Algebraic and topological aspects of feedback stabilization

In this paper we give essentially complete results concerning various algebraic and topological aspects of feedback stabilization. In particular, we give necessary and sufficient conditions for a given transfer function matrix to have a right-coprime or a left-coprime factorization, and exhibit a large class of transfer function matrices that have both. We give the most general set of feedback stability criteria available to date, and derive a characterization of all compensators that stabilize a given plant. We define what is meant by "proper" and "strictly proper" in an abstract setting and show that (i) every strictly proper plant can be stabilized by a proper compensator, and (ii) every compensator that stabilizes a strictly proper plant must be proper. We then define a topology for unstable plants and compensators, and show that it is the weakest topology in which feedback stability is a robust property.