Complete Results on Control and Filtering of Discrete Systems with Time Scales

The paper provides complete results of the feedback control design problem for a wide class of discrete-time systems possessing fast and slow modes. The mode-separation is expressed in terms of an inequality relating norms of system sub-matrices. The slow and fast subsystems are considered to be completely controllable and observable. A systematic two-stage procedure is developed which enables designing separate gain matrices for the fast and slow subsystems based on H_∞ and H₂ optimization criteria and using linear matrix inequalities. It is established that the composite control yields first-order approximations to the behavior of the discrete system. The theoretical analysis is extended to designing of Kalman filters and linear quadratic Gaussian controllers. It is shown that the design procedure eventually reduces to solving pure-slow and pure-fast reduced-order Kalman filters followed by pure-slow and pure-fast reduced-order discrete-time algebraic Riccati equations. Typical applications are considered to illustrate the design procedure.