Skip to main content
SpringerLink
Log in

Necessary and sufficient conditions for normalization and sliding mode control of singular fractional-order systems with uncertainties

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

The sliding mode control (SMC) problem for a normalized singular fractional-order system (SFOS) with matched uncertainties was investigated. Firstly, SFOS was normalized under constrained conditions. Then, the linear sliding mode (SM) function was designed using a fractional-order (FO) positive definite matrix and a linear matrix inequality (LMI). The SM controller was subsequently constructed based on switching laws. Finally, the feasibility of the method was evaluated using a numerical example.

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

Similar content being viewed by others

References

  1. Podlubny I. Fractional Differential Equations. NewYork: Academic Press, 1999

    MATH  Google Scholar 

  2. Petráš I. Modeling and numerical analysis of fractional-order Bloch equations. Comput Math Appl, 2011, 61: 341–356

    Article  MathSciNet  Google Scholar 

  3. Sabatier J, Farges C, Trigeassou J C. Fractional systems state space description: some wrong ideas and proposed solutions. J Vib Control, 2014, 20: 1076–1084

    Article  MathSciNet  Google Scholar 

  4. Kilbas A A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier, 2006

    MATH  Google Scholar 

  5. Wang Z. A numerical method for delayed fractional-order differential equations. J Appl Math, 2013, 2013: 1–7

    MathSciNet  Google Scholar 

  6. Wang Z, Huang X, Zhou J P. A numerical method for delayed fractional-order differential equations: based on G-L definition. Appl Math Inf Sci, 2013, 7: 525–529

    Article  MathSciNet  Google Scholar 

  7. Zhang B Y, Xu S Y, Ma Q, et al. Output-feedback stabilization of singular LPV systems subject to inexact scheduling parameters. Automatica, 2019, 104: 1–7

    Article  MathSciNet  Google Scholar 

  8. Feng Y, Yagoubi M. Robust Control of Linear Descriptor Systems. Berlin: Springer, 2017

    Google Scholar 

  9. Xu S Y, Lam J. Robust Control and Filtering of Singular Systems. Berlin: Springer, 2006

    Google Scholar 

  10. Dai L Y. Singular Control Systems. Berlin: Springer, 1989

    Google Scholar 

  11. N’Doye I, Darouach M, Zasadzinski M, et al. Robust stabilization of uncertain descriptor fractional-order systems. Automatica, 2013, 49: 1907–1913

    Article  MathSciNet  Google Scholar 

  12. Kaczorek T. Singular fractional linear systems and electrical circuits. Int J Appl Math Comput Sci, 2011, 21: 379–384

    Article  MathSciNet  Google Scholar 

  13. Yu Y, Jiao Z, Sun C Y. Sufficient and necessary condition of admissibility for fractional-order singular system. Acta Autom Sin, 2013, 39: 2160–2164

    Article  MathSciNet  Google Scholar 

  14. Zhang X F, Chen Y Q. Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order α: the 0 < α < 1 case. ISA Trans, 2018, 82: 42–50

    Article  Google Scholar 

  15. Lin C, Chen B, Shi P, et al. Necessary and sufficient conditions of observer-based stabilization for a class of fractionalorder descriptor systems. Syst Control Lett, 2018, 112: 31–35

    Article  Google Scholar 

  16. Shiri B, Baleanu D. System of fractional differential algebraic equations with applications. Chaos Soliton Fract, 2019, 120: 203–212

    Article  MathSciNet  Google Scholar 

  17. Ji Y D, Qiu J Q. Stabilization of fractional-order singular uncertain systems. ISA Trans, 2015, 56: 53–64

    Article  Google Scholar 

  18. Wei Y, Tse P W, Yao Z, et al. The output feedback control synthesis for a class of singular fractional order systems. ISA Trans, 2017, 69: 1–9

    Article  Google Scholar 

  19. Liu S, Zhou X F, Li X Y, et al. Asymptotical stability of Riemann-Liouville fractional singular systems with multiple time-varying delays. Appl Math Lett, 2017, 65: 32–39

    Article  MathSciNet  Google Scholar 

  20. Dassios I K, Baleanu D I. Caputo and related fractional derivatives in singular systems. Appl Math Comput, 2018, 337: 591–606

    MathSciNet  MATH  Google Scholar 

  21. Wei Y H, Wang J C, Liu T Y, et al. Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state. J Franklin Inst, 2019, 356: 1975–1990

    Article  MathSciNet  Google Scholar 

  22. Meng B, Wang X H, Wang Z. Synthesis of sliding mode control for a class of uncertain singular fractional-order systems-based restricted equivalent. IEEE Access, 2019, 7: 96191–96197

