Algebraic Geometric Aspects of Feedback Stabilization

Abstract

This paper develops a theory of feedback stabilization for SISO transfer functions over a general integral domain which extends the well-known coprime factorization approach to stabilization. Necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained. These conditions are then refined in the special cases of unique factorization domains (UFDs), Noetherian rings, and rings of fractions. It is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring. This interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant. The set of transfer functions is topologized so as to restrict to the graph topology of Vidyasagar [IEEE Trans. Automatic Control, AC-29 (1984), pp. 403–418], when the ring is a Bezout domain. It is shown that stability of a feedback system is robust in this topology when the ring is a UFD. This theory is then applied to the problem of stabilization of multidimensional systems. The above stabilizability criterion is interpreted geometrically in terms of affine varieties in $\mathbb{C}''$ when the stability region is the complement of a compact polynomially convex domain $\Gamma $. This criterion restricts to the well-known result for two-dimensional systems when $\Gamma $ is the unit polydisc; it also allows the resolution of an open problem of Guiver [Multidimensional Systems Theory, D. Reidel, 1985]. Finally it is shown that while feedback stabilizability is robust, it is not, however, a generic property.