Determination of the characteristic curves of a nonlinear first order system from Fourier analysis

Abstract

Based on Fourier analysis, we develop an expression for modeling and simulating nonlinear first order systems. This expression is associated to a nonlinear first order differential equation y= f(x) + g(x) x′, where x= x(t) is the dynamical variable, y= y(t) is the driving force, and the f and g functions are the characteristic curves which are associated to dissipative and memory elements, respectively. The model is obtained from the sinusoidal response, specifically by calculating the Fourier analysis of y(t) for x(t) = A1sin (ωt) + A, where the model parameters are the Fourier coefficients of the response, and the values of A, A1 and A1′=A1ω. The same expression is used for two kinds of time-domain simulations: to calculate other driving force y^ based on a dynamical variable x^ ; and, to calculate the dynamical variable x^ based on a driving force y^. In both cases, the dynamical variable must remain in the range x^ ∈ [A- A1, A+ A1]. By analyzing this expression, we found an equivalence between the Fourier coefficients and the polynomial regressions of the characteristic curves of f and g. This result allows us to obtain the system modeling and simulation based on the amplitude and phase Fourier spectrum obtained from the Fast Fourier Transform (FFT) of the sampled yn version of y(t). It is shown that this technique has a low computational complexity, and it is expected to be suitable for real-time applications for system modeling, simulation and control, in particular when the explicit expressions of the characteristic curves are unknown. Fourier analysis is a fundamental tool in electronics, mathematics and physics, but to the best of the author’s knowledge, no work has found this clear evidence of the connection between the Fourier analysis and a first order differential equation. The aim of this work is to initiate a systematic study on this topic.