Automatica 44 (2008) 3036–3045 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Discrete time sliding mode control with application to induction motors ✩ B. Castillo-Toledo a , S. Di Gennaro b,∗ , A.G. Loukianov a , J. Rivera c a CINVESTAV del IPN – Unidad Guadalajara, Av. Cientifíca, Col. El Bajío, Zapopan, 45010, Jalisco, Mexico b Department of Electrical and Information Engineering, and Center of Excellence DEWS, University of L’Aquila, Poggio Di Roio, 67040 L’Aquila, Italy c Centro Universitario de Ciencias Exactas e Ingenierías de la Universidad de Guadalajara, Av. Revolución, Col. Olimpica, Guadalajara, 44430, Jalisco, Mexico article info Article history: Received 10 March 2005 Received in revised form 30 November 2007 Accepted 5 May 2008 Available online 6 November 2008 Keywords: Discrete time control Sliding mode control Parameter uncertainty Observers Induction motors abstract This work deals with a sliding mode control scheme for discrete time nonlinear systems. The control law synthesis problem is subdivided into a finite number of subproblems of lower complexity, which can be solved independently. The sliding mode controller is designed to force the system to track a desired reference and to eliminate unwanted disturbances, compensating at the same time matched and unmatched parameter variations. Then, an observer is designed to eliminate the need of the state in the controller implementation. This design technique is illustrated determining a dynamic discrete time controller for induction motors. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction The advent of digital technology and its widespread use have revolutionized the computer-based implementation of advanced control schemes. In fact, recent advances in digital microprocessor technology have given considerable credit to digital control systems, exhibiting relatively low operational cost, flexibility in implementation, simple and functional interactive communication among several control loops. At the same time, there has been a growing interest in the design of controllers based on the digital model of the system. When the system is continuous, the first step is to obtain an accurate sampled model. This has motivated an interesting research activity in the area of discrete time control and has determined the development of digital control methods. Starting from the first studies on the sampling of continuous time nonlinear systems (Monaco & Normand-Cyrot, 1985, 1988), many tools have been developed in the last two decades to control ✩ Work supported by Consejo Nacional de Ciencia y Tecnología (Conacyt, México) under grants 46538 and 46069Y, by the Consiglio Nazionale delle Ricerche (C.N.R., Italy), and by the Ministero degli Affari Esteri (M.A.E., Italy). This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Torkel Glad under the direction of Editor Hassan K. Khalil. ∗ Corresponding author. Tel.: +39 862434461; fax: +39 6233233142. E-mail addresses: toledo@gdl.cinvestav.mx (B. Castillo-Toledo), digennar@ing.univaq.it (S. Di Gennaro), louk@gdl.cinvestav.mx (A.G. Loukianov), jorge.rivera@cucei.udg.mx (J. Rivera). digital and sampled nonlinear systems, see for instance Monaco and Di Giamberardino (1996), Monaco and Normand-Cyrot (1997, 2001, 2007) and references therein. The aim of this work is to give a further contribution in this field. The main objective is to design a discrete time feedback controller which ensures stability and achieves a specified transient response for discrete time nonlinear systems. Moreover, to provide a certain robust stability margin against bounded uncertainties, a simple approach is used here, based on the sliding mode approach (Utkin, 1993). Sliding mode control is a particular type of variable structure control, designed to drive and constrain the system state to lie within a neighborhood of a switching function. The advantages of the sliding mode technique are well known. First, this method enables the decomposition of the design problem into two independent subproblems: (a) selection of discontinuity surfaces with the desired sliding motion, and (b) determination of a control law to force the sliding mode along this manifold. This allows the suppression of the effects of matched parameter uncertainties and disturbances, and total invariance is obtained when the motion of the system is in sliding mode. In this paper, an iterative procedure is proposed to design a discrete time sliding mode control law for a class of nonlinear systems. This controller complies with the bounds on the control resources, and is such that the system state is driven toward a certain sliding manifold and stays there for all sampled time instants, avoiding chattering. Furthermore, a discrete time observer is designed to estimate the non-measurable states and perturbation. A separation principle is then applied to verify the 0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.05.009