The generalized PID controller for stochastic systems

The P, PD, PI and PID controllers (the PID family) are successfully applied controllers to many applications. In a previous paper by the author the linear quadratic regulator (LQR) theory has been used to formulate tracking problems and to show those cases when their solution gives the PID controllers. Namely, problems have been stated whose solution leads to the PD, PI and PID controllers. This enabled avoiding heuristics and gives a systematic approach to explanation for the good performance of the PID controllers. The main contribution of the results in is that it shows for what problems the PID controllers are the optimal controllers and for which they are not. In this paper we apply the approach that has been developed in to a more general case. Namely, in it was assumed that the reference trajectory is generated by a deterministic linear system evolving from its deterministic initial conditions. It was previously assumed that the reference trajectory is generated by a stochastic linear system driven by stochastic process with perfect measurements. This gave the generalized PID controllers. Here we assume that the reference trajectory is generated by a linear system driven by a stochastic process, its initial conditions are random and the measurements are noisy. The result is the generalized stochastic PID controller. These controllers converge for noise free measurement to the generalized PID controllers.