International Journal of Dynamics and Control https://doi.org/10.1007/s40435-018-0399-x On improved delay-range-dependent stability condition for linear systems with time-varying delay via Wirtinger inequality Rupak Datta 1 · Baby Bhattacharya 1 · Abanishwar Chakrabarti 2 Received: 6 November 2017 / Revised: 4 January 2018 / Accepted: 22 January 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract This paper studies the problem of delay-range-dependent stability analysis for the continuous-time linear systems with time- varying delay. A new and appropriate Lyapunov–Krasovskii (L–K) functional is constructed. To estimate the quadratic integral terms coming out from the derivative of L–K functional, utilize the well-known Wirtinger integral inequality together with the reciprocal convex lemma. Then, an improved delay-range-dependent stability condition is being established in terms of linear matrix inequalities (LMIs) in such a way that it can be effectively solved by using existing software (LMI toolbox in MATLAB). The delay upper bound results obtained by the developed stability condition are found to be less conservative than other recent results. Furthermore, the proposed stability criterion use the less number of decision variables and give the consistent delay bound results compared to some other methods. Two numerical examples are given to illustrate the effectiveness of the obtained stability condition compared to some recently published stability methods. Keywords Delay-range-dependent stability · Linear matrix inequality (LMI) · Lyapunov–Krasovskii functional · Wirtinger integral inequality · Reciprocal convex lemma 1 Introduction In many applications, one assumes that the future state of the dynamical system is determined exclusively by the present state of the system and is independent of the past state information. The system which includes past states infor- mation along with present states is termed as time-delay systems. Time-delays are inbuilt in many physical systems such as transport, communication, manufacturing process and long transmission line in pneumatic systems [1]. The presence of time-delay causes oscillation or even instability and it is strongly necessary to study the stability problem for time-delay systems before implementing various control strategies. Thus, stability analysis for dynamic systems is one of the important research area in control society. Stability assessment of time-delay systems using Lya- punov second method are broadly divided into two notions, B Rupak Datta rupak.kls@gmail.com 1 Department of Mathematics, National Institute of Technology, Agartala 799046, India 2 Department of Electrical Engineering, National Institute of Technology, Agartala 799046, India (i) delay-dependent stability [2–18] and (ii) delay inde- pendent stability [19]. Recently, the extension of delay- dependent stability into delay-range-dependent stability has also been reported in [13,20–25]. The delay-dependent sta- bility is less conservative than the delay independent stability because more information on time-delay is contained in the former. Thus, more attention in the field of delay depen- dent stability has attracted for systems with time varying delay [6,20–22,26] and [14,15,25,27–30]. The main research issue of the stability analysis for time delay systems is to achieve the maximum delay upper bounds for a given lower bound that guarantees the asymptotic stability of the con- cerned systems as large as possible and the resulting stability criterion was derived in an LMI [31] framework. Based on the above literature, there are mainly two key points to reduce the conservativeness and they are (i) constructing an appropriate Lyapunov–Krasovskii (L–K) functional, and (ii) approximating the integral terms arising out of from the time derivative of L–K functional by using a tighter bounding inte- gral inequality. For the construction of L–K functional, the most effective methods have delay-partitioning L–K func- tional approach in [3,22,25] and augmented L–K functionals in [30,32]. In [5], it is mentioned that the delay partitioning of L–K functional will always give improved results at the 123