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New Robust Stability Criteria for Lur’e Systems with Time-varying Delays and Sector-bounded Nonlinearities

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  • Control Theory and Applications
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Abstract

This paper deals with a robust stability problem for uncertain Lur’e systems with time-varying delays and sector-bounded nonlinearities. An improved delay-dependent robust stability criterion is proposed via a modified Lyapunov-Krasovskii functional (LKF) approach. Firstly, a modified LKF consisting of delay-dependent matrices and double-integral items under two delay subintervals is constructed, thereby making full use of the delay and its derivative information. Secondly, the stability criteria can be expressed as convex linear matrix inequality (LMI) via the properties of quadratic function application. Thirdly, to further reduce the conservatism of stability criteria, the quadratic generalized free-weighting matrix inequality (QGFMI) is used. Finally, some numerical examples, including the Lur’e system and the general linear time-delayed system, are presented to show the improvement of the proposed approach.

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References

  1. J. Li, Q. Zhang, D. Zhai, and Y. Zhang, “Delay-dependent H∞ control for descriptor Markovian jump systems with time-varying delay,” Journal of Applied Mathematics, vol. 2013, Article ID: 545636, 2013.

  2. J. Feng, J. Lam, and G. Yang, “Optimal partitioning method for stability analysis of continuous/discrete delay systems,” International Journal of Robust and Nonlinear Control, vol. 25, pp. 559–574, 2015.

    MathSciNet  MATH  Google Scholar 

  3. L. Ding, Y. He, M. Wu, and Z. Zhang, “A novel delay partitioning method for stability analysis of interval time-varying delay systems,” Journal of the Franklin Institute, vol. 354, pp. 1209–1219, 2017.

    MathSciNet  MATH  Google Scholar 

  4. X. Zhang, Q. Han, A. Seuret, and F. Gouaisbaut, “An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay,” Automatica, vol. 84, pp. 221–226, 2017.

    MathSciNet  MATH  Google Scholar 

  5. J. Chen, J. Park, and S. Xu, “Stability analysis of continuous-time systems with time-varying delay using new Lyapunov-Krasovskii functionals,” Journal of the Franklin Institute, vol. 355, pp. 5957–5967, 2018.

    MathSciNet  MATH  Google Scholar 

  6. D. Liao, S. Zhong, J. Cheng, C. Zhao, X. Zhang, and Y. Yu, “A new result on stability analysis for discrete system with interval time-varying delays,” Advances in Difference Equations, vol. 2019, pp. 123–129, 2019.

    MathSciNet  MATH  Google Scholar 

  7. C. Jeong, P. Park, and S. Kim, “Improved approach to robust stability and H∞ performance analysis for systems with an interval time-varying delay,” Applied Mathematics and Computation, vol. 218, pp. 10533–10541, 2012.

    MathSciNet  MATH  Google Scholar 

  8. J. Sun, Q. Han, J. Chen, and G. Liu, “Less conservative stability criteria for linear systems with interval time-varying delays,” International Journal of Robust and Nonlinear Control, vol. 25, pp. 475–485, 2015.

    MathSciNet  MATH  Google Scholar 

  9. T. Lee, J. Park, and S. Xu, “Relaxed conditions for stability of time-varying delay systems,” Automatica, vol. 75, pp. 11–15, 2017.

    MathSciNet  MATH  Google Scholar 

  10. T. Lee and J. Park, “A novel Lyapunov functional for stability of time-varying delay systems via matrix-refinedfunction,” Automatica, vol. 80, pp. 239–242, 2017.

    MathSciNet  MATH  Google Scholar 

  11. W. Qian, M. Yuan, L. Wang, X. Bu, and J. Yang, “Stabilization of systems with interval time-varying delay based on delay decompsing approach,” ISA Transactions, vol. 70, pp. 1–6, 2017.

    Google Scholar 

  12. T. Lee and J. Park, “Improved stability conditions of time-varying delay systems based on new Lyapunov functionals,” Journal of the Franklin Institute, vol. 355, pp. 1176–1191, 2018.

    MathSciNet  MATH  Google Scholar 

  13. K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proceedings of the Conference of the IEEE Industrial Electronics, Sydney, Australia, 2010.

