Distributed Nash equilibrium seeking for high-order integrator dynamics subject to disturbances of unknown bounds

In this paper, we study the problem of Nash equilibrium seeking of N-player games for high-order integrator dynamics with disturbances. Three features of our results are worth mentioning. First, our results apply to jointly strongly connected networks, which can be disconnected at every time instant. Second, our disturbances can be any bounded time function with the bound unknown, which include discontinuous disturbances that cannot be handled by the existing results. Third, in addition to two standard assumptions on the cost functions, the existing results also rely on the satisfaction of an inequality involving some Lipschitz constant of the pseudogradients of the cost functions and the smallest nonzero eigenvalue of the Laplacian of the communication graph. In contrast, our results hold without relying on the satisfaction of this inequality. To achieve our objective, we have developed an approach by integrating the distributed estimator, some nonlinear control technique, and adaptive control technique. Our design is illustrated by one example of force-actuated robots in sensor networks.