Constrained coalition games for USV swarm confrontation

The gist This paper proposes a distributed Nash equilibrium seeking strategy for uncertain Euler-Lagrange systems subject to disturbances with unknown bounds in coalition games, which is significant because it provides a robust method for multi-agent systems with complex constraints.

Problem Formulation and Constraints

The problem considers a non-cooperative coalition game where agents cooperatively minimize a shared cost function while competing against other coalitions Coalition games synthesize distributed optimization principles [3] and distributed non-cooperative game paradigms [4]. The feasible set of the j-th agent in the i-th coalition is given by Xij = xij ∈ omegaij (xij, xi,−j) ∈ omegai ∩ Xi For the i-th coalition, its objective can be expressed as min xi Ji(xi, x−i), s.t. xi ∈ omegai ∩ Xi The definition of NE is given as Definition 1. An action profile x⋆ = (x⋆i, x⋆−i) is an NE of the non-cooperative coalition game a if and only if Ji(x⋆i, x⋆−i) ≤ Ji(xi, x⋆−i), ∀xi: (xi, x⋆−i) ∈ omega ∩ X, i ∈ N Assumption 1 states that for all i ∈ N and j ∈ Vi, fij (xi,x−i) is continuously differentiable and strongly convex with respect to xi for any given x−i ∈ Qk∈N∖k omegak

Proposed Distributed NE Seeking Strategy

The proposed distributed strategy combines adaptive techniques with sign functions to handle model uncertainties and disturbances with unknown bounds in the EL systems To deal with the constraints under local information, an adaptive method is employed to update the Lagrange multipliers associated with local constraints, while distributed primal-dual dynamics with consensus terms are harnessed to resolve those linked to coupling constraints Meanwhile, average consensus and leader-following protocols are leveraged to estimate the gradient of coalition’s cost functions and the actions of other agents, respectively The control input uij is designed as uij =Υij (xij, x˙ ij, ¨xˆij, ˙xˆij)ˆµij − γeij − sgn(eij)ˆdij

Convergence Analysis and Robustness

Under standard convexity and graph-connectivity assumptions, global convergence of the closed-loop EL system to the NE is established The convergence analysis involves defining a Lyapunov function candidate V = 1/2 kz¯k squared + X N i=1 Vi, where Vi = Vi1 + Vi2 + (¯v q i)⊤Piv¯q i By applying Barbalat’s Lemma [30], η(t) asymptotically converges to the NE x⋆ The proof shows that if Assumptions 1-4 hold, there exist sufficiently large parameters α, β, γ, and κ for algorithm (4), (5), (6), (7), and (8) such that limt→+∞ kx(t) − x⋆k = 0

Application to USV Swarm Confrontation

A USV swarm confrontation case study is presented to highlight the flexibility of the coalitiongame framework The model uses a point-mass model where the dynamics of the j-th USV in the i-th swarm is given by x˙ ij =Ψij (φij)νij, Mijν˙ij = − Πij (νij)νij − Φijνij + τij + dij The cost functions are formulated as weighted sums of task objectives, such as J1j = X 4 l=1 ˆl R jlJ iv l (x), ∀j ∈ miv 1 The coupling constraints for maintaining the integrity of the supply lines are formally expressed as X j∈V1(x usv 1j − x line R) ≤ X j∈V1 g˜1j, X k∈V2(x line B − x usv 2k) ≤ X k∈V2 g˜2k Theorem 2 states that under Assumptions 3 and 4, if each USV updates its state according to algorithm (57), (5), (6), (7), and (8), all states asymptotically converge to the NE The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE

Conclusion

In this paper, we proposed a distributed NE seeking strategy tailored for uncertain EL systems subject to disturbances with unknown bounds, while accounting for both local and coupling constraints within coalition games In future work, we will develop distributed online NE seeking methods for timevarying coalition games The proposed algorithm is capable of handling both local and coupling constraints In comparison with the constraints in [9] and [14], the constraints in this paper are more general In addition, different from [18], [23]–[25], less information of disturbances is required The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig. 3(b) The simulation results reveal that both the red and blue sides have reached an equilibrium, which corresponds to the NE The trajectories of the Lagrange multipliers for the local constraints are presented in Fig. 3(a), while those for the coupling constraints are shown in Fig.

1. This system can solve Nash equilibrium problems in coalition games involving multiple Euler-Lagrange (EL) systems with both local and coupling constraints under unknown disturbances with unknown bounds by employing a distributed strategy combining adaptive techniques with sign functions. This allows for robust optimization where agents only have access to its own action, cost function, and constraint parameters.

2. The improved AI system can perform complex tasks like USV swarm confrontation scenarios, including formation tracking, encirclement, and interception. It achieves this by modeling the confrontation as a coalition game framework where task objectives are formulated as weighted sums of local and coupling constraints (e.g., J1j = X4 l=1 ˆl RjlJ iv l(x)).

3. The system can maintain logistical sustainability by incorporating coupling constraints that reflect supply line integrity, expressed as X j∈V1 (x usv 1j − x line R) ≤ X j∈V1 g˜1j and similar terms for the blue swarm. This ensures that the USVs adhere to geographical constraints while optimizing their cooperative objectives.

4. The system can achieve asymptotic convergence to the Nash equilibrium (NE) of the game, as shown by Theorem 2, provided sufficient gains are chosen. The convergence is guaranteed by analyzing a Lyapunov function candidate whose time derivative is shown to be less than or equal to zero, leading to limt→+∞ kx(t) − x⋆k = 0.