    Article  Google Scholar 

  23. Utkin V I, Poznyak A S. Adaptive sliding mode control. In: Advances in Sliding Mode Control. Berlin: Springer, 2013

    MATH  Google Scholar 

  24. Dong S L, Chen C L P, Fang M, et al. Dissipativity-based asynchronous fuzzy sliding mode control for T-S fuzzy hidden Markov jump systems. IEEE Trans Cybern, 2019}. doi: 10.1109/TCYB.2019.291929

    Google Scholar 

  25. Meng B, Wang X H. Adaptive synchronization for uncertain delayed fractional-order hopfield neural networks via fractional-order sliding mode control. Math Problem Eng, 2018, 2018: 1–8

    Article  MathSciNet  Google Scholar 

  26. Meng B, Wang Z C, Wang Z. Adaptive sliding mode control for a class of uncertain nonlinear fractional-order Hopfield neural networks. AIP Adv, 2019, 9: 065301

    Article  MathSciNet  Google Scholar 

  27. Pisano A, Tanelli M, Ferrara A. Switched/time-based adaptation for second-order sliding mode control. Automatica, 2016, 64: 126–132

    Article  MathSciNet  Google Scholar 

  28. Edwards C, Shtessel Y B. Adaptive continuous higher order sliding mode control. Automatica, 2016, 65: 183–190

    Article  MathSciNet  Google Scholar 

  29. Pérez-Ventura U, Fridman L. Design of super-twisting control gains: a describing function based methodology. Automatica, 2019, 99: 175–180

    Article  MathSciNet  Google Scholar 

  30. Castanos F, Hernández D, Fridman L M. Integral sliding-mode control for linear time-invariant implicit systems. Automatica, 2014, 50: 971–975

    Article  MathSciNet  Google Scholar 

  31. Obeid H, Fridman L M, Laghrouche S, et al. Barrier function-based adaptive sliding mode control. Automatica, 2018, 93: 540–544

    Article  MathSciNet  Google Scholar 

  32. Guo Y, Lin C, Chen B, et al. Necessary and sufficient conditions for the dynamic output feedback stabilization of fractional-order systems with order 0 < α < 1. Sci China Inf Sci, 2019, 62: 199201

    Article  MathSciNet  Google Scholar 

  33. Wei Y H, Chen Y Q, Cheng S S, et al. Completeness on the stability criterion of fractional order LTI systems. Fract Calc Appl Anal, 2017, 20: 159–172

    MathSciNet  MATH  Google Scholar 

  34. Cobb D. Controllability, observability, and duality in singular systems. IEEE Trans Autom Control, 1984, 29: 1076–1082

    Article  MathSciNet  Google Scholar 

  35. Chen Y Q, Ahn H S, Xue D Y. Robust controllability of interval fractional order linear time invariant systems. Signal Process, 2006, 86: 2794–2802

    Article  Google Scholar 

  36. Aguila-Camacho N, Duarte-Mermoud M A, Gallegos J A. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul, 2014, 19: 2951–2957

    Article  MathSciNet  Google Scholar 

  37. Wu Z G, Dong S L, Shi P, et al. Reliable filter design of Takagi-Sugeno fuzzy switched systems with imprecise modes. IEEE Trans Cybern, 2019. doi: 10.1109/TCYB.2018.2885505

    Google Scholar 

  38. Dong S L, Fang M, Shi P, et al. Dissipativity-based control for fuzzy systems with asynchronous modes and intermittent measurements. IEEE Trans Cybern, 2019. doi: 10.1109/TCYB.2018.2887060

    Google Scholar 

  39. Hu X H, Xia J W, Wei Y L, et al. Passivity-based state synchronization for semi-Markov jump coupled chaotic neural networks with randomly occurring time delays. Appl Math Comput, 2019, 361: 32–41

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 61573008), Natural Science Foundation of Shandong Province (Grant No. ZR2016FM16), and Post-Doctoral Applied Research Projects of Qingdao (Grant No. 2015122). The authors would like to thank the anonymous reviewers for their valuable suggestions.

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Bo Meng, Xinhe Wang, Ziye Zhang & Zhen Wang

Authors
  1. Bo Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinhe Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ziye Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, B., Wang, X., Zhang, Z. et al. Necessary and sufficient conditions for normalization and sliding mode control of singular fractional-order systems with uncertainties. Sci. China Inf. Sci. 63, 152202 (2020). https://doi.org/10.1007/s11432-019-1521-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11432-019-1521-5

Keywords

Search

Navigation