Skip to main content
SpringerLink
Log in

A new modulating functions-based non-asymptotic state estimation method for fractional-order systems with MIMO

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In the present article, a new modulating functions-based state estimation method is advanced both for integer and non-integer-order systems with Multiple Inputs and Multiple Outputs in a noisy environment. It can be used to non-asymptotically and robustly estimate not only the states of considered systems but also their fractional derivatives. Different from the existing modulating functions methods which are based on an input–output differential equation, the proposed method is simpler which is directly applied to the fractional-order Brunovsky’s observable canonical form to obtain algebraic integral formulas for the sought variables. In particular, the proposed method is more accurate. For this, a family of modulating functions are introduced and built with a design parameter. Then, error analysis in discrete noisy case is addressed. Finally, the robustness and accuracy of the method are verified by numerical simulations.

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
Fig. 9

Similar content being viewed by others

References

  1. Chen, X., Xi, L., Zhang, Y.N., Ma, H., Huang, Y.H., Chen, Y.Q.: Fractional techniques to characterize non-solid aluminum electrolytic capacitors for power electronic applications. Nonlinear Dyn. 98(3), 1–17 (2019)

    Google Scholar 

  2. Ding, C.S., Cao, J.Y., Chen, Y.Q.: Fractional-order model and experimental verification for broadband hysteresis in piezoelectric actuators. Nonlinear Dyn. 98(3), 3143–3153 (2019)

    Article  Google Scholar 

  3. Sheng, Y.Z., Zhang, Z., Xia, L.: Fractional-order sliding mode control based guidance law with impact angle constraint. Nonlinear Dyn. 106, 425–444 (2021)

    Article  Google Scholar 

  4. Wang, R.M., Zhang, Y.N., Chen, Y.Q., Chen, X., Xi, L.: Fuzzy neural network-based chaos synchronization for a class of fractional-order chaotic systems: an adaptive sliding mode control approach. Nonlinear Dyn. 100, 1275–1287 (2020)

    Article  MATH  Google Scholar 

  5. Wei, Y., Wang, J., Liu, T., Wang, Y.: Sufficient and necessary conditions for stabilizing singular fractional order systems with partially measurable state. J. Franklin Inst. 356(4), 1975–1990 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, J., Wei, Y., Liu, T., Li, A., Wang, Y.: Fully parametric identification for continuous time fractional order Hammerstein systems. J. Franklin Inst. 357(1), 651–666 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bagley, R.L.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30(1), 133–155 (1998)

    Article  MATH  Google Scholar 

  8. Ai, Z.Y., Zhao, Y.Z., Liu, W.J.: Fractional derivative modeling for axisymmetric consolidation of multilayered cross-anisotropic viscoelastic porous media-sciencedirect. Comput. Math. Appl. 79(5), 1321–1334 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, L., Chen, Y., Cheng, G., Barrière, T.: Numerical analysis of fractional partial differential equations applied to polymeric visco-elastic Euler-Bernoulli beam under quasi-static loads. Chaos, Solitons Fractals 140, 110255 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kaczorek, T., Rogowski, K.: Fractional linear systems and electrical circuits. Springer-Verlag, Berlin, Germany (2014)

    MATH  Google Scholar 

  11. Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59(5), 1586–1593 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yin, C., Huang, X., Chen, Y., Dadras, S., Zhong, S.M., Cheng, Y.: Fractional-order exponential switching technique to enhance sliding mode control. Appl. Math. Model. 44, 705–726 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sheng, D., Wei, Y., Cheng, S., Yong, W.: Adaptive backstepping state feedback control for fractional order systems with input saturation. IFAC-PapersOnLine 50(1), 6996–7001 (2017)

