PD Control Second Order Type 1 Step response and Root locus

Cuthbert Nyack

The applets here show the step and ramp response of a type 1 second order system with PD control.
The continuous system is discretised by a back difference approximation to the derivative and a ZOH transformation of the system transfer function. The continuous system is always stable.
eg parameters (10.0, 3.0, 0.0, 0.1, 2.0). This shows the step response with P control. This response is unsatisfactory because of the damped oscillations. Adding D control can remove the transients. eg increasing Td from 0.0 to 1.0 damps out the transient and changing the complex conjugate poles to real poles.
eg parameters (K, 3.0, 0.0, 0.2, 2.0). System without PD control. For K =0.1 locus is very close to the unit circle and crosses unit circle when = 10.0. Increasing Td to 0.8 pulls the locus away from the unit circle because of the introduced zero and keeps the system stable for K up to 30.8.

Gif image below show how the applet should appear.

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