergence Analysis
  9. USV Swarm Confrontation Model
  10. USV Model
  11. Task Model
  12. Coalition Game Model
  13. Numerical Example
  14. Conclusion

Key Result

Lemma 1

Define a function $V(x): \mathbb{R}^m \mapsto \mathbb{R}$ as $V(x) = \frac{1}{2} \left( \|x - \mathcal{P}_{\Omega}(\tilde{x})\|^2 - \|x - \mathcal{P}_{\Omega}(x)\|^2 \right)$, $\forall x, \tilde{x} \in \mathbb{R}^m$. Then the following statements hold:

Figures (3)

  • Figure 1: Communication graphs. (a) Communication graph $\mathcal{G}_1$ of the USVs in the red swarm. (b) Communication graph $\mathcal{G}_2$ of the USVs in the blue swarm. (c) Communication graph $\bar{\mathcal{G}}$ of all the USVs.
  • Figure 2: Simulation results of USV formation control.
  • Figure 3: Simulation results of USV formation control.

Theorems & Definitions (14)

  • Lemma 1: clarke2008nonsmooth
  • Definition 1
  • Lemma 2
  • Lemma 3: Ortega1998
  • Remark 1
  • Lemma 4: Gadjov8354898
  • Lemma 5
  • proof
  • Theorem 1
  • proof
  • ...and 4 more