Computation of convergence radius and error bounds of Volterra series for multiple input systems with an analytic nonlinearity in state

In this paper, the Volterra series decomposition of a class of multiple input time-invariant systems, analytic in state and affine in inputs is addressed. Computable bounds for the non-local-in-time convergence of the Volterra series to a trajectory of the system are given for infinite norms (Bounded Input Bounded Output results) and for specific weighted norms adapted to some “fading memory systems” (exponentially decreasing input-output results). This work extends results previously obtained for polynomial single input systems. Besides the increase in combinatorial complexity, a major difference with the single input case is that inputs may play different roles in the system behavior. Two types of inputs (called “principal” and “auxiliary”) are distinguished in the convergence process to improve the accuracy of the bounds. The method is illustrated on the example of a frequency-modulated Duffing´s oscillator.