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The optimality of PID controllers

The PID family of controllers is very widely used and successfully applied to many applications. The facts of their successful application, good performance, easiness of tuning are sufficient reason for their use, although their structure is justified by heuristics. In this paper by the use of LQR theory we formulate a control-tracking problem and show those cases when their solution gives the PID family of controllers, thus avoiding heuristics and giving a systematic approach to explanation for the performance of the PID controllers. The PID controller architecture is optimal for linear quadratic tracking problem of second order systems with no zero. The reference trajectory is generated by a system identical to the plant.