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![]() ![]() ![]() | and | (A - B C)
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![]() ![]() ![]() |
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![]() ![]() ![]() |
A
m![]() ![]() | |
B
-
m![]() ![]() |
-c +
C
m![]() | |
(FAC) |
n![]() ![]() |
n![]() ![]() |
n![]() ![]() |
n![]() ![]() | |
-g +
C
m![]() |
m![]() ![]() ![]() ![]() |