[Paper Review] Feedback Stabilization over Commutative Rings: Further study of the coordinate-free approach

Kazuyoshi Mori, Kenichi Abe|ArXiv.org|Feb 22, 1999

Advanced Topics in Algebra4 citations

TL;DR

This paper develops a coordinate-free approach to feedback stabilization for MIMO systems over commutative rings, generalizing Sule's results by introducing two criteria: one based on projective modules generated from causal plants, and another using generalized elementary factors. It establishes stabilizability conditions without requiring coprime factorizations or strict causality, with application to discrete finite-time delay systems.

ABSTRACT

This paper is concerned with the coordinate-free approach to control systems. The coordinate-free 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.

Motivation & Objective

• To generalize Sule's feedback stabilizability results for MIMO systems over commutative rings using a coordinate-free framework.
• To eliminate the need for coprime factorizations of the plant, a key limitation in traditional factorization approaches.
• To develop stabilizability criteria applicable to both causal and non-strictly causal plants.
• To introduce generalized elementary factors as a tool for assessing stabilizability via coprimeness conditions.
• To demonstrate the applicability of the framework through a discrete finite-time delay system example.

Proposed method

• Formulates feedback stabilizability in terms of the projective property of modules generated from the plant's transfer functions over a total ring of fractions.
• Introduces generalized elementary factors as ideals derived from the plant's transfer function matrix.
• Establishes a stabilizability criterion based on the coprimeness of these generalized elementary factors.
• Applies the module-theoretic approach to handle non-strictly causal systems, avoiding assumptions on strict causality.
• Uses the algebraic structure of commutative rings and their total rings of fractions to generalize control-theoretic results.
• Employs a coordinate-free framework that avoids explicit state-space representations or parameterizations.

Experimental results

Research questions

• RQ1 How can feedback stabilizability be characterized without relying on coprime factorizations of the plant?
• RQ2 What algebraic conditions on modules generated by the plant ensure stabilizability over commutative rings?
• RQ3 Can the concept of generalized elementary factors be used to determine stabilizability in a coordinate-free setting?
• RQ4 How does the proposed framework extend Sule’s results to non-strictly causal MIMO systems?
• RQ5 What is the role of projective modules in characterizing stabilizable systems over commutative rings?

Key findings

• A causal MIMO plant is feedback stabilizable if and only if the module generated by its transfer functions is projective over the total ring of fractions.
• The stabilizability of a causal plant is equivalent to the coprimeness of its generalized elementary factors.
• The proposed criteria generalize Sule’s results by removing the requirement for coprime factorizations and strict causality.
• The framework applies to systems with finite-time delays, as demonstrated in the discrete-time example.
• The coordinate-free approach provides a unified algebraic characterization of stabilizability using module theory and ideal-theoretic conditions.
• The results are valid for total rings of fractions of commutative rings, extending the scope beyond polynomial or rational function rings.