PID control of uncertain nonlinear stochastic systems with state observer

The classical proportional-integral-derivative (PID) controller is ubiquitous in engineering systems that are typically nonlinear with various uncertainties, including random noise. However, most of the literature on PID control focused on linear deterministic systems. Thus, a theory that explains the rationale of the linear PID when dealing with nonlinear uncertain stochastic systems and a concrete design method that can provide explicit formulas for PID parameters are required. Recently, we have demonstrated that the PID controller can globally stabilize a class of second-order nonlinear uncertain stochastic systems, where the derivative of the system output is assumed to be obtainable, which is generally unrealistic in practical applications. This has motivated us to present some theoretical results on PID control with a state observer for nonlinear uncertain stochastic systems. Specifically, a five-dimensional parameter manifold can be explicitly constructed, within which the three PID parameters and two observer gain parameters can be arbitrarily selected to globally stabilize nonlinear uncertain stochastic systems, as long as some knowledge about the unknown nonlinear drift and diffusion terms is available.