    Article  Google Scholar 

  14. Tian, Y., Wang, Z.B., Liu, D.Y., Boutat, D., Liu, H.R.: Non-asymptotic estimation for fractional integrals of noisy accelerations for fractional order vibration systems. Automatica 135, 109996 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  15. N’Doye, I., Zasadzinski, M., Darouach, M., Radhy, N. E.: Observer-based control for fractional-order continuous-time systems. In: 48th Conference on Decision and Control (CDC). IEEE, Shanghai PR China, pp. 1932–1937 (2009)

  16. N’Doye, I., Laleg-Kirati, T.M., Darouach, M., Voos, H.: Adaptive observer for nonlinear fractional-order systems. Int. J. Adapt. Control Signal Process. 31(3), 314–331 (2017)

  17. Fliess, M., Sira-Ramírez, H.: An algebraic framework for linear identification. Esaim Control Optimisation and Calculus of Variations 9, 151–168 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shinbrot, M.: On the analysis of linear and nonlinear dynamical systems from transient-response data. Rozhledy 62(3), 205–211 (1954)

    Google Scholar 

  19. Pin, G., Assalone, A., Lovera, M., Parisini, T.: Non-asymptotic kernel-based parametric estimation of continuous-time linear systems. IEEE Trans. Autom. Control 61(2), 360–373 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Fliess, M., Sira-Ramírez, H.: Reconstructeurs d’état. C.R. Math. 338(1), 91–96 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, D.Y., Laleg-Kirati, T.M., Perruquetti, W., Gibaru, O.: Non-asymptotic state estimation for a class of linear time-varying systems with unknown inputs. In IFAC Proceedings, pp. 3132–3138 (2014)

  22. Jouffroy, J., Reger, J.: Finite-time simultaneous parameter and state estimation using modulating functions. In 2015 IEEE Conference on Control Applications (CCA), pp. 394–399 (2015)

  23. Pin, G., Chen, B., Parisini, T.: Robust deadbeat continuous-time observer design based on modulation integrals. Automatica 107, 95–102 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wei, Y.Q., Liu, D.Y., Boutat, D.: Innovative fractional derivative estimation of the pseudo-state for a class of fractional order linear systems. Automatica 99, 157–166 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wei, X., Liu, D.Y., Boutat, D.: Non-asymptotic pseudo-state estimation for a class of fractional order linear systems. IEEE Trans. Autom. Control 62(3), 1150–1164 (2017)

    Article  MATH  Google Scholar 

  26. Wei, Y.Q., Liu, D.Y., Boutat, D., Liu, H.R., Lv, C.W.: Modulating functions based differentiator of the pseudo-state for a class of fractional order linear systems. J. Comput. Appl. Math. 384, 113161 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wei, Y.Q., Liu, D.Y., Boutat, D., Liu, H.R., Wu, Z.H.: Modulating functions based model-free fractional order differentiators using a sliding integration window. Automatica 130, 109679 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Podlubny, I.: Fractional Differential Equations. Academic Press, New York, USA (1999)

    MATH  Google Scholar 

  29. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Van, M.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  30. Wu, G.C., Kong, H., Luo, M., Fu, H., Huang, L.L.: Unified predictor-corrector method for fractional differential equations with general kernel functions. Fractional Calculus and Applied Analysis 25, 648–667 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jiang, Y.W., Zhang, B.: Comparative study of Riemann-Liouville and Caputo derivative definitions in time-domain analysis of fractional-order capacitor. IEEE Trans. Circuits Syst. II Express Briefs 67(10), 2184–2188 (2020)

  32. Freeborn, T.J., Maundy, B., Elwakil, A.S.: Measurement of supercapacitor fractional-order model parameters from voltage-excited step response. IEEE Journal on Emerging and Selected Topics in Circuits and Systems 3(3), 367–376 (2013)

    Article  Google Scholar 

  33. Petráš, Ivo: Fractional-order nonlinear systems: modeling, analysis and simulation. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  34. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)

  35. Concepción, A.M., Chen, Y.Q., Xue, D.Y., Feliu-Batlle, V.: Fractional-order Systems and Controls: Fundamentals and Applications. Springer, Amsterdam (2010)

    MATH  Google