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By D. Atherton

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For many control engineering situations this may be satisfactory, as for example in the consideration of torque saturation in a motor. Many situations do occur, however, where nonlinearities do not have odd symmetry and also even when they do the operation may not be about the odd symmetric axis. Thus, to handle the general situation for a single valued nonlinearity with characteristic n(x) one needs to consider an input consisting of a sinusoid plus bias, that is c + a cos i and evaluate both the sinusoidal and bias output of the nonlinearity.

5. Also a quantized characteristic can be modeled from relay characteristics with dead zone and no hysteresis in parallel. 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.

9. 3 at which point instability occurred, with the output from the relay locking at one of its limit levels of ±1. 60 An Introduction to Nonlinearity in Control Systems Stability and Limit Cycles using the DF It can be seen from the gains marked that the complex poles cross the imaginary axis for a gain of 18 and for that value of gain the real pole is at -1. However for a gain of 9 (half of 18) the real pole will be in the rhs of the s plane so that the system is unstable. 9 then the gain when the complex roots cross the axis will be twice that for the real root crossing the axis.

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An Introduction to Nonlinearity in Control Systems by D. Atherton

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