Kalman Canonical Decomposition of Linear Time-Varying Systems

In this paper, a Kalman canonical decomposition of finite-dimensional linear time-varying continuous-time systems is revisited. Two types of Kalman canonical decompositions are considered; one is collectively dubbed the intervalwise Kalman canonical decomposition and the other is collectively dubbed the global Kalman canonical decomposition. Each type is classified into subtypes resulting from variants of controllabilities (i.e., controllability, reachability, and influenceability) and observabilities (i.e., observability, determinability, and visibility). This paper first investigates the invariance properties of subspaces resulting from variants of controllabilities and observabilities. This paper then gives a self-contained exposition of each Kalman canonical decomposition on an interval. The main result is to prove the existence of intervals on each of which each Kalman canonical decomposition exists. Moreover, the relations among different Kalman canonical decompositions are extensively studied and the inconsistency between the works by Youla and Weiss is resolved.