  14. A. Seuret and F. Gouaisbaut, “Wirtinger-based integral inequality: Application to time-delay systems,” Automatica, vol. 49, pp. 2860–2866, 2013.

    MathSciNet  MATH  Google Scholar 

  15. P. Park, W. Lee, and S. Lee, “Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems,” Journal of the Franklin Institute, vol. 352, pp. 1378–1396, 2015.

    MathSciNet  MATH  Google Scholar 

  16. M. Park, O. Kwon, and J. Ryu, “Advanced stability criteria for linear systems with time-varying delays,” Journal of the Franklin Institute, vol. 355, pp. 520–543, 2018.

    MathSciNet  MATH  Google Scholar 

  17. F. Long, C. Zhang, L. Jiang, Y. He, and M. Wu, “Stability analysis of systems with time-varying delay via improved Lyapunov-Krasovskii functionals,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019. DOI: https://doi.org/10.1109/TSMC.2019.2914367

  18. C. Shen, Y. Li, X. Zhu, and W. Duan, “Improved stability criteria for linear systems with two additive time-varying delays via a novel Lyapunov functional,” Journal of Computational and Applied Mathematics, vol. 363, pp. 312–324, 2020.

    MathSciNet  MATH  Google Scholar 

  19. X. Duan, F. Tang, and W. Duan, “Improved robust stability criteria for uncertain linear neutral-type systems via novel Lyapunov-Krasovskii functional,” Asian Journal of Control, vol. 2019, 2019. DOI: https://doi.org/10.1002/asjc.2142

  20. A. Seuret and F. Gouaisbaut, “Hierarchy of LMI conditions for the stability analysis of time-delay systems,” Systems Control Letter, vol. 81, pp. 1–7, 2015.

    MathSciNet  MATH  Google Scholar 

  21. P. Park, W. Lee, and S. Lee, “Improved stability criteria for linear systems with interval time-varying delays: generalized zero equalities approach,” Applied Mathematics and Computation, vol. 292, pp. 336–348, 2017.

    MathSciNet  MATH  Google Scholar 

  22. S. Kim, “Further results on stability analysis of discrete-time systems with time-varying delays via the use of novel convex combination coefficients,” Applied Mathematics and Computation, vol. 261, pp. 104–113, 2015.

    MathSciNet  MATH  Google Scholar 

  23. W. Qian, L. Wang, and M. Chen, “Local consensus of nonlinear multiagent systems with varying delay coupling,” IEEE transactions on systems, man, and cybernetics. Part B, Cybernetics, vol. 48, pp. 2462–2469, 2017.

    Google Scholar 

  24. W. Lee, S. Lee, and P. Park, “A combined first- and second-order reciprocal convexity approach for stability analysis of systems with interval time-varying delays,” Journal of the Franklin Institute, vol. 353, pp. 2104–2116, 2016.

    MathSciNet  MATH  Google Scholar 

  25. J. Chen, J. Park, and S. Xu, “Stability analysis for neural networks with time-varying delay via improved techniques,” IEEE Transactions on Cybernetics, vol. 49, pp. 4495–4500, 2018.

    Google Scholar 

  26. H. Zeng, Y. He, M. Wu, and J. She, “Free-matrix-based integral inequality for stability analysis of systems with time-varying delay,” IEEE Transactions on Automatic Control, vol. 60, pp. 2768–2772, 2015.

    MathSciNet  MATH  Google Scholar 

  27. C. Zhang, Y. He, L. Jiang, M. Wu, and H. Zeng, “Stability analysis of systems with time-varying delay via relaxed integral inequalities,” Systems and Control Letters, vol. 92, pp. 52–61, 2016.

    MathSciNet  MATH  Google Scholar 

  28. C. Zhang, Y. He, L. Jiang, and M. Wu, “Notes on stability of time-delay systems: Bounding inequalities and augmented Lyapunov-Krasovskii functionals,” IEEE Transactions on Automatic Control, vol. 62, pp. 5331–5336, 2017.

    MathSciNet  MATH  Google Scholar 

  29. H. Xu, C. Zhang, L. Jiang, and J. Smitha, “Stability analysis of linear systems with two additive time-varying delays via delay-product-type Lyapunov functional,” Applied Mathematical Modelling, vol. 45, pp. 955–964, 2017.

