# Measuring the higher order sinusoidal input describing functions of a non-linear plant operating in feedback

TL;DR: In this article, two measuring techniques are presented for measuring the higher order sinusoidal input describing functions (HOSIDF) of a non-linear plant operating in feedback.

Abstract: In this paper two measuring techniques are presented for measuring the higher order sinusoidal input describing functions (HOSIDF) of a non-linear plant operating in feedback. The HOSIDF relate the magnitude and phase of the harmonics of the periodic response of a non-linear system to the magnitude and phase of the sinusoidal excitation. In a controlled system the harmonics generated by the non-linear system will be fed back to the input changing the sinusoidal excitation into a harmonic excitation. The first method applies linear techniques to compensate the bias caused by the harmonic components in the excitation signal. The second method uses a modified repetitive control scheme to suppress the harmonic components in the excitation signal. The effectiveness of both methods is tested in simulations of a mass subjected to Coulomb friction, Stribeck effect and hysteresis in the pre-sliding regime. The friction forces are modeled with the modified Leuven friction model. The results are compared with the HOSIDF simulated under open-loop condition. Both methods prove able to produce reliable results.

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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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### "Measuring the higher order sinusoid..." refers methods in this paper

...The learning filter L can be designed with the ZPETC algorithm [Tomizuka, 1987] and the resulting phase delay of l samples is absorbed in the two delay blocks....

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