Stability of linear systems with periodic feedback and sampled-data systems

Abstract

Abstract A linear multivariable time-invariant system with three periodic state feedback strategies—dynamic, static, and of the instantaneous state—is considered. For each of these feedback strategies the stability question of an appropriate closed-loop system in relation to its dependence upon the repetition frequency is examined. The proposed approach to the question is based on exponential matrix representations of the dynamics of the considered closed-loop systems. It is shown that at sufficiently large values of the repetition frequency the stability question may be decided on the basis of the stability of a certain matrix which is common for all the considered feedback strategies. It is also shown that peculiar to discrete-time control, the dead-beat stability phenomenon may appear only for a dynamic or static feedback strategy and it may be achieved when the repetition frequency is small enough. The proposed approach results in some new characterization of the stability property for multivariable sampled-data systems.