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I am confused about how adding a D (which adds a zero to the complete system) decreases the speed of the system. But when we normally add a zero to the system, it causes the system to overshoot.

The same goes for the I part of the PID. Normally when we add a pole to the system, it has less overshoot, but at the same time the integrator increases the overshoot!

How can I make sense out of this inverse relation?

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I think your confusion stems from the fact that you don't make any difference between adding pole/zero as cascade to a generic system (i.e. no loop) and adding pole/zero to a PID controller in a loop. In a loop, the behavior of phase lead/lag does matter a lot.

Then, just to be precise, an integrator adds up a very particular pole (in the origin), while the derivative part adds up not only a zero (in the origin) but also a high-frequency pole to make D feasible.

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