Phase LagLead 0T1 Transient, Frequency Analysis and Root locus

Cuthbert Nyack

The Transfer function of the LagLead Compensator is.

This compensator has the benefit of both the Lag and Lead Compensators and gives improved performance compared with individual Lag or Lead Compensators. The Lead component increases the phase margin and the Lag contribution increases the gain margin where the gain margin is defined. The speed of the transient response is a compromise between that of the Lead and Lag compensators. TL moves the pole-zero pair of the Lead component, TG does the same for the pole-zero pair of the Lag component and β moves both poles in different directions.

Different algorithms are used to obtain suitable values of β, TL and TG all of which require assumptions based on experience. Here a more graphical approach is used. This compensator is similar to a PID controller with the effect of β similar to K, TL similar to Td and TG similar to Ti. Increasing β from 1 is similar to reducing K, TL damps out transients at early times(not as early as Td does) and Ti affects response at intermediate times, not up to infinity as Ti does.

For Kv = 20, T1 = 0.5, it is convenient to start with β = 10.0 and adjust TL and TG alternately until a suitable set is found. Eg TL = 1.63, TG = 1.5 produces a phase margin of 60.04°, a crossover frequency of 4.8193rad/s with slope 25.79dB/dec, upper 3dB frequency of 7.703rad/s and a closed loop frequency response with peak 0.041dB.
Other possibilities are ;-
β = 8.0, TL = 1.25, TG = 1.65, phase margin = 60.44°. This produces a double peaked closed loop frequency response(increase vgain to 5(2dB/div) to see).
β = 12.0, TL = 1.99, TG = 1.42, phase margin = 60.08°. This produces a double peaked closed loop frequency response(increase vgain to 5 to see).
β = 6.0, TL = 0.97, TG = 4.0, phase margin = 60.03°. This produces a double peaked closed loop frequency response(increase vgain to 5 to see).

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.

For the parameter values (K, 1.5, 1.5, 8.0, 10.0, 1.0, 1.0), the poles remain real for K up to ~ 17.2. Larger values of K produce complex conjugate poles with the angle they make to the -ve real axis increasing. For the parameter values (10.0, 1.5, 1.5, 8.0, β, 1.0, 1.0), the poles remain real for β > ~7.6. Smaller values of β result in the poles becoming complex conjugate with the angle made with the -ve real axis increasing faster than they do with the Lag compensator.

For the parameter values (20.0, 1.5, TL, 10.0, 10.0, 1.0, 1.0), Varying TL for small TL produces a change in the locus similar to Td eg TL = 0.3.
For the parameter values (20.0, 1.5, 10.0, TG, 10.0, 1.0, 1.0), Varying TG for small TG moves to poles to the right with the poles crossing the imaginary axis for TG slightly less than 0.6.

Applet below shows the root locus over a wider range of the s plane. This can be used to design a LagLeag Compensator using root locus techniques.

eg to get the dominant closed poles at -3.5 ± j5 the parameters (20, 0.5, 1.4, 0.7, 10.0, 0.5, 1.0) can be used. With these parameters the crossover frequency is 4.58rad/s with slope 24.93dB/dec, the phase margin is 56.51°, the 3dB frequency is 7.484rad/s and the peak of the closed loop frequency response is 0.697dB.

Gif images below shows how applet should appear.

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