Stability Regions for Saturated Linear Systems via Conjugate Lyapnov Functions Tingshu Hu * , Zongli Lin†, Rafal Goebel * , Andrew R. Teel * Abstract—We use a recently developed duality theory for linear differential inclusions (LDIs) to enhance the stability analysis of systems with saturation. Based on the duality theory, the condition of stability for a LDI in terms of one Lyapunov function can be easily derived from that in terms of a Lyapunov function conjugate to the original one in the sense of convex analysis. This paper uses a particular conjugate pair, the convex hull of quadratics and the maximum of quadratics, along with their dual relationship, for the purpose of esti- mating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a LDI system with an effective approach which enables optimization on the local LDI description. 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 that the estimation of the domain of attraction by this paper’s methods drastically improve those by the earlier methods. Keywords: Stability, saturation, linear differential inclusions, conjugate Lyapunov functions, duality. I. Introduction One practical way to approach nonlinear systems, that can also be applied to 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). The practice of approaching nonlinear uncertain time varying systems through LDIs can be traced back to the early development of the absolute stability the- ory, where the nonlinearity and uncertainties were described with conic sectors (see, e.g.,[1], [15], [21]). The effectiveness of the LDI approach depends on two factors: how close the LDI description is to the original system, and what tools are used for analyzing the approx- imate system. One of such tools is a common Lyapunov function for all the member systems of a LDI. In the early development, the search for such a function was often restricted to quadratic functions. For example, the circle criterion for absolute stability actually gives the necessary and sufficient condition for the existence of a common quadratic Lyapunov function for two or more linear systems (such situation was later referred to as quadratic stability). * Department of Electrical and Computer Engineering, Univer- sity of California, Santa Barbara, CA 93106-9560, U.S.A. Email: {tingshu,rafal,teel}@ece.ucsb.edu. Work supported in part by NSF under grant ECS-0324679 and by AFOSR under grant F49620-03-01-0203. †Department of Electrical and Computer Engineering, University of Virginia, P. O. Box 400743, Charlottesville, VA 22904-4743, U.S.A. Email: zl5y@virginia.edu. Work supported in part by NSF under grant CMS- 0324329. The theory based on quadratic Lyapunov functions was completed with the advancement of the LMI optimization technique [3]. While the search for a common Lyapunov function has been justified in an array of recent literature (e.g.,[6], [14]), evidence has accumulated to indicate that the search should be widened beyond quadratic forms (see, e.g. [5], [6], [11], [16], [22]). Recent years have witnessed an extensive search for nonquadratic and/or homogeneous Lyapunov functions, among which are piecewise quadratic Lyapunov functions ([12], [20]), polyhedral Lyapunov functions ([2], [4]), and homogeneous polynomial Lyapunov functions (HPLFs) ([5], [11], [22]). An important contribution was made in [7], making avail- able in the LDI framework tools based on duality, similar to those that are well-appreciated in linear systems. The exponential stability of an LDI was shown to be equivalent to that of its dual LDI (which is given by transposes of the matrices describing the original one); similar equivalences were given for dissipativity and stabilizability/detectability properties. What made such results possible was relying on convex (not necessarily quadratic) Lyapunov and storage functions, and their conjugates in the sense of convex analysis. For example, if V (x) is a Lyapunov function for a given LDI, then its conjugate V * (x) is a Lyapunov function for the dual. Such relationship shows in particular that some types of Lyapunov functions naturally appear as conjugates of functions of another type. For instance, a HPLF of degree m induces a homogeneous Lyapunov function of degree m/(m−1). As is demonstrated by examples, known numerical tools applied to one type of Lyapunov functions can yield strikingly different stability results from those obtained for the conjugate type. This is one of the strengths of the duality theory – it doubles the number of tools one can use. In this paper, we will employ a particular pair of con- jugate functions: the convex hull of a family of quadratic functions V c (x) and the pointwise maximum of a family of quadratic functions V * c (x). Their conjugacy relationship follows from general principles of convex analysis, see [17], details can be also found in [7]. These two functions have quite different properties. In what follows, we will often refer to V c (x) as the convex hull function, and to V * c (x) as the max function. We note that V c (x) was successfully used, without the duality framework, in stability analysis in [8], where it was referred to as the composite quadratic function. The functions V c (x), V * c (x) and their dual relationship 43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas 0-7803-8682-5/04/$20.00 ©2004 IEEE FrC13.3 5499