On dynamic feedback compensation and compactification of systems

Abstract

This paper introduces a compactification of the space of proper $p \times m$ transfer functions with a fixed McMillan degree $n$. Algebraically, this compactification has the structure of a projective variety and each point of this variety can be given an interpretation as a certain autoregressive system in the sense of Willems. It is shown that the pole placement map with dynamic compensators turns out to be a central projection from this compactification to the space of closed-loop polynomials. Using this geometric point of view, necessary and sufficient conditions are given when a strictly proper or proper system can be generically pole assigned by a complex dynamic compensator of McMillan degree $q$. ©1994 Society for Industrial and Applied Mathematics