Phase Lead 0T1 Transient, Frequency Analysis and Root locus

Cuthbert Nyack

The Transfer function of the Lead Compensator is.

This function has a lead and a lag factor with the 3dB frequency of the Lead factor being lower than that of the Lag factor. The Transfer function therefore produces a phase lead which increases the phase margin and improves the stability of the system. Ideally the maximum phase lead should occur at the gain crossover frequency of the system plus compensator. The maximum phase lead f _m occurs at w_m given by:-

This means that the geometric mean of the pole and zero 3dB frequencies should occur at the new gain crossover frequency. On the negative side, this compensator reduces the gain margin where the gain margin can be defined. The Applets below shows how the transient response, frequency response and root locus are affected by the compensator. Adjusting TL moves both the zero and the pole while α moves the pole independently of the zero. The lead compensator damps out the transient oscillations but unlike the lag compensator, it speeds up the response.

The conventional approach used to design a lead compensator uses the following steps:-
Consider the default parameters (20.0, 0.5, 0.3, 0.12, 0.2, 1.0).
The data on the applet shows :-
The frequency of max phase lead w_m is 9.622rad/s and the max phase lead f _m is 51.786°.
The uncompensated cross over frequency cofu is ~6.17rad/s with slope ~38.17dB/Decade.
The uncompensated phase margin upm is 17.95° while the uncompensated gain margin is undefined because the phase does not become less than -180°.
For the given value of a the lead gain dBLead is 9.206dB and the estimated new crossover frequency cofe is 10.65rad/sec with phase margin 10.62°.
The compensated system crossover frequency cofc is 11.4rad/s with a slope of 23.86 dB/dec.
The phase margin of the compensated system cpm is 61.33°.

To get a phase margin of ~60° change a to 0.13 to get a f _m of 50.345°. This together with the uncompensated phase margin of 10.84° at the estimated new crossover frequency will give a total phase margin of ~ 60°.
TL is now adjusted to make w_m( frequency with maximum phase lead) equal to cofc the new cross over frequency. Within the resolution of the applet TL = 0.27 gives w_m = 10.27rad/s and cofc = 10.55rad/s with a compensated crossover frequency of 61.06°.
The slope at the new crossover frequency is 23.91dB/dec down from 38.17dB/dec.
The 3dB bandwidth is 16.55rad/s up from 9.659rad/s and the peak of the closed loop response is 0.838dB, down from ~10dB.

eg.
For Kv = 20.0, T1 = 0.3, Phase Margin = 50°, Set TL = 0.18, a = 0.28.
For Kv = 20.0, T1 = 0.6, Phase Margin = 50°, Set TL = 0.26, a = 0.21.
For Kv = 15.0, T1 = 0.6, Phase Margin = 50°, Set TL = 0.29, a = 0.25.
For Kv = 20.0, T1 = 0.5, Phase Margin = 60°, Set TL = 0.26, a = 0.14.

The applet below show the Nyquist plot of a system with a lead compensator. The information is similar to the Bode plot but shown differently.
The parameters on the plot show the values of the curves where the "x" is located. The position of the "x" is changed by w on the applet.
Setting the parameters to (20.0, 0.5, 0.28, 0.1, 0.2, 6.2 or 11) shows that the compensator changes the phase margin by ~47° and an increase in crossover frequency by ~ 5rad/s.

The applet below show the root locus for the system.
For the parameter values (K, 0.5, 0.3, 0.12, 0.2, 1.2), increasing K from zero first produces real poles up to K ~ 0.7. Furthur increase in K does not produce large overshoot because the closed loop poles move furthur to the left. For the parameter values (10.0, 0.5, 0.3, α, 0.2, 1.0), increasing α up to ~0.18 moves the poles furthur to the left, thereafter they start moving to the right.
For parameters (0.0, 0.5, 0.3, α, 0.2, 1.0). setting α = 1 show the locus of the uncompensated system. Reducing α pulls the locus to the left. At α = 0.16 it comes back to the real axis and forms a circle for smaller α.
This is similar to what happens when Td in a PD control system is increased.

Design by the root locus method requires finding TL and a which produces a given w_n (shown as Mag on applet) and angle with the real axis.
eg for Kv = 15, T1 = 0.5, Mag = ~6, angle ~ 35°. Set TL = 0.23 and a = 0.1. With these parameters the crossover frequency is 7.578rad/s with slope 24.37dB/dec, the 3dB frequency is 10.75rad/s and the phase margin is 65.05°.

Gif images below shows how applet should appear.

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