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

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

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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.

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