A second order discrete sliding mode observer for the variable structure control of a semi-batch reactor Mohamed Mihoub à , Ahmed Said Nouri, Ridha Ben Abdennour National Engineering School of Gabes (ENIG), 6029 Gabes, Tunisia article info Article history: Received 21 March 2010 Accepted 25 June 2011 Available online 16 July 2011 Keywords: Sliding mode Observers Discrete-time systems Chattering Stability analysis Robust control abstract This paper concerns the chattering elimination from discrete sliding mode observers. The dilemma chattering-precision, that characterizes the first order sliding mode observer in case of relatively large parameter variations and/or external disturbances, is discussed and the influence of the discontinuous term amplitude on the estimation performance is shown by a simulation example. The proposed second order discrete sliding mode observer is, then, introduced. The stability of a closed loop control system based on the proposed observer is analyzed. A real time application of the proposed observer incorporated into a 2-DSMC control loop of a chemical reactor is presented. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Some process controller strategies use the entire state vector in producing the control signal (Glad, 1987). Since, in practice, it is not always possible to measure the state vector, either a design method based solely upon the output information is required or a suitable estimate of the state vector has to be constructed for use in the original control law. This paper considers the latter approach. Sliding mode (SM) systems are notorious for their robustness with respect to parameter variations and external disturbances (see Garcia-Gabin, Zambrano, & Camacho, 2009; Lopez & Nouri, 2006; Mihoub, Nouri, & Ben Abdennour, 2008a; Shahraz & Bozorgmehry Boozarjomehry, 2009; Tannuri, Agostinho, Morishita, & Moratelli, 2010). In the continuous-time domain, many applications of the SM theory to the state estimation were reported. Walcott and Zak (1987) designed an observer which has the output error being fed back linearly and used a Lyapunov approach to prove stability. Utkin (1992) designed a simple observer, with only the discontinuous term being fed back through an appropriate gain. Edwards and Spurgeon (1992) and Edwards, Spurgeon, and Patton (2000) proposed a canonical form for sliding mode observer design subject to certain conditions relating to the input and output distribution matrixes, and also the invariant zeros of the system. Pin Tan and Edwards (2000) proposed a sliding mode observer based on the LMI approach. Boukhobza and Barbot (1998) and Fridman, Shtessel, Edwards, and Yan (2008) exploited the high order sliding mode approach for the observation of non-linear systems. Kalsi, Lian, Hui, and Zak (2010) proposed an observer based on the high gain approach. In the discrete-time domain, some discrete sliding mode obser- vers can be found in the literature. Takahashi and Peres (1996) and Aitken and Schwartz (1996) have proposed a discrete sliding mode observer (DSMO) for linear systems in the presence of matched uncertainties. In Caminhas, Takahashi, Peres, and Tavares (1996), an extension that allows the observation of systems with ‘‘non- matching’’ non-linearities is proposed. Koshkouei and Zinober (1996) introduced an observer exploiting the Lyapunov min–max direct method. Haskara, Ozguner, and Utkin (1998) presented a sliding mode observer based on the equivalent control. Lee and Lee (1999) proposed an observer based on the time delay control concept. Thein (2002) proposed a sliding mode observer with a stability proof in case of matched and unmatched uncertainties. This last observer has an attractive boundary layer, outside of which the DSMO has convergent quasi-sliding mode, and within which the estimation’s error have a finite bound. Veluvolu, Soh, Cao, and Liu (2006) proposed an observer for SISO non-linear uncertain systems, where Taylor’s series expansion is used to discretize the system. Resendiz, Yu, and Fridman (2008) exploited the neural approach to reduce the chattering of the discrete sliding mode observer of mechanical systems velocity. A common pro- blem of these approaches is the chattering phenomenon which was more or less reduced. In this work, the considerable decrease of the classical DSMO state estimation’s performances, in case of relatively large parameters’ variations, non-linearities and/or external disturbances, is shown. A solution to this problem is, then, proposed and experimented on an esterification semi-batch reactor. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/conengprac Control Engineering Practice 0967-0661/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2011.06.012 à Corresponding author. Tel.: þ216 75 392 100. E-mail addresses: mihoubmed4@yahoo.fr (M. Mihoub), AhmedSaid.nouri@enig.rnu.tn (A.S. Nouri), Ridha.benabdennour@enig.rnu.tn (R. Ben Abdennour). Control Engineering Practice 19 (2011) 1216–1222