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B. Smith-McMillan Forms

B.6 Matrix Fraction Descriptions (MFD)

A model structure that is related to the Smith-McMillan form is that of a matrix fraction description (MFD). There are two types, namely a right matrix fraction description (RMFD) and a left matrix fraction description (LMFD).

We recall that a matrix \ensuremath{\mathbf{G}(s)} and its Smith-McMillan form \ensuremath{\mathbf{G^{SM}}(s)} are equivalent matrices. Thus, there exist two unimodular matrices, \ensuremath{\mathbf{L}(s)} and \ensuremath{\mathbf{R}(s)}, such that


\begin{displaymath} \ensuremath{\mathbf{G^{SM}}(s)} =\ensuremath{\mathbf{L}(s)}\ensuremath{\mathbf{G}(s)}\ensuremath{\mathbf{R}(s)}\end{displaymath} (B.6.1)

This implies that if \ensuremath{\mathbf{G}(s)} is an $m\times m$ proper transfer-function matrix, then there exist a $m\times m$ matrix \ensuremath{\mathbf{\tilde{L}}(s)} and an $m\times m$ matrix \ensuremath{\mathbf{\tilde{R}}(s)}, such as


\begin{displaymath} \ensuremath{\mathbf{G}(s)} =\ensuremath{\mathbf{\tilde{L}}(... ...nsuremath{\mathbf{G^{SM}}(s)}\ensuremath{\mathbf{\tilde{R}}(s)}\end{displaymath} (B.6.2)

where \ensuremath{\mathbf{\tilde{L}}(s)} and \ensuremath{\mathbf{\tilde{R}}(s)} are, for example, given by


\begin{displaymath}\ensuremath{\mathbf{\tilde{L}}(s)} =[\ensuremath{\mathbf{L}(s... ...ath{\mathbf{\tilde{R}}(s)} =[\ensuremath{\mathbf{R}(s)} ]^{-1} \end{displaymath} (B.6.3)

We next define the following two matrices:


\begin{displaymath}\ensuremath{\mathbf{N}(s)}\stackrel{\rm\triangle}{=}\diag(\epsilon_1(s),\ldots,\epsilon_r(s),0,\dots,0) \end{displaymath} (B.6.4)


\begin{displaymath}\ensuremath{\mathbf{D}(s)}\stackrel{\rm\triangle}{=}\diag(\delta_1(s),\ldots,\delta_r(s),1,\dots,1) \end{displaymath} (B.6.5)

where $ \ensuremath{\mathbf{N}(s)} $ and $\ensuremath{\mathbf{D}(s)} $ are $m\times m$ matrices. Hence, \ensuremath{\mathbf{G^{SM}}(s)} can be written as


\begin{displaymath} \ensuremath{\mathbf{G^{SM}}(s)} = \ensuremath{\mathbf{N}(s)} [\ensuremath{\mathbf{D}(s)} ]^{-1} \end{displaymath} (B.6.6)

Combining (B.6.2) and (B.6.6), we can write


\begin{displaymath} \ensuremath{\mathbf{G}(s)} =\ensuremath{\mathbf{\tilde{L}}(... ...suremath{\mathbf{G_N}(s)} [\ensuremath{\mathbf{G_D}(s)} ]^{-1} \end{displaymath} (B.6.7)

where


\begin{displaymath} \ensuremath{\mathbf{G_N}(s)}\stackrel{\rm\triangle}{=}\ensu... ...uremath{\mathbf{\tilde{R}}(s)} ]^{-1}\ensuremath{\mathbf{D}(s)}\end{displaymath} (B.6.8)

Equations (B.6.7) and (B.6.8) define what is known as a right matrix fraction description (RMFD) .

It can be shown that $\ensuremath{\mathbf{G_D}(s)} $ is always column-equivalent to a column proper matrix $\ensuremath{\mathbf{P}(s)} $. (See definition B.7). This implies that the degree of the pole polynomial $p_p(s)$ is equal to the sum of the degrees of the columns of $\ensuremath{\mathbf{P}(s)} $.

We also observe that the RMFD is not unique, because, for any nonsingular $m\times m$ matrix $\ensuremath{\mathbf{\ensuremath{\boldsymbol{\Omega}}}(s)} $, we can write $ \ensuremath{\mathbf{G}(s)} $ as


\begin{displaymath}\ensuremath{\mathbf{G}(s)} = \ensuremath{\mathbf{G_N}(s)}\ens... ...suremath{\mathbf{\ensuremath{\boldsymbol{\Omega}}}(s)} ]^{-1} \end{displaymath} (B.6.9)

where $\ensuremath{\mathbf{\ensuremath{\boldsymbol{\Omega}}}(s)} $ is said to be a right common factor. When the only right common factors of $\ensuremath{\mathbf{G_N}(s)} $ and $\ensuremath{\mathbf{G_D}(s)} $ are unimodular matrices, then, from definition definition B.7, we have that $\ensuremath{\mathbf{G_N}(s)} $ and $\ensuremath{\mathbf{G_D}(s)} $ are right coprime. In this case, we say that the RMFD $(\ensuremath{\mathbf{G_N}(s)} ,\ensuremath{\mathbf{G_D}(s)} )$ is irreducible.

It is easy to see that when a RMFD is irreducible, then

  • $s=z$ is a zero of $ \ensuremath{\mathbf{G}(s)} $ if and only if $\ensuremath{\mathbf{G_N}(s)} $ loses rank at $s=z$; and

  • $s=p$ is a pole of $ \ensuremath{\mathbf{G}(s)} $ if and only if $\ensuremath{\mathbf{G_D}(s)} $ is singular at $s=p$. This means that the pole polynomial of $ \ensuremath{\mathbf{G}(s)} $ is $p_p(s)=\det (\ensuremath{\mathbf{G_D}(s)} )$.

Remark B.1 A left matrix fraction description (LMFD) can be built similarly, with a different grouping of the matrices in (B.6.7). Namely,


\begin{displaymath}\ensuremath{\mathbf{G}(s)} =\ensuremath{\mathbf{\tilde{L}}(s)... ...verline{G}_D}(s)} ]^{-1}\ensuremath{\mathbf{\overline{G}_N}(s)}\end{displaymath} (B.6.10)

where


\begin{displaymath} \ensuremath{\mathbf{\overline{G}_N}(s)}\stackrel{\rm\triang... ...math{\mathbf{D}(s)} [\ensuremath{\mathbf{\tilde{L}}(s)} ]^{-1} \end{displaymath} (B.6.11)

$\Box \Box \Box $

The left and right matrix descriptions have been initially derived starting from the Smith-McMillan form. Hence, the factors are polynomial matrices. However, it is immediate to see that they provide a more general description. In particular, $\ensuremath{\mathbf{G_N}(s)} $, $\ensuremath{\mathbf{G_D}(s)} $, $\ensuremath{\mathbf{\overline{G}_N}(s)} $ and $\ensuremath{\mathbf{\overline{G}_N}(s)} $ are generally matrices with rational entries. One possible way to obtain this type of representation is to divide the two polynomial matrices forming the original MFD by the same (stable) polynomial.

An example summarizing the above concepts is considered next.

Example 2.2 Consider a $2\times 2$ MIMO system having the transfer function


\begin{displaymath} \ensuremath{\mathbf{G}(s)} =\begin{bmatrix}\displaystyle{\f... ...ac{1}{s+2}} & \displaystyle{\frac{2}{(s+1)(s+2)}}\end{bmatrix} \end{displaymath} (B.6.12)

B.2.1 Find the Smith-McMillan form by performing elementary row and column operations.
B.2.2 Find the poles and zeros.
B.2.3 Build a RMFD for the model.

Solution

B.2.1 We first compute its Smith-McMillan form by performing elementary row and column operations. Referring to equation (B.6.1), we have that


\begin{displaymath} \ensuremath{\mathbf{G^{SM}}(s)} =\ensuremath{\mathbf{L}(s)}... ...{0} & \displaystyle{\frac{s^2+3s+18}{(s+1)(s+2)}} \end{bmatrix}\end{displaymath} (B.6.13)

with


\begin{displaymath}\ensuremath{\mathbf{L}(s)} = \begin{bmatrix}\displaystyle{\fr... ...{8}} \\ [2mm] \displaystyle{0} & \displaystyle{1} \end{bmatrix}\end{displaymath} (B.6.14)
B.2.2 We see that the observable and controllable part of the system has zero and pole polynomials given by


\begin{displaymath}p_z(s)=s^2+3s+18; \hspace{20mm}p_p(s)=(s+1)^2(s+2)^2 \end{displaymath} (B.6.15)

which, in turn, implies that there are two transmission zeros, located at $-1.5 \pm j3.97$, and four poles, located at $-1,-1,-2$ and $-2$.

B.2.3 We can now build a RMFD by using (B.6.2). We first notice that


\begin{displaymath} \ensuremath{\mathbf{\tilde{L}}(s)} =[\ensuremath{\mathbf{L}... ...{8}} \\ [2mm] \displaystyle{0} & \displaystyle{0} \end{bmatrix}\end{displaymath} (B.6.16)

Then, using (2.24), with


\begin{displaymath} \ensuremath{\mathbf{N}(s)} =\begin{bmatrix}\displaystyle{1}... ...m] \displaystyle{0} & \displaystyle{(s+1)(s+2)} \end{bmatrix}\end{displaymath} (B.6.17)

the RMFD is obtained from (B.6.7), (B.6.16), and (B.6.17), leading to


\begin{displaymath}\ensuremath{\mathbf{G_N}(s)} =\begin{bmatrix}\displaystyle{4}... ...ystyle{s+1} & \displaystyle{\frac{s^2+3s+18}{8}} \end{bmatrix} \end{displaymath} (B.6.18)

and


\begin{displaymath}\ensuremath{\mathbf{G_D}(s)} =\begin{bmatrix}\displaystyle{1}... ...m]\displaystyle{0} & \displaystyle{(s+1)(s+2)} \end{bmatrix} \end{displaymath} (B.6.19)


\begin{displaymath}=\begin{bmatrix}\displaystyle{(s+1)(s+2)} & \displaystyle{\fr... ... \\ \displaystyle{0} & \displaystyle{(s+1)(s+2)} \end{bmatrix} \end{displaymath} (B.6.20)

These can then be turned into proper transfer-function matrices by introducing common stable denominators.

$\Box \Box \Box $