I was studying about PD controllers, which made use of adding a zero at such location which gave the required settling time(Ts) while keeping the overshoot same as it was in the uncompensated system. My question is can we achieve the same by adding a pole instead of a zero? Because even the pole will reshape the root locus. If not then why ? Is it something to do with the changes that happens to the response of a system when zeroes and poles are added?
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$\begingroup$ I think adding some response plots and a root locust plot would make this question clearer. $\endgroup$– Chris_abcCommented Apr 3 at 12:32
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$\begingroup$ To reduce settling time and overshoot, effectively means the following desired movements of CL poles: faster CL dominant pole, or higher damping ratio of CL dominant pole-pair. So in root locus the CL pole-pair must be moved towards real axis and/or to the left. Adding zeros will attract the poles and can accomplish this. Besides the poles, it's important to realize that zeros also contribute to overshoot and ringing! And any zero placed in the forward path will appear in the CL transfer function (whereas if placed in the feedback path, it won't!)... $\endgroup$– Pete WCommented Apr 3 at 12:56
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$\begingroup$ ... In some circumstances, e.g. a single dominant pole (often at the origin), and for a particular (limited!) range of gain, adding a pole rather than a zero can also draw the dominant pole to the left, in the pattern where they come together east-west then diverge going north-south. The important thing is that once they start moving north/south, more gain will degrade damping ratio. The room for playing around in this manner, is more limited though $\endgroup$– Pete WCommented Apr 3 at 12:58
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1$\begingroup$ @PeteW: to shift the phase of a resonance peak by 90 degrees, so you can have over-unity gain through the resonance and keep the loop stable. Seriously -- I was handed a loop that did that; it took me two weeks of failing to make it work better without the lag filter before I gave up and did what I was being told. $\endgroup$– TimWescottCommented Apr 6 at 19:38
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1$\begingroup$ It was a dynamically-tuned torque balancing gyro. You drove the gyro wheel to stay centered in its case, and inferred the rates from the amount of current to the torquers. It had a honkin' big resonance at the gyro's nutation frequency, 180 degrees phase shift. If you applied the usual rule of staying well below the resonance the loop bandwidth would be worthless, even with notch filters to help things out. $\endgroup$– TimWescottCommented Apr 6 at 22:44
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1 Answer
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Yes, you can reshape a root-locus with a pole.
Usually you don't get the result you want -- usually the increase in phase lag means that you end up with loop performance slower than you would without the lag. If you're putting a pure lag into a controller it's usually to kill the high-frequency gain well above the unity-gain frequency of the loop, or it's because you have a noisy measurement and you need to use low-pass filtering to keep the controller from saturating on the noise.
If you play around with root-locus plots for a bit you'll see that zeros tend to attract root loci, while poles tend to repel them. Since your poles need to end up being stable, this means that you tend to end up with low-frequency zeros because you need them, and high-frequency poles because that's the price you have to pay to get a zero.
