PID Control Second Order Type 0 Step response and Bode Plot
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.
Return to main page
Return to page index
COPYRIGHT © 2006 Cuthbert Nyack.