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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 , .
(i) and are said to be the zero polynomial and the pole polynomial of , respectively, where (B.5.1)

and where , , , and , , , are the polynomials in the Smith-McMillan form, of .

Note that and are monic polynomials.

(ii) The zeros of the matrix are defined to be the roots of , and the poles of are defined to be the roots of .
(iii) The McMillan degree of is defined as the degree of .

In the case of square plants (same number of inputs as outputs), it follows that is a simple function of and . Specifically, we have (B.5.2)

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.