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

Abstract

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)SαQo(A, c) where Qc(A, b) andQo (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.

On coprimeness of two polynomials in the framework of conjugate product

“…In addition, it was shown in [7] that two polynomials and are coprime if and only if where B is the companion matrix of with the coefficients of in the top row. In [8], an identity was established on coprimeness, controllability and observability.…”