Skip to main content
SpringerLink
Log in

Impulsive Functional Observer Design for Fractional-Order Nonlinear Systems Satisfying Incremental Quadratic Constraints

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper considers the impulsive functional observer design problem for fractional-order nonlinear systems satisfying incremental quadratic constraints. The incremental quadratic constraints can describe many commonly encountered nonlinearities in the existing literature. Firstly, a fractional-order nonlinear system with incremental quadratic constraints and output nonlinearities is presented and the definition of incremental quadratic constraints is introduced. Secondly, an impulsive functional observer is designed to guarantee that the error system is exponentially stable. Thirdly, based on matrix partition, an approach to obtain the gains of the observer is proposed. Meanwhile, the sufficient conditions can be obtained via solving linear matrix inequalities. Subsequently, an algorithm is given to illustrate the steps of impulsive functional observer design. Finally, an example about fractional-order chaotic system is provided to illustrate the effectiveness of the designed controller.

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

Similar content being viewed by others

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

References

  1. B. Açkmeşe, M. Corless, Observers for systems with nonlinearities satisfying incremental quadratic constraints. Automatica 47(7), 1339–1348 (2011)

    Article  MathSciNet  Google Scholar 

  2. M. Arcak, P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities. Autom. Control IEEE Trans. Autom. Control 46(7), 1146–1150 (2001)

    Article  MathSciNet  Google Scholar 

  3. M. Ayati, H. Khaloozadeh, Designing a novel adaptive impulsive observer for nonlinear continuous systems using LMIs. IEEE Trans. Circuits Syst. I Regul. Pap. 59(1), 179–187 (2012)

    Article  MathSciNet  Google Scholar 

  4. E.A. Boroujeni, H.R. Momeni, Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems. Signal Process. 92(10), 2365–2370 (2012)

    Article  Google Scholar 

  5. A. Chakrabarty, M.J. Corless, G.T. Buzzard, S.H. Żak, A.E. Rundell, State and unknown input observers for nonlinear systems with bounded exogenous inputs. IEEE Trans. Autom. Control 62(11), 5497–5510 (2017)

    Article  MathSciNet  Google Scholar 

  6. K. Chen, R.N. Tang, C. Li, P.N. Wei, Robust adaptive fractional-order observer for a class of fractional-order nonlinear systems with unknown parameters. Nonlinear Dyn. 94(1), 415–427 (2018)

    Article  Google Scholar 

  7. W.H. Chen, D.X. Li, X.M. Lu, Impulsive functional observers for linear systems. Int. J. Control Autom. Syst. 9(5), 987–992 (2011)

    Article  Google Scholar 

  8. W.H. Chen, Y. Wu, X.Z. Wei, Adaptive impulsive observers for nonlinear systems: revisited. Automatica 61, 232–240 (2015)

    Article  MathSciNet  Google Scholar 

  9. Y.G. Chen, S.M. Fei, K.J. Zhang, Stabilization of impulsive switched linear systems with saturated control input. Nonlinear Dyn. 69(3), 793–804 (2012)

    Article  MathSciNet  Google Scholar 

  10. Y.G. Chen, Z.D. Wang, B. Shen, H.L. Dong, Exponential synchronization for delayed dynamical networks via intermittent control: dealing with actuator saturations. IEEE Trans. Neural Netw. Learn. Syst. 30(4), 1000–1013 (2019)

    Article  MathSciNet  Google Scholar 

  11. M. Corless, L. DAlto, Incremental quadratic stability. Numer. Algebra 3(1), 175–201 (2013)

    MathSciNet  Google Scholar 

  12. M. Darouach, Linear functional observers for systems with delays in state variables. IEEE Trans. Autom. Control 50(2), 228–233 (2005)

    Article  MathSciNet  Google Scholar 

  13. S. Djennoune, M. Bettayeb, U. Saggaf, Impulsive observer with predetermined finite convergence time for synchronization of fractional-order chaotic systems based on Takagi–Sugeno fuzzy model. Nonlinear Dyn. 98(2), 1331–1354 (2019)

