Recent results in realisation theory of systems with discrete-time bilinear and multilinear response maps

In the paper, a review of recent results in the realisation theory of systems having multilinear input/output response functions is given. Such systems form a subclass of the more general class of polynomial systems, which in turn are a very special class of nonlinear system. It is shown that, in the bilinear case, the problem has essentially been resolved by two important results: an extension of the uniqueness result for canonical realisations of linear systems to the polynomial case, and a computable criterion for the quasireachability of realisations of bilinear response functions. These results, together with those of Kalman´s early investigations into multilinear response functions, lead to an explicit canonical realisation procedure. In the general multilinear case, however, only a few results are available, and the many questions which remain to be resolved are described.