Skip to main content
SpringerLink
Log in

Further results on delay-dependent state feedback \(H_\infty \) control of linear parameter varying time-delay systems

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This paper is concerned with the \(H_\infty \) control problem for linear parameter varying time-delay systems subject to \(L_2\)-norm bounded disturbances. Based on the Lyapunov-Krasovskii functional method, a new delay-dependent sufficient condition is derived for designing a state feedback \(H_\infty \) controller. In general, the coupling between decision variables and system matrices causes the results to be quite conservative. In order to obtain synthesis condition in terms of linear matrix inequalities, the well-known Young’s relation is employed to linearize the bilinear terms which unavoidably emerge in controller design for linear parameter varying time-delay systems. The proposed method can provide a controller with a larger delay range and a better disturbance attenuation effect. The efficiency and validity of the proposed control scheme are verified by the simulation results and comparisons.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Ramakrishnan K, Ray G (2015) Improved results on delay-dependent stability of LFC systems with multiple time-delays. J Control Autom Electr Syst 26(3):235

    Article  Google Scholar 

  2. Liu X, Zhang K (2018) Stabilization of nonlinear time-delay systems: distributed-delay dependent impulsive control. Syst Control Lett 120:17

    Article  MathSciNet  MATH  Google Scholar 

  3. Mahmoud MS (2009) Improved robust stability and feedback stabilization criteria for time-delay systems. IMA J Math Control Inf 26(4):451

    Article  MathSciNet  MATH  Google Scholar 

  4. da Cruz Figueredo LF, Ishihara JY, Borges GA, Bauchspiess A (2013) Delay-dependent robust stability analysis for time-delay T–S fuzzy systems with nonlinear local models. J Control Autom Electr Syst 24(1–2):11

    Google Scholar 

  5. Chen M, Sun J (2020) Delay-dependent conditions for finite time stability of linear time-varying systems with delay. Asian J Control 22(2):924

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Dong Y, Liang S, Guo L, Wang W (2018) Exponential stability and stabilization for uncertain discrete-time periodic systems with time-varying delay. IMA J Math Control Inf 35(3):963

    Article  MathSciNet  MATH  Google Scholar 

  8. Tlili AS (2021) \(H_\infty \) optimization-based stabilization for nonlinear disturbed time delay systems. J Control Autom Electr Syst 32(1):96

    Article  Google Scholar 

  9. Wu F, Packard A, Balas G (1995) In: Proceedings of 1995 34th IEEE conference on decision and control, vol 1. IEEE, pp 188–193

  10. Witte J, Balini H (2010) In: Scherer CW in proceedings of the 2010 American control conference. IEEE, pp 2194–2199

  11. He B, Yang M (2006) Robust LPV control of diesel auxiliary power unit for series hybrid electric vehicles. IEEE Trans Power Electron 21(3):791

    Article  Google Scholar 

  12. He X, Zhao J (2012) Robust LPV control of diesel auxiliary power unit for series hybrid electric vehicles. Appl Math Comput 218(9):5508

    MathSciNet  Google Scholar 

  13. Calafiore GC, Dabbene F (2009) Observer design with guaranteed RMS gain for discrete-time LPV systems with Markovian jumps. Int J Robust Nonlinear Control IFAC-Affili J 19(6):676

    Article  MathSciNet  MATH  Google Scholar 

  14. Briat C, Sename O, Lafay JF (2010) Memory-resilient gain-scheduled state-feedback control of uncertain LTI/LPV systems with time-varying delays. Syst Control Lett 59(8):451

    Article  MathSciNet  MATH  Google Scholar 

  15. Briat C, Sename O, Lafay JF (2008) Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach. IFAC Proc Vol 41(2):4946

    Article  Google Scholar 

  16. Wang J, Shi P, Wang J (2008) In: 2008 3rd international conference on innovative computing information and control. IEEE, pp 344–344

  17. Weiwei Q, Bing H, Gang L, Pengtao Z (2016) Robust model predictive tracking control of hypersonic vehicles in the presence of actuator constraints and input delays. J Franklin Inst 353(17):4351

    Article  MathSciNet  MATH  Google Scholar 

  18. Blanchini F, Casagrande D, Miani S, Viaro U (2016) Stable LPV realisation of the Smith predictor. Int J Syst Sci 47(10):2393

    Article  MathSciNet  MATH  Google Scholar 

  19. Yin H, Gao J, Liu Z (2017) A parameter dependent controller design approach for delayed LPV system. Asian J Control 19(1):391

    Article  MathSciNet  MATH  Google Scholar 

  20. Hu Y, Duan G, Tan F (2017) Finite-time control for LPV systems with parameter-varying time delays and exogenous disturbances. Int J Robust Nonlinear Control 27(17):3841

    MathSciNet  MATH  Google Scholar 

  21. Salavati S, Grigoriadis K, Franchek M (2019) Reciprocal convex approach to output-feedback control of uncertain LPV systems with fast-varying input delay. Int J Robust Nonlinear Control 29(16):5744

