J-Lossless Factorizations

Assume that G is a stable and invertible transfer function. If G-1 has a stabilizing J-lossless conjugator Θ, then 6.1 $$ H\text{ : = }G^{ - 1} \Theta $$ is stable. Due to Lemma 5.3, the zeros of H coincide with those of G-1 which are stable from the assumption that G is stable. Hence, H-1 is also stable. Writing the relation (6.1) as $$ G\text{ = }\Theta H^{ - 1} , $$ we see that G is represented as the product of a J-lossless matrix Θ and a unimodular matrix H-1 . This is a factorization of G which is of fundamental importance in H∞ control theory.