A unified structural interpretation of some well-known stability-test procedures for linear systems

A number of well-known stability-test procedures for continuous-and discrete-time systems are re-examined in a unified manner, leading to well-defined network-theoretic interpretations. The representation and network interpretation are based on the fact that the stability of any linear system (scalar or multivariable) is equivalent to the stability of a related all-pass system, which in turn can always be synthesized as a cascade of (scalar or matrix) two-pair all-pass (lossless) networks. The original system of interest is stable if and only if each all-pass two-pair is stable (and hence "lossless bounded real"). As a result of this interpretation, a number of related issues, such as enumeration of unstable poles, prematured terminations, and singularity situations can all be approached in a unified manner, based only on "two-pair extraction formulas." In addition, the network interpretation also leads to direct test procedures for testing relative stability, and the stability of multi-input, multi-output systems.