Standard forms associated with the state-space model of a system are presented in Chapter 8 of Control System Professional. The Smith standard form of a polynomial matrix and the McMillan standard form of a rational polynomial matrix, presented here, can reveal further important system properties. A linear continuous time-invariant system described by the state-space equations where x(t) is an n × 1 vector of (state) variables, u(t) is a p × 1 vector of input functions, and y(t) is a q × 1 vector of outputs, can equally be described by the corresponding transfer-function representation where Any rational q × p matrix G(s) can be reduced to its McMillan form, as follows. Let d(s) be the monic least common denominator of G(s), and let G(s) be expressed as where N(s) is a q × p polynomial matrix. By means of unimodular matrices L(s) and R(s), that is, nonsingular matrices whose determinants are independent of the variable s, N(s) can be reduced to its Smith form where where the  i(s) are the invariant polynomials of N(s).
The McMillan form of G(s) is then given by where M(s) is the result of dividing the Smith form of N(s) by d(s), and canceling out all common factors that exist between numerator and denominator elements on the leading diagonal of M(s). Consider the transfer-function matrix The Smith form of N(s) can be determined, as follows:
Let fi (s) be the monic highest common factor of all i × i minors of N(s), that is, here Then, the diagonal elements i(s) in the Smith form of N(s) are determined as where the fi(s) are called the determinantal divisors of N(s), and f0(s)=1, always. So, the Smith form of N(s) is and the McMillan form of G(s) is |
