Abstract: Based on the Smith-McMillan normal form of a transfer function matrix，a kind of minimal realizations of a proper rational transfer function matrix are discussed.The minimal realization of each invariant corresponds to a cycle system.If the rank of the Smith-McMillan normal form of a transfer function matrix is ｒ,then the minimal realization corresponds to a direct sum of ｒ　cycle systems.Especially,the controllability canonical realizations and the observability canonical realizations are obtained when a transfer function matrix is the form of column vector or row vector and when the system matrix is the form of block circulant matrix,which are the improvement of current conclusions.

Key words: control theory and control engineer, time-invariant linear system, transfer function matrix, invariant factors, minimal realization