By D. Atherton
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Extra info for An Introduction to Nonlinearity in Control Systems
If the stability criterion indicates instability (stability) for a point on C (a) with a < a0 and stability (instability) for a point on C(a) with a > a0, then the limit cycle is stable (unstable). This result is known as the Loeb intersection criterion and in terms of the approximate DF analysis, is a necessary, but not a sufficient criterion for a stable limit cycle. To investigate the stability of a limit cycle one has to examine the perturbation equation. 5) q (D) dx (t) + p (D) n' (x * (t)) dx (t)) = 0 This is a time varying equation for the perturbation, dx (t) , which must be stable for the limit cycle to be stable.
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. Typically the functions G (j~) and N (a) are plotted separately on Bode, Nyquist, or Nichols diagrams. 2) although here it should be remembered that the equation is appropriate only for s .
The latter is very straightforward when the relay has no dead zone as its output is a square wave of amplitude ! h , which has a fundamental component of amplitude 4h/r , either in phase with the input for D = 0 , or lagging when Δ is finite. 20 for no hysteresis, D = 0 ; for no dead zone, d = 0 ; and for an ideal relay, D = d = 0 . 1. Unfortunately this description is not correct for any input signal, x, and it has been used to obtain erroneous results for x a random signal. 11, one for single valued nonlinearities and the other for double valued nonlinearities.