A local approach to solving the inverse minimax control problem for discrete-time systems

Abstract

A local approach is used to solve the inverse problems of minimax control and of worst-case disturbance for linear discrete-time systems. It is shown that minimax control and worst-case disturbance with respect to a quadratic performance index are always those for some local criterion defined by a one-step increment of a state quadratic function along a system trajectory and current values of the control and the disturbance, while the inverse statement generally speaking is not true. Using a solution to the inverse problem of worst-case disturbance, which is of interest itself, the necessary and sufficient condition expressed in a frequency domain are derived under which a given local worst-case disturbance and a given local minimax control will be the worst-case disturbance and the minimax control respectively for some quadratic performance index with a suitable non-negative weight on the state. In addition, the set of all linear state feedbacks corresponding to minimax controls is revealed to be a subset of the set of all stable feedbacks corresponding to optimal controls in the absence of disturbances.