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?
Anonymous
The answer to both questions is no. Stability of the controller is not required for closed-loop stability, even though it is often desired. In fact, as the student correctly points out, if we want asymptotic tracking of constant references and the system being controlled does not have a pole at zero, then the controller must have a pole at zero, that is it must be unstable!
Unstable controllers are tricky to work with it. As you might suspect, one should not turn an unstable controller on without resetting the controller just before closing the loop. In the case of a simple integrator that means making sure that the integrator state is zero. Likewise, one should make sure that the loop never opens while the controller is in operation. That includes ensuring that the control signal does not saturate, which can cause integrator wind up, as seen in Section 4.3.
Finally, it may be necessary for a controller to have poles with strictly positive real part. Think about a system with one real open-loop pole trapped between two real zeros on the right-hand side of the complex plane. A quick root-locus sketch, see Chapter 6, shows that a stabilizing controller must sport an unstable pole to offer that lone pole an opportunity to branch out of the right-hand side.
Closing the loop with unstable controllers, or unstable systems for that matter, often requires sophisticated engineering solutions. Think about how do you get yourself upright: before turning on your “inverted pendulum” stabilizing controller you might have used all sorts of other devices to help you stand up, for examples, you might have used your arms or other nearby support, all of which were necessary to get your balancing started but are not needed once the loop gets going.