A Class of Bounded and Partially Bounded Nonlinear Controllers for First and Second Order Dynamical Systems

Abstract

This letter proposes structurally simple, bounded and partially bounded nonlinear controllers that offer satisfactory performance, demonstrated by their application to first and second-order dynamical systems. This is done by taking advantage of the properties of a particular class of bounded sector nonlinear functions that can be parameterized in bound value and slope. In contrast to the classical methods of saturated control, the proposed controllers' design can be defined as an explicit summation of sector nonlinear functions, whose Lyapunov global stability proof can be straightforwardly demonstrated for the single and double integrator dynamics. Thus, the proposed approach derives nonlinear controllers where each term is bounded (or partially bounded) by design. Although the stability proof is provided for single and double integrator dynamics, one of the controllers is tested in a first-order nonlinear system and another in a nonlinear second-order system, both to achieve tracking. The numerical results evidence good performance even for large initial errors, and without the further introduction of auxiliary dynamics, such as compensation terms or feedback linearization. This is done by only tuning the gains of each term, while maintaining boundedness (or partial boundedness) properties on the control input.