Generalized Hermite Matrices and Complete Invariants of Strict System Equivalence
D. Hinrichsen, D. Prätzel-Wolters · SIAM Journal on Control and Optimization · 1983
A complete list of invariants for reachable system matrices \[ \Sigma (s) = \left[ {\begin{array}{*{20}c} {P(s)} & { - Q(s)} \\ {V(s)} & {W(s)} \\ \end{array} } \right] \]with respect to strict system equivalence (s.s.e.) is determined by polynomial methods. The polynomial input-output pairs $(u,y)$ for which there exists a polynomial vector z such that $Pz = Qu$ and $y = Vz + Wu$ form a $K[s]$-module $\mu (\Sigma )$. It is shown that the unique basis matrix of $\mu (\Sigma )$ in Hermite form yields a complete set of discrete (“Hermite indices”) (resp. continuous) invariants of s.s.e. The Hermite invariants are characterized in state space terms, and a realization of $\Sigma (s)$ in Hermite canonical form is presented. Nice orders and generalized Hermite forms are introduced in order to develop a framework that encompasses Hermite invariants and Kronecker invariants. Hermite’s theorem is generalized to these matrices. Finally, nice orders are used to single out unique representatives among all minimal bases of a given full submodule $M \subset \mathbb{K} [s]^m $ and Forney’s echelon form is characterized in this framework.