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24) n -a where Tn (x/a) = cos 6 n cos- (x/a) @ and Dn (x/a) = sin 6 n cos- (x/a) @ 1 1 45 An Introduction to Nonlinearity in Control Systems The Describing Function Since for single valued nonlinearities bn = 0 the more important result is for an. 7 Sine plus Bias DF and the IDF So far only DFs for nonlinearities with odd symmetry have been considered. 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.

11, one for single valued nonlinearities and the other for double valued nonlinearities. 5 Nonlinear Models and DFs Nonlinear models, which consist of linear segments with multiple break points, can often be modeled from the nonlinearities in (b) and (c) above and linear gains, usually connected in parallel. 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.

1) where G (j~) = GC (j~) G1 (j~) . This condition means that the first harmonic is balanced around the closed loop assuming its passage through the nonlinearity is accurately described by N(a). 1) will yield both the frequency ~ and amplitude a of an assumed sinusoidal limit cycle. 1) with the choice affected to some extent by the problem. For example, relevant factors might be whether the nonlinearity is single or double valued or whether G (j~) is available from a transfer function G (s) or as measured frequency response data.

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