Automatica 44 (2008) 2034–2039 www.elsevier.com/locate/automatica Brief paper Control design for a class of nonlinear continuous-time systems ✩ Eugˆ enio B. Castelan a , Sophie Tarbouriech b,∗ , Isabelle Queinnec b a DAS-CTC-UFSC, P.O. Box 476, 88040-900 Florian´ opolis, SC, Brazil b LAAS-CNRS, University of Toulouse, 7 Avenue du Colonel Roche, 31077 Toulouse Cedex 4, France Received 4 August 2006; received in revised form 3 July 2007; accepted 22 November 2007 Available online 5 March 2008 Abstract This paper addresses the control design problem for a certain class of continuous-time nonlinear systems subject to actuator saturations. The system under consideration consists of a system with two nested nonlinearities of different type: saturation nonlinearity and cone-bounded nonlinearity. The control law investigated for stabilization purposes depends on both the state and the cone-bounded nonlinearity. Constructive conditions based on LMIs are then provided to ensure the regional or global stability of the system. Different points, like other approaches issued from the literature, are quickly discussed. An illustrative example allows to show the interest of the approach proposed. c  2008 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Saturations; Nonlinear feedback design; Nested nonlinearities; LMIs 1. Introduction The stability and stabilization problems of dynamical systems subject to nonlinearities is of interest due to the fact that such systems include a wide variety of practical systems and devices, like servo systems, flexible systems, etc. Indeed, smooth and non-smooth nonlinearities often occur in real control process, due to physical, technological, safety constraints or imperfections, even inherent characteristic of considered controlled systems (Kapila & Grigoriadis, 2002; Kokotovic & Arcak, 2001; Tarbouriech, Garcia, & Glattfelder, 2007). In the current paper, we consider a particular class of nonlinear systems consisting of a linear system affected by a state-dependent nonlinearity belonging to a general class of sectors and subject to amplitude saturation in the input. This class of systems includes as a special case the system without saturation studied in the context of absolute stability in de Oliveira, Geromel, and Hsu (2002) through the use ✩ Partially supported by CNPq/Brazil. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor teD Iwasaki under the direction of Editor Roberto Tempo. ∗ Corresponding author. E-mail addresses: eugenio@das.ufsc.br (E.B. Castelan), tarbour@laas.fr (S. Tarbouriech), queinnec@laas.fr (I. Queinnec). of suitable Lur’e–Lyapunov functions. This class of systems without saturation but with uncertain parameters in the matrices is also studied in Montagner et al. (2007). The main objective of this paper is to design a saturating control law resulting from both the system states and the nonlinearity, through constant feedback gain matrices (Arcak & Kokotovic, 2001; Arcak, Larsen, & Kokotovic, 2003). The problem of the design of suitable feedback control gains is investigated. Thus, both regional and global stabilization results are proposed by considering a quadratic candidate Lyapunov function. Differently from Arcak et al. (2003), the objective of designing nonlinear feedback is to enlarge the region of stability of the closed-loop system subject to nested nonlinearities. Hence, this current paper can be viewed as a work complementary to Arcak et al. (2003) and Montagner et al. (2007). Based on the application to the current case of the modified sector conditions proposed in Tarbouriech, Prieur, and Gomes da Silva (2006), the conditions developed to address stabilization (in a regional or global context) appear under LMI forms and can directly be cast into convex optimization problems. Some discussions about the LDI-based approach, developed in Hu and Lin (2001), are provided. The numerical example intends to show that the considered control law can guarantee a larger closed-loop basin of attraction than in the case of a classical saturating state feedback. 0005-1098/$ - see front matter c  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.11.013