On cascades of bilinear systems and generating series of weighted Petri nets

It has been established in the literature that the cascade interconnection of two bilinear systems does not in general produce another bilinear system. The goals of this paper are two-fold. First, an alternative proof of the sufficient condition for preserving bilinearity under cascades due to Ferfera is presented which is much simpler than the original. Then it is shown that the well known correspondence between rational series and formal power series recognized by weighted finite-state automata can be generalized to produce a correspondence between the generating series of cascades of bilinear systems and a class of weighted Petri nets.