On the Capability of PID Control for Nonlinear Uncertain Systems

TL;DR

This paper investigates the theoretical capability of PID controllers to stabilize second-order nonlinear uncertain systems, establishing conditions under which global stabilization is possible, and highlighting limitations for higher-order or faster-growing nonlinear systems.

Contribution

It provides a theoretical foundation showing PID control can stabilize certain nonlinear uncertain systems, specifically second-order systems with Lipschitz nonlinearities.

Findings

PID controllers can globally stabilize second-order nonlinear uncertain systems.

Stability results do not extend to higher-order systems or those with superlinear growth.

Theoretical insights bridge classical PID control with nonlinear system stabilization.

Abstract

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.