Structure at infinity and defect of transfer matrices with time-varying coefficients, with application to exact model-matching

• Published in: Linear algebra and its applications Vol. 698; pp. 189 - 219
• Main Author: Bourlès, Henri
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2024; Elsevier
• Subjects: Defect; Engineering Sciences; Mathematics; Model matching; Time-varying; Transfer matrix; 93B25; Transfer matrix; Model matching; 93C05; Time-varying; 93N10; Defect; defect model matching time-varying transfer matrix AMS subject classification 93B25 93N10 93C05; defect; time-varying; transfer matrix AMS subject classification 93B25; model matching
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2024.06.009

We study the structure at infinity of transfer matrices with time-varying coefficients. Such transfer matrices have their entries in a skew field F of rational fractions, i.e. of quotients of skew polynomials. Any skew rational fraction is the quotient of two proper ones, the latter forming a ring Fpr (a subring of F) on which a “valuation at infinity” is defined. A transfer matrix G has both a “generalized degree” and a valuation at infinity, the sum of which is the opposite of the “defect” of G. The latter was first defined by Forney in the time-invariant case to be the difference between the total number of poles and the total number of zeros of G (poles and zeros at infinity included and multiplicities accounted for). In our framework, which covers both continuous- and discrete-time systems, the classic relation between the defect and Forney's left- and right-minimal indices is extended to the time-varying case. The exact model-matching problem is also completely solved. These results are illustrated through an example belonging to the area of power systems.