A model reduction approach for discrete-time linear time-variant systems with delayed inputs

A model reduction approach is presented for discrete-time linear time-variant input-delayed systems. According to this proposed approach, a dynamical variable is constructed by taking advantage of the current state and historical information of input. It is revealed that the behavior of this dynamical variable is governed by a discrete-time linear delay-free system. It is worth noting that the presented variable transformation does not require the system matrix to be invertible. Based on the reduced delay-free models, stabilizing control laws can be easily obtained for the original delayed system. For the case with a single input delay, the constructed variable is an exact prediction for the future state, and thus the stabilizing control law could be designed by replacing the future state with its prediction. Finally, three discrete-time periodic systems with delayed input are employed to illustrate how to utilize the presented model reduction approaches.