Second-Order Uniform Exact Sliding Mode Control with Uniform Sliding Surface Emmanuel Cruz-Zavala, Jaime A. Moreno and Leonid Fridman Abstract— In this paper we propose an uniform sliding mode controller for a second order uncertain system providing convergence to an arbitrary small vicinity of origin in finite time, which can be bounded by some constant independent from initial conditions and uncertainties. With this aim a nonlinear sliding surface is proposed ensuring during the sliding motions an uniform convergence of the trajectories to any arbitrary small vicinity of origin in finite time bounded by some constant independent from initial conditions on the surface. I. INTRODUCTION The problem of robust prescribed time stabilization is one of the actual tasks in modern control theory. For example, controlling hybrid systems with strictly positive dwell time, it is preferably for control task to provide the robust exact system stabilization before the next of switching or impulse takes place. A reasonable class of controllers providing both: finite time convergence and insensitivity with respect to matched uncertainties/disturbances are sliding mode con- trollers (see, for example [15]). Traditional sliding mode control design consists of two steps, [15]: (a) design of the sliding surface ensuring de- sired behavior of system without uncertainties; (b) design of discontinuous controllers enforcing the sliding motions and compensation of matched uncertainties. The main dis- advantage of such methodology is the so called chattering phenomenon restricting the possibilities of usage of first order sliding mode controllers to the hardware, where the switching is a natural mode of work. A super-twisting controller allows to adjust the chattering problem in the system with Lipschitz continuous uncertain- ties/disturbances [10]. It opened the door for usage of the properties of sliding modes to practically all continuous controllers. It is necessary to remark that for both classical and super-twisting control design methodologies the conver- gence time grows together with initial conditions, i.e., (a) the time of convergence to the sliding surface grows together with initial conditions; (b) even if the trajectory is starting on the sliding surface the convergence time to the given vicinity of origin on sliding surface is also growing together with initial conditions on the surface. The uniform super- twisting based differentiator and observer for mechanical E. Cruz-Zavala and Jaime A. Moreno are with the Coordinaci´ on El´ ectrica y Computaci´ on, Instituto de Ingenier´ ıa, Universidad Na- cional Aut´ onoma de M´ exico, 04510 M´ exico D.F., Mexico, Email: emitacz@yahoo.com.mx, JMorenoP@ii.unam.mx L. Fridman is with the Dept. Control Autom´ atico CINVESTAV-IPN, AP- 14-740 M´ exico D.F., M´ exico, on leave on Departamento de Ingenier´ ıa de Control y Rob´ otica, Divisi´ on de Ingenier´ ıa El´ ectrica, Facultad de Ingeniera UNAM, Email: lfridman@unam.mx systems was designed in [3], [4], ensuring the convergence of differentiator/observer in finite time bounded by some constant independent from initial conditions. For the case of a sliding mode controller, the methodology of these papers can ensure the uniform convergence to the sliding surface only but does not ensure the uniform convergence to origin for the system trajectories on the sliding surface. On the other hand, the use of nonlinear sliding surfaces instead of linear surfaces of a classical sliding mode control design have proved to enhance the desired performance in closed-loop of the system with sliding mode control algorithms, which can not be always achieved only with linear switching surfaces, (see [2], [1], [14] and references there). In this paper we propose an exact sliding mode controller for a second order system providing the convergence of the system trajectories to any arbitrary small vicinity of origin in finite time upper bounded by some constant in- dependent from initial conditions and some class of uncer- tainties/disturbances. To achieve this aim: (a) a nonlinear sliding surface is suggested ensuring during sliding motions uniform convergence of the trajectories to any arbitrary small vicinity of origin in finite time upper bounded by some constant independent from initial conditions on the surface; (b) an absolutely continuous super-twisting based controller is suggested providing convergence of the trajectories to the sliding surface in a finite time upper bounded by another constant independent from initial conditions and for a special class of uncertainties/disturbances. The methodology used in this paper is based on Lyapunov functions proposed in [12], [3], [5], [13], [4]. In the following section we introduce some basic concepts which are useful for the understanding of the paper. A. Basic Definitions Consider the following non autonomous dynamic system ˙ x = f (x)+ w(x, t ), (1) where x ∈ R n are the system states, the functions f : R n → R n , w : R × R n → R n , w(x, t ) ∈ W nv is an uncertainty/disturbance and the class W nv functions represents a family of non vanishing perturbations at the origin. The functions f (·) and w(·, ·) ensure the existence of solutions to the system (1) in the sense of Filippov (1988). Denote a solution trajectory of (1) with the initial condition x(t 0 )= x 0 and t 0 ∈ [0, ∞) by x s (t , x 0 , t 0 ). Let B μ = {x : ||x|| < μ } be a ball centered at the origin with radius μ > 0 and let T (x 0 , μ ) be the convergence time 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 12-15, 2011 978-1-61284-799-3/11/$26.00 ©2011 IEEE 4616