Automatica 41 (2005) 1949 – 1956 www.elsevier.com/locate/automatica Brief paper Conjugate Lyapunov functions for saturated linear systems  Tingshu Hu a , ∗ , Rafal Goebel b , Andrew R. Teel c , Zongli Lin d a Department of Electrical and Computer Engineering, University of Massachusetts, Lowell, MA 01854, USA b 3518 NE 42 Street Seattle, WA 98105, USA c Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106-9560, USA d Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743, USA Received 11 October 2004; received in revised form 18 February 2005; accepted 12 May 2005 Available online 24 August 2005 Abstract Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper’s methods.  2005 Elsevier Ltd. All rights reserved. Keywords: Saturation; Domain of attraction; Linear differential inclusions; Conjugate Lyapunov functions; Duality 1. Introduction One practical way to study nonlinear systems, and also hybrid, switched, or uncertain time-varying linear ones, is to obtain an approximate description of a given system in terms of linear differential/difference inclusions (LDIs). Such a practice can be traced back to the early development of the absolute stability theory, where the nonlinearity and uncer- tainties were described with conic sectors and the systems were treated with tools adapted from those for linear sys- tems (see, e.g., Aizerman & Gantmacher, 1964; Narendra & Taylor, 1973). The effectiveness of the LDI approach  This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Faryar Jabbari under the direction of Editor Roberto Tempo. Work supported in part by NSF under grant ECS-0324679, AFOSR under grant F49620-03- 01-0203, and NSF under grant CMS-0324329. ∗ Corresponding author. Tel.: +1 978 9344374; fax: +1 978 9343027. E-mail addresses: tingshu@gmail.com (T. Hu), rafal@ece.ucsb.edu (R. Goebel), teel@ece.ucsb.edu (A.R. Teel), zl5y@virginia.edu (Z. Lin). 0005-1098/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.05.021 depends on two factors: how close the LDI approximation is, and what tools are used for analyzing it. One of the tools is a common Lyapunov function for all the member systems of the LDI. In the early development, the search for such a function was often restricted to quadratics. In fact, the circle criterion for absolute stability gives a necessary and sufficient condition for the existence of a common quadratic Lyapunov function for all convex combinations of two linear systems. The theory based on quadratic Lyapunov functions for LDIs was completed by the LMI optimization technique (see, e.g., Boyd, El Ghaoui, Feron, & Balakrishnan, 1994). While the search for a common Lyapunov function has been justified (e.g., Dayawansa & Martin, 1999; Molchanov, 1989), evidence has accumulated to indicate that the search should be widened beyond quadratic forms (see, e.g., Chesi, Garulli, Tesi, & Vicino, 2003; Dayawansa & Martin, 1999; Jarvis-Wloszek & Packard, 2002; Power & Tsoi, 1973; Ze- lentsovsky, 1995). Recent years have witnessed an extensive search for nonquadratic and/or homogeneous Lyapunov functions, among which are piecewise quadratic functions