On stability tests for continuous and discrete-time linear systems

In order to ensure the stabiltity of an n-th order linear system there are tests (due to Hurwitz and Schur) to check whether the roots of the denominator polynomials of the transfer functions in the continuous and the discretetime case are in the left complex half-plane or within the unit circle, respectively. In this contribution, the parallel treatment developed for both cases leads to a simple and insightful proof for the classical stability tests. Instead of looking at the location of the roots of a polynomial as a purely mathematical problem, a systems approach is used that determines whether the covariance matrices of the associated linear systems in state space are positive definite. The result is that the three critical constraints for stability (given by Jury [3]) are simply found from the determinants of the generating matrices for the covariance. These critical conditions, which are a subset of the n+1 stability constraints, are sufficient if one starts with a stable system and all parameters are varied, for example in an adaptive environment.