By D. Atherton

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**Extra resources for An Introduction to Nonlinearity in Control Systems**

**Example text**

5 Modelling of an ideal dead zone characteristic from a unit gain and ideal saturation Since it is easily shown that the DF of two nonlinearities in parallel is equal to the sum of their individual DFs, the DFs of linear segmented characteristics with multiple break points can easily be written down from the DFs of simpler characteristics. Several procedures are available for obtaining approximations for the DF of a given nonlinearity either by numerical integration or by evaluation of the DF of an approximating nonlinear characteristic defined, for example, by a quantized characteristic, linear segmented characteristic or Fourier series.

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 .

13) is that they easily yield proofs of some interesting properties of the DF for symmetrical odd nonlinearities. These include the following: 1. For a double-valued nonlinearity the quadrature component Nq (a) is proportional to the area of the nonlinearity loop, that is: Nq (a) = - (1/a r) (area of nonlinearity loop) 2 2. For two single-valued nonlinearities na (x) and n b (x) , with na (x) < n b (x) for all 0 < x < b, then Na (a) < N b (a) for input amplitudes a less than b. 39 An Introduction to Nonlinearity in Control Systems The Describing Function 3.

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