    MathSciNet  MATH  Google Scholar 

  30. X. Xie, D. Yue, H. Zhang, and C. Peng, “Control synthesis of discrete-time T-S fuzzy systems: Reducing the conservatism whilst alleviating the computational burden,” IEEE Transactions on Cybernetics, vol. 47, pp. 2480–2491, 2016.

    Google Scholar 

  31. X. Xie, D. Yue, and C. Peng, “Relaxed real-time scheduling stabilization of discrete-time Takagi-Sugeno fuzzy systems via an alterable-weights-based ranking switching mechanism,” IEEE Transactions on Fuzzy Systems, vol. 26, pp. 3808–3819, 2018.

    Google Scholar 

  32. J. Li and Q. Zhang, “Fuzzy reduced-order compensatorbased stabilization for interconnected descriptor systems via integral sliding modes,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, pp. 752–765, 2017.

    Google Scholar 

  33. H. Shen, F. Li, H. Yan, H. Karimi, and H. Lam, “Finite-time event-triggered H∞ control for T-S fuzzy Markov jump systems,” IEEE Transactions on Fuzzy Systems, vol. 26, pp. 3122–3135, 2018.

    Google Scholar 

  34. A. Lurie, Some Nonlinear Problem in the Theory of Automatic Control, H. M. Stationary Office, London, 1957.

    Google Scholar 

  35. K. Ramakrishnan and G. Ray, “An improved delay-dependent stability criterion for a class of Lur’e systems of neutral-type,” Journal of Dynamic Systems, Measurement, and Control, vol. 134, no. 1, 011008, Jan. 2012.

    Google Scholar 

  36. W. Duan, B. Du, Y. Li, C. Shen, X. Zhu, X. Li, and J. Chen, “Improved sufficient LMI conditions for the robust stability of time-delayed neutral-type Lur’e systems,” International Journal of Control, Automation and Systems, vol. 16, pp. 2343–2353, 2018.

    Google Scholar 

  37. W. Duan, Y. Li, J. Chen, and L. Jiang, “New results on stability analysis of uncertain neutral-type Lur’e systems derived from a modified lyapunov-krasovskii functional,” Complexity, vol. 2019, Article ID: 1706264, 2019.

  38. M. Wu, Z. Feng, Y. He, and J. She, “Improved delay-dependent absolute stability and robust stability for a class of nonlinear systems with a time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 20, pp. 694–702, 2010.

    MathSciNet  MATH  Google Scholar 

  39. Y. Wang, Y. Xue, and X. Zhang, “Less conservative robust absolute stability criteria for uncertain neutral-type Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 353, pp. 816–833, 2016.

    MathSciNet  MATH  Google Scholar 

  40. H. Huang, H. Li, and J. Zhong, “Master-slave synchronization of general Lur’e systems with time-varying delayand parameter uncertainty,” International Journal of Bifurcation and Chaos, vol. 16, pp. 281–294, 2006.

    MathSciNet  MATH  Google Scholar 

  41. Q. Han and D. Yue, “Absolute stability of Lur’e systems with time-varying delay,” IET Control Theory and Applications, vol. 1, pp. 854–859, 2007.

    MathSciNet  Google Scholar 

  42. Q. Han, “On designing time-varying delay feedback controllers for master-slave synchronization of Lur’e systems,” IEEE Transactions on Circuits and Systems I: Regular Paper, vol. 54, pp. 1573–1583, 2007.

    MATH  Google Scholar 

  43. J. Park, S. Lee, and P. Park, “An improved stability criteria for neutral-type Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 355, pp. 5291–5309, 2018.

    MathSciNet  MATH  Google Scholar 

  44. F. Long, C. Zhang, Y. He, L. Jiang, Q. Wang, and M. Wu, “Stability analysis of Lur’e systems with additive delay components via a relaxed matrix inequality,” Applied Mathematics and Computation, vol. 328, pp. 224–242, 2018.

    MathSciNet  MATH  Google Scholar 

  45. S. Xiao, X. Liu, C. Zhang, and H. Zeng, “Further results on absolute stability of Lur’e systems with a time-varying delay,” Neurocomputing, vol. 207, pp. 823–827, 2016.

