Distributed Nash equilibrium seeking for aggregative games of linear systems subject to unknown disturbances

In this paper, we address the distributed Nash equilibrium seeking problem for aggregative games of $N$ players subject to unknown disturbances over strongly connected networks. Compared with existing works, the general linear dynamics, general directed and strongly connected networks, as well as unknown disturbances are tackled simultaneously in the aggregative games. First, by introducing certain coordinate transformation and feedback linearization method, we develop a distributed gradient-based Nash equilibrium seeking law. A dynamic average consensus dynamics is designed to deal with the challenge by unbalance of general strongly connected networks. By the graph-related property and converse Lyapunov theorem, we establish the global exponential stability of a linear system and a class of nonlinear systems, respectively. Then, we propose a gain design method to obtain the stability of the nonlinear closed-loop system, which is not in the lower triangular form. Inspired by the output regulation theory, we design an internal model and an adaptive dynamics to tackle the unknown disturbances. Resorting to the perturbation theory and the internal model principle, we demonstrate that distributed Nash equilibrium seeking for aggregative games of $N$ players with general linear systems subject to unknown disturbances over strongly connected networks can be achieved. Finally, the effectiveness of the proposed distributed Nash equilibrium seeking approaches are verified by their applications to some simulation examples.