Best Linear Approximation revisited: Random Gain Approach

Abstract

Measurement sciences counts numerous applications wherein one wishes to characterize the dynamics of a system. In case this characterization is intended non-parametrically the Frequency Response Function (FRF) is instrumental to achieve this goal. In the presence of nonlinearities of the system, the linear dynamics are accessible through the FRF which is known as the best linear approximation (BLA) of the nonlinear system. Once the BLA is determined and a periodic excitation was applied, one may describe the nonlinear distortion and the measurement noise characteristics. The description of the nonlinear distortion and measurement noise remains currently restricted to establishing its variance of both sources of error.In this paper, we go a step further in the characterization of the stochastic properties of the nonlinear distortion. Indeed, we show that the for a class of nonlinear systems, the nonlinear distortion acts as a stochastic gain contribution to the measured BLA depending on the input signal’s root mean square (RMS) value. Moreover we reveal that the random gain follows a log-normal distribution which allows improved uncertainty bounds on the FRF in the presence of nonlinear distortions.