A unified Lyapunov-like characterization for predefined time synchronization of nonlinear systems

In this paper, we present a new design for predefined time stable Lyapunov-like characterizations. Our approach combines several previously proposed Lyapunov characterizations and provides a unified idea for designing predefined time stable dynamic systems, based on which the framework of the predefined time sliding mode controller (PTSMC) is designed. First, we define Class-Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ Class }-\mathcal {Z}$$\end{document} functions and use them to design a unified form of Lyapunov-like characterization; moreover, we prove that it satisfies the predefined time stability by using the comparison principle. Second, a class of predefined time stable dynamic systems is developed based on this constructed function, and its predefined time stability properties are confirmed by Lyapunov theory. Finally, the dynamic system is used to construct a sliding mode controller, and a universal framework for PTSMCs is designed. By applying the above to chaos synchronization, numerical simulations demonstrate the universality and the feasibility of this unified control framework.