Abstract— A novel super-twisting adaptive sliding mode controller is proposed. A drift uncertain term is assumed to be bounded with unknown boundary. The proposed Lyapunov- based approach consists in using dynamically adapted control gains that ensure the establishment, in a finite time, of a second order sliding mode. Finite convergence time is estimated. A numerical example confirms the efficacy of the proposed adaptive super-twisting control. I. INTRODUCTION LIDING mode control is one of the best choices for controlling perturbed systems with matched disturbances/uncertainties [1,2]. The price for achieving the robustness/insensitivity to these disturbances is control chattering [1,2,3]. The traditional ways for avoiding chattering are as follows: (a) Replacing the discontinuous control function by “saturation” or “sigmoid ones” [4,5]. This approach yields continuous control and chattering elimination. However, it constrains the sliding system’s trajectories not to the sliding surface but to its vicinity loosing the robustness to the disturbances. (b) Using the higher order sliding mode control techniques [6-8, 12, 13]. This approach allows driving to zero the sliding variable and its consecutive derivatives in the presence of the disturbances/ uncertainties increasing the accuracy of the sliding variable stabilization. One of the most powerful second order continuous sliding mode control algorithms is the super-twisting control law (STW) that handles a relative degree equal to one. It generates the continuous control function that drives the sliding variable and its derivative to zero in finite time in the presence of the smooth matched disturbances with bounded gradient, when this boundary is known. Since STW algorithm contains a discontinuous function under the integral, chattering is not eliminated but attenuated. The main drawback of STW control algorithm is the requirements to know the boundaries of the disturbance gradient. In many practical cases this boundary cannot be easily estimated. The overestimating of the disturbance Y. B. Shtessel is with the University of Alabama in Huntsville, Huntsville, AL 35899 USA (tel: +(256) 824-6164; fax: +(256)824-6803; e- mail: shtessel@eng.uah.edu). J. A. Moreno is with Electrica Y Computacion, Instituto de Ingeneria, UNAM, D.F., Mexico, (e-mail: jmorenop@iingen.unam.mx) F. Plestan.is with Ecole Centrale de Nantes-IRCCyN, Nantes, France (e-mail: Franck.Plestan@irccyn.ec-nantes.fr). L. M. Fridman is with Division de Ingenieria Electrica, Facultad de Ingenieria, UNAM, 04510, D.F., Mexico, (e-mail: lfridman@unam.mx). A. S. Poznyak is with CINVESTAV-IPN, DF, Mexico boundary yields to larger than necessary control gains, while designing the STW control law. Contribution. In this work we propose the novel adaptive STW control law that continuously drives the sliding variable and its derivative to zero in the presence of the bounded disturbance with the unknown boundary. The finite convergence time is estimated. The proof is based on recently proposed Lyapunov function [9, 10] that is used for the derivation of the novel adaptive STW control algorithm. II. PROBLEM FORMULATION Consider a single-input uncertain nonlinear system (,) (,) x fxt hxtu = +  (1) where n x ∈ \ is a state vector, u ∈ \ is a control function, (,) n f xt ∈ \ is a differentiable, partially known vector-field. Assume that (A1) A sliding variable (,) xt σ σ = ∈ \ is designed so that the system’s (1) desirable compensated dynamics are achieved in the sliding mode (,) 0 xt σ σ = = . (A2) The system’s (1) input-output ( u σ → ) dynamics are of a relative degree one, and the internal dynamics are stable. Therefore, the input-output dynamics can be presented (,) (,) 1 (,) () (,) (,) (,) , (,) (,) xt bxt fxt hxu t x x xt bxtu x t bxtu u b xt ϕ σ σ σ σ σ ϕ σ ϕ ω ω ω − ∂ ∂ ∂ = + + → ∂ ∂ ∂ = + → = + = ↔ =      (2) The solution of system (2) is understood in the sense of Filippov [11]. Assume that (A3) the function (,) bxt ∈ \ is known and not equal to zero x ∀ and [ ) 0, t ∈ ∞ (A4) the function (,) xt ϕ ∈ \ is bounded 1/2 (,) xt ϕ δσ ≤ (3) where the finite boundary 0 δ > exists but is not known. The problem is to drive the sliding variable σ and its derivative σ  to zero in finite time in the presence of the bounded perturbation with the unknown boundary by means of continuous control. The classical SMC and the second order sliding mode (SOSM) controllers, including the continuous STW control algorithm, can robustly handle such problem if the boundary of the perturbation is known. The main disadvantage of the classical SMC is introducing control chattering, while Super-twisting Adaptive Sliding Mode Control: a Lyapunov Design Yuri B. Shtessel, Jaime A. Moreno, Franck Plestan, Leonid M. Fridman, Alexander S. Poznyak S 49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA 978-1-4244-7744-9/10/$26.00 ©2010 IEEE 5109