# Non-parametric identification of higher order sinusoidal output describing functions

TL;DR: The HOSODF can be defined for the class of causal, stable, time invariant non-linear systems which give a sinusoidal response to a specific harmonic excitation of non- linear systems.

Abstract: In this paper the concept of the Higher Order Sinusoidal Output Describing Functions (HOSODF) is presented. HOSODF can be defined for the class of causal, stable, time invariant non-linear systems which give a sinusoidal response to a specific harmonic excitation. The HOSODF relate the magnitude and phase of the individual harmonics, which together compose that specific input signal, to the sinusoidal output signal of such a system. HOSODF are the dual of the Higher Order Sinusoidal Input Describing Functions (HOSIDF). Like the HOSIDF, the HOSODF are the results of an extension of linear techniques towards non-linear systems analysis. Using the HOSODF, the non-linear systems under investigation can be modeled as a cascade of the HOSODF and a Virtual Harmonics Compressor (VHC). The VHC is defined as a non-linear component which transforms a harmonic input signal y(t) into a sinusoidal output signal y(t) with frequency ω, amplitude â and phase φ. This input signal y(t) consists of an infinite amount of harmonics of the output signal y(t) with frequency nω, amplitude â and phase nω with n = 0, 1, ...∞. Special attention is paid to the non-parametric identification of the HOSODF. The identification requires control of the frequency and amplitude of the sinusoidal output of the system within its domain of possible sinusoidal output signals. This specific state of these non-linear systems can be reached by incorporating the system under test in a feedback loop. In this loop the desired sinusoidal output is defined as the control objective of a dedicated repetitive controller consisting of a memory loop with positive feedback. The design of the learning filter required for stability is also addressed. As a spinoff of the identification technique, the authors see opportunities for advanced non-linear control of shaker systems aimed at sinusoidal excitation of non-linear systems.

...read more

##### Citations

115 citations

27 citations

20 citations

10 citations

### Cites methods from "Non-parametric identification of hi..."

...This is done as well in [16] by application of a describing function approach....

[...]

6 citations

##### References

2,278 citations

1,397 citations

### "Non-parametric identification of hi..." refers methods in this paper

...The learning filterL can be designed with the ZPETC algorithm [31] and the resulti ng phase delay ofl samples is absorbed in the two delay blocks....

[...]

1,227 citations

### "Non-parametric identification of hi..." refers background in this paper

...• Describing Functions The Describing Function concept extends the FRF in the way th t identification of amplitude dependency becomes possible [19, 20]....

[...]

1,120 citations

### "Non-parametric identification of hi..." refers background in this paper

...• Identification of the Generalized Frequency Response Funct ion For the class of causal, stable, time-invariant, non-linea r systems with fading memory 1, the convolution integral description of the linear system can be generalize d to an infinite series called the Volterra series [9, 10, 11]....

[...]

844 citations