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I just watched this MatLab video on the basics of how PI/PID controllers work and for the most part it makes sense. They started with an example with just a P controller and showed that, while it worked fine for a really simple case, in other cases you wind up having to make gain arbitrarily large in order to minimize the error, which is obviously not an ideal situation. They then explained how adding an integrator path to the system can help fix this because it keeps a running total of the error w.r.t. time, stating that, since the integral of a nonzero value is always nonzero, the integrator path will keep feeding nonzero values to the controller input until the error is zero (or, in real-life, as opposed to ideal circuits, presumably until the error is small enough that the system can't distinguish it from zero).

My question then is, why not just get rid of the proportional branch entirely and just use the integral branch as the controller? If the integrator is going to cause the error to go to zero anyway, what purpose does the proportional branch even serve? Does it cause the error to decrease more quickly? If so, how? Or is there some reason that using just an integrator by itself as a controller wouldn't work?

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There are several questions, but maybe I can make sense.

The integrator needs an error to operate. To operate more quickly it need a big error. As the error decreases the rate of decrease also slows.

Even when the error is decreasing, the integrator will still keep adjusting so the chance of over shoot and oscillation is possible. You really want the I to take over from the P.

The proportional path responds immediately to the error so that we don't have to wait for the integrator to catch up. When the error is zero the proportional path has no effect, so the integrator can take over and reduce the error to zero.

The derivative path is the fastest. It slows the errror's rate of change before the integrator has a chance to start.

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