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
(n,
n)
to
(m,
m).
These requirements combine to imply that
each of the expressions of (FAC) below is zero.
The problem
Minimal factors exist if and only if
there exist
m ,
m ,
n ,
n ,
a, b, c, e,
f and g
such that
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 |