This paper presents a method for estimating a linear time-varying approximation of a general class of nonlinear time-varying systems. It starts from noisy measurements of the response of the nonlinear time-varying system to a special class of periodic excitation signals. These measurements are subject to measurement noise, process noise and a trend. The proposed method is a two-step procedure. First, the disturbing noise variance is quantified. Next, using this knowledge, the linear time-varying dynamics are estimated together with the nonlinear time-varying distortions. The latter are split into even and odd contributions. As a result, the signal-to-nonlineardistortion ratio is quantified. It allows one to decide whether or not a linear approximation is justifiable for the application at hand. The two-step algorithm is fully automatic in the sense that the user only has to choose upper bounds on the number of basis functions used for modeling the response signal. The obtained linear time-varying approximation is the best in the sense that the difference between the actual nonlinear response and the response predicted by the linear approximation is uncorrelated with the input. Therefore, it is called the best linear time-varying approximation (BLTVA). Finally, the theory is validated on a simulation example, and illustrated on two measurement examples: the cristallographic pitting corrosion of aluminum, and copper electrorefining.

The class of nonlinear time-varying (NLTV) systems includes all possible systems and, hence, is difficult to identify. Still, when the nonlinearities are not too strong then, depending on the application, a linear model might be sufficient for approximating the true response. To quantify the approximation error of the linear model, detecting and quantifying the nonlinear behavior is of key importance. In this paper we propose a fully automated procedure for detecting, classifying and quantifying the nonlinear distortions in the response, possibly subject to a trend, of a specific class of NLTV systems to odd random phase multisine excitations. The result is a measurement of the timevarying frequency response function together with uncertainty bounds due to noise and nonlinear distortions. The user only has to specify four integer numbers: an upper bound on (i) the degree on the time-domain polynomial modelling of the trend, (ii) the degree of the frequency-domain polynomial basis function and (iii) the number of frequency-domain hyperboliclike basis functions, all used for modeling the output spectrum; and (iv) a quality measure -called degrees-of-freedom -of the noise variance estimate. Guidelines are provided for obtaining reasonable values for these upper bounds.

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