A connection between multi-linear and Volterra systems

The Volterra system is a non-linear system with the structure of the Volterra series. The Volterra series is attractive from the system-theoretic point of view, since it enables to obtain the output of a class of non-linear systems in terms of the input explicitly rather than involving input–output coupling terms and allows substantial simplifications for the numerical simulation. The Volterra system allows to derive the stability condition as well, i.e. obtain a bound on the output for a given bound on the input function, especially for the bilinear system. The bilinear system possesses the Volterra series. This paper derives the Volterra series formalism from the multi-linear system involving the coupling term attributed to (k-1)(k-1)th order non-linearity and output function, where 1