Bilinear system interconnections and generating series of weighted Petri nets

This paper has three objectives. The first objective is to provide a simpler proof concerning a sufficient condition due to Ferfera under which bilinearity is preserved for cascade interconnections. The next objective is to provide a specific counterexample to show that this condition does not apply to the feedback connection. The final objective is to show 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 cascaded and feedback connected bilinear systems and a class of weighted Petri nets.