You are here : Control System Design  Index  Book Contents  Appendix B  Section B.4 B. SmithMcMillan FormsB.4 SmithMcMillan Form for Rational MatricesA straightforward application of Theorem B.1 leads to the following result, which gives a diagonal form for a rational transferfunction matrix:
Theorem 2.2 (SmithMcMillan form)
Let be an matrix transfer function, where are rational scalar transfer functions:
where is an polynomial matrix of rank and is the least common multiple of the denominators of all elements . Then, is equivalent to a matrix , with
where is a pair of monic and coprime polynomials for . Furthermore, is a factor of and is a factor of . ProofWe write the transferfunction matrix as in (B.4.1). We then perform the algorithm outlined in Theorem B.1 to convert to Smith normal form. Finally, canceling terms for the denominator leads to the form given in (B.4.2).
We use the symbol to denote , which is the SmithMcMillan form of the transferfunction matrix . We illustrate the formula of the SmithMcMillan form by a simple example.
Example B.1 Consider the following transferfunction matrix
We can then express in the form (B.4.1):
The polynomial matrix can be reduced to the Smith form defined in Theorem B.1. To do that, we first compute its greatest common divisors:
This leads to
From here, the SmithMcMillan form can be computed to yield