    Article  Google Scholar 

  14. F.F. Du, J.G. Lu, Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities. Appl. Math. Comput. 375, 1–17 (2020)

    MathSciNet  Google Scholar 

  15. M. Ekramian, F. Sheikholeslam, S. Hosseinnia, M.J. Yazdanpanah, Adaptive state observer for Lipschitz nonlinear systems. Syst. Control Lett. 62(4), 319–323 (2013)

    Article  MathSciNet  Google Scholar 

  16. L. Etienne, S. Gennaro, J.P. Barbot, Periodic event-triggered observation and control for nonlinear Lipschitz systems using impulsive observers. Int. J. Robust Nonlinear Control 27(18), 4363–4380 (2017)

    Article  MathSciNet  Google Scholar 

  17. Q.L. Han, Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica 41(12), 2171–2176 (2005)

    Article  MathSciNet  Google Scholar 

  18. R. Horn, C. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)

    Book  Google Scholar 

  19. T.T. Hu, Z. He, X.J. Zhang, S.M. Zhong, Leader-following consensus of fractional-order multi-agent systems based on event-triggered control. Nonlinear Dyn. 99(3), 2219–2232 (2019)

    Article  Google Scholar 

  20. G.J. Huang, S.X. Luo, L.N. Wei, W.H. Chen, Impulsive observer design for a class of one-sided Lipschitz system. J. Guangxi Univ. (Nat. Sci. Edit.) 40(6), 1588–1596 (2015)

    Google Scholar 

  21. O. Jaramillo, B. Castillo-Toledo, S.D. Gennaro, Impulsive observer-based stabilization for a class of Lipschitz nonlinear systems with time-varying uncertainties. J. Frankl. Inst. 357(17), 12518–12537 (2020)

    Article  MathSciNet  Google Scholar 

  22. H.Y. Jia, Q. Tao, Z.Q. Chen, Analysis and circuit design of a fractional-order Lorenz system with different fractional orders. Syst. Sci. Control Eng. 2(1), 745–750 (2014)

    Article  Google Scholar 

  23. T.H. Jia, Y.N. Pan, H.J. Liang, H.K. Lam, Event-based adaptive fixed-time fuzzy control for active vehicle suspension systems with time-varying displacement constraint. IEEE Trans. Fuzzy Syst. (2021). https://doi.org/10.1109/TFUZZ.2021.3075490

    Article  Google Scholar 

  24. A. Jmal, O. Naifar, A.B. Makhlouf, N. Derbel, M.A. Hammami, Sensor fault estimation for fractional-order descriptor one-sided Lipschitz systems. Nonlinear Dyn. 91(3), 1713–1722 (2018)

    Article  Google Scholar 

  25. N. Kalamian, H. Khaloozadeh, M. Ayati, Adaptive state-dependent impulsive observer design for nonlinear deterministic and stochastic dynamics with time-delays. ISA Trans. 98, 87–100 (2020)

    Article  Google Scholar 

  26. Y.H. Lan, W.J. Li, Y. Zhou, Y.P. Luo, Non-fragile observer design for fractional-order one-sided Lipschitz nonlinear systems. Int. J. Autom. Comput. 10(4), 296–302 (2013)

    Article  Google Scholar 

  27. H.J. Liang, L.G. Liu, T.W. Huang, H.K. Lam, B.W. Wang, Cooperative fault-tolerant control for networks of stochastic nonlinear systems with nondifferential saturation nonlinearity. IEEE Trans. Syst. Man Cybern. Syst. (2020). https://doi.org/10.1109/TSMC.2020.3020188

    Article  Google Scholar 

  28. L.P. Liu, X.Y. Cao, Z.M. Fu, S.Z. Song, H. Xing, Guaranteed cost finite-time control of fractional-order nonlinear positive switched systems with D-perturbations via MDADT. J. Syst. Sci. Complex. 32(3), 857–874 (2019)

    Article  MathSciNet  Google Scholar 

  29. L.P. Liu, X.Y. Cao, Z.M. Fu, S.Z. Song, H. Xing, Positive impulsive observer design of fractional-order positive switched systems with mode-dependent average dwell time. Trans. Inst. Meas. Control 41(5), 1340–1347 (2018)

