This research effort has demonstrated that, except for very special cases, the 'self-tuning regulator' will not converge to the optimal. Despite this, it has been shown that if the order of the controller is large enough then one can still obtain self-tuning to the optimal. This work yields insights into the way in which a system needs to be parameterized in order for it to be controlled by the class of adaptive algorithms called 'self-tuning'.