# Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses

TL;DR: In this paper, an extension to higher-order describing functions is realized by introducing the concept of the harmonics generator, which relates the magnitude and phase of the higher harmonics of the periodic response of the system to the magnitude of a sinusoidal excitation.

Abstract: For high-precision motion systems, modelling and control design specifically oriented at friction effects is instrumental. The sinusoidal input describing function theory represents an approximative mathematical framework for analysing non-linear system behaviour. This theory, however, limits the description of the non-linear system behaviour to a quasi-linear amplitude-dependent relation between sinusoidal excitation and sinusoidal response. In this paper, an extension to higher-order describing functions is realised by introducing the concept of the harmonics generator. The resulting higher-order sinusoidal input describing functions (HOSIDFs) relate the magnitude and phase of the higher harmonics of the periodic response of the system to the magnitude and phase of a sinusoidal excitation. Based on this extension two techniques to measure HOSIDFs are presented. The first technique is FFT based. The second technique is based on IQ (in-phase/quadrature-phase) demodulation. In a simulation, the measurement techniques have been tested by comparing the simulation results to analytically derived results from a known (backlash) non-linearity. In a subsequent practical case study both techniques are used to measure the changes in dynamic behaviour as a function of drive level due to friction in an electric motor. Both methods prove successful for measuring HOSIDFs.

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### Cites methods from "Higher-order sinusoidal input descr..."

..., the describing function and harmonic balance in the frequency domain [22, 23] and the averaging method in the time domain [24–27], etc....

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### Additional excerpts

...Finally, the Higher Order Sinusoidal Input Describing Functions (HOSIDF) [16,18,26] are used to analyze nonlinear effects in the following....

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...where Hkðo0,gÞ 2 C is the kth order Higher Order Sinusoidal Input Describing Function (HOSIDF) [16,26]....

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##### References

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...Some approaches addressed the describing function analysis [5,6] to replace a non-linear element with a quasi-linear descriptor which gain is a function of input amplitude....

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### "Higher-order sinusoidal input descr..." refers background in this paper

...This structure, however, is not a Hammerstein model since the second block is not necessarily linear [14]....

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