Temporal Linear System Structure

Piecewise constant rank systems and the differential Kalman decomposition are introduced in this note. Together these enable the detection of temporal uncontrollability/unreconstructability of linear continuous-time systems. These temporal properties are not detected by any of the four conventional Kalman decompositions. Moreover piecewise constant rank systems admit the state dimension to be time-variable. As demonstrated in this note linear continuous-time systems with variable state dimensions enable the well rounded realization theory suggested already by Kalman. The differential Kalman decomposition introduced in this note is associated with differential controllability and differential reconstructability. The system structure obtained from the differential Kalman decomposition may be interpreted as the temporal linear system structure. This note 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. Also, this note reveals how the differential Kalman decomposition relates to the conventional ones.