    Google Scholar 

  46. B. Liu and X. Jia, “New absolute stability criteria for uncertain Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 355, pp. 4015–4031, 2018.

    MathSciNet  MATH  Google Scholar 

  47. W. Duan, Y. Li, and J. Chen, “Further stability analysis for time-delayed neural networks based on an augmented lyapunov functional,” IEEE Access, vol. 7, pp. 104655–104666, 2019.

    Google Scholar 

  48. W. Kwon, B. Koo, and S. Lee, “Novel Lyapunov-Krasovskii functional with delay-dependent matrix for stability of time-varying delay systems,” Applied Mathematics and Computation, vol. 320, pp. 149–157, 2018.

    MathSciNet  MATH  Google Scholar 

  49. Y. He and M. Wu, “Absolute stability for multiple delay general Lur’e control systems with multiple nonlinearities,” Journal of Computational and Applied Mathematics, vol. 159, pp. 241–248, 2003.

    MathSciNet  MATH  Google Scholar 

  50. J. Kim, “Further improvement of Jensen inequality and application to stability of time-delayed systems,” Automatica, vol. 64, pp. 121–125, 2016.

    MathSciNet  MATH  Google Scholar 

  51. I. Petersen, “A stabilization algorithm for a class of uncertain linear systems,” Systems and Control Letters, vol. 8, pp. 351–357, 1987.

    MathSciNet  MATH  Google Scholar 

  52. W. Duan, B. Du, Z. Liu, and Y. Zou, “Improved stability criteria for uncertain neutral-type Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 351, pp. 4538–4554, 2014.

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Electrical Engineering, Yancheng Institute of Technology, Yancheng Jiangsu, 224051, China

    Wenyong Duan & Jian Chen

  2. Undergraduate’s Office, Yancheng Biological Engineering Higher Vocational Technology School, Yancheng, Jiangsu, China

    Yan Li

  3. School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu, China

    Baozhu Du

Authors
  1. Wenyong Duan
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  2. Yan Li
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  3. Jian Chen
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  4. Baozhu Du
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Correspondence to Baozhu Du.

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recommended by Editor PooGyeon Park. This work is supported partially by the National NSF of China under Grant no. 61603325, the NSF of Jiangsu Province under Grant no. BK20160441, Outstanding Young Teacher of Jiangsu’ Blue Project’, the Talent Introduction Project of Yancheng Institute of Technology under Grant no. xj201516 and the Yellow Sea Rookie of Yancheng Institute of Technology.

Wenyong Duan received his Ph.D. degree in control science and engineering from the School of Automation, Nanjing University of Science and Technology, Nanjing, Jiangsu Province, China, in 2014. He joined the School of Electrical Engineering, YanCheng Institute of Technology in October 2014. His areas of interest include stability analysis and robust control of time-delay systems, singular systems and complex dynamic network systems.

Yan Li received her master’s degree in the School of Science, Nanjing University of Science and Technology, China in 2013. She has been a lecturer in Undergraduate Office, Yancheng Biological Engineering Higher Vocational Technology School, China since Sep. 2016. Her areas of interest include stability analysis of time-delay systems.

Jian Chen received his Ph.D. degrees from the University of Liverpool in 2015. He has been a lecturer in the School of Electrical Engineering, Yancheng Institute of Technology, China since Sep. 2016. His current research interests include wind power system control.

Baozhu Du received his B.S. in information and computing science, and M.S. degrees in operational research and cybernetics from Northeastern University, Shenyang, Liaoning Province, China, in 2003 and 2006, respectively. She obtained a Ph.D. degree in Mechanical Engineering from The University of Hong Kong in 2010. She joined Nanjing University of Science and Technology in April 2011, taking a lectureship in School of Automation. Her current research interests include stability analysis and robust control/filter theory of time-delay systems, positive systems, Markovian jump systems, and networked control systems.

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Duan, W., Li, Y., Chen, J. et al. New Robust Stability Criteria for Lur’e Systems with Time-varying Delays and Sector-bounded Nonlinearities. Int. J. Control Autom. Syst. 19, 596–606 (2021). https://doi.org/10.1007/s12555-019-0195-1

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