    Article  Google Scholar 

  30. J.Q. Lu, H. Daniel, J.D. Cao, A unified synchronization criterion for impulsive dynamical networks. Automatica 46(7), 1215–1221 (2010)

    Article  MathSciNet  Google Scholar 

  31. J.H. Luo, State-feedback control for fractional-order nonlinear systems subject to input saturation. Math. Probl. Eng. 2014, 8 (2014)

    MathSciNet  MATH  Google Scholar 

  32. K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)

    MATH  Google Scholar 

  33. N. Oucief, M. Tadjine, S. Labiod, Adaptive observer-based fault estimation for a class of Lipschitz nonlinear systems. Arch. Control Sci. 26(2), 245–259 (2016)

    Article  MathSciNet  Google Scholar 

  34. R. Rajamani, Observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control 43(3), 397–401 (1998)

    Article  MathSciNet  Google Scholar 

  35. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, New York, 1993)

    MATH  Google Scholar 

  36. M.V. Thuan, D.C. Huong, N.H. Sau, Q.T. Ha, Unknown input fractional-order functional observer design for one-side Lipschitz time-delay fractional-order systems. Trans. Inst. Meas. Control 41(15), 4311–4321 (2019)

    Article  Google Scholar 

  37. X.J. Wu, S.L. Shen, Chaos in the fractional-order Lorenz system. Int. J. Comput. Math. 86(7), 1274–1282 (2009)

    Article  MathSciNet  Google Scholar 

  38. T. Zhan, S.P. Ma, Reduced-order observer design with unknown input for fractional order descriptor nonlinear systems. Trans. Inst. Meas. Control 41(13), 3705–3713 (2019)

    Article  Google Scholar 

  39. H.Z. Zhang, W. Zhang, Y.N. Zhao, M.M. Ji, Adaptive state observers for incrementally quadratic nonlinear systems with application to chaos synchronization. Circuits Syst. Signal Process. 39(3), 1290–1306 (2020)

    Article  Google Scholar 

  40. W. Zhang, Y.N. Zhao, M. Abbaszadeh, Exponential observers for discrete-time nonlinear systems with incremental quadratic constraints. J. Control Autom. Electr. Syst. 477–482 (2019)

  41. Y.J. Zhao, X.S. Cai, C. Lin, J.F. Zhang, L.B. Liu, Functional observer design for nonlinear systems with incremental quadratic constraints. Int. J. Syst. Sci. 52(5), 1097–1105 (2021)

    Article  MathSciNet  Google Scholar 

  42. Y.N. Zhao, W. Zhang, H.S. Su, J.Q. Yang, Observer-based synchronization of chaotic systems which satisfy incremental quadratic constraints and its application in secure communication. IEEE Trans. Syst. Man Cybern. Syst. 50(12), 5221–5232 (2020)

    Article  Google Scholar 

  43. Y.N. Zhao, W. Zhang, W.Y. Zhang, F. Song, Exponential reduced-order observers for nonlinear systems satisfying incrementally quadratic constraints. Circuits Syst. Signal Process. 37, 3725–3738 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China [Grant Number 61773350]; the Scientific and Technological Innovation Leaders in Central Plains [Grant Number 194200510012]; and the Science and Technology Innovative Teams at the University of Henan Province [Grant Number 18IRTSTHN011].

Author information

Authors and Affiliations

  1. School of Information Engineering, Henan University of Science and Technology, Luoyang, 471023, Henan, China

    Leipo Liu, Yilin Shang, Yifan Di & Zhumu Fu

  2. College of Physics and Electronic Information Engineering, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Xiushan Cai

Authors
  1. Leipo Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yilin Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yifan Di
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhumu Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiushan Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leipo Liu.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, L., Shang, Y., Di, Y. et al. Impulsive Functional Observer Design for Fractional-Order Nonlinear Systems Satisfying Incremental Quadratic Constraints. Circuits Syst Signal Process 41, 3130–3152 (2022). https://doi.org/10.1007/s00034-021-01940-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-021-01940-1

Keywords

Search

Navigation