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B. Smith-McMillan Forms

B.4 Smith-McMillan Form for Rational Matrices

A straightforward application of Theorem B.1 leads to the following result, which gives a diagonal form for a rational transfer-function matrix:

Theorem 2.2 (Smith-McMillan form)
Let $\mathbf{G}(s)=[G_{ik}(s)]$ be an $m\times m$ matrix transfer function, where $G_{ik}(s)$ are rational scalar transfer functions:

			  \mathbf{G}(s)=\frac{\mathbf{\ensuremath{\boldsymbol{\Pi}} }(s)}{D_G(s)}
			  \end{displaymath} (B.4.1)

where $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$ is an $m\times m$ polynomial matrix of rank $r$ and $D_G(s)$ is the least common multiple of the denominators of all elements $G_{ik}(s)$.

Then, $\mathbf{G}(s)$ is equivalent to a matrix \ensuremath{\mathbf{M}(s)}, with

			  \mathbf{M}(s)=\diag \left(\frac{\epsilon_1(s)}{\delta_1(s)},\ldots,
			  \end{displaymath} (B.4.2)

where $\{\epsilon_i(s),\delta_i(s)\}$ is a pair of monic and coprime polynomials for $i=1,2,\ldots,r$.

Furthermore, $\epsilon_i(s)$ is a factor of $\epsilon_{i+1}(s)$and $ \delta_i(s)$ is a factor of $\delta_{i-1}(s)$.


We write the transfer-function matrix as in (B.4.1). We then perform the algorithm outlined in Theorem B.1 to convert $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$ to Smith normal form. Finally, canceling terms for the denominator $D_G(s)$ leads to the form given in (B.4.2).

$\Box \Box \Box $

We use the symbol $\mathbf{G}^{SM}(s)$ to denote $\ensuremath{\mathbf{M}(s)} $, which is the Smith-McMillan form of the transfer-function matrix $\mathbf{G}(s)$ .

We illustrate the formula of the Smith-McMillan form by a simple example.

Example B.1 Consider the following transfer-function matrix

			  \mathbf{G}(s)=\begin{bmatrix}\displaystyle{\frac{4}{(s+1)(s...{2}{s+1}}&\displaystyle{\frac{-1}{2(s+1)(s+2)}}\end{bmatrix} \end{displaymath} (B.4.3)

We can then express $\mathbf{G}(s)$ in the form (B.4.1):

\begin{displaymath}\mathbf{G}(s)=\frac{\mathbf{\ensuremath{\boldsymbol{\Pi}} }(s...
			  ...+2)&-\dfrac{1}{2}\end{bmatrix}; \hspace{5mm}D_G(s)=(s+1)(s+2)
			  \end{displaymath} (B.4.4)

The polynomial matrix $\mathbf{\ensuremath{\boldsymbol{\Pi}} } (s)$ can be reduced to the Smith form defined in Theorem B.1. To do that, we first compute its greatest common divisors:

			    \end{displaymath} (B.4.5)

\begin{displaymath}\chi_1=gcd\left\{4;-(s+2);2(s+2);-\frac{1}{2} \right\}=1
			  \end{displaymath} (B.4.6)

			    \end{displaymath} (B.4.7)

This leads to

			  \end{displaymath} (B.4.8)

From here, the Smith-McMillan form can be computed to yield

			  0&\displaystyle{\frac{s+3}{s+2}}\end{bmatrix}\end{displaymath} (B.4.9)