Copyright © IFAC System, Structure and Control Oaxaca, Mexico, USA, 8-10 December 2004 ElSEVIER IFAC PUBLICATIONS www.elsevier.comllocalelifac HIGHER ORDER SLIDING MODE OBSERVER-BASED CONTROL R. Castro-Linares', A. Glumineau+, S. Laghrouche+, F. Plestan+ • Corresponding author CINVESTA V-IPN, Department of Electrical Engineering Av. IPN No. 2508, Col. San Pedro Zacatenco 073600 Mexico , D.F., Mexico Email: rcastro@mail.cinvestav.mx + IRCCyN - Ecole Centrale de Nantes, Nantes, CNRS, BP 92101 , 1 rue de la Noli, 44321 Nantes cedex 03, France Abstract: The aim of this paper is to present a new high-order sliding mode con- troller for uncertain SISO minimum-phase nonlinear systems. The main properties of this controller arc that the convergence time is finite and the uncertainties robustness is ensured. The controller design combines standard equivalent sliding mode control with linear quadratic (LQ) onc over a finite time interval with a fixed final state (Lewis, 1986) . The advantages of the algorithm are that no exact differentiation is necessary if the value of sliding mode order is equal to the relative degree p and the discontinuous gain is minimized via a preliminary nonlinear feedback. Furthermore, the upper bound of the convergence time is chosen a priori and the control law can be adjusted via two weighting matrices. Copyright © 2004 IFAC Keywords: High order sliding mode , uncertain system, observer 1. INTRODUCTION Standard sliding mode (SM) features are high accu- racy and robustness with respect to various inter- nal and external disturbances. The SM controller mainly forces the state via discontinuous feedback to move on a prescribed manifold, called the slid- ing manifold. A specific problem involved by this technique is the chattering effect, i. e. dangerous high-frequency vibrations of the controlled system. To overcome this problem, one can replace the "sign" function in a small vicinity of the surface by a smooth approximation, by using either ob- servers (Utkin, 1992), or generalized sliding mode controllers (Sira-Rarnirez, 1992). Recently, a new approach called "higher order sliding mode" has been proposed (Bartolini et al., 1998),(Emelyanov 481 et al., 1993), (Levant, 1993). Instead of influencing the first sliding manifold time derivative, the "sign" function is acting on its higher time derivative. Keeping the main advantages of the standard sliding mode control, the chattering effect is eliminated and higher order precision is provided. Let {O'(x,t)} be the sliding variable, with x E mn the state variable. In the case of the rlh order sliding mode , the idea is to satisfy the set of constraint conditions O'(x , t) = a(x , t) = .. . = 0'(r-2)(x, t) = q(r-l )(x , t) = 0 1 , where rEIN (Levant, 1993). In the case of the second order sliding mode con- trol, i.e. r = 2, some solutions have been pro- I All over the paper , [u(- )](k) denotes the eh time derivative of the function u(·). This notation is also applied for every function.