Classical state space realization of the rational NARMAX model

Rational models have been gradually adopted in various applications of nonlinear systems in the area of the systems identification and control because they have the advantage of modeling certain types of discontinuous functions and even severe non-linearilities using only a very few parameters. Based on the principle of integrability of certain ℌk subspaces of one-forms, this work presents a new procedure to compute the subspace ℌs+2 of one-forms so that a class of rational NARMAX model, starting from an input-output nonlinear rational difference equation, can be realized via the classical state space representation in case that the systems is realizable. The procedure is noticeable less time-consuming and therefore it will tell us whether or not the system has a classical state space representation in a faster way. This allows to simplify the step-by-step algorithm given in terms of sequence of subspaces and the non-commutative skew polynomial ring procedure. Three examples are included to show the performance of this simple approach.