Please cite this article as: Z. Li, H. Yan, H. Zhang et al., Stability analysis of linear systems with time-varying delay via intermediate polynomial-based functions. Automatica (2019) 108756, https://doi.org/10.1016/j.automatica.2019.108756. Automatica xxx (xxxx) xxx Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Technical communique Stability analysis of linear systems with time-varying delay via intermediate polynomial-based functions ✩ Zhichen Li a , Huaicheng Yan a,b,∗ , Hao Zhang b , Yan Peng c , Ju H. Park d , Yong He e a Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of Education, East China University of Science and Technology, Shanghai, 200237, China b Shanghai Institute of Intelligent Science and Technology, Tongji University, Shanghai, 200092, China c School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China d Department of Electrical Engineering, Yeungnam University, Gyeongsan 38541, South Korea e School of Automation, China University of Geosciences, Wuhan, 430074, China article info Article history: Received 17 December 2018 Received in revised form 7 June 2019 Accepted 20 November 2019 Available online xxxx Keywords: Time delay systems Stability analysis Fractional order intermediate polynomial Variable parameters abstract This note is devoted to stability analysis for linear system with time-varying delay. By advisably introducing slack matrices, novel fractional order intermediate polynomial-based functions (IPFs) are proposed. Then, the stability condition is derived for the time delay system. From configuration on fractional order polynomials, the relationships among system states are taken into account and consolidated via slack variables, and the characteristics for integral inequality are synthetically considered, while avoiding higher order time delays. More remarkably, adjusting tunable parameters also contributes to reduction of conservatism. The comparisons of computational complexity and stability region on a well known numerical example are provided to validate the advantages of the resulting stability criterion. © 2019 Elsevier Ltd. All rights reserved. 1. Introduction Time delay is a natural phenomenon in practical systems (Frid- man & Dambrine, 2009; Li, Yan, Zhang, Zhan, & Huang, 2019b; Yan, Tian, Li, Zhang, & Li, 2019), such as power system (Schiffer, Dörfler, & Fridman, 2017), robotic airship (Wang et al., 2019), and vehicle suspension system (Li, Jing, & Karimi, 2014). Time delay brings about sensitive dependence of system performance and stability on delay intervals (Li, Yan, Zhang, Zhan, & Huang, 2019a; Zhang & Han, 2015). In addition, the input delay idea is extensively used to model variable sampling intervals (Fridman, 2010) and network-induced delays (Freirich & Fridman, 2016; Liu, Fridman, & Johansson, 2015; Zhang & Han, 2013). Accordingly, stability analysis of time delay systems has received considerable attentions. ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor André L. Tits. ∗ Corresponding author at: Key Laboratory of Advanced Control and Opti- mization for Chemical Process of Ministry of Education, East China University of Science and Technology, Shanghai, 200237, China. E-mail addresses: zcli@ecust.edu.cn (Z. Li), hcyan@ecust.edu.cn (H. Yan), zhang_hao@tongji.edu.cn (H. Zhang), pengyan@shu.edu.cn (Y. Peng), jessie@ynu.ac.kr (J.H. Park), heyong08@cug.edu.cn (Y. He). As is well known, general Lyapunov–Krasovskii functionals (LKFs) with more information on time delay are helpful for re- ducing conservatism (Li et al., 2014, 2019a; Yan et al., 2019). A relaxed stability condition is suggested that some Lyapunov matrices need not to be positive definite (Lee, Park, & Xu, 2017). Dividing the delay interval by delay-central-point (DCP) method, novel LKF is introduced with delay-dependent matrices (Fridman, Shaked, & Liu, 2009). Moreover, a generalized delay partition- ing approach is developed in Yue, Tian, and Zhang (2009) with N > 2 equally spaced subintervals, in which the number of decision variables grows dramatically as the delay partitioning segments increase. In Liu and Li (2015), an optimal delay division approach is proposed by using a variable parameter. However, this adjustable parameter can only vary within the delay range. In Lee and Park (2017), the matrix-refined-functions (MRFs) are pro- posed in single integral augmentation form to provide impressive flexibility. Furthermore, by introducing a couple of orthogonal polynomials, new auxiliary polynomial-based functions (APFs) are presented to produce single integral with the 1st order scalar function (Li et al., 2019b). Based on a novel double integral in- equality, an improved inequality-based functions (IBFs) approach is given to offer both of single and double integrals (Li et al., 2019a). It is noted that only the single integrals are considered in Lee and Park (2017), and the matrices in Li et al. (2019a) and Li et al. (2019b) are incomplete with some zero components for https://doi.org/10.1016/j.automatica.2019.108756 0005-1098/© 2019 Elsevier Ltd. All rights reserved.