Abstract
Recently, stability analysis of time-delay systems has received much attention. Rich results have been obtained on this topic using various approaches and techniques. Most of those results are based on Lyapunov stability theories. The purpose of this article is to give a broad overview of stability of linear time-delay systems with emphasis on the more recent progress. Methods and techniques for the choice of an appropriate Lyapunov functional and the estimation of the derivative of the Lyapunov functional are reported in this article, and special attention is paid to reduce the conservatism of stability conditions using as few as possible decision variables. Several future research directions on this topic are also discussed.
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References
Liu G P, Rees D, Chai S C, Nie X Y. Design, simulation and implementation of networked predictive control systems. Measurement & Control, 2005, 38: 17–21
Liu G P, Mu J, Rees D. Design and stability analysis of networked control systems with random communication time delay using the modified MPC. International Journal of Control, 2006, 79: 288–297
Liu G P, Xia Y, Rees D, Hu W S. Design and stability criteria of networked predictive control systems with random network delay in the feedback channel. IEEE Transactions on Systems, Man and Cybernetics–Part C, 2007, 37(2): 173–184
Sun J, Chen J, Gan MG. A necessary and sufficient stability criterion for networked predictive control systems. Science China Technological Sciences, 2016, 59(1): 2–8
Li M, Sun J, Dou L, Stability of an improved dynamic quantized system with time-varying delay and packet losses. IET Control Theory & Applications, 2015, 9(6): 988–995
Sun J, Chen J. Networked predictive control for systems with unknown or partially known delay. IET Control Theory & Applications, 2014, 8(18): 2282–2288
Michiels W, Niculescu S-I, Moreau L. Using delays and time-varying gains to improve the static output feedback stabilizability of linear systems: a comparison. IMA Journal of Mathematical Control and Information, 2004, 21(4): 393–418
Salarieh H, Alasty A. Delayed feedback control of chaotic spinning disk via minimum entropy approach. Nonlinear Analysis: Theory, Methods and Applications, 2008, 69(10): 3273–3280
Zhang B L, Huang Z W, Hang Q L. Delayed non-fragile H ∞ control for offshore steel jacket platforms. Journal of Vibration and Control, 2015, 21(5): 959–974
El’sgol’ts L E, Norkin S B. Introduction to the theory and applications of differential equations with deviating arguments. Mathematics in Science and Engineering, Vol 105. New York: Academic Press, 1973
Kolmanovskii V B, Nosov V R. Stability of functional differential equations. Mathematics in Science and Engineering, Vol 180. New York: Academic Press, 1986
Kolmanovskii V B, Myshkis A. Applied Theory of Functional Differential Equations. Boston: Kluwer Academic Publishers, 1992
Dugard L, Verriest E I. Stability and Control of Time-delay Systems, London: Springer-Verlag, 1998
Kolmanovskii V, Myshkis A. Applied theory of functional differential equations. Dordrecht: Kluwer, 1999
Niculescu SI. Delay Effects on Stability: A Robust Control Approach (Lecture Notes in Control and Information Sciences). London: Springer–Verlag, 2001
Boukas E K, Liu Z K. Deterministic and Stochastic Time Delay Systems. Boston: Birkhäuser, 2002.
