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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:


\begin{displaymath} \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


\begin{displaymath} \mathbf{M}(s)=\diag \left(\frac{\epsilon_1(s)}{\delta_1(s)},\ldots, \frac{\epsilon_r(s)}{\delta_r(s)},0,\dots,0\right) \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 $\delta_{i-1}(s)$.

Proof

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

\begin{displaymath} \mathbf{G}(s)=\begin{bmatrix}\displaystyle{\frac{4}{(s+1)(s... ...ac{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:


\begin{displaymath}\chi_0=1 \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)


\begin{displaymath}\chi_2=gcd\{2s^2+8s+6\}=s^2+4s+3=(s+1)(s+3) \end{displaymath} (B.4.7)

This leads to

\begin{displaymath}\overline{\epsilon}_1=\frac{\chi_1}{\chi_0}=1;\hspace{10mm} \overline{\epsilon}_2=\frac{\chi_2}{\chi_1}=(s+1)(s+3) \end{displaymath} (B.4.8)

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


 \begin{displaymath} \mathbf{G^{SM}}(s)= \begin{bmatrix}\displaystyle{\frac{1}{(s+1)(s+2)}}&0\\ 0&\displaystyle{\frac{s+3}{s+2}}\end{bmatrix}\end{displaymath} (B.4.9)