PID Control Second Order Type 0 Step and Ramp response

Cuthbert Nyack

The applets below show the step and ramp response for a type 0 second order 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. Increasing Td to 0.3 removes the overshoot. With Td = 0.4 decreasing Ti to 1.4 removes the offset. 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(from the orange curve).

Gif images below show how the applets should appear.

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