Feedback Stabilization over Commutative Rings: Further Study of the Coordinate-Free Approach

Abstract

This paper is concerned with the coordinate-free approach to control systems. The coordinatefree approach is a factorization approach but does not require the coprime factorizations of the plant. We present two criteria for feedback stabilizability for MIMO systems in which transfer functions belong to the total rings of fractions of commutative rings. Both of them are generalizations of Sule's results in [SIAM J. Control Optim., 32-6, 1675-1695(1994]. The first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal. It shows that if the plant is stabilizable, the modules are projective. The other criterion is expressed in terms of ideals called generalized elementary factors. This gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors. As an example, a discrete finite-time delay system is considered. ). 1 rings. Further, we will not use the theory of algebraic geometry.The paper is organized as follows. In § 2, we give mathematical preliminaries, set up the feedback stabilization problem, present the previous results, and define the causality of the transfer functions. In § 3, we state our main results. As a preface to our main results, we also introduce there the notion of the generalized elementary factor of a plant. In § 4, wegive intermediate results which we will utilize in the proof of the main theorem. In § 5, we prove our main theorem. In § 6, we discuss the causality of the stabilizing controllers. Also, in order to make the contents clear, we present examples concerning a discrete finite-time delay system in § 3, 4, 5 in series. Preliminaries.In the following we begin by introducing the notations of commutative rings, matrices, and modules, commonly used in this paper. Then we give in order the formulation of the feedback stabilization problem, the previous results, and the causality of transfer functions. Notations.Commutative Rings. In this paper, we consider that any commutative ring has the identity 1 different from zero. Let R denote a commutative ring. A zerodivisor in R is an element x for which there exists a nonzero y such that xy = 0. In particular, a zerodivisor x is said to be nilpotent if x n = 0 for some positive integer n. The set of all nilpotent elements in R, which is an ideal, is called the nilradical of R. A nonzerodivisor in R is an element which is not a zerodivisor. The total ring of fractions of R is denoted by F (R).The set of all prime ideals of R is called the prime spectrum of R and is denoted by Spec R. The prime spectrum of R is said to be irreducible as a topological space if every non-empty open set is dense in Spec R.We will consider that the set of stable causal transfer functions is a commutative ring, denoted by A. Further, we will use the following three kinds of rings of fractions:

A parameterization of stabilizing controllers over commutative rings

“…Then we give the formulation of the feedback stabilization problem and the previous results. We also review the construction method of a stabilizing controller presented in [6].…”

“…is unique up to an isomorphism with respect to any choice of fractions AB −1 of X ( B −1 A of X) (Lemma 2.1 of [6]). Therefore, for a matrix X over R, we denote by T X,R and W X,R the modules M r ([ A t B t ] t ) and M c ( A B ), respectively.…”

“…In such models, we cannot employ the Youla-Kučera parameterization directly. In this paper we will give a parameterization of the stabilizing controllers over commutative rings by using the results given by Sule [3], and Mori and Abe [5,6].…”

“…This approach to the stabilization theory is called "coordinate-free approach" in the sense that the coprime factorizability of transfer matrices is not required. Recently, Mori and Abe in [5,6] have generalized his theory over commutative rings under the assumption that the plant is causal. They have introduced the notion of the generalized elementary factor, which is a generalization of the elementary factor introduced by Sule [3], and have given the necessary and sufficient condition of the feedback stabilizability.Since the stabilizing controllers are not unique in general, the choice of the stabilizing controllers is important for the resulting closed loop.…”

“…In such models, we cannot employ the Youla-Kučera parameterization directly. In this paper we will give a parameterization of the stabilizing controllers over commutative rings by using the results given by Sule [3], and Mori and Abe [5,6].Here we briefly outline how the parameterization, which is different from the Youla-Kučera parameterization, will be obtained. Let H(P, C) denote the transfer matrix of ‡ Part of this work was done at Institut de Recherche en Cybernétique de Nantes, France and the author thanks this institution for its hospitality and support.…”