Chiasson J, Loiseau J J. Applications of Time Delay Systems (Lecture Notes in Control and Information Sciences). London: Springer–Verlag, 2007
Fridman E. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. System & Control Letters, 2001, 43: 309–319
Fridman E, Shaked U. An improved stabilization method for linear systems with time-delay. IEEE Transactions on Automatic Control, 2002, 47(11): 1931–1937
Fridman E, Orlov Y. Exponential stability of linear distributed parameter systems with time-varying delays. Automatica, 2009, 45(2): 194–201
Fridman E, Shaked U, Liu K. New conditions for delay-derivativedependent stability. Automatica, 2009, 45(11): 2723–2727
Liu K, Fridman E. Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica, 2012, 48(1): 102–108
Han Q L. Robust stability of uncertain delay-differential systems of neutral type. Automatica, 2002, 38(4): 719–723
Han Q L. On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty. Automatica, 2004, 40(6): 1087–1092
Jiang X, Han Q L. On H ∞ control for linear systems with interval time-varying delay. Automatica, 2005, 41(12): 2099–2106
Jiang X, Han Q L. Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica, 2006, 42(6): 1059–1065
Gao H J, Wang C H. Comments and further results on “A descriptor system approach to H ∞ control of linear time-delay systems”. IEEE Transactions on Automatic Control, 2003, 48(3): 520–525
Gao H J, Wang C H. A delay-dependent approach to robust H ∞ filtering for uncertain discrete-time state-delayed systems. IEEE Transactions on Signal Processing, 2004, 52(6): 1631–1640
Gao H J, Meng X Y, Chen T W. Stabilization of networked control systems with a new delay characterization. IEEE Transactions on Automatic Control, 2008, 53(9): 2142–2148
Lam J, Gao H J, Wang C H. Stability analysis for continuous systems with two additive time-varying delay components. Systems & Control Letters, 2007, 56(1): 16–24
Yue D, Han Q L, Peng C. State feedback controller design of networked control systems. IEEE Transactions on Circuits and Systems—II: Express Briefs, 2004, 41: 640–644
Yue D, Han Q L, Lam J. Network-based robust H ∞ control of systems with uncertainty. Automatica, 2005, 41(6): 999–1007
Xu S Y, Lam J. Improved delay-dependent stability criteria for timedelay systems. IEEE Transactions on Automatic Control, 2005, 50(3): 384–387
Xu S Y, Lam J, Zou Y. Further results on delay-dependent robust stability conditions of uncertain neutral systems. International Journal of Robust and Nonlinear Control, 2005, 15(5): 233–246
Li X, De Souza C E. Delay-dependent robust stability and stabilisation of uncertain linear delay systems: a linear matrix inequality approach. IEEE Transactions on Automatic Control, 1997, 42(8): 1144–1148
Shao H Y. Improved delay-dependent stability criteria for systems with a delay varying in a range. Automatica, 2008, 44(12): 3215–3218
Shao H Y. New delay-dependent stability criteria for systems with interval delay. Automatica, 2009, 45(3): 744–749
Chen W H, Zheng W X. Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica, 2007, 43(1): 95–104
Mahmoud M S. Resilient L 2 − L ∞ filtering of polytopic systems with state delays. IET Control Theory & Applications, 2007, 1(1): 141–154
Zhang XM, Han Q L. Abel lemma-based finite-sum inequality and its application to stability analysis for linear discrete time-delay systems. Automatica, 2015, 57: 199–202
Liu K, Fridman E, Johansson K H, Xia Y. Generalized Jensen inequalities with application to stability analysis of systems with distributed delays over infinite time-horizons. Automatica, 2016, 69: 222–231
Chen Y G, Fei S M, Gu Z, Li Y M. New mixed-delay-dependent robust stability conditions for uncertain linear neutral systems. IET Control Theory Applications, 2014, 8(8): 606–613
Zhang B Y, Lam J, Xu S Y. Stability analysis of distributed delay neural networks based on relaxed Lyapunov-Krasovskii functionals. IEEE Transactions on Neural Networks and Learning Systems, 2015, 26(7): 1480–1492
