14.1 Background

Theorem([BGKvD]) A minimal factorization

PICT

of a system [A,B,C, 1] corresponds to projections P1 and P2 satisfying P1 + P2 = 1,

AP2  = P2AP2    (A  - BC)P1  =  P1(A - BC)P1
(14.1)

provided the state dimension of the [A,B,C, 1] system is d1 + d2. (which has the geometrical interpretation that A and A - BC have complimentary invariant subspaces).

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

(m1, m2) :  Statespace of the product  - →   Statespace of the original
(14.2)

which intertwines the original and the product system. Also minimality of the factoring is equivalent to the existence of a two sided inverse (n1T ,n 2T )T to (m 1,m2). These requirements combine to imply that each of the following expressions is zero.