  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.
