PID Control Second Order Type 0 Step response and Root Locus

Cuthbert Nyack
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.

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COPYRIGHT 2006 Cuthbert Nyack.