Poles, zeros, and feedback: A module point of view

The treatment of poles and zeros for feedback systems and their transfer function matrices can be greatly complicated when these matrices have nonzero kernels or incomplete images and when the poles and zeros themselves are repeated, and may therefore display a variety of invariant structures. One way to overcome these difficulties is to regard poles and zeros as vector spaces equipped with operators, and thus to invoke the methodology of module theory. For example, with such means, it is possible to give a precise algebraic generalization to the adage 'the poles of a feedback compensator become zeros of the closed-loop system'. More importantly, it is possible to study the effects of feedback upon the creation of decoupling zeros and system poles by interaction of plant and compensator. The authors give an up-to-date accounting of the use of module methods to study spaces of poles and zeros, insofar as they are related to the use of feedback.<>