Peng D, Hua C C. Improved approach to delay-dependent stability and stabilisation of two-dimensional discrete-time systems with interval time-varying delays. IET Control Theory Applications, 2015, 9(12): 1839–1845
Li J, Chen Z H, Cai D S, Zhen W, Huang Q. Delay-Dependent stability control for power system with multiple time-delays. IEEE Transactions on Power Systems, 2016, 31(3): 2316–2326
Ding L, He Y, Wu M, Ning C. Improved mixed-delay-dependent asymptotic stability criteria for neutral systems. IET Control Theory Applications, 2015, 9(14): 2180–2187
Zhang C K, He Y, Jiang L, Wu M, Zeng H B. Delay-variationdependent stability of delayed discrete-time systems. IEEE Transactions on Automatic Control, 2015
Boyd S, Ghaoui L E, Feron E, Balakrishnan V. Linear Matrix Inequality in Systems and Control Theory (SIAM Studies in Applied Mathematics). Philadelphia, PA: SIAM, 1994
Hale J K, Lunel S M V. Introduction to functional differential equations. New York: Springer, 1993
Gu K, Kharitonov V L, Chen J. Stability of Time-delay Systems, Boston: Brikhäuser, 2003
Fridman E. Introduction to Time-delay Systems: Analysis and Control (Systems and Control: Foundations and Applications). Springer, 2014
Wu M, He Y, She J H, Liu G P. New delay-dependent stability criteria for robust stability of time-varying delay systems. Automatica, 2004; 40(8): 1435–1439
He Y, Wang Q G, Lin C, Wu M. Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems. International Journal of Robust and Nonlinear Control, 2005, 15(18): 923–933
He Y, Wang Q G, Xie L, Lin C. Further improvement of freeweighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 2007, 52(2): 293–299
Sun J, Liu G P, Chen J. Delay-dependent stability and stabilization of neutral time-delay systems. International Journal of Robust and Nonlinear Control, 2009, 19(12): 1364–1375
Sun J, Liu G P. On improved delay-dependent stability criteria for neutral time-delay systems. European Journal of Control, 2009, 15(6): 613–623
Sun J, Chen J, Liu G P, Rees D. Delay-dependent robust H ∞ filter design for uncertain linear systems with time-varying delay. Circuits, Systems, and Signal Processing, 2009, 28(5): 763–775
Sun J, Liu G P, Chen J, Rees D. Improved stability criteria for neural networks with time-varying delay. Physics Letters A, 2009, 373(3): 342–348
Sun J, Liu G P. A new delay-dependent stability criterion for timedelay systems. Asian Journal of Control, 2009, 11(4): 427–431
Sun J, Chen J, Liu G P, Rees D. Delay-range-dependent and raterange- dependent stability criteria for linear systems with time-varying delays. In: Proceedings of IEEE Conference on Decision and Control. 2009, 251–256
Sun J, Chen J, Liu G P, Rees D. On robust stability of uncertain neutral systems with discrete and distributed delays. In: Proceedings of American Control Conference. 2009, 5469–5473
Sun J, Liu G P, Chen J, Rees D. Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica, 2010, 46(2): 466–470
Sun J, Liu G P, Chen J, Rees D. Improved stability criteria for linear systems with time-varying delay. IET Control Theory & Applications, 2010, 4(4): 683–689
Chen J, Sun J, Liu G P, Rees D. New delay-dependent stability criteria for neural networks with time-varying interval delay. Physics Letters A, 2010, 374(43): 4397–4405
Qian W, Cong S, Li T, Fei S M. Improved stability conditions for systems with interval time-varying delay. International Journal of Control, Automation, and Systems, 2012, 10(6): 1146–1152
Lee WI, Jeong C, Park P G. Further improvement of delay-dependent stability criteria for linear systems with time-varying delays. In: Proceedings of the 12th International Conference on Control, Automation and Systems. 2012, 1300–1304
Liu J, Hou ZW. New stability analysis for systems with interval timevarying delay based on Lyapunov functional method. Journal of Information & Computational Science, 2014, 11(6): 1843–1851
He Y, Wang Q G, Lin C, Wu M. Delay-dependent stability for systems with time-varying delay. Automatica, 2007, 43(2): 371–376
Sun J, Han Q L, Chen J, Liu G P. Less conservative stability criteria for linear systems with interval time-varying delays. International Journal of Robust and Nonlinear Control, 2015, 25(40): 475–485
