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Delay-Dependent Stabilization of Time-Delay Systems with Nonlinear Perturbations

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Abstract

This paper deals with the stability and stabilization problems of time-delay systems with nonlinear perturbations. The perturbations are modelled by a nonlinear function of current and/or delayed states. By utilizing Lyapunov–Krasovskii functional, sufficient conditions are obtained in terms of LMIs for the different types of perturbations. The stabilization problem is originally non-convex because of the coupling between the Lyapunov matrices and the controller gains. In order to decouple decision variables, the Young’s relation is used in a judicious manner. Numerical examples are given to demonstrate that the proposed method can provide a state feedback controller for a larger delay range than existing methods.

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Data Availability Statement

The datasets generated during the current study are available from the corresponding author on reasonable request.

References

  1. M. ApS, The MOSEK optimization toolbox for MATLAB manual. Version 9.0. (2019). http://docs.mosek.com/9.0/toolbox/index.html

  2. T. Binazadeh, H. Gholami, Finite-time robust passive control of uncertain discrete time-delay systems using output feedback: application on chua’s circuit. Circuits, Systems, and Signal Processing pp. 1–27 (2019)

  3. M. Charqi, N. Chaibi, M. Ouahi, E.H. Tissir, Delay-dependent admissibility and control of discrete-time switched singular systems with time-delay. Int. J. Syst. Control Commun. 11(2), 178–199 (2020)

    Article  Google Scholar 

  4. F. Chen, S. Kang, S. Qiao, C. Guo, Exponential stability and stabilization for quadratic discrete-time systems with time delay. Asian J. Control 20(1), 276–285 (2018)

    Article  MathSciNet  Google Scholar 

  5. Y. Chen, A. Xue, S. Zhou, R. Lu, Delay-dependent robust control for uncertain stochastic time-delay systems. Circuits Syst. Signal Process. 27(4), 447–460 (2008)

    Article  MathSciNet  Google Scholar 

  6. D. Cong Huong, V. Thanh Huynh, H. Trinh, Interval functional observers for time-delay systems with additive disturbances. Int. J. Adapt. Control Signal Process. 34(9): 1281–1293 (2020)

  7. O. Druzhinina, N. Sedova, Analysis of stability and stabilization of cascade systems with time delay in terms of linear matrix inequalities. J. Comput. Syst. Sci. Int. 56(1), 19–32 (2017)

    Article  MathSciNet  Google Scholar 

  8. A. Ech-charqy, M. Ouahi, E.H. Tissir, Delay-dependent robust stability criteria for singular time-delay systems by delay-partitioning approach. Int. J. Syst. Sci. 49(14), 2957–2967 (2018)

    Article  MathSciNet  Google Scholar 

  9. Q.L. Han, Robust stability for a class of linear systems with time-varying delay and nonlinear perturbations. Comput. Math. Appl. 47(8–9), 1201–1209 (2004)

    Article  MathSciNet  Google Scholar 

  10. C. Hua, P.X. Liu, X. Guan, Backstepping control for nonlinear systems with time delays and applications to chemical reactor systems. IEEE Trans. Industr. Electron. 56(9), 3723–3732 (2009)

    Article  Google Scholar 

  11. J.J. Hui, X.Y. Kong, H.X. Zhang, X. Zhou, Delay-partitioning approach for systems with interval time-varying delay and nonlinear perturbations. J. Comput. Appl. Math. 281, 74–81 (2015)

    Article  MathSciNet  Google Scholar 

  12. D.C. Huong, V.T. Huynh, H. Trinh, Interval functional observers design for time-delay systems under stealthy attacks. IEEE Trans. Circuits Syst. I Regul. Pap. 67(12), 5101–5112 (2020)

    Article  MathSciNet  Google Scholar 

  13. D.C. Huong, M.V. Thuan, On reduced-order linear functional interval observers for nonlinear uncertain time-delay systems with external unknown disturbances. Circuits Syst. Signal Process. 38(5), 2000–2022 (2019)

    Article  MathSciNet  Google Scholar 

  14. D.C. Huong, H. Trinh et al., Distributed functional interval observers for nonlinear interconnected systems with time-delays and additive disturbances. IEEE Syst. J. (2020)

  15. D.C. Huong, H. Trinh et al., On static and dynamic triggered mechanisms for event-triggered control of uncertain systems. Circuits Syst. Signal Process. 39(10), 5020–5038 (2020)

    Article  Google Scholar 

  16. D. Kharrat, H. Gassara, A. El Hajjaji, M. Chaabane, Adaptive fuzzy observer-based fault-tolerant control for Takagi-Sugeno descriptor nonlinear systems with time delay. Circuits Syst. Signal Process. 37(4), 1542–1561 (2018)

    Article  MathSciNet  Google Scholar 

  17. O.M. Kwon, M.J. Park, J.H. Park, S.M. Lee, E.J. Cha, Improved delay-dependent stability criteria for discrete-time systems with time-varying delays. Circuits Syst. Signal Process. 32(4), 1949–1962 (2013)

    Article  MathSciNet  Google Scholar 

  18. C.H. Lee, S.H. Lee, M.J. Park, O.M. Kwon, Stability and stabilization criteria for sampled-data control system via augmented lyapunov-krasovskii functionals. Int. J. Control Autom. Syst. 16(5), 2290–2302 (2018)

    Article  Google Scholar 

  19. W. Li, Z. Feng, W. Sun, J. Zhang, Admissibility analysis for Takagi-Sugeno fuzzy singular systems with time delay. Neurocomputing 205, 336–340 (2016)

    Article  Google Scholar 

  20. X. Lin, K. Liang, H. Li, Y. Jiao, J. Nie, Finite-time stability and stabilization for continuous systems with additive time-varying delays. Circuits Syst. Signal Process. 36(7), 2971–2990 (2017)

    Article  Google Scholar 

  21. P.L. Liu, New results on stability analysis for time-varying delay systems with non-linear perturbations. ISA Trans. 52(3), 318–325 (2013)

    Article  Google Scholar 

  22. P.L. Liu, New results on delay-range-dependent stability analysis for interval time-varying delay systems with non-linear perturbations. ISA Trans. 57, 93–100 (2015)

    Article  Google Scholar 

  23. Z.G. Liu, Y.Q. Wu, Universal strategies to explicit adaptive control of nonlinear time-delay systems with different structures. Automatica 89, 151–159 (2018)

    Article  MathSciNet  Google Scholar 

  24. J. Lofberg, YALMIP: A toolbox for modeling and optimization in MATLAB. In: 2004 IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508), pp. 284–289. IEEE (2004)

  25. K. Moezzi, A.G. Aghdam, Adaptive robust control of uncertain neutral time-delay systems. In: 2008 American Control Conference, pp. 5162–5167. IEEE (2008)

  26. A. Mukherjee, A. Sengupta, Delay-dependent stabilization of Lipschitz nonlinear systems. Sādhanā 43(9), 148 (2018)

    Article  MathSciNet  Google Scholar 

  27. I. Nejem, M.H. Bouazizi, F. Bouani, \(H_\infty \) dynamic output feedback control of LPV time-delay systems via dilated linear matrix inequalities. Trans. Inst. Meas. Control. 41(2), 552–559 (2019)

    Article  Google Scholar 

  28. S.K. Nguang, Robust stabilization of a class of time-delay nonlinear systems. IEEE Trans. Autom. Control 45(4), 756–762 (2000)

    Article  MathSciNet  Google Scholar 

  29. F.S. de Oliveira, F.O. Souza, Improved delay-dependent stability criteria for linear systems with multiple time-varying delays. International Journal of Control pp. 1–9 (2020)

  30. L. Pan, J. Cao, Stochastic quasi-synchronization for delayed dynamical networks via intermittent control. Commun. Nonlinear Sci. Numer. Simul. 17(3), 1332–1343 (2012)

    Article  MathSciNet  Google Scholar 

  31. S. Pourdehi, P. Karimaghaee, Stability analysis and design of model predictive reset control for nonlinear time-delay systems with application to a two-stage chemical reactor system. J. Process Control 71, 103–115 (2018)

    Article  Google Scholar 

  32. K. Ramakrishnan, G. Ray, Delay-range-dependent stability criterion for interval time-delay systems with nonlinear perturbations. Int. J. Autom. Comput. 8(1), 141–146 (2011)

    Article  Google Scholar 

  33. C. Scherer, S. Weiland, Linear matrix inequalities in control. Lecture Notes, Dutch Institute for Systems and Control, Delft, The Netherlands 3(2) (2000)

  34. H. Shao, Novel delay-dependent stability results for neural networks with time-varying delays. Circuits Syst. Signal Process. 29(4), 637–647 (2010)

    Article  MathSciNet  Google Scholar 

  35. J. Sun, J. Zhang, Further stability analysis of time-delay systems with nonlinear perturbations. Math. Probl. Eng. 2017(2017)

  36. W. Wang, S.K. Nguang, S. Zhong, F. Liu, Novel delay-dependent stability criterion for time-varying delay systems with parameter uncertainties and nonlinear perturbations. Inf. Sci. 281, 321–333 (2014)

    Article  MathSciNet  Google Scholar 

  37. Y. Wu, H. Zhang, G. Li, D. Sun, Y. Li, Novel robust stability condition for uncertain systems with interval time-varying delay and nonlinear perturbations. Int. J. Autom. Control 14(1), 98–114 (2020)

    Article  Google Scholar 

  38. A. Zemouche, A. Alessandri, A new LMI condition for decentralized observer-based control of linear systems with nonlinear interconnections. In: 53rd IEEE Conference on Decision and Control, pp. 3125–3130. IEEE (2014)

  39. W. Zhang, X.S. Cai, Z.Z. Han, Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations. J. Comput. Appl. Math. 234(1), 174–180 (2010)

    Article  MathSciNet  Google Scholar 

  40. X.M. Zhang, M. Wu, J.H. She, Y. He, Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41(8), 1405–1412 (2005)

    Article  MathSciNet  Google Scholar 

  41. Z. Zhang, C. Lin, B. Chen, New stability and stabilization conditions for T-S fuzzy systems with time delay. Fuzzy Sets Syst. 263, 82–91 (2015)

    Article  MathSciNet  Google Scholar 

  42. X. Zhou, H. Zhang, X. Hu, J. Hui, T. Li, Improved results on robust stability for systems with interval time-varying delays and nonlinear perturbations. Math. Probl. Eng. 2014(2014)

  43. Z. Zuo, Y. Wang, New stability criterion for a class of linear systems with time-varying delay and nonlinear perturbations. IEE Proc. Control Theory Appl. 153(5), 623–626 (2006)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Electrical and Computer Engineering, Babol Noshirvani University of Technology, Babol, Iran

    Majid Shahbazzadeh & Seyed Jalil Sadati

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  1. Majid Shahbazzadeh
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  2. Seyed Jalil Sadati
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Shahbazzadeh, M., Sadati, S.J. Delay-Dependent Stabilization of Time-Delay Systems with Nonlinear Perturbations. Circuits Syst Signal Process 41, 684–699 (2022). https://doi.org/10.1007/s00034-021-01810-w

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