By D. Atherton
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Extra resources for An Introduction to Nonlinearity in Control Systems
It has also been shown theoretically that the estimate for the limit cycle frequency, ~ , is more accurate than for the amplitude, a. Further, as is often not clearly pointed out, a is the estimate for the fundamental component in the limit cycle not the peak amplitude. When the DF method does not predict a limit cycle it will be correct if the C (a) and G (j~) loci are not near to intersecting. If this is not the case, however, one can assume a possible limit cycle at the nonlinearity input with amplitude a and frequency ~ corresponding to the near intersection values of C (a) = G (j~) .
This can be done in several ways with a particularly elegant and convenient engineering one, since it can be done from knowledge of the open loop frequency response locus, being due to Nyquist. On the other hand the problem of determining precisely the stability of an autonomous nonlinear feedback system, even one simply containing one linear dynamic element and one nonlinearity, has not been solved, even though it has exercised the minds of mathematicians and engineers for many years. The DF provides an approximate method for answering the problem and will be addressed in this chapter, whilst some exact approaches, known as absolute stability methods, will be considered in chapter 8.
It is therefore important to have some idea of the validity and accuracy of any DF result. If the DF method predicts a limit cycle then its validity can normally be checked by assuming this sinusoidal signal as the nonlinearity input and evaluating the signal fed back to the nonlinearity, assuming the loop open at the nonlinearity input. If the percentage distortion in this signal is less than 5% then the DF prediction should be valid. It has also been shown theoretically that the estimate for the limit cycle frequency, ~ , is more accurate than for the amplitude, a.
An Introduction to Nonlinearity in Control Systems by D. Atherton