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

Cuthbert Nyack
The applets here show the step response and Bode plot of a type 1 second order system with Lag Compensation.
The continuous system is discretised by the bilinear transform to get D(z) and a ZOH transformation of the system transfer function.

eg parameters (20.0, 0.5, 10.0, 1.0, 0.01, 0.2, 1.0) show the response of the uncompensated system.

eg parameters (20.0, 0.5, 10.0, 10.0, 0.1, 1.0, 1.0) show the response with lag compensation and reasonable agreement between the continuous and discrete systems. The phase margin of the cotinuous system is 48.5° and of the sampled system is 44.0°. The sampled system phase drops below -180° at ~5.5rad/s with a sampling frequency of 62.83rad/s.

eg parameters (20.0, 0.5, 10.0, 10.0, 1.0, 4.0, 1.0). Significant damped oscillation on the discrete response. The sampled system phase margin has dropped to 10.9°. The sampled system phase drops below -180° at ~1.8rad/s with a sampling frequency of 6.283rad/s.

eg parameters (20.0, 0.54, 10.0, 10.0, 1.68, 5.0, 1.0). Discrete system oscillatory. Phase margin has dropped to ~0°, changing Ts show the closed loop poles are keeping close to the unit circle.
eg parameters (20.0, 0.6, 10.0, 10.0, 1.68, 5.0, 1.0). Discrete system unstable with growing oscillation. Phase margin now -ve at -1.7°.



Applet below show the same information with the frequency axis being linear from 0 to half of the sampling frequency.


Gif images below show how the applets should appear.

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COPYRIGHT 2006 Cuthbert Nyack.