Game-based Adaptive Fuzzy Optimal Bipartite Containment of Nonlinear Multi-agent Systems

Existing adaptive optimal consensus approaches for high-order nonlinear multi-agent systems (MASs) are limited by their complicated and computation-intensive identifier-actorcritic structure and ignore the conflict of interest between agents. This article proposes a graphical game-based adaptive fuzzy optimal bipartite containment scheme that removes these restrictions. The optimal containment is formulated as an N-player game over the communication topology for high-order nonlinear MASs by defining a cost function that integrates the agent's control inputs, including those of its neighbors, and the local tracking errors. To seek the Nash equilibrium (NE), integral reinforcement learning (IRL) is adopted, which does not involve the system drift dynamic. This approach eliminates the need for an identifier network and simplifies the control scheme using adaptive critic learning. To drive the online learning mechanism, the Bellman residual error is utilized, and a fuzzy logic system (FLS) is used to approximate the optimal value functions of the critic networks. The updating laws incorporate an experience stack, resulting in an easy-to-check persistence excitation condition. It is proven that the synchronization error is uniformly ultimately bounded (UUB), and the bipartite containment of the outputs of followers is achieved. An illustrative example is presented to verify the effectiveness of the developed control scheme