Towards a theoretical foundation of PID control for uncertain nonlinear systems

As is well-known, the classical PID control plays a dominating role in various control loops of industrial processes. However, a theory that can explain the rationale why the linear PID can successfully deal with the ubiquitous uncertain nonlinear dynamical systems and a method that can provide explicit design formulae for the PID parameters are still lacking. This paper is a continuation of the authors recent endeavor towards establishing a theoretical foundation of PID for nonlinear uncertain systems. In contrary to most of the existing literature on linear or affine nonlinear systems, we will consider a class of non-affine nonlinear uncertain systems, and will show that a three dimensional parameter set can be constructed explicitly, such that whenever the PID parameters are chosen from this set, the closed-loop systems will be globally stable and the regulation error will converge to zero exponentially fast, under some suitable conditions on the system uncertainties. Moreover, we will also consider the simpler PI and PD control, and provide a necessary and sufficient condition for the choice of the PI parameters for a class of one dimensional non-affine uncertain systems, by applying the Markus-Yamabe theorem in differential equations. These theoretical results show explicitly that the ubiquitous PID control does indeed have strong robustness with respect to both the system nonlinear uncertainties and the selection of the controller parameters. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the application of PID control in nonlinear uncertain systems. By constructing a three-dimensional parameter set and selecting PID parameters based on certain conditions, the global stability of the closed-loop systems and the convergence of the regulation error to zero can be achieved. In addition, the study also examines the simpler PI and PD control, providing necessary and sufficient conditions for the selection of PI parameters for a class of one-dimensional non-affine uncertain systems. These theoretical results explicitly demonstrate the strong robustness of PID control to both the nonlinear uncertainties of the system and the selection of controller parameters.