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12. 5 enters the loop at the nonlinearity output. 5. This is due to the integrator in the loop, the input of which must have an average value of zero in the steady state.

K1 and k2’ Thus, since use of the DF provides an approximate method for the determination of limit cycles, it also provides an approximate stability test for our simple feedback loop. 1, the nonlinearity n(x) is replaced by its DF, N(a). 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.

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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