Sun J, Chen J. Stability analysis of static recurrent neural networks with interval time-varying delay. Applied Mathematics and Computation, 2013, 221(15): 111–120
Qian W, Liu J, Fei SM. New augmented Lyapunov functional method for stability of uncertain neutral systems with equivalent delays. Mathematics and Computers in Simulation, 2012, 84: 42–50
Gu K. Discretized LMI set in the stability problem of linear uncertain time-delay systems. International Journal of Control, 1997, 68(4): 923–934
Gu K, Han Q L, Luo A C J, Niculescu S I. Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficients. International Journal of Control, 2001, 74(7): 737–744
Gu K. An improved stability criterion for systems with distributed delays. International Journal of Robust and Nonlinear Control, 2003, 13(9): 819–831
Kharitonov V L, Niculescu S I. On the stability of linear systems with uncertain delay. IEEE Transactoins on Automatic Control, 2003, 48(1): 127–132
Kharitonov V L, Zhabko A P. Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems. Automatica, 2003, 39(1): 15–20
Fridman E, Niculescu S I. On complete Lyapunov-Krasovskii functional techniques for uncertain systems with fast-varying delays. International Journal of Robust Nonlinear Control, 2008, 18(3): 364–374
He Y, Liu G P, Rees D. New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Transactions on Neural Network, 2007, 18(1): 310–314
He Y, Wu M, Liu G P, She J H. Output feedback stabilization for a discrete-time systems with a time-varying delay. IEEE Transactions on Automatic Control, 2008, 53(10): 2372–2377
Fridman E, Shaked U. Delay-dependent stability and H ∞ control: constant and time-varying delays. International Journal of Control, 2003, 76(1): 48–60
Park P. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Transactions on Automatic Control, 1999, 44(4): 876–877
Moon Y S, Park P, Kwon W H, Lee Y S. Delay-dependent robust stabilization of uncertain state-delayed systems. International Journal of Control, 2001, 74(14): 1447–1455
WuM, He Y, She J H. Stability Analyssi and Robust Control of Timedelay Systems. London: Springer, 2010.
Xu S, Lam J. A survey of linear matrix inequality techniques in stability analysis of delay systems. International Journal of Systems Science, 2008, 39(12): 1095–1113
He Y, Wu M, She J H, Liu G P. Parameter-dependent Lyapunov for stability of time-delay systems with polytopic-type uncertainties. IEEE Transactions on Automatic Control, 2004, 49(5): 828–832
He Y,Wu M, She J H. An improved H ∞ filter design for systems with time-varying interval delay. IEEE Transactions on Signal Processing, 2006, 53(11): 1235–1239
He Y, Wu M, She J H. An improved global asymptotic stability criterion for delayed cellular neural networks. IEEE Transactions on Neural Network, 2006, 17(1): 250–252
He Y, Wu M, She J H. Delay-dependent exponential stability of delayed neural networks with time-varying delay. IEEE Transactions on Circuits and Systems—II: Express Briefs, 2006, 53(7): 553–557
Wu M, He Y, She J H. New delay-dependent stability criteria and stabilising method for neutral systems. IEEE Transactions on Automatic Control, 2004, 49(12): 2266–2271
Zhang X M, Wu M, She J H, He Y. Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica, 2005, 41(8): 1405–1412
Zeng H B, He Y, Wu M, She J H. Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Transactions on Automatic Control, 2015, 60(10): 2768–2772
Gu K. An integral inequality in the stability problem of time-delay systems. In: Proceedings of the 39th IEEE Conference on Decision and Control. 2000, 2805–2810
Gahinet P, Apkarian P. A linear matrix inequality approach to H ∞ control. International Journal of Robust and Nonlinear Control, 1994, 4(4): 421–448
Gouaisbaut F, Peaucelle D. A note on stability of time delay systems. In: Proceedings of the 5th IFAC Symposium on Robust Control Design. 2006
Gouaisbaut F, Peaucelle D. Delay-dependent robust stability of time delay systems. In: Proceedings of the 5th IFAC Symposium on Robust Control Design. 2006
