Canonical forms for linear systems (Q794964)

Canonical forms associated to time-invariant linear systems in the state- space description \(\dot x(t)=Ax(t)+Bu(t)\) or \(x(k+1)=Ax(k)+Bu(k)\) are studied. Given \(\Sigma_{nm}=\{(A,B)\in {\mathbb{C}}^{n.n}\times {\mathbb{C}}^{n.m}\}\) and denoting by \(S_{AB}\) the orbit of the Gl(n)- action on \(\Sigma_{nm}\), i.e. \(Gl(n)\times \Sigma_{nm}\to \Sigma_{nm},\quad(H,(A,B))\mapsto(HAH^{-1},NB),\) a canonical form on \(\Sigma_{nm}\) is a map \(c:\Sigma_{nm}\to \Sigma_{nm}\) such that \(c(A,B)\in S_{AB},\quad c(A,B)=c(\hat A,\hat B)\Leftrightarrow S_{AB}=S_{\hat A\hat B}.\) These canonical forms are discussed in particular for three special cases where a homogeneous interpretation is given in terms of (\(\prec)\)-minimal orbit elements, where the total order (\(\prec)\) is defined by the author. These cases are: completely uncontrollable systems \((B=0)\), where the orbits \(S_ A\) are parametrized by the Jordan canonical form; integrators \((A=0)\), where the orbits \(R_ B\) contain row echelon matrices, and reachable (A,B)-pairs (i.e. \(rank[B AB...A^{n-1}B]=n).\) New canonical forms for the general Gl(n)-action on \(\Sigma_{nm}\) are also derived, using the (\(\prec)\) order.