Canonical forms of singular 1D and 2D linear systems (Q2702523)

This paper is devoted to canonical forms of singular 1D and 2D discrete linear systems. For 1D systems of the form NEWLINE\[NEWLINEEx_{i+1}= Ax_i+ Bu_i,\quad y_i= Cx_iNEWLINE\]NEWLINE a canonical form \((\overline E,\overline A,\overline B,\overline C)\) has been defined, and sufficient conditions for the existence of a pair of matrices \(P\), \(Q\) transforming \((E,A,B,C)\) to this canonical form (i.e. such that \(\overline E= PEQ\), \(\overline A= PAQ\), \(\overline B= PB\), \(\overline C= CQ\)) is established (Theorem 3). Similarly, for 2D systems in Roesser form NEWLINE\[NEWLINEE\left[\begin{matrix} x^h_{i+ 1j}\\ x^\nu_{i+ 1j}\end{matrix}\right]= A\left[\begin{matrix} x^h_{ij}\\ x^\nu_{ij}\end{matrix}\right]+ Bu_{ij},\quad y_{ij}= C\left[\begin{matrix} x^h_{ij}\\ x^\nu_{ij}\end{matrix}\right]NEWLINE\]NEWLINE a canonical form has been proposed, and necessary (Theorem 4) and sufficient (Theorem 5) conditions have been derived for the existence of a pair of block-diagonal matrices NEWLINE\[NEWLINEP= \left[\begin{matrix} P_1 & 0\\ 0 & P_2\end{matrix}\right],\quad Q= \left[\begin{matrix} Q_1 & 0\\ 0 & Q_2\end{matrix}\right]NEWLINE\]NEWLINE transforming a system to its canonical form. In both the 1D and the 2D cases, explicit procedures for constructing the matrices \(P\), \(Q\) are provided.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00062].