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| Received:March 29, 2006Revised:May 25, 2006 |
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| An identity concerning controllability observability and coprimeness of linear systems and its applications |
| Bin ZHOU;Guangren DUAN;Shenmin SONG |
| (Center for Control Theory and Guidance Technology,
Harbin Institute of Technology, Harbin Heilongjiang 150001,
China) |
| 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)
and Qo(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. |
| Key words: Controllability Observability Resultant matrix Coprime Commuting
matrix equation Realization |
