Strict Lyapunov Functions for the Super-Twisting Algorithm

• Published in: IEEE transactions on automatic control Vol. 57; no. 4; pp. 1035 - 1040
• Main Authors: Moreno, J. A., Osorio, M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.04.2012; Institute of Electrical and Electronics Engineers
• Subjects: Adaptative systems; Applied sciences; Computer science; control theory; systems; Control system analysis; Control system synthesis; Control theory. Systems; Convergence; Discontinuous systems; Exact sciences and technology; Lyapunov functions; Lyapunov methods; Modelling and identification; Observers; robust observers; Robustness; second order sliding modes (SOSM); Symmetric matrices; Trajectory; second order sliding modes (SOSM); Discontinuity; Stability; Sliding mode; Variable structure system; Control synthesis; robust observers; Linear time invariant system; Lyapunov functions; Observer; Function derivative; Robustness; Discontinuous systems; System identification; Quadratic form; Lyapunov function
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2012.2186179

A method to construct a family of strict Lyapunov functions, i.e., with negative definite derivative, for the super-twisting algorithm, without or with perturbations, is provided. This second order sliding modes algorithm is widely used to design controllers, observers and exact differentiators. The proposed Lyapunov functions ascertain finite time convergence, provide an estimate of the convergence time, and ensure the robustness of the finite-time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm. Since the Lyapunov functions and their derivatives are quadratic forms, the operation with them is as simple as for linear time invariant systems.