An identity concerning controllability observability and coprimeness of linear systems and its appli

• 作 者: Bin ZHOU Guangren DUAN Shenmin SONG
• 机 构: Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin Heilongjia
• 出 处: 控制理论与应用（英文版）
• 发表日期: 2007年2期
• 关键词: Controllability Observability Resultant matrix Coprime Commuting matrix equation

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.