A Lyapunov-Like Characterization of Predefined-Time Stability

• Published in: IEEE transactions on automatic control Vol. 65; no. 11; pp. 4922 - 4927
• Main Authors: Jimenez-Rodriguez, Esteban, Munoz-Vazquez, Aldo Jonathan, Sanchez-Torres, Juan Diego, Defoort, Michael, Loukianov, Alexander G.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Automatic; Autonomous systems; Control stability; Convergence; Dynamic stability; Dynamical systems; Engineering Sciences; Estimation; Manifolds; Nonlinear control systems; predefined-time stability; Robust control; Settling; sliding mode (SM) control; Stability analysis; Stability criteria; stability of nonlinear systems; Theorems; Time functions; Uncertainty analysis; Upper bounds; "Manifolds"; "Convergence"; "Dynamical systems"; "Estimation"; "Stability criteria"; "Autonomous systems"
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2020.2967555

This article studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefined-time stability. The origin of a dynamical system is predefined-time stable if it is fixed-time stable, and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow us to demonstrate the equivalence between previous Lyapunov theorems for predefined-time stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound, which is useful when analyzing uncertain dynamical systems. Therefore, the proposed results constitute a general framework for analyzing the predefined-time stability, and they also unify a broad class of systems that present the predefined-time stability property. On the other hand, the proposed framework is used to design robust controllers for affine control systems, which induce predefined-time stability (predefined-time ultimate boundedness of the solutions) w.r.t. to some desired manifold. A simulation example is presented to show the behavior of a developed controller, especially regarding the settling time estimation.