Design Of Distributed Robust Nash Equilibrium Seeking Method

With the rapid development of game theory and its wide application in various fields,distributed Nash equilibrium seeking problem has attracted much attention from many scholars in recent years.Most of the existing distributed Nash equilibrium seeking algorithms do not take unknown system dynamics and disturbances into consideration.However,the inevitable unknown dynamics and disturbances in practical systems may destroy system stability and invalidate the controllers designed for ideal systems.It can be seen that the study of distributed robust Nash equilibrium seeking method has very important practical application value.Inspired by the above observation,this thesis has done the following works:First,it studies the distributed Nash equilibrium seeking problem for game where firstand second-order players are affected by time-varying disturbances.To solve the problem,a distributed robust Nash equilibrium seeking method is designed based on a synthesis of gradient method,leader-following consensus protocols and disturbance observers.Through Lyapunov stability analysis,it is proved that the method can drive the actions of first-and second-order players to converge to an arbitrarily small neighborhood of the Nash equilibrium.Second,since there are so many factors that can affect players’ dynamics in reality,it is far from enough to study time-varying disturbances alone.For this reason,distributed Nash equilibrium seeking problem where first-and second-order players’ dynamics are subjected to time-varying disturbances and unmodeled terms is further studied.To solve the problem,a distributed robust Nash equilibrium seeking method is designed based on gradient method,leader-following consensus protocols and unknown dynamics estimators.Through Lyapunov stability analysis,it is proved that the method can drive the actions of first-and second-order players to converge to an arbitrarily small neighborhood of the Nash equilibrium.Finally,noticing that unknown dynamics estimator based distributed Nash equilibrium seeking method results in a bounded convergent result.Distributed Nash equilibrium seeking problem is further studied with asymptotic convergence guarantees.To solve the problem,a distributed robust Nash equilibrium seeking method is designed based on sliding mode control.Through Lyapunov stability analysis,it is proved that the method can drive the actions of second-order player to globally and exponentially converge to the Nash equilibrium.Noticing that in practical engineering applications,the bounds of time-varying disturbances can hardly be obtained,an adaptive parameter adjusting mechanism is further introduced the proposed method.Through Lyapunov stability analysis,it is proved that the method can guarantee global asymptotic convergence to the Nash equilibrium.