On the bilinearity of cascaded bilinear systems

When two bilinear state space systems are interconnected in a cascade fashion, the resulting input-output map may not have a bilinear realization. In 1972, Brockett asked under what conditions bilinearity is 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. As this result is not widely known, the first goal of this paper is to present the idea in a broad context and to supply a concise, self- contained proof. The approach taken here follows largely from the original and is based on the theory of rational transductions. A second goal of the paper is to explore generalizations of Ferfera´s sufficient condition.