Automatica 79 (2017) 214–217 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Technical communique A new discrete-time stabilizability condition for Linear Parameter-Varying systems ✩ Amit Prakash Pandey, Maurício C. de Oliveira Department of Mechanical and Aerospace Engineering, UC San Diego, La Jolla, CA 92093, USA article info Article history: Received 3 October 2016 Received in revised form 2 December 2016 Accepted 11 January 2017 keywords: Parameter-varying systems Stabilizability Time-varying systems Linear matrix inequalities abstract We introduce a new condition for the stabilizability of discrete-time Linear Parameter-Varying (LPV) systems in the form of Linear Matrix Inequalities (LMIs). A distinctive feature of the proposed condition is the ability to handle variation in both the dynamics as well as in the input matrix without resorting to dynamic augmentation or iterative procedures. We show that this new condition contains the existing poly-quadratic stabilizability result as a particular case. A numerical example illustrates the results. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction and motivation Consider the class of time-varying discrete-time linear systems of the form x(k + 1) = A(ξ(k))x(k) + B(ξ(k))u(k), (1) where x ∈ R n and the matrices A(ξ(k)) and B(ξ(k)) are assumed to depend affinely on the time-varying parameter ξ(k), which takes values in the unit simplex Ξ =  ξ ∈ R N + : N  i=1 ξ i = 1  . The affine assumption means that matrices A(ξ(k)) and B(ξ(k)) can be written as A(ξ(k)) = N  i=1 ξ i (k)A i , B(ξ(k)) = N  i=1 ξ i (k)B i . In this paper, we are concerned with stabilizability by a gain- scheduled controller of the form: u(k) = K (ξ(k)) x(k) (2) ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor André L. Tits. E-mail addresses: appandey@ucsd.edu (A.P. Pandey), mauricio@ucsd.edu (M.C. de Oliveira). using Linear Matrix Inequalities (LMIs). There are various examples of practical applications of the above model, including spacecraft control (Calloni, Corti, Zanchettin, & Lovera, 2012; Corti, Dardanelli, & Lovera, 2012), active suspension systems (Do, da Silva, Sename, & Dugard, 2011; Do, Sename, & Dugard, 2010), and the many other applications in Mohammadpour and Scherer (2012) and related references. To the best of our knowledge, the most general necessary and sufficient stabilizability conditions for this class of time-varying systems that can still be expressed as LMIs are the ones from Daafouz and Bernussou (2001), which we reproduce in the next lemma. Lemma 1 (Daafouz & Bernussou, 2001). Consider the time-varying discrete-time linear system of the form (1). Assume that B i = B for all i = 1,..., N . The following statements are equivalent: (a) System (1) is poly − quadratically stabilizable; (b) There exist matrices X i , L i and Q i ≻ 0,i = 1,..., N , such that  X i + X T i − Q i X T i A T i + L T i B T A i X i + BL i Q j  ≻ 0, (3) for all i, j = 1,..., N. Furthermore, if inequalities (3) are feasible the gain-scheduled state- feedback controller (2) with gains K (ξ(k)) = N  i=1 ξ i (k)K i , K i = L i X −1 i , (4) poly-quadratically stabilizes the system (1). http://dx.doi.org/10.1016/j.automatica.2017.02.006 0005-1098/© 2017 Elsevier Ltd. All rights reserved.