Generalized Super-Twisting Observer for Nonlinear Systems I. Salgado  I. Chairez.  J. Moreno  L. Fridman  A. Poznyak   Automatic Control Department, CINVESTAV-IPN, A.P. 14.740, C.P. 07360 Mexico, D.F. (e-mail: isalgado@ ctrl.cinvestav.mx).  Bioprocesses Department , UPIBI-IPN, Av. Acueducto C.P. 07340, Mexico D.F. (e-mail: ichairez@ctrl.cinvestav.mx)  Engineering Control Department, UNAM, Coyoacan, C.P. 04510, Mexico D.F. Abstract: In this paper it is proposed a novel Lyapunov based design of a generalized Super- Twisting Observer for a class of 2-dimensional nonlinear system. The observer can deal with systems whose states are composed of bounded nonlineaer functions.This is the main di/erence with the classical Super-Twisting observer, in which the second state is only the derivative of the rst state. Working with a Strong Lyapunov Function it can be shown su¢cient conditions to properly choose the observer gains to ensure nite time convergence to the real states. The obsrver is tested in a mathematical model regarding to the reduced Glucose-Insulin process. The numerical results have shown a better performance of the observer with lineal compensators in comparison to the classical Super-Twisting Observer. The gains for the observer are designed in order to compensate a more general class of perturbations that appear in the suggested glucose-insuline nonlinear model. Keywords: Sliding Modes, Super-Twisting Observer, Nonlinear systems 1. INTRODUCTION Sliding modes are well-known for their robustness against perturbations and uncertainties in the mathematical de- scription of several phisycal systems. In most cases, sliding modes are obtained by mean of the injection of a non- linear discontinuous term. In general this discontinuous term is depending on the output error. This framework may be used to construct robust controlling or observing algorithms. Discontinuous injection must be designed in such a way that system trajectories are enforced to remain in a submanifold contained in the estimation error space (the so-called sliding surface). For both, the control and the observation problem, the resulting motion is referred to as the sliding mode (Utkin (1992)). One additional pos- sitve characteristic using this discontinuous term regards to the rejection of external matched disturbances (Tan and Edwards (2001)). Classical Sliding-Mode Observers (Utkin (1992), Walcott and Zak (1987)) estimate robustly the state when the per- turbations/measurement map when the sliding surface has relative degree (RD) one with respect to the system input. However, the disturbance cannot be reconstructed exactly. The observer and controller design based on the second- order sliding modes (SOSM) approach has been consid- ered as an interesting topic by many researchers within the last decade (see Sthessel et al. (2003), Sira-Ramirez (2004), Punta (2006) and the references therein). Some attractive features of SOSM compared to the classical rst- order sliding modes are widely recognized: higher accuracy motions, chattering reduction, nite-time convergence for systems with relative degree two (Levant (2005), Boiko et al. (2007), etc.). In order to perform this disturbance reconstruction task, a Second Order Sliding Mode algorithm, the so-called Super- Twisting Algorithm (STA), has been proposed recently (Davila et al. (2005)) for second-order (mechanical) non- linear systems. The STA robustly reconstructs, in nite- time, the states, if the perturbation is of relative degree two (RD=2), or reconstructs the perturbation itself, when it is of relative degree one (RD=1). Besides, the sliding mode observers are widely used due to the nite-time convergence, robustness with respect to uncertainties and the possibility of uncertainty estimation (see, for example, the bibliography in the recent tutori- als Barbot et al. (2002), Edwards and Spurgeon (1998), and Poznyak (2001)). In particular, asymptotic observers (Shtessel and Shkolnikov (2003)) and the asymptotic ob- server for systems with Coulomb friction (Alvarez et al. (2000) and Orlov et al. (2003)) were designed based on the second-order sliding-mode. These observers requires the assumption on the so-called separation principle due to the asymptotic convergence of the estimated values to the real ones. In general, the convergence of all these algorithms (high order sliding modes) was proved using very complex geo- metrical conditions Levant (2005). Just a couple of years ago, the Lyapunov methodology was succesfully applied to show how and why these algorithms converge in nite time. Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 14353