Computation of convergence radius and error bounds of Volterra series for single input systems with a polynomial nonlinearity

In this paper, the Volterra series decomposition of a class of time invariant system, polynomial in the state and affine in the input, with an exponentially stable linear part is analyzed. A formal recursive expression of Volterra kernels of the input-to-state system is derived and the singular inversion theorem is used to prove the non-local-in-time convergence of the Volterra series to a trajectory of the system, to provide an easily computable value for the radius of convergence and to compute a guaranteed error bound for the truncated series. These results are available for infinite norms (Bounded Input Bounded Output results) and also for specific weighted norms adapted to some so-called ¿fading memory systems¿ (exponentially decreasing input-output results). The method is illustrated on two examples including a Duffing´s Oscillator.