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B. Smith McMillan Forms

B.5 Poles and Zeros

The Smith-McMillan form can be utilized to give an unequivocal definition of poles and zeros in the multivariable case. In particular, we have:

Definition B.11 Consider a transfer-function matrix , $\mathbf{G}(s)$.
(i) $p_z(s)$ and $p_p(s)$ are said to be the zero polynomial and the pole polynomial of $\mathbf{G}(s)$, respectively, where


\begin{displaymath}p_z(s)\stackrel{\rm\triangle}{=}\epsilon_1(s)\epsilon_2(s)\cd... ...krel{\rm\triangle}{=}\delta_1(s)\delta_2(s)\cdots \delta_r(s) \end{displaymath} (B.5.1)

and where $ \epsilon_1(s)$, $\epsilon_2(s)$, $\ldots$, $\epsilon_r(s)$ and $ \delta_1(s)$, $\delta_2(s)$, $\ldots$, $\delta_r(s)$ are the polynomials in the Smith-McMillan form, $\mathbf{G}^{SM}(s)$ of $\mathbf{G}(s)$.

Note that $p_z(s)$ and $p_p(s)$ are monic polynomials.

(ii) The zeros of the matrix $\mathbf{G}(s)$ are defined to be the roots of $p_z(s)$, and the poles of $\mathbf{G}(s)$ are defined to be the roots of $p_p(s)$.
(iii) The McMillan degree of $\mathbf{G}(s)$ is defined as the degree of $p_p(s)$.

In the case of square plants (same number of inputs as outputs), it follows that $ \det[\mathbf{G}(s)]$ is a simple function of $p_z(s)$ and $p_p(s)$. Specifically, we have


\begin{displaymath}\det[\mathbf{G}(s)]=K_\infty \frac{p_z(s)}{p_p(s)} \end{displaymath} (B.5.2)

Note, however, that $p_z(s)$ and $p_p(s)$ are not necessarily coprime. Hence, the scalar rational function $ \det[\mathbf{G}(s)]$is not sufficient to determine all zeros and poles of $\mathbf{G}(s)$. However, the relative degree of $ \det[\mathbf{G}(s)]$ is equal to the difference between the number of poles and the number of zeros of the MIMO transfer-function matrix.