B. Smith McMillan Forms
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 , .
In the case of square plants (same number of inputs as outputs), it follows that is a simple function of and . Specifically, we have
Note, however, that and are not necessarily coprime. Hence, the scalar rational function is not sufficient to determine all zeros and poles of . However, the relative degree of is equal to the difference between the number of poles and the number of zeros of the MIMO transfer-function matrix.