  This paper aims to accommodate games in which the players' dynamics are
subject to un-modeled and disturbance terms. The un-modeled and disturbance
terms are regarded as extended states for which observers are designed to
estimate them. Compensating the players' dynamics with the observed values, the
control laws are designed to achieve the robust seeking of the Nash equilibrium
for networked games. Firstly, we consider the case in which the players'
dynamics are subject to bounded disturbances only. In this case, the seeking
strategy is developed by employing a smooth observer based on the
Proportional-Integral (PI) control. By utilizing the designed strategy, we show
that the players' actions would converge to a small neighborhood of the Nash
equilibrium. Moreover, the ultimate bound can be adjusted to be arbitrarily
small by tuning the control gains. Then, we further consider the case in which
both an un-modeled term and a disturbance term coexist in the players'
dynamics. In this case, we adapt the idea from the Robust Integral of the Sign
of the Error (RISE) method in the strategy design to achieve the asymptotic
seeking of the Nash equilibrium. Both strategies are analytically investigated
via the Lyapunov stability analysis. The applications of the proposed methods
for a network of velocity-actuated vehicles are discussed. Finally, the
effectiveness of the proposed methods is verified via conducting numerical
simulations.
