The determination of structural properties of a linear multivariable system by operations of system similarity. 1. Strictly proper systems

Abstract

Abstract Structural properties of a linear multivariable system have been defined as invariants under various transformation groups using the theory of singular pencils of matrices or using a geometric approach. In this paper it is shown that for a strictly proper system in state-space form, these structural properties can be identified by bringing the system matrix to a particular form under system similarity. This allows a structural analysis to be incorporated in existing computer-aided design methods for multivariable control systems. In relation to these structural properties, Rosenbrock's definition of system zeros is investigated and it is shown how this definition fits into the framework of pencil theory and of the geometric approach. This leads to a precise structural interpretation of the relation between Rosenbrock's system zeros and the invariant zeros defined in accordance with pencil theory. The structural analysis presented in this paper also allows us to identify the zeros at infinity and to characterize their nature irrespective of restrictive assumptions such as system invertibility or equal number of inputs and outputs A simple algorithm is presented to extract the structural properties of a multi-variable system via elementary matrix operations under system similarity.