Abstract
This article deals with the problem of delay-dependent finite-time stability (FTS) for delayed discrete-time systems with nonlinear perturbations. First, based on a Lyapunov–Krasovskii Functional, delay-dependent FTS conditions are provided by introducing some free-weighting matrices. Then, a new reduced free-matrix-based inequality is established to estimate the single summation term. The dimensions of these free matrices integral in our results are less than those obtained in the literature. This reduction in the number of variables does not mean that our method is a particular case but simply that our approach is completely different from the others and therefore our method is more effective. Thus, less conservative design conditions are obtained in this paper in terms of linear matrix inequalities (LMIs) and solved using MATLAB’s LMI toolbox to achieve the desired performance. The purpose of this paper is to derive sufficient conditions that ensure the finite-time stability of the discrete-time system. Finally, numerical examples are examined to show the advantage and effectiveness of the proposed results.
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References
Abolpour R, Dehghani M, Talebi HA (2021) A non-conservative state feedback control methodology for linear systems with state delay. Int J Syst Sci 52(12):2549–2563
Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. In: Livre, studies in applied mathematics. SIAM
Cai Z, Hu G (2021) Stability analysis of drilling inclination system with time-varying delay via free-matrix-based Lyapunov–Krasovskii functional. J Adv Comput Intell Intell Inf 25(6):1031–1038
Chen G, Yang Y (2015) Robust finite-time stability of fractional order linear time-varying impulsive systems. Circuits Syst Signal Process 34(4):1325–1341
Chen J, Lu J, Xu S (2016) Summation inequality and its application to stability analysis for time-delay systems. IET Control Theory Appl 10(4):391–395
El Fezazi N, Lamrabet O, El Haoussi F, Tissir EH (2020) New observer-based controller design for delayed systems subject to input saturation and disturbances. Iran J Sci Technol Trans Electr Eng 44(3):1081–1092
El Fezazi N, Tissir EH, El Haoussi F, Bender FA, Husain AR (2019) Controller synthesis for steer-by-wire system performance in vehicle. Iran J Sci Technol Trans Electr Eng 43(4):813–825
El Haoussi F, Tissir EH, Tadeo F, Hmamed A (2011) Delay-dependent stabilisation of systems with time-delayed state and control: application to a quadruple-tank process. Int J Syst Sci 42(1):41–49
Gu K, Kharitonov VL, Chen J (2003) Stability of time-delay systems. In: Livre, control engineering. Springer
Hedayati Khodayari M, Pariz N, Balochian S (2022) Stabilizer design for an under-actuated autonomous underwater vehicle in a descriptor model under unknown time delay and uncertainty. Trans Inst Meas Control 44(2):484–496
Idrissi S, Tissir EH, Boumhidi I, Chaibi N (2013) New delay dependent robust stability criteria for TS fuzzy systems with constant delay. Int J Control Autom Syst 11(5):885–892
Idrissi S, Tissir EH (2012) Delay dependent robust stability of T-S fuzzy systems with additive time varying delays. Appl Math Sci 6(1):1–12
Jiang X, Han QL, Yu X (2005) Stability criteria for linear discrete-time systems with interval-like time-varying delay. In: IEEE American control conference, Portland, pp 2817–2822
Kamenkov G (1953) On stability of motion over a finite interval of time. J Appl Math Mech 17(2):529–540
Kang W, Zhong S, Shi K, Cheng J (2016) Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Trans 60:67–73
Lin X, Liang K, Li H, Jiao Y, Nie J (2017) Finite-time stability and stabilization for continuous systems with additive time-varying delays. Circuits Syst Signal Process 36(7):2971–2990
Liu H, Shi P, Karimi HR, Chadli M (2016) Finite-time stability and stabilisation for a class of nonlinear systems with time-varying delay. Int J Syst Sci 47(6):1433–1444
Liu Y, Ma W, Mahmoud MS, Lee SM (2015) Improved delay-dependent exponential stability criteria for neutral-delay systems with nonlinear uncertainties. Appl Math Model 39(10–11):3164–3174
Nam PT, Trinh H, Pathirana PN (2015) Discrete inequalities based on multiple auxiliary functions and their applications to stability analysis of time-delay systems. J Franklin Inst 352(12):5810–5831
Park P, Ko JW, Jeong C (2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1):235–238
Saravanakumar R, Datta R, Cao Y (2022) New insights on fuzzy sampled-data stabilization of delayed nonlinear systems. Chaos Solitons Fractals 154:111654
Seuret A, Gouaisbaut F, Fridman E (2015) Stability of discrete-time systems with time-varying delays via a novel summation inequality. IEEE Trans Autom Control 60(10):2740–2745
Shen JC, Chen BS, Kung FC (1991) Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach. IEEE Trans Autom Control 36(5):638–640
Stojanovic SB, Debeljkovic DL, Dimitrijevic N (2012) Finite-time stability of discrete-time systems with time-varying delay. Chem Ind Chem Eng Q CICEQ 18(4–1):525–533
Xie W (2008) Improved delay-independent \(H_2\) performance analysis and memoryless state feedback for linear delay systems with polytopic uncertainties. Int J Control Autom Syst 6(2):263–268
Yan X, Song X, Wang X (2018) Global output-feedback stabilization for nonlinear time-delay systems with unknown control coefficients. Int J Control Autom Syst 16(4):1550–1557
Zhang XM, Han QL (2015) Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems. Automatica 57:199–202
Zhang Z, Zhang Z, Zhang H, Zheng B, Karimi HR (2014) Finite-time stability analysis and stabilization for linear discrete-time system with time-varying delay. J Franklin Inst 351(6):3457–3476
Zhang CK, He Y, Jiang L, Wu M, Zeng HB (2015) Delay-variation-dependent stability of delayed discrete-time systems. IEEE Trans Autom Control 61(9):2663–2669
Zhang XM, Han QL, Ge X (2017) A novel finite-sum inequality-based method for robust \(H_\infty\) control of uncertain discrete-time Takagi-Sugeno fuzzy systems with interval-like time-varying delays. IEEE Trans Cybern 48(9):2569–2582
Zhong Q, Cheng J, Zhao Y (2015) Delay-dependent finite-time boundedness of a class of Markovian switching neural networks with time-varying delays. ISA Trans 57:43–50
Zuo Z, Li H, Wang Y (2013) New criterion for finite-time stability of linear discrete-time systems with time-varying delay. J Franklin Inst 350(9):2745–2756
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Teresa Alvarez is funded by Agencia Estatal de Innovación, MICInn, PID2020-112871RB-C21 and PID2021-123654OB-C31.
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El Akchioui, N., El Fezazi, N., Farkous, R. et al. Finite-Time Stability for Discrete-Time Systems with Time-Varying Delays and Nonlinear Perturbations Using Relaxed Summation Inequality. Iran J Sci Technol Trans Electr Eng 48, 433–443 (2024). https://doi.org/10.1007/s40998-023-00669-8
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DOI: https://doi.org/10.1007/s40998-023-00669-8