Controllers gone unstable

In order to obtain asymptotic tracking, we need $S(0)=0$, which means we’d need one of the poles of $G K$ to be zero. If $G$ has no poles at zero, then $K$ must be the transfer-function to have the pole at zero. But in order to be asymptotically stable, we need the real parts of the poles to be less than zero, correct? So does it matter if our controller is not asymptotically stable? Is there a way to design a controller that is both asymptotically stable AND provides asymptotic tracking?

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