Finite Volterra-series realizations and input-output approximations of non-linear discrete-time systems

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Abstract
It is shown that a separability property of a given finite family of stationary kernels turns out to be necessary and sufficient for the realization of the associated functional by means of a polynomial affine system (i.e. a system that is polynomial in the input and affine in the state). Moreover, the discrete Volterra kernels associated with the input-output map of a non-linear analytic discrete-lime system, initialized at an equilibrium point, are shown to possess a separability property. On this basis, we state an approximation result for the given input-output map by considering the first kernels of the discrete Volterra series. Two explicit constructions of the approximating polynomial affine systems are proposed.