In this paper, a lower bound of maximal dimensions of commutable matrix spaces (CMS) is given. It is found that the linear dependence of a group of one to one commutable matrices is related to whether some equations i...In this paper, a lower bound of maximal dimensions of commutable matrix spaces (CMS) is given. It is found that the linear dependence of a group of one to one commutable matrices is related to whether some equations in system can be eliminated. The corresponding relation is given. By introducing conceptions of eliminating set and eliminating index, we give an estimation of upper bound of maximal dimensions of CMS. For special cases n=5,6, the further estimation of maximal dimensions of CMS is presented.展开更多
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）=Qe（A,b）Sα Qo（...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）=Qe（A,b）Sα Qo（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 S,~ 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.展开更多
基金Supported by the Youth Mainstay Teacher Foundation of HunanProvince Educational Committee
文摘In this paper, a lower bound of maximal dimensions of commutable matrix spaces (CMS) is given. It is found that the linear dependence of a group of one to one commutable matrices is related to whether some equations in system can be eliminated. The corresponding relation is given. By introducing conceptions of eliminating set and eliminating index, we give an estimation of upper bound of maximal dimensions of CMS. For special cases n=5,6, the further estimation of maximal dimensions of CMS is presented.
基金the Chinese Outstanding Youth Foundation(No. 69925308)Program for Changjiang Scholars and Innovative Research Team in University.
文摘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）=Qe（A,b）Sα Qo（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 S,~ 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.
