Robust Strategies for Trajectory Tracking of Nash Linear Quadratic Games

This paper deals with the design of feedback strategies for a novel class of multi-player linear quadratic differential games. To this end the classical definition of Nash equilibrium solution is extended to deal with problems in which the goal of the i -th player is the tracking of a pre-specified trajectory in the state-space by adopting a state feedback control policy. In this context a sufficient condition for the existence of the optimal Nash equilibrium solution of each player is provided. Then the more general case in which the players do not know exactly the model of the game is considered; in this case each player assumes a different behaviour of the game by modelling the system dynamics by means of a linear dynamic system subject to norm bounded uncertainties. The proposed strategies guarantee to each player a given performance bound and, to be evaluated, require the solution of a set of coupled Riccati Differential Equations containing certain multiplier functions. A numerical example is provided to show the effectiveness of the proposed method.