The likely situation is that the process that the loop controls is non-linear.
The solution is either to take advantage on gain scheduling, where the gain of the loop is changed based on which operating region the loop is trying to control. This solution only works for the automation systems which support gain scheduling or built-in adaptive modeling.
John notes that you can select control valves with non-linear characteristics which offset the non-linearity of the process. He describes it:
The control valve characteristic is used to compensate for the process gain to achieve an approximate installed linear process.
He does note it is difficult to exactly get the non-linearities to cancel one another out, but that at least you can come close in approximating a linear constant gain process and be in better shape than before you made the change.