Discrete-Time Robust Hierarchical Linear-Quadratic Dynamic Games

Abstract

In this paper, the theory of robust min-max control is extended to hierarchical and multiplayer dynamic games for linear quadratic discrete time systems. The proposed game model consists of one leader and many followers, while the performance of all players is affected by disturbance. The Stackelberg-Nash-saddle equilibrium point of the game is derived and a necessary and sufficient condition for the existence and uniqueness of such a solution is obtained. In the infinite time horizon, it is shown that the solution of the Riccati equation is upper bounded under a condition that is called individual controllability. By assuming such a condition and using a time-varying Lyapunov function the input-to-state stability of the hierarchical dynamic game is achieved, considering the optimal feedback strategies of the players and an arbitrary disturbance as the input.