On the irreducible Jordan form realization and the degree of a rational matrix

A new method for realizing a rational transfer-function matrix, in the factored form, into an irreducible Jordan canonical form state equation is presented. The method consists of two steps. First, the irreducible Jordan form structure of the internal dynamics of the system is determined simply from the ranks of an augmented coefficient matrix and its submatrices, without actually having to construct a realization. Second, the augmented coefficient matrix is decomposed in a simple manner to obtain the final realization. The construction procedure is straightforward and allows one to choose explicitly the element values with a high degree of freedom. As a natural consequence of the irreducible realization, the rank of the augmented coefficient matrix associated with a pole is defined as the degree of the pole in question. This new definition is equivalent to the McMillan/Duffin-Hazony/Kalman degree. Using this definition, many well-known properties of the degree of a rational matrix are readily established.