Suplin V, Fridman E, Shaked U. H ∞ control of linear uncertain timedelay systems–a projection approach. IEEE Transactions on Automatic Control, 2006, 51(4): 680–685
Zhang X M, Han Q L. Novel delay-derivative-dependent stability criteria using new bounding techniques. International Journal of Robust and Nonlinear Control, 2013, 23(13): 1419–1432
Briat C. Convergence and equivalence results for the Jensen’s inequality—application to time-delay and sampled-data systems. IEEE Transactions on Automatic Control, 2011, 56(7): 1660–1665
Seuret A, Gouaisbaut F. Wirtinger-based integral inequality: application to time-delay systems. Automatica, 2013, 49(9): 2860–2866
Seuret A, Gouaisbaut F, Fridman E. Stability of systems with fastvarying delay using improved wirtinger’s inequality. In: Proceedings of the 52nd IEEE Conference on Decision and Control. 2013, 946–951
Ji X F, Su H Y. A note on equivalence between two integral inequalities for time-delay systems. Automatica, 2015, 53: 244–246
Gyurkovics E. A note onWirtinger-type integral inequalities for timedelay systems. Automatica, 2015, 61: 44–46
Park M, Kwon O, Park JH, Lee S, Cha E. Stability of time-delay systems via Wirtinger-based double integral inequality. Automatica, 2015, 55: 204–208
Hien L V, Trinh H. Refined Jensen-based inequality approach to stability analysis of time-delay systems. IET Control Theory & Applications, 2015, 9(14): 2188–2194
Park P G, Lee W, Lee S Y. Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. Journal of the Franklin Institute, 2015, 352(4): 1378–1396
Kim J H. Further improvement of Jensen inequality and application to stability of time-delayed systems. Automatica, 2016, 61: 121–125
Park P, Ko J W. Stability and robust stability for systems with a timevarying delay. Automatica, 2007, 43(10): 1855–1858
Kim J H. Note on stability of linear systems with time-varying delay. Automatica, 2011, 47(9): 2118–2121
Park P, Ko JW, Jeong C. Reciprocally convex approach to stability of systems with time-varying delays. Automatica, 2011, 47(1): 235–238
Lee WI, Park P. Second-order reciprocally convex approach to stability of systems with interval time-varying delays. Applied Mathematics and Computation, 2014, 229: 245–253
Zhu X L, Wang Y, Yang G H. New stability criteria for continuoustime systems with interval time-varying delay. IET Control Theory & Applications, 2010, 4(6): 1101–1107
Tang M, Wang Y W, Wen C. Improved delay-range-dependent stability criteria for linear systems with interval time-varying delays. IET Control Theory & Applications, 2012, 6(6): 868–873
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61104097, 61321002, 61120106010, 61522303, and U1509215), Program for Changjiang Scholars and Innovative Research Team in University (IRT1208), ChangJiang Scholars Program, Beijing Outstanding PhD Program Mentor (20131000704), Program for New Century Excellent Talents in University (NCET-13-0045), and Beijing Higher Education Young Elite Teacher Project.
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Jian Sun received his PhD Degree from the Institute of Automation, Chinese Academy of Sciences, China in 2007. He is currently a professor in the School of Automation, Beijing Institute of Technology, China. His current research interests include networked control systems, timedelay systems, security of CPSs, and robust control. He is the awardee of the NSFC Excellent Young Scholars Program in 2015.
Jie Chen is a professor at School of Automation, Beijing Institute of Technology, China. His research interest covers complex system multi-objective optimization and decision, constrained nonlinear control, and optimization methods.
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Sun, J., Chen, J. A survey on Lyapunov-based methods for stability of linear time-delay systems. Front. Comput. Sci. 11, 555–567 (2017). https://doi.org/10.1007/s11704-016-6120-3
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DOI: https://doi.org/10.1007/s11704-016-6120-3