Frequency response functions for nonlinear rational models

Generalized frequency response function matrix for MIMO non-linear systems

“…The output of the system under the harmonic excitation of equation (9) becomes (Zhang et al 1995) y…t †ˆX N 1 nˆ1 X ‰all perm: of R freq: taken n at a timeŠ X ‰all perm: of…”

“…n †t . The procedure of computing`H ' by solving equation (11) was derived using an extraction operator°n‰:Š by Zhang et al (1995). For a given expression, the operator " n ‰:Š for SISO systems involves the execution of the following steps:…”

“…An alternative approach suggested by Billings and co-workers (Billings and Tsang 1989, Peyton Jones and Billings 1989, Billings and Peyton Jones 1990, Zhang et al 1995 involves estimating a parametric NARMAX (Leontaritis and Billings 1985) model and subsequently deriving the frequency response functions from this model using harmonic probing.…”

New Connections Between Frequency Response Functions for a Class of Nonlinear Systems*

“…The GFRFs generally require direct identification, identification of the corresponding Volterra kernels [Boyd et al, 1983], or identification of an underlying nonlinear model [Peyton Jones and Billings, 1989, Billings and Peyton Jones, 1990, Zhang et al, 1995. Hence, Theorem 1 significantly simplifies the identification process and provides useful insight in the mechanism that generates the GFRF.…”

Analysis of Output Frequencies of Nonlinear Systems

“…In Lang and Billings [6], an expression for the output frequency response of the nonlinear systems was derived in a manner that reveals how the underlying nonlinear mechanisms operate on the input spectra to produce the system output frequency response, when the system is excited by the general input (2) The result is given by (3) where and represent the Fourier transforms of the system output and input, represents the system th-order output frequency response, and (4) is known as the th-order GFRF, is the maximum order of the system nonlinearity (5) denotes the integration of over the -dimensional hyperplane , and reveals the way in which the input spectrum makes a contribution, of degree ,to the output frequency component . Equation (3) is a natural extension of the well-known linear relationship (6) to the nonlinear case, and compared with other results [11], [12], provides additional insight into the composition of the output frequency response of nonlinear systems.…”