On the reduced hermite and reduced schur-cohn matrix relationships

Linear time-invariant systems are considered as aggregates of subsystems, In this paper, a transformation which connects the matrices of the Hermite and Schur-Cohn criteria for root distribution is obtained. Based on this transformation, the relationship between the reduced Hermite and reduced Schur-Cohn criteria is also obtained. Furthermore, the transformation which connects the Lienard-Chipart stability criterion and the simplified deter-minantal stability criterion for the unit circle is derived. Finally, the connection between the inner form of the Hermite and Schur-Cohn criteria is established. The importance of the various transformations lies in the fact that their existence implies that a proof of one left-half-plane stability criterion immediately yields a proof of a corresponding unit circle criterion, and conversely. The various matrix transformations are obtained from the matrix transformation linking vectors of the coefficients of two polynomials whose respective root distributions are studied and which are related by a bilinear transformation of the underlying variables. The symmetry and skew symmetry of the matrix transformation are utilized to obtain the various transformations derived in this paper.which again may be composed of subsystems at a lower level, and 80 on. No special ateucture is supposed for the system, apart from that imposed by the method of interconnection. Some basic results are established, and applications made to analogue computers and to electrical networks. In particular it is shown that whe~ a proper a-por-t impedance matrix can be realized (without ideal transformers) .only by using more reactive elements than the McMillan degree would suggest, still the realization may have least order.