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By D. Atherton

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Condition (a) can be checked from linear methods using the sector bounds of the nonlinearity provided any input does not cause a bias at the nonlinearity input and (b) by the DF method. The DF method, however, can only do this approximately since it assumes any limit cycle will be sinusoidal at the input to the nonlinearity, which will never be quite true in practice. This means that the DF method could indicate incorrectly either the existence or non-existence of a limit cycle. It is therefore important to have some idea of the validity and accuracy of any DF result.

This means that the DF method could indicate incorrectly either the existence or non-existence of a limit cycle. 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.

12) and using the fact that n(x) is odd, one has a1 = (4/a) # a 4 x p (x) dx 0 The integral nn = # x p (x) dx is known as the n 3 n th -3 distribution with p (x) = (1/r) (a - x ) 2 moment of the probability density function and for the sinusoidal , nn , has the value 2 - 1/2 0 for n odd nn = * a n (n - 1) (n - 3) 1 .... for n even n (n - 2) 2 2 4 2 Therefore N (a) = (4/a ) 1 . 3 . 1 a = 3a /4 as before. 5 , the nonlinearity output waveform y (i) is as shown in the same figure. 2 Saturation nonlinearity, input sinusoidal and output waveforms 41 An Introduction to Nonlinearity in Control Systems The Describing Function Again, because of the symmetry of the nonlinearity the fundamental of the output can be evaluated from the integral over a quarter period, so that for a linear regime slope of m and saturation at δ gives N (a) = 4 ar # y (i) sin idi r/2 0 which for a > d gives N (a) = 4 ; ar # a # md sin idi E r/2 2 ma sin idi + a 0 with a = sin- d/a .

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An Introduction to Nonlinearity in Control Systems by D. Atherton


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