Generalized Hermite Matrices and Complete Invariants of Strict System Equivalence
Generalized Hermite Matrices and Complete Invariants of Strict System Equivalence
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广义Hermite矩阵和严格系统等价的完全不变量
DOI：
10.1137/0321017
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发表时间：
1983
影响因子：
2.2
通讯作者：
D. Prätzel
中科院分区：
文献类型：
--
作者：
D. Hinrichsen;D. Prätzel
关于严格系统等价（s.s.e.），用多项式方法确定了可达系统矩阵\[ \Sigma (s) = \left[ {\begin{array}{*{20}c} {P(s)} & { - Q(s)} \\ {V(s)} & {W(s)} \\ \end{array} } \right] \]的不变量的完整列表。多项式输入输出对$(u,y)$，其中存在一个多项式向量z，使得$Pz = Qu$和$y = Vz + Wu$形成一个$K[s]$ -模块$\mu (\Sigma )$。证明了在Hermite形式下$\mu (\Sigma )$的唯一基矩阵产生一组完整的离散（“Hermite指数”）（见图2）。用状态空间项来表示Hermite不变量，并给出了$\Sigma (s)$的Hermite正则化实现。为了建立一个包含Hermite不变量和Kronecker不变量的框架，引入了Nice阶和广义Hermite形式。赫米特定理推广到这些矩阵。最后，nice order用于在所有最小值中挑选出唯一的代表。
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 bas...