A new class of nonlinear PID controllers with robotic applications

This paper introduces a new class of simple nonlinear PID controllers and provides a formal treatment of their stability analysis. These controllers are comprised of a sector-bounded nonlinear gain in cascade with a linear fixed-gain P, PD, PI, or PID controller. Three simple nonlinear gains are proposed: the sigmoidal function, the hyperbolic function, and the piecewise–linear function. The systems to be controlled are assumed to be modeled or approximated by second-order transfer functions, which can represent many robotic applications. The stability of the closed-loop systems incorporating nonlinear P, PD, PI, and PID controllers are investigated using the Popov stability criterion. It is shown that for P and PD controllers, the nonlinear gain is unbounded for closed-loop stability. For PI and PID controllers, simple expressions are derived that relate the controller gains and system parameters to the maximum allowable nonlinear gain for stability. A numerical example is given for illustration. The stability of partially-nonlinear PID controllers is also discussed. Finally, the nonlinear PI controller is implemented as a force controller on a robotic arm and experimental results are presented. These results demonstrate the superior performance of the nonlinear PI controller relative to a fixed-gain PI controller. © 1998 John Wiley & Sons, Inc. 15: 161–181, 1998