    Article  MathSciNet  MATH  Google Scholar 

  22. Nguyen CM, Pathirana PN, Trinh H (2019) Robust observer and observer-based control designs for discrete one-sided Lipschitz systems subject to uncertainties and disturbances. Appl Math Comput 353:42

    MathSciNet  MATH  Google Scholar 

  23. Zhang B, Zhou S, Xu S (2008) Gain-scheduled \(H_\infty \)-output feedback control for parameter-varying systems with delays. IMA J Math Control Inf 25(2):251

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang Y, Yang F, Han QL (2014) \(H_\infty \) control of LPV systems with randomly multi-step sensor delays. Int J Control Autom Syst 12(6):1207

    Article  Google Scholar 

  25. Sun M, Jia Y, Du J, Yuan S (2008) Delay-dependent \(H_\infty \) control for LPV systems with time delays. Int J Syst Control Commun 1(2):256

    Article  Google Scholar 

  26. Li F, Zhang X (2012) A delay-dependent bounded real lemma for singular LPV systems with time-variant delay. Int J Robust Nonlinear Control 22(5):559

    Article  MathSciNet  MATH  Google Scholar 

  27. Zope R, Mohammadpour J, Grigoriadis K, Franchek M (2012) In control of linear parameter varying systems with applications. Springer, New York, pp 279–299

    Book  Google Scholar 

  28. Nejem I, Bouazizi MH, Bouani F (2017) In: International conference on electrical engineering and control applications. Springer, pp 60–71

  29. Rosa TE, Frezzatto L, Morais CF, Oliveira RC (2018) \(H_\infty \) static output-feedback gain-scheduled control for discrete LPV time-delay systems. IFAC-PapersOnLine 51(26):137

    Article  Google Scholar 

  30. Zope R, Mohammadpour J, Grigoriadis K, Franchek M (2012) In: 2012 American control conference (ACC). IEEE, pp 775–780

  31. Ramezanifar A, Mohammadpour J, Grigoriadis KM (2014) Output-feedback sampled-data control design for linear parameter-varying systems with delay. Int J Control 87(12):2431

    Article  MathSciNet  MATH  Google Scholar 

  32. Wu F, Grigoriadis KM (2001) LPV systems with parameter-varying time delays: analysis and control. Automatica 37(2):221

    Article  MathSciNet  MATH  Google Scholar 

  33. Hu Y, Duan G (2019) \(H_\infty \) finite-time control for LPV systems with parameter-varying time delays and external disturbance via observer-based state feedback. J Franklin Inst 356(12):6303

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

    Article  Google Scholar 

  35. Huang J, Pan X, Hao X, Putra W (2020) Dynamic output feedback \(H_\infty \) control for linear parameter-varying systems with time-delay, international journal of control, automation and systems. pp 1–13

  36. Su X, Jin H, Shen W, Gu Y (2014) Delay-range-dependent control for automatic mooring positioning system with time-varying input delay. Shock Vib 2014

  37. Xie W (2005) \(H2\) gain scheduled state feedback for LPV system with new LMI formulation. IEE Proc-Control Theory Appl 152(6):693

    Article  Google Scholar 

  38. Boyd S, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory (SIAM)

  39. Zhang XM, Wu M, She JH, He Y (2005) Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica 41(8):1405

    Article  MathSciNet  MATH  Google Scholar 

  40. Zemouche A, Alessandri A (2014) in 53rd IEEE Conference on Decision and Control (IEEE), pp. 3125–3130

  41. Fridman E (2014) Introduction to time-delay systems: Analysis and control. Springer, New York

    Book  MATH  Google Scholar 

  42. Lofberg J (2004) in 2004 IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508) (IEEE), pp. 284–289

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

  44. Zhang X, Tsiotras P, Knospe C (2002) Stability analysis of LPV time-delayed systems. Int J Control 75(7):538

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang F, Grigoriadis KM (2005) in Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005. (IEEE), pp. 1532–1537

Download references

Funding

There was no funding.

Author information

Authors and Affiliations

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

    Majid Shahbazzadeh & Seyed Jalil Sadati

Authors
  1. Majid Shahbazzadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seyed Jalil Sadati
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors contributed to the design and implementation of the research, analysis of the results, and writing of the article.

Corresponding author

Correspondence to Seyed Jalil Sadati.

Ethics declarations

Conflict of interest

We declare that there is no conflict of interest in the publication of this article, and that there is no conflict of interest with any other author or institution for the publication of this article.

Ethical statements

We hereby declare that this manuscript is the result of our independent creation under the reviewers’ comments. Except for the quoted contents, this manuscript does not contain any research achievements that have been published or written by other individuals or groups.

Data availability

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

Code availability

Not applicable.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shahbazzadeh, M., Sadati, S.J. Further results on delay-dependent state feedback \(H_\infty \) control of linear parameter varying time-delay systems. Int. J. Dynam. Control 10, 1847–1857 (2022). https://doi.org/10.1007/s40435-022-00924-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-022-00924-6

Keywords

Search

Navigation