Double integrator, PD control: stability

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 20:59

Summary:

This video is the first in a series to present the theory that justifies the behaviour we had found by ‘trial and error’ when tuning a PID controller for a ‘double integrator’ process $2∕s^2$, see video [dintpid2tunEN] for such hand-tuning ideas.

In this first video on the underlying theory, the generic closed-loop equations $e(s)=\frac{1}{1+GK}r(s)-\frac{G}{1+GK}{d}_u(s)$ are obtained, and $G=2∕s^2$ is substituted; the controller is also substituted in there, several versions actually, as follows.

First, a proportional controller is tested, checking that the loop CANNOT be stabilized, having sustained oscillations (i.e., they do not decay to zero) with $K_p>0$.

Then, a PD controller is tested checking that, if both proportional and derivative constants are positive, then it CAN be stabilized. With ‘little’ derivative action there are oscillations, but if the intensity of the derivative action is increased, then there are a pair of real poles (overdamped response).

The analysis continues in the video [dintteoerrEN], where the steady-state error is analyzed with PD and PID controls. The design of controllers by pole assignment will be discussed in later videos of the case study.

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