Stability of Lurie Time-Varying Systems with Time-Varying Delay Feedbacks

  • Shuli Guo
  • Lina Han


In this chapter, the stability problems of Lurie nonlinear system with time-varying plant and time-varying actuator under time-varying delay feedback are presented. First, absolute stability criteria, which we define as solvable matrix inequalities, are outlined by constructing a Lyapunov–Razumikhn functional. Second, we analyze these solvable matrix inequalities, and give their sufficient and necessary solvable conditions that are easily computed in practical systems. Third, based on our norm definition, solvable conditions and time-delay bound estimations can be obtained from two cases of solvable norm inequalities. More importantly, by analyzing all obtained procedures, we present the optimal combination method about a pair of \((\tau ,Q)\). Finally, an interested numerical example is presented to illustrate the above results effectively.



This work is Supported by National Key Research and Development Program of China (2017YFF0207400).


  1. 1.
    Bliman PA. Lyapunov-Krasovskii functionals and frequency domain: delay-independent absolute stability criteria for delay systems. Int J Robust Nonlinear Control. 2001;11(8):771–88.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bliman PA. Stability criteria for delay systems with sector-bounded nonlinearities. In: American control conference, proceedings of the IEEE, vol. 1; 2001. p. 402–7.Google Scholar
  3. 3.
    Cao J, Zhong S, Hu Y. Delay-dependent condition for absolute stability of Lurie control systems with multiple time delays and nonlinearities. J Math Anal Appl. 2006;338(1):497–504.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gao JF, Pan HP, Ji XF. A new delay-dependent absolute stability criterion for Lurie systems with time-varying delay. Acta Automatica Sinica. 2010;36(6):845–50.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gu K, Chen J, Kharitonov VL. Stability of time-delay systems. Berlin: Springer; 2003.CrossRefGoogle Scholar
  6. 6.
    Guo SL, Chu TG, Huang L. Absolute stability of the origin of Lurie-type nonlinear systems with MIMO bounded time-delays. J Eng Math. 2002;19(3):1–6.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hale JK. Introduction to functional differential equations, vol. 99. Berlin: Springer; 1993.zbMATHGoogle Scholar
  8. 8.
    Hale JK, Lunel SMV. Effects of small delays on stability and control. In: Operator theory and analysis. Birkhauser, Basel; 2001. p. 275–301.Google Scholar
  9. 9.
    Han QL. Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica. 2005;41(12):2171–6.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lur’e AI. Some non-linear problems in the theory of automatic control. HM Stationery Office; 1957.Google Scholar
  11. 11.
    Malek-Zavarei M, Jamshidi M. Time-delay systems: analysis, optimization and applications. Amsterdam: Elsevier Science Inc; 1987.Google Scholar
  12. 12.
    Mukhija P, Kar IN, Bhatt RKP. Delay-distribution-dependent robust stability analysis of uncertain Lurie systems with time-varying delay. Acta Automatica Sinica. 2012;38(7):1100–6.MathSciNetGoogle Scholar
  13. 13.
    Qiu F, Zhang Q. Absolute stability analysis of Lurie control system with multiple delays: an integral-equality approach. Nonlinear Anal Real World Appl. 2011;12(3):1475–84.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tian J, Zhong S, Xiong L. Delay-dependent absolute stability of Lurie control systems with multiple time-delays. Appl Math Comput. 2007;188(1):379–84.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang JZ, Duan ZS, Yang Y, Huang L. Analysis and control of nonlinear systems with stationary sets: time-domain and frequency-domain methods. Singapore: World Scientific; 2009.CrossRefGoogle Scholar
  16. 16.
    Xue M, Fei S, Li T, Pan J. Delay-dependent absolute stability criterion for Lurie system with probabilistic interval time-varying delay. J Control Theory Appl. 2012;10(4):477–82.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yakubovich VA, Leonov GA, Gelig AK. Stability of stationary sets in control systems with discontinuous nonlinearities. Singapore: World Scientific; 2004.CrossRefGoogle Scholar
  18. 18.
    Yin C, Zhong SM, Chen WF. On delay-dependent robust stability of a class of uncertain mixed neutral and Lure dynamical systems with interval time-varying delays. J Franklin Inst. 2010;347(9):1623–42.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Yu P. Absolute stability of nonlinear control systems, vol. 25. Berlin: Springer; 2008.Google Scholar
  20. 20.
    Yu L. On the absolute stability of a class of time-delay systems. Acta Automatica Sinica. 2003;5(29):780–4.MathSciNetGoogle Scholar
  21. 21.
    Yu L, Han QL, Yu S, Gao J. Delay-dependent conditions for robust absolute stability of uncertain time-delay systems. In: 42nd IEEE conference on decision and control, 2003. Proceedings, vol. 6; 2003. p. 6033–6037.Google Scholar
  22. 22.
    Zhu S, Zhang C, Cheng Z, Feng JE. Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems. IEEE Trans Autom Control. 2007;52(5):880–5.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Key Laboratory of Complex System Intelligent Control and Decision, School of AutomationBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Cardiovascular Internal Medicine of Nanlou Branch, National Clinical Research Center for Geriatric DiseasesChinese PLA General HospitalBeijingChina

Personalised recommendations