DOI QR코드

DOI QR Code

Stability and Robust H Control for Time-Delayed Systems with Parameter Uncertainties and Stochastic Disturbances

  • Kim, Ki-Hoon (School of Electrical Engineering, Chungbuk National University) ;
  • Park, Myeong-Jin (School of Electrical Engineering, Chungbuk National University) ;
  • Kwon, Oh-Min (School of Electrical Engineering, Chungbuk National University) ;
  • Lee, Sang-Moon (School of Electronics Engineering, Daegu University) ;
  • Cha, Eun-Jong (Department of Biomedical Engineering, School of Medicine, Chungbuk National University)
  • Received : 2014.02.10
  • Accepted : 2015.09.01
  • Published : 2016.01.01

Abstract

This paper investigates the problem of stability analysis and robust H controller for time-delayed systems with parameter uncertainties and stochastic disturbances. It is assumed parameter uncertainties are norm bounded and mean and variance for disturbances of them are known. Firstly, by constructing a newly augmented Lyapunov-Krasovskii functional, a stability criterion for nominal systems with time-varying delays is derived in terms of linear matrix inequalities (LMIs). Secondly, based on the result of stability analysis, a new controller design method is proposed for the nominal form of the systems. Finally, the proposed method is extended to the problem of robust H controller design for a time-delayed system with parameter uncertainties and stochastic disturbances. To show the validity and effectiveness of the presented criteria, three examples are included.

Keywords

Stochastic disturbances;Parameter uncertainties;Time-varying delays;Lyapunov method

References

  1. Y. He, Q.G. Wang, L. Xie, C. Lin, “Further improvement of free-weighting matrices technique for systems with time-varying delay,” IEEE Transactions on Automatic Control, vol. 52, pp. 293-299, 2007. https://doi.org/10.1109/TAC.2006.887907
  2. P.G. Park, J. Wan Ko, “Stability and robust stability for systems with a time-varying delay,” Automatica, vol. 43, pp. 1855-1858, 2007. https://doi.org/10.1016/j.automatica.2007.02.022
  3. W. Qian, H.R. Karimi, “New delay-dependent stability conditions for time-varying delay systems,” Mathematical Problems in Engineering, vol. 40, pp. 1435-1439, 2013.
  4. W. Qian, S. Cong, T. Li, S. Fei, “Improved stability conditions for systems with interval time-varying delay,” International Journal of Control, Automation, and Systems, vol. 10, pp. 1146-1152, 2012. https://doi.org/10.1007/s12555-012-0609-9
  5. O.M. Kwon, M.J. Park, J.H. Park, S.M. Lee, E.J. Cha, “New delay-partitioning approaches to stability criteria for uncertain neutral systems with time-varying delays,” Journal of the Franklin Institute, vol. 349, pp. 2799-2823, 2012. https://doi.org/10.1016/j.jfranklin.2012.08.013
  6. J.H. Kim, “Note on stability of linear systems with time-varying delay,” Automatica, vol. 47, pp. 2118-2121, 2011. https://doi.org/10.1016/j.automatica.2011.05.023
  7. M.J. Park, O.M. Kwon, J.H. Park, S.M. Lee, “A new augmented Lyapunov-Krasovskii functional approach for stability of linear systems with time-varying delays” Applied Mathematics and Computation, vol. 217, pp.7197-7209, 2011. https://doi.org/10.1016/j.amc.2011.02.006
  8. P.G. Park, J.W. Ko, C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, pp. 235-238, 2011. https://doi.org/10.1016/j.automatica.2010.10.014
  9. A. Seuret, F. Gouaisbaut, “Wirtinger-based integral inequality: Application to time-delay systems,” Automatica, vol. 49, pp. 2860-2866, 2013. https://doi.org/10.1016/j.automatica.2013.05.030
  10. K. Gu, “An integral inequality in the stability problem of time-delay systems,” in Proc. the 39th IEEE Conference on Decision and Control, pp. 2805-2810, 2000.
  11. J. Liu, J. Zhang, “Note on stability of discrete-time time-varying delay systems,” IET Control Theory Appl., vol. 6, pp. 335-339, 2012. https://doi.org/10.1049/iet-cta.2011.0147
  12. X.L. Zhu, G.H. Yang, T. Li, C. Lin, L. Guo, “LMI stability criterion with less variables for time-delay systems,” International Journal of Control, Automation and Systems, vol. 7, pp. 530-535, 2009. https://doi.org/10.1007/s12555-009-0404-4
  13. S.H. Kim, P. Park, and C.K. Jeong, “Robust $H_\infty$ stabilisation of networks control systems with packet analyser,” IET Contr. Theory Appl., vol.4, no. 9, pp. 1828-1837, 2010. https://doi.org/10.1049/iet-cta.2009.0346
  14. E. Fridman, U. Shaked, “An improved stabilization method for linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 47, pp. 1931-1937, 2002. https://doi.org/10.1109/TAC.2002.804462
  15. X.M. Zhang, M. Wu, J.H. She, Y. He, “Delay-dependent stabilization of linear systems with time-varying state and input delays,” Automatica, vol. 41, pp. 1405-1412, 2005. https://doi.org/10.1016/j.automatica.2005.03.009
  16. J. Sun, G.P. Liu, J. Chen , “Delay-dependent stability and stabilization of neutral time-delay systems,” International Journal of Robust and Nonlinear Control, vol. 19, pp. 1364-1375, 2009. https://doi.org/10.1002/rnc.1384
  17. O. M. Kwon, J. H. Park, “Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays,” Applied Mathematics and Computation, vol. 208, pp. 58-68, 2009. https://doi.org/10.1016/j.amc.2008.11.010
  18. O.M. Kwon, M.J. Park, J.H. Park, S.M. Lee, E.J. Cha, “Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov-Krasovskii functional,” Journal of the Franklin Institute, vol. 350, pp. 521-540, 2013. https://doi.org/10.1016/j.jfranklin.2012.12.013
  19. M.J. Park, O.M. Kwon, J.H. Park, S.M. Lee, K.H. Kim, “Randomly occurring leader-following consensus criterion for multi-agent systems with communication delay” in Proc. Control, Automation and Systems (ICCAS), 2012 12th International Conference on, pp. 883-886, 2012.
  20. W. Qian, S. Cong, Y. Sun, S. Fei, “Novel robust stability criteria for uncertain systems with time-varying delay,” Applied Mathematics and Computation, vol. 215, pp. 866-872, 2009. https://doi.org/10.1016/j.amc.2009.06.022
  21. M. Wu, Y. He, J.H. She, G.P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, vol. 40, pp. 1435-1439, 2004. https://doi.org/10.1016/j.automatica.2004.03.004
  22. S. Xu, J. Lam, Y. Zou, “Further results on delay-dependent robust stability conditions of uncertain neutral systems,” International Journal of Robust and Nonlinear Control, vol. 15, pp. 233-246, 2005. https://doi.org/10.1002/rnc.983
  23. Y. He, M. Wu, J.H. She, G.P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Systems and Control Letters, vol. 51, pp. 57-65, 2004. https://doi.org/10.1016/S0167-6911(03)00207-X
  24. T. Li, L. Guo, Y. Zhang, “Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 18, pp. 1372-1387, 2008. https://doi.org/10.1002/rnc.1280
  25. E. Fridman, U. Shaked, “Delay-dependent stability and H control: constant and time-varying delays,” International Journal of Control, vol. 76, pp. 48-60, 2003. https://doi.org/10.1080/0020717021000049151
  26. H. Gao, C. Wang, “Comments and further results on “A descriptor system approach to H control of linear time-delay systems”,” IEEE Transactions on Automatic Control, vol. 48, pp. 520-525, 2003. https://doi.org/10.1109/TAC.2003.809154
  27. H. Yan, H. Zhang, M.Q.H. Meng, “Delay-range-dependent robust H control for uncertain systems with interval time-varying delays,” Neurocomputing, vol. 73, pp. 1235-1243, 2010. https://doi.org/10.1016/j.neucom.2010.01.004
  28. C.E. De Souza, X. Li, “Delay-dependent robust H control of uncertain linear state-delayed systems,” Automatica, vol. 35, pp. 1313-1321, 1999. https://doi.org/10.1016/S0005-1098(99)00025-4
  29. J. Yao, Z.H. Guan, G. Chen, D.W.C. Ho, “Stability, robust stabilization and H control of singular-impulsive systems via switching control,” Systems and Control Letters, vol. 55, pp. 879-886, 2006. https://doi.org/10.1016/j.sysconle.2006.05.002
  30. G. Zames, “Feedback and optimal sensitivity: Model reference transformations, multiplicative semi norms, and approximate inverses,” IEEE Transactions on Automatic Control, vol. 26, pp. 301-320, 1981. https://doi.org/10.1109/TAC.1981.1102603
  31. H. Gao, J. Wu, P. Shi, “Robust sampled-data H control with stochastic sampling,” Automatica, vol. 45, pp. 1729-1736, 2009. https://doi.org/10.1016/j.automatica.2009.03.004
  32. Z. Wang, D.W.C. Ho, Y. Liu, X. Liu, “Robust H control for a class of nonlinear discrete time-delay stochastic systems with missing measurements,” Automatica, vol. 45, pp. 684-691, 2009. https://doi.org/10.1016/j.automatica.2008.10.025
  33. Z. Wang, Y. Wang, Y. Liu, “Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays,” IEEE Transactions on Neural Networks, vol. 21, pp. 11-25, 2010. https://doi.org/10.1109/TNN.2009.2033599
  34. J. Hu, Z. Wang, H. Gao, L.K. Stergioulas, “Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities,” IEEE Transactions on Industrial Electronics, vol. 29, pp. 3008-3015, 2012.
  35. S. Boyd, El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics: Philadelphia, SIAM, 1994.
  36. M.C. de Oliveira, R.E. Skelton, Stability tests for Constrained Linear systems: Berlin, Springer-Verlag, 2001.
  37. Y. Wang, L. Xie, C.E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems and Control Letters, vol. 19, pp. 139-149, 1992. https://doi.org/10.1016/0167-6911(92)90097-C

Cited by

  1. Influence Analysis of a Higher-Order CSI Effect on AMD Systems and Its Time-Varying Delay Compensation Using a Guaranteed Cost Control Algorithm vol.7, pp.4, 2017, https://doi.org/10.3390/app7040313
  2. Observer-based resilient finite-time control of blood gases model during extra-corporeal circulation vol.12, pp.4, 2018, https://doi.org/10.1049/iet-syb.2017.0083