PID Control Second Order Type 0 Step response and Bode Plot

Cuthbert Nyack

Applets show the Bode plot and closed loop frequency response of a system with PID control.
The sampled controller transfer function is derived by using a trapezoidal approximation to the integral and a back difference approximation to the derivative.
The sampled system transfer function is derived as a ZOH equivalent.

eg parameters (8.0, 1.001, 0.5, 50.0, 0.0, 0.05, 1.0). Because of the large integral time this effectively shows the system with P control. The response has both overshoot and offset. The phase of the sampled system is -180° at ~ 10 rad/s when the gain is ~ -35dB. The gain reaches 0dB at ~ 4rad/s when the phase is ~ -140°. Increasing Td to 0.3 removes the overshoot. With Td = 0.4 decreasing Ti to 1.4 removes the offset. With parameters (8.0, 1.001, 0.5, 1.4, 0.3, 0.05, 1.0) the phase reaches -180° at ~31rad/s hen the gain is ~ -35dB. The gain reaches 0dB at 5.5rad/s when the phase is ~ - 100°. The system can become unstable if Ts is made large, eg increasing Ts from 0.05 to 0.37 causes the system to oscillate at ~ 5rad/s ( ~ 0.29w _s).

The applet below show the closed loop frequency response on a linear scale from 0 to half the sampling frequency.

The applet below show the magnitude and phase response of the continuous and digitised controller.

Gif images below show how the applets should appear.

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