Stability Analysis of the Newton-Raphson Controller for a Class of Differentially Flat Systems

TL;DR

This paper investigates the stability of the Newton-Raphson controller for a class of nonlinear differentially flat systems, providing stability conditions and demonstrating its effectiveness in regulation and tracking tasks through simulations.

Contribution

It extends stability analysis of the Newton-Raphson controller to nonlinear systems, offering verifiable criteria and demonstrating practical effectiveness.

Findings

Stability within a neighborhood of the origin for output regulation.

Successful regulation and tracking on inverted pendulum and bicycle models.

Potential for future control applications.

Abstract

The Newton-Raphson Controller, established on the output prediction and the Newton-Raphson algorithm, is shown to be effective in a variety of control applications. Although the stability condition of the controller for linear systems has already been established, such condition for nonlinear systems remains unexplored. In this paper, we study the stability of the Newton-Raphson controller for a class of differentially flat nonlinear systems in the context of output regulation and tracking control. For output regulation, we prove that the controlled system is stable within a neighborhood of the origin if the corresponding flat system and output predictor satisfy a verifiable stability criterion. A semi-quantitative analysis is conducted to determine the measure of the domain of attraction. For tracking control, we prove that the controller is capable of driving the outputs to the external reference signals using a specific selection of controller parameters. Simulation results show that the controller achieves regulation and tracking respectively on the inverted pendulum and the kinematic bicycle, suggesting a potential in future control applications.