Phase Lead 0T1T2 Transient, Frequency Analysis and Root locus

Cuthbert Nyack
Applets below illustrate the application of the Lead Compensator to a third order Type 1 system. Since this is a 3 pole system, then it can become unstable.
The lead compensator does not work very well for this system and the applets illustrate why. For the Bode plot the phase drops rapidly because the system is third order so the phase margin at the estimated compensated frequency has a significant negative value. To get a sufficiently large phase lead, very small values of a must be used.

eg for T1 = 1, T2 = 0.5, the uncompensated system has a phase margin of ~ 40° for Kv slightly less than 0.8. For Kv = 1.9, TL = 1.97, a = 0.05 can give a phase margin of ~ 40°. Here the phase margin of the uncompensated system at the new crossover frequency is -24.8°. However even here there is still significant transients. The reason for the transient can be seen in the root locus applet below. For these numbers, the dominant closed loop poles make an angle of ~ 73° with the real axis.
Tolerable results can be obtained for smaller Kv. eg (1.0, 1.0, 0.5, 2.18, 0.1, 1.4, 1.0).

Gif images below shows how applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.