PID Control Second Order Type 0 Step response and Root Locus
Applets show the root locus and step 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 response is dominated by a pair of complex conjugate
poles at ~ +0.92 ± j0.2. 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 dominant poles are
now real at ~ +0.19 and ~ +0.7.
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 ~ 0.3w s. The dominant poles have moved to the unit circle at
~ -0.31 ± j0.95.
The applet below show the a magnified view of the right hand
side of the unit circle.
Gif images below show how the applets should appear.
Return to main page
Return to page index
COPYRIGHT © 2006 Cuthbert Nyack.