On the Capability of PID Control for Nonlinear Uncertain Systems
  It is well-known that the classical PID controller is by far the most widely used ones in industrial processes, despite of the remarkable progresses of the modern control theory over the past half a century. It is also true that the existing theoretical studies on PID control mainly focus on linear systems, although most of the practical control systems are inherently nonlinear with uncertainties. Thus, a natural question is: can we establish a theory on PID controller for nonlinear uncertain dynamical systems? This paper will initiate an investigation on this fundamental problem, showing that any second order uncertain nonlinear dynamical systems can be stabilized globally by the PID controller as long as the nonlinearity satisfies a Lipschitz condition. We will also demonstrate that this result can be generalized neither to systems with order higher than 2, and nor to systems with nonlinear growth rate faster than linear in general. 
Cheng Zhao, Lei Guo
math.OC