Controllability and observability of linear time-invariant compartmental models

The problems of complete controllability and complete observability of linear time-invariant compartmental models with general input-output structures are considered. The analysis of these problems is based largely on graph-theoretic methods and the properties of polynomial matrices. Specifically, statements on complete controllability and complete observability are proven by using graphical constructions which do not change the basic properties of controllability and observability. The major results of this paper are: 1) A unique decomposition of the digraph of a compartmental model into sources, sinks, and transits. 2) A theorem which states that a compartmental model with closed sinks is completely observable if and only if the matrix [
C^{T}, A^{T}
] is full rank, and a corollary which provides a sufficient condition for complete observability of all other linear time-invariant compartmental models. 3) A theorem which states that a single sink compartmental model is completely controllable if and only if a compartment from each source is controlled. 4) A controllability algorithm which provides sets of excitations which are sufficient for complete controllability of any linear time-invariant compartmental model.