Poles and zeros of linear multivariable systems : a survey of the algebraic, geometric and complex-variable theory

This survey is concerned with the definition and main properties of sets of specific values of complex frequency, called poles and zeros, which may be associated with a matrix-valued function of a complex frequency variable. The main emphasis is on the physical interpretation of invariant zeros in terms of the general zero-output behaviour of a linear dynamical system. A discussion is given of the relationship between this definition of a zero and the various other forms adopted in the current literature. The relationship is considered between poles and zeros defined by algebraic means and the standard complex variable theory of algebraic functions. It is shown that the poles and zeros of a square matrix-valued function of a complex variable G(s) are the same as the poles and zeros of an associated algebraic function g(s).