Feedback invariants for linear dynamical systems over a principal ideal domain
type: Item
instance of: scholarly article
title: Feedback invariants for linear dynamical systems over a principal ideal domain
zbMATH Open document ID: 0824.93015
DOI: 10.1016/0024-3795(93)00153-Q
author: M. Pilar Perez
author: Jose Angel Hermida Alonso
author: T. Sánchez-Giralda
published in: Linear Algebra and its Applications
publication date: 1995-10-25T00:00:00Z
review text: The paper studies the action of the feedback group on linear reachable systems \(\Sigma = (F,G)\) defined over a principal ideal domain. Let \((F,G)\) be a pair of matrices of size \(n \times n\) and \(n \times m\) with entries in a principal ideal domain \(R\). The feedback group acting on \((F,G)\) is generated by the following three types of transformations \((F,G) \to (PFP^{-1}, PG)\), \((F,G) \to (F,GQ)\), \((F,G) \to (F + GK,G)\) for invertible matrices \(P\), \(Q\) and for \(m \times n\) matrices \(K\) over \(R\). A pair \((F', G')\) is feedback equivalent to \((F,G)\) if it can be obtained from \((F,G)\) by one element of the feedback group. When \(n = 2\) and \((F,G)\) is reachable, i.e., the columns of \([G FG]\) generate \(R^ 2\), the pair \((F, G)\) is shown to be feedback equivalent to a pair of the form \[ \widehat {F} = \left [ \begin{smallmatrix} 0 & 0 \\ f & 0 \end{smallmatrix} \right],\quad \widehat {G} = \left[ \begin{smallmatrix} 1 & 0 &\ldots & 0\\ 0 & d &\ldots & 0\end{smallmatrix} \right], \] where \(d\) and \(f\) are coprime elements of \(R\). Here \(d\) is uniquely determined by \((F,G)\) upto multiplication by units in \(R\) but \(f\) is not such an invariant. The pair \(\{f, d\}\) however has the following property: Two reachable pairs \((F,G)\) and \((F', G')\) are feedback equivalent if and only if \(d' = vd\) and \(f' = uh^ 2 f \pmod d\) for units \(u\), \(v\) and an element \(h\) of \(R\), where \(\{f, d\}\) and \(\{f', d'\}\) are associated with \((F, G)\) and \((F', G')\), respectively. If \(\widetilde {\mathbf f}\) denotes the equivalence class in the quotient set \((R/ (d))^*\) obtained by the equivalence relation ``\(f \sim f'\) if and only if \(f' = uh^ 2 f \pmod d\) for some \(h\) and for some unit \(u\) in \(R\)'', then \(\{\widetilde {{\mathbf f}}, d\}\) is a complete set of invariants of \((F, G)\) under the action of the feedback group. This result is applied to obtain a characterization of reachable pairs \((F,G)\) with \(n…
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 93B52
zbMATH DE Number: 753859
zbMATH Keywords: systems over rings
zbMATH Keywords: feedback invariants
zbMATH Keywords: feedback group
zbMATH Keywords: principal ideal domain
zbMATH Keywords: reachable pairs
MaRDI profile type: Publication
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Wikidata QID: Q127610199
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description: scientific article; zbMATH DE number 753859