Fast Fixed-time Convergence in Nonlinear Dynamical Systems

TL;DR

This paper introduces a method for achieving rapid fixed-time convergence in nonlinear dynamical systems by manipulating the derivative of a quadratic function, with applications to control law design for linear plants.

Contribution

It presents new conditions for fixed-time convergence based on the derivative of a quadratic function and applies these to control law design using backstepping.

Findings

Achieves fixed-time convergence with specific derivative conditions.

Demonstrates the method's effectiveness through numerical simulations.

Provides a comparison showing improvements over existing solutions.

Abstract

A fast convergence in a fixed-time of solutions of nonlinear dynamical systems, for which special requirements are satisfied on the derivative of a quadratic function calculated along the solutions of the system, is proposed. The conditions for the system solutions to converge to zero and to a given region within a fixed-time are obtained. To achieve fast convergence, a negative power is applied to the derivative of a quadratic function within a specific time interval during the evolution of the system. The application of the proposed results to the design of control laws for arbitrary order linear plants using the backstepping method is considered. All the main results are accompanied by numerical modelling and a comparison of the proposed solutions with some existing ones.