A new class of Lyapunov functions for the constrained stabilization of linear systems Aldo Balestrino a , Andrea Caiti a , Sergio Grammatico a a Department of Energy and Systems Engineering, University of Pisa, Largo Lucio Lazzarino 1, 56122 Pisa, Italy. Abstract The constrained stabilization of linear uncertain systems is investigated via the set-theoretic framework of control Lyapunov R-functions. A novel composition rule allows the design of a composite control Lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones, generalizing both polyhedral and truncated ellipsoidal control Lyapunov functions. The feasibility test of the proposed smooth control Lyapunov functions can be casted into matrix inequalities conditions. The constrained linear quadratic control is addressed as an application. Key words: Generated Lyapunov functions; Lyapunov methods; Feedback stabilization; Constraints; Uncertain linear systems. 1 Introduction The state-feedback stabilization of constrained uncer- tain linear systems, covering saturations of the control inputs, state constraints and model uncertainties, is equivalent to the design of a robust Control Lyapunov Function (CLF). Since the particular choice of the can- didate CLF also provides an estimation of the controlled invariant set [Blanchini, 1999], the exact solution con- sists in providing the largest controlled invariant region of the state space, according to both state and control constraints [Balestrino et al., 2011a]. In general, non- trivial classes of candidate CLFs are required to shape the maximal controlled invariant set. For instance, Polyhedral CLFs (PCLFs) are a universal class of func- tions for the stabilizability of uncertain linear systems [Blanchini, 1995], or equivalently Linear Differential In- clusions (LDIs). PCLFs can be smoothed with standard norms [Blanchini and Miani, 1999] in order to obtain an everywhere differentiable smoothed PCLF that can be used together with nonlinear gradient-based continuous controllers [Petersen and Barmish, 1987]. Recently, the class of Truncated Ellipsoids (TEs) [O’Dell and Misawa, 2002, Thibodeau et al., 2009] has been proposed as can- ⋆ E-mail addresses: {a.balestrino,a.caiti}@dsea.unipi.it (A. Balestrino, A. Caiti); grammatico.sergio@gmail.com (S. Grammatico). Corresponding author: S. Grammatico. Tel.: +39 0502217384; fax: +39 0502217333. didate LFs and CLFs for constrained uncertain linear systems to provide a good approximation of the maxi- mal controlled invariant region with a reduced number of parameters [O’Dell and Misawa, 2002]. In [Thibodeau et al., 2009] a linear state-feedback control is designed by solving a Bilinear Matrix Inequality (BMI), maximizing the volume of the estimated controlled invariant set. The main contribution of this paper is the definition of a novel composition rule for merging two different CLFs, allowing the design of a non-homothetic smooth CLF with the following properties: a) the external level set exactly shapes the maximal controlled invariant set; b) the inner sublevel sets can be made arbitrarily close to any given choice of smooth ones. This properties allow to define a stabilizing nonlinear gradient-based control law that is continuous everywhere inside the maximal controlled invariant set. The results of [Balestrino et al., 2010, 2011a,b], where a basic composition rule is intro- duced, are extended to the class of constrained uncer- tain linear systems by deriving the more general class of so called Control Lyapunov R-Functions (CLRFs). Moreover, a Linear Matrix Inequality (LMI) feasibility test for the candidate CLRF is here proposed. As in [Chesi and Hung, 2008, Hu and Blanchini, 2010], the synthesis condition is obtained via BMIs. CLRFs can smooth both PCLFs and TEs in a non-homothetic way and they can be made everywhere differentiable. The novel smoothing technique follows from the frame- work of R-functions, referred in the next section. In Sections 3 and 4 the main results are provided. Section 5 Preprint submitted to Automatica 15 June 2012 CONFIDENTIAL. Limited circulation. For review only Preprint submitted to Automatica Received June 15, 2012 12:02:09 PST