Temporal linear system structure: The discrete-time case

A Kalman decomposition for linear time-varying discrete-time systems is introduced that detects temporal uncontrollability/unreconstructability that is not detected by any of the four conventional Kalman decompositions. This new discrete-time Kalman decomposition is associated with j-step reachability and k-step observability. The system structure obtained from the Kalman decomposition may be interpreted as the temporal linear system structure. This paper reveals that the difference between controllability and reachability as well as reconstructability and observability is entirely due to changes of the temporal linear system structure. Finally this paper reveals how our Kalman decomposition relates to the conventional ones and how temporal discrete-time linear system properties relate to their well established non temporal counterparts.