任何东西说空格实现，这在这篇论文被显示出(一， b ， c )给定的转移函数 T =(β( s ))/(α( s ))与α( s ) monic 和暗淡(A)= deg (α( s ))，满足身份β( A )= Q_c (一， b )S_ αQ _o (一， c )在哪儿 Q_c (一， b )并且 Q_o (一， c )是可控制性矩阵和矩阵三元组的 observability 矩阵(一， b ， c )，分别地并且S_α 是一个非退化的对称的矩阵。如此的身份给在州的空格描述和对的转移函数描述之间的一种深关系单个输入单个产量(SISO ) 线性系统。作为一个直接结论，如果并且仅当分子和转移功能的分母是警察白霜，我们到达任何转移功能的一条认识是最小的著名结果。如此的结果也被扩大到 SISO 描述符线性系统大小写。作为一应用程序，变换矩阵方程斧子 = X A 的一个完全的答案被建议并且线性系统被考虑的多输入多产量(MIMO ) 的最小的实现。
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机译：It is shown in this paper that any state space realization (A, b, c) of a given transfer function T(s) = β(s)/α(s)with α(s) monic and dim(A) = deg(α(s)), satisfies the identity β(A) = Qc(A, b)SαQο(A, c) where Qc(A, b) and Qο(A, c) are the controllability matrix and observability matrix of the matrix triple (A, b, c), respectively, and Sα is a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output (SISO) linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications,a complete solution to the commuting matrix equation AX = XA is proposed and the minimal realization of multi-input multi-output (MIMO) linear system is considered.
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