On the Rationality of the Composition Product: A Survey

When two bilinear state space systems are interconnected in a cascade fashion, the resulting input-output map may not have a bilinear realization. So in 1972, Brockett asked under what conditions is bilinearity preserved under composition. In 1979, Ferfera produced the least restrictive sufficient condition which is presently known using formal power series representations of the input-output maps. The primary intent of this paper is to supply a self-contained proof of Ferfera´s sufficient condition for preserving rationality under composition, or equivalently, preserving bilinearity under composition. The proof provided here, which in the large follows the original, is somewhat less complicated as it employs different tools from the theory of rational transductions.