Why is PID suboptimal in case of non-linearities? And also what problems may arise if the differential part is missing in a PID?
From the theoretical point of view, the D part of the PID controller is not realizable because we cannot calculate the D part (that is, the derivative of the error) for the given present and past error (note that "realization" is defined under the condition that the derivative is not available and only the present and past error are available). But, from practical point of view, the D part of the PID controller can be realized by backward numerical derivative with the sampling time of the PID controller. Of course, using the numerical derivative with the nonzero sampling time is an approximation of the D part.
Even though the D part of the PID controller is approximately realizable, the ideal PID controller should not used if the sampling time is small because the output of the PID controller severely fluctuates, resulting in shortening the life of actuators such as valves because the sensitivity of the numerical derivative to noises is inversely proportional to the magnitude of the sampling time. So, commercial PID controllers usually suppress the effects of noises by adding a low-pass filter to the D part