Lead Compensator Second Order Type 1, Step Response and Bode Plot

Cuthbert Nyack
The applets below show the effect of sampling on the step response and bode plot of a type 1 second order system with Lead Compensation. The function of the compensator is to increase the phase margin at the gain crossover frequency without too large an increase in the gain crossover frequency.

eg parameters (20.0, 0.5, 0.3, 1.0, 0.02, 0.2, 1.0).
With a = 1, the response is that of the uncompensated system. The response shows damped oscillations, a gain crossover frequency(cofk) ~ 6.16rad/s and a phase margin ~ 17.9°. The sampled system phase margin is 16.1°, the sampling frequency is 628.3rad/s and the sampled phase becomes less than -180° at ~ 19rad/s.

With parameters (20.0, 0.5, 0.28, 0.1, 0.01, 0.2, 1.0), the transient response has improved, the gain crossover frequency has increased to ~ 11.06rad/s, the maximum phase advance of ~54.9° occurs at 11.29rad/s and the new phase margin of the cotinuous system ~ 65.1°. At this sampling frequency 628.3rad/s, the phase margin of the sampled system is ~ 61.9°. Reducing the sampling frequency to 83.8rad/s reduces the phase margin to ~ 40°

Reducing the sampling frequency to 34.9rad/s shows the phase margin dropping to ~0 and the system becoming oscillatory at ~ 12rad/s. The pole of the continuous Lead compensator is at 35.7rad/s. eg (20.0, 0.5, 0.28, 0.1, 0.18, 1.0, 1.0)

The applet below shows the linear frequency response from zero to half the sampling frequency.
eg parameters (20.0, 0.5, 0.42, 0.56, 0.11, 0.5, 1.0) show the sampled system becoming oscillatory at ~ 0.135 ws.

Gif images below show how the applets should appear.

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COPYRIGHT 2006 Cuthbert Nyack.