Double integrator, PD control: position and velocity errors (setpoint tracking and disturbance rejection)

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 13:51

Summary:

This video is a continuation of the video [dintteostEN]; in that video the closed loop poles of a double integrator with PD control were analyzed (it was stable, oscillatory or not depending on the amount of derivative action); in the present video, the error in steady state is analyzed.

Applying the final value theorem with a control law $K_{P}e+K_D\frac{de}{dt}$ and $G(s)=2∕s^2$, the following situations are analyzed:

• Error with step ($e_{pos}$) and ramp ($e_{vel}$) setpoint changes. Both are zero with the control $K_P+K_{D}s$.

• Error with input disturbance step (non-zero final deviation) and ramp (infinite final deviation) increments.

The results are verified in simulation.

Note: if the derivative action were implemented as $K_D\frac{dy}{dt}$ (output derivative instead of error derivative), then the position error with step reference change would be zero, but not the velocity error (stationary error with ramp reference); this fact is stated, without proof, for brevity.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.