The use of frequency transmission concepts in linear multivariate system analysis

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Abstract
The classical work by Nyquist and Bode and its extensions to the multivariable case are based on a frequency-response approach to feedback systems. Modern state-space techniques, on the other hand, make use of an internal system description as dictated by a set of first-order linear differential equations. Relevant to both approaches is the problem of frequency propagation which in this paper is seen as the means of unifying the external and internal system description. The inextricable association of frequency transmission with the concept of (A, B)-invariant subspaces and controllability subspaces empowers this approach with the ability to effect a penetrating study of a number of aspects of linear system behaviour such as the restriction of the state vector in given subspaces and the characterization of such subspaces, the characterization of multivariate zeros and the categorization of systems into degenerate and non-degenerate. In this paper the study of the transmission of a particular frequency, or a set of particular frequencies, leads to the definition of monofrequency transmission subspaces and multifrequency transmission subspaces respectively. Futher investigation of the properties of these subspaces yields insight into the geometric structure of linear multivariate systems and suggests techniques of practical use to the feedback designer. One such technique discussed in this paper concerns the placement of eigenvectors and provides the geometric means for the derivation of solutions to the pole-placement problem.