Robust H_∞ tracking: A game theory approach

This paper investigates the problem of finite‐time‐horizon robust H_∞ tracking, for linear continuous time‐varying systems, from the game theory point of view. Three tracking problems are considered, depending on whether the reference signal to be tracked is perfectly known in advance, measured on line, or previewed in a fixed interval of time ahead. No a priori knowledge of a dynamic model for the reference signal is assumed, and the parameters of the system are not completely known. A game is defined where, given the specific information on the tracking signal, the controller plays against nature that can choose any initial condition, any bounded energy disturbance input and measurement noise, and any set of parameters in a prescribed bounded region. A standard quadratic pay‐off function is defined where the energy of the tracking error signal is weighted against the energy of the disturbance, the noise signal, and the Euclidean norm of the initial condition. Conditions for the existence of a saddle‐point equilibrium in this zero‐sum game are not easy to find. We, therefore, augment the state‐space description of the system to convert the parameter uncertainty into exogenous bounded energy signals. An augmented game is then defined on the new perfectly known system, and it is shown that its saddle‐point equilibrium solution, if it exists, guarantees a prescribed H_∞‐norm performance of the tracker, in the original system, for all possible parameters. Necessary and sufficient conditions for the existence of a saddle‐point solution to the augmented game are determined. H_∞‐tracking controllers, which guarantee the prescribed performance level for all possible parameters, are derived for both the state and the output feedback cases.