Technical communique: Uniquely connecting frequency domain representations of given order polynomial Wiener-Hammerstein systems

The notion of frequency response functions has been generalized to nonlinear systems in several ways. However, a relation between different approaches has not yet been established. In this paper, frequency domain representations for nonlinear systems are uniquely connected for a class of nonlinear systems. Specifically, by means of novel analytical results, the generalized frequency response function (GFRF) and the higher order sinusoidal input describing function (HOSIDF) for polynomial Wiener-Hammerstein systems are explicitly related, assuming the linear dynamics are known. Necessary and sufficient conditions for this relation to exist and results on the uniqueness and equivalence of the HOSIDF and GFRF are provided. Finally, this yields an efficient computational procedure for computing the GFRF from the HOSIDF and vice versa.