Convergence without oscillation to Nash equilibria in non-cooperative games with quadratic payoffs

In this paper, a Nash equilibrium seeking problem for an N-players static non-cooperative game with quadratic payoff function is considered, in which each player seeks its maximum quadratic payoff. Unlike most classical game methods for seeking Nash equilibria, in this method, the model of payoff functions and the actions of other players do not need to be known. In the presented algorithm, we adjust the classical extremum seeking control so that the amplitude of sinusoidal excitation signal locally exponentially converges to zero. Consequently, this method can both increase the speed of convergence dramatically and eliminate the adverse effects of steady-state oscillation by reducing the amplitude of sinusoidal excitation signal. The details of stability analysis and proof of this method is provided. Also, at the end a numerical example and simulation which shows the effectiveness and superiority of this approach is presented.