Feedback invariants for linear dynamical systems over a principal ideal domain
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 \(n = 2\) having the feedback cyclization property. A pair \((F,G)\) is said to have feedback cyclization property if there exists a feedback equivalent pair \((F', G')\) to \((F,G)\) and a vector \(w \in R^ m\) such that \((F', G'w)\) is reachable. It is shown that a reachable \((F,G)\) has feedback cyclization property if and only if \(f = uh^ 2 \pmod d\) for a unit \(u\) and an element \(h\) in \(R\). Using this result the cases where \(R\) is the ring of integers and \(R\) is the polynomial ring are studied in terms of the cardinal of the quotient set \((R/ (d))^*\). For the general case of arbitrary \(n\), a similar feedback classification is given for the class of reachable pairs \((F,G)\) such that \(G\) has at most one nonunit invariant factor.
scientific article; zbMATH DE number 753859
scientific article