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A derivative controller \$C_{d} = s-z_{d}\$ adds a zero to a system, placed at \$s=z_{d}\$. I know that a derivative controller is not used alone but for the same of the argument let's say it is.

In every source I have found an integral controller is defined to be of the form \$C_{i} = 1/p_{i}\$ and adds a pole at the origin of a system.

Why an integral controller always adds a pole at the origin and isn't of the form \$C_{i} = 1/(s-p_{i})\$ so it can place a pole anywhere?

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Because an integrator, by definition, has a pole at \$s = 0\$. If the pole has a finite value, then the element is a low-pass filter.

There is a thing called a leaky integrator, where you place the pole at some \$s = -a\$ and \$a\$ is a few orders of magnitude lower in frequency than the anticipated closed-loop bandwidth of your system: this is done to make analog implementations happy, and to some extent for integrator anti-windup. I don't favor using it at all, but some people love it (and when I was actively consulting, it seemed that any time I said "I never use that technique" the next week I'd be working on a project that demanded it).

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