Automatica 42 (2006) 1745 – 1751 www.elsevier.com/locate/automatica Brief paper Robust MPC of constrained discrete-time nonlinear systems based on approximated reachable sets  J.M. Bravo a , T. Alamo b, ∗ , E.F. Camacho b a Departamento de Ingeniería Electrónica, Sistemas Informáticos y Automática, Universidad de Huelva, Carretera Huelva-La Rábida, Palos de la Frontera, 21071 Huelva, Spain b Departamento de Ingeniería de Sistemas y Automática, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Received 18 April 2005; received in revised form 10 February 2006; accepted 8 May 2006 Available online 7 July 2006 Abstract A robust MPC for constrained nonlinear systems with uncertainties is presented. Outer bounds of the reachable sets of the system are used to predict the evolution of the system under uncertainty. A method that uses zonotopes to represent the approximated reachable sets is proposed. The closed-loop system is ultimately bounded thanks to a contractive constraint that drives the system to a robust invariant set.  2006 Elsevier Ltd. All rights reserved. Keywords: Predictive control; Nonlinear systems; Constrained control; Robust stability 1. Introduction Model predictive control is a control strategy that has been widely adopted in industry (Qin & Badwell, 2003) and academia (Camacho & Bordons, 1999). The reason for this success is the ability to deal with constraints and multivariable systems. A survey about model predictive control can be found in Mayne, Rawlings, Rao, and Scokaert (2000) where sufficient conditions to guarantee asymptotic stability are given. The problem of robust nonlinear model predictive control is addressed in this paper. When uncertainties are present, they should be taken into account in the computation of the control law in order to guarantee robust stability. Some authors have formulated this problem as in Michalska and Mayne (1993) where a dual-mode receding horizon controller is proposed and robustness under decaying additive uncertainties is achieved by a proper choice of the terminal region. In Magni, Nijmeijer, and van der Shaft (2001) a robust MPC strategy based on an  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tor Arne Johansen under the direction of Editor Frank Allgöwer. ∗ Corresponding author. Tel.: +034-605631404. E-mail addresses: caro@uhu.es (J.M. Bravo), alamo@cartuja.us.es (T. Alamo), eduardo@cartuja.us.es (E.F. Camacho). 0005-1098/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.05.003 H ∞ cost function is presented. In Limon, Bravo, Alamo, and Camacho (2005) a robust nonlinear predictive controller based on reachable sets is presented. Natural interval extension is used to bound the uncertain evolution of the system. A linear difference inclusion of the original nonlinear system is used by some authors to implement a robust control ( Angeli, Casavola, & Mosca, 2002; Cannon, Deshmukh, & Kouvaritakis, 2002; Casavola, Famularo, & Franze, 2002; Kothare, Balakr- ishnan, & Morari, 1996). In Langson, Chryssochoos, Rakovic, and Mayne (2004) a tube predictive controller is proposed for linear constrained systems with additive uncertainty. In this paper, a new robust MPC for nonlinear systems with parametric uncertainty is proposed. To improve the re- sults obtained in Limon et al. (2005), the new approach relies on a prediction method that uses zonotopes ( Alamo, Bravo, & Camacho, 2005; Kühn, 1998) to bound the reachable sets. The paper is organized as follows: In Section 2, the class of nonlinear uncertain systems under consideration is introduced. In Section 3, an outer bound of the uncertain trajectory of the system is presented. The proposed robust nonlinear MPC controller is introduced in Section 4. The stability of the closed- loop system is analyzed in Section 5. An illustrative example is given in Section 6. The paper draws to a close with a section of conclusions.