Transfer function conditions for stability

A feedback system is defined to be internally stable if all the eigenvalues of the composite state-space representation of the closed-loop system lie in the left plane. Recently, it has been established [1, Theorem 4.3.6, p. 154] that this internal stability is equivalent to external stability of an appropriate system with virtual inputs injected into each subsystem. The proof in [1] uses matrix fraction descriptions. In this paper, a new state proof of this result is given. The proof is simple and demonstrates the validity of this important transfer function test from the point of view of controllability and observability.