On stability tests for continuous and discrete-time linear systems

TL;DRAbstract

Abstract — In order to ensure the stabiltity of an n-th or-der 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 discrete-time case are in the left complex half-plane or within the unit circle, respectively. In this contribution, the parallel treat-ment developed for both cases leads to a simple and insight-ful proof for the classical stability tests. Instead of looking at the location of the roots of a polynomial as a purely mathe-matical problem, a systems approach is used that determines whether the covariance matrices of the associated linear sys-tems 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 matri-ces for the covariance. These critical conditions, which are a subset of the n+1 stability constraints, are sufficient if one starts wit

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