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Extra info for Applied Optimal Control: Optimization, Estimation and Control
12 it would appear that phase-plane contours may only follow a clockwise path - and this is indeed the general case. There are periodic equilibrium points, or singularities, at zero frequency error and at a phase error sufficient to give zero phase detector output. For analogue multiplier phase detectors, for example, these singularities occur at 8e = tr/2 + nn and alternate between points of stable and unstable equilibrium, corresponding to negative and positive loop gain, respectively. A point of unstable equilibrium is termed a saddle point where, of course, operation cannot be sustained for any length of time.
13. Now, suppose at some time after the initial frequency step, t 1 say, the instantaneous phase and frequency errors, q,; and are observed (measured, in practice). Then, without any knowledge of the original excitation, we are asserting that it is possible to exactly determine the future behaviour of the transients. It is possible to show that this is true by finding the necessary combination of phase and frequency steps which, if applied at time t t 1, would give rise to the phase and frequency errors, q,; and ru;.
2 0 0 ~ ~ "' " ~ " '\ r--r = -r 3td2 9e (rads) ' "" "\ t = 00 "\ .............. 11 It is interesting to note that the frequency error when the phase crosses 2n rads is slightly lower (around 12%) than the initial frequency error. This is the reason why the second stage of the transient results in a smaller 55 Loop Basics peak phase error with respect to the final, settled, value and is the basis of acquisition in PLLs in general. 05nemn then many cycle slips might have occurred, though the system would have still eventually converged to a state of stable equilibrium.