The describing functions method serves quite well for harmonic oscillators, because the nonlinearity represented with a describing fiunction responds to the harmonic input with a harminic output. In the case of a relaxation oscillator, you must expect for amplitude errors even unacceptable. This is because energy in the relaxation oscillator is distributed among overtones and not concentrated mostly at the fundamental frequency as in the harmonic oscillator. So, an answer to your question may be either No or Yes.

Sr. Y. Shmaliy, thank you for replying.

Let me emphasize again that my question concerns harmonic oscillators only - in particular, oscillators with a frequency-determining network that does NOT suppress harmonics sufficiently (like allpass networks).

Dear Lutz

Thank you for an interesting question

Please take a look here:

http://www.facstaff.bucknell.edu/mastascu/econtrolhtml/Nonlinear/NonLinear2DescFcn.htm

Best regards

ERIK

Hello Erik, nice to hear from you again.

Thank you for the link you have provided - however it does not answer my question since - of course - severe damping of harmonics is assumed.

Excerpt:

"As the Fourier Series square wave is input to G(s), G(s) acts as a low-pass filter.

That implies that signals will be attenuated as they pass through G(s).

The higher frequency signals will be attenuated more.

If G(s) has a good drop-off, the output of G(s) may well be pretty much the fundamental of the Fourier Series. Higher frequency terms will be attenuated, possible pretty severely."

Dear Lutz, the method was developed for harmonic oscillators irrespective of what goes on outside the oscillator closed-loop. So, I guess you can use it. Note that the approximation error is lower if the quality factor is higher and the nonlinearity is not strong. For more detail you may take a look at my book on Continuous-Time Systems, Springer, 2007 availavle in researchgate.

Prof. Dr. Y. Shmaliy,

thank you very much for your reply and, in particular, for the information about your book. This book matches nearly up to 100% my interest and the area of my work. Thank you again. I will keep you - and all other who are interested - informed in case of a certain progress in my activities.

Best regards

Lutz vW

Dear Lutz, nice to hear that you have approached a solution in such a way. It will certainly be interesting to know further progress. Please keep me informed! Bests, Yuriy

Dear Lutz, the question is interesting indeed. Beside the accuracy of prediction of the frequency and amplitude of self-oscillations by this method, there is an aspect concerning the existence of the oscillations (of which you note in your original question). Khalil, for example, in his book mentions 2 possibilities: when the DF method predicts the existence of the limit cycle but it does not exist in the system, and when the DF method says that a limit cycle does not exist but oscillations occur in the system. However, despite my long-term interest and involvement in this area of control, I don't know such examples of the discrepancy between qualitative predictions of the DF method and actual occurrence of self-oscillations. I would be really interested in seeing them if someone can point out.

Regarding the accuracy of computation by the DF method - it would probably be hard to estimate it in advance. However, you can improve the accuracy. There are a few methods available. I also uploaded my recent publication concerning this problem (and your question prompted me to do it), which you can see here on Research Gate at https://www.researchgate.net/publication/248382558_Finding_exact_periodic_solutions_in_Lure_systems_through_periodic_signal_mapping?ev=prf_pub

Best regards, Igor

Article Finding exact periodic solutions in Lure systems through per...

Dear Igor, thank you for your reply - and special thanks for the link to your publication.

[Quote: However, despite my long-term interest and involvement in this area of control, I don't know such examples of the discrepancy between qualitative predictions of the DF method and actual occurrence of self-oscillations. I would be really interested in seeing them if someone can point out.]

I don`t know if you know the article as given below (it tells something about limitations/failure of the DF method)

http://www.ingelec.uns.edu.ar/asnl/Materiales/Cap04Extras/FnDesc/Engelberg.pdf

Best Regards

Lutz

Dear Lutz, thank you for noting this. I actually didn't mean systems that have a phase lag smaller than 180 deg. Such systems in particular may provide a sliding mode and what some authors call "asymptotic" sliding mode, as well as higher-order sliding mode. These cases are analyzed by a number of researches, and also in my book "Discontinuous control systems" (http://www.springer.com/birkhauser/engineering/book/978-0-8176-4752-0) using the DF and LPRS methods. And it is shown where the DF method doesn't work in predicting a limit cycle.

I meant cases with linear part having larger than 180 deg phase lag and not necessarily a relay nonlinearity. Thanks for correcting me.

Best regards, Igor

Just some addition to my previous comment. Of course, I mean maximum possible phase lag of 180 deg, which happens when the frequency approaches infinity.

By the way, I would not interpret the 1st example (section V) in the referenced article in such a way that the DF method predicts a limit cycle. Because the linear part includes a multiple integrator, the phase balance is not satisfied. Intersections with the negative part of the real axis happen at phase angles -540 deg, -900 deg, etc.

Regards, Igor

@Lutz Sir, thanks for starting discussion on such a good concept. This is indeed very helpful and informative.