Automatica 46 (2010) 610–614 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Technical communique A delay-partitioning approach to the stability analysis of discrete-time systems ✩ Xiangyu Meng a,b,∗ , James Lam b , Baozhu Du b , Huijun Gao a a Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China b Department of Mechanical Engineering, University of Hong Kong, Hong Kong article info Article history: Received in revised form 23 August 2009 Accepted 18 November 2009 Available online 16 December 2009 Keywords: Asymptotic stability Discrete-time systems Delay systems abstract This paper revisits the problem of stability analysis for linear discrete-time systems with time-varying delay in the state. By utilizing the delay partitioning idea, new stability criteria are proposed in terms of linear matrix inequalities (LMIs). These conditions are developed based on a novel Lyapunov functional. In addition to delay dependence, the obtained conditions are also dependent on the partitioning size. We have also established that the conservatism of the conditions is a non-increasing function of the number of partitions. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction As an important and fundamental problem, stability analysis has been at the forefront of the research on time-delay systems in recent years. Compared with continuous-time systems with time delays (Papachristodoulou, 2004; Peet, Papachristodoulou, & Lall, 2009), discrete-time systems with state delay have a strong background in engineering applications, among which network- based control has been well recognized to be a typical example. However, little effort has been made towards investigating the stability of discrete time-delay systems (Chen, Guan, & Lu, 2003). So far, a few approaches have been proposed to solve discrete- time systems with time delay. For a constant delay, a delay system can be converted to a delay-free one by using the so-called lifting or state-augmentation approach (Xu, Lam, & Zhang, 2002), while systems with time-varying delays have been transformed into switched systems in Hetel, Daafouz, and Iung (2008) and Xia, Liu, Shi, Rees, and Thomas (2007), so that classic results can be applied to analyze the problems of stability. Solving the problem of stability without performing model transformation to the original system is another research direction. In Lee and Kwon (2002), a delay-dependent stability condition is presented for discrete- time systems with unknown constant delay. Improved delay- dependent conditions are provided in Xu, Lam, and Zou (2005), ✩ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor André L. Tits. ∗ Corresponding address: University of Alberta, 9107 - 116 Street 2nd Floor ECERF Bldg., T6G 2V4 Edmonton, Canada. Tel.: +1 780 4924875; fax: +1 780 4921811. E-mail addresses: xmeng2@ece.ualberta.ca (X. Meng), james.lam@hku.hk (J. Lam). which has been established that the proposed conditions are less conservative. For time-varying delay, a stability condition is proposed in Gao, Lam, Wang, and Wang (2004) by using Moon’s inequality (Moon, Park, Kwon, & Lee, 2001), which is dependent on the minimum and maximum delay bounds. However, some useful terms are ignored in Gao et al. (2004); the method in Gao and Chen (2007) improved the result in Gao et al. (2004) by defining new Lyapunov functions and by using bounding inequalities for cross products between two vectors. When revisiting this problem, we find that the results reported in the literature still leave much room for improvement. The choices of specific Lyapunov functionals and bounding techniques are the origin of conservatism. In Gouaisbaut and Peaucelle (2006), artificial fractioning of the delay is introduced to give a sequence of LMI conditions for continuous- time systems. The same idea is also introduced to study the stability of continuous systems with multiple time-varying delay components (Du, Lam, Shu, & Wang, 2009). Another related approach can be found in Gu, Kharitonov, and Chen (2003), in which the discretized functional is used. This motivated us to carry out the present study. In this paper, for the first time, we utilize the delay partitioning idea to solve the problem of stability analysis for linear discrete systems with time-varying delay in the state. The aim of this paper is to provide tractable conditions for stability analysis, which have significantly reduced conservatism. This reduced conservatism benefits from the fact that the free-weighting matrix approach (Wu, He, She, & Liu, 2004) is employed and the delay partitioning idea is adopted. In addition to delay dependence, the obtained conditions are also dependent on the partitioning part. The approach developed has much lower computational complexity than those using full state augmentation, and our method is 0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.12.004