Opamp precision PID loop without capacitors?
You only need to adjust the RC product to tune a PID controller, so there should be no problem if you have fixed (or discretely switched) capacitor values and adjustable resistors. You might need to switch capacitors of you have no idea of the plant characteristics and have to accommodate a huge range of possible parameters.

But more and more this kind of thing is done entirely in the digital domain. In particular, slower loops are excellent candidates,