last modified Sep 11, 1996
Theorem([BGKvD]) Minimal factorizations of a system [A, B, C, 1] correspond to projections P and P satisfying P + P = 1,
|A P = P A P||and||(A - B C) P = P (A - BC) P.|
We begin by giving the algebraic statement of the problem. Suppose that these factors exist. By the Youla-Tissi statespace isomorphism theorem, there is map
|(m, m): Statespace of the product Statespace of the original|
which intertwines the
original and the product system.
Also minimality of the factoring is equivalent to
the existence of a two-sided inverse
These requirements combine to imply that
each of the expressions of (FAC) below is zero.
Minimal factors exist if and only if
a, b, c, e,
f and g
the following polynomials are zero.
|A m - m a - m f c||A m - m e|
|B - m b - m f||-c + C m|
|(FAC)||n m - 1||n m - 1|
|n m||n m|
|-g + C m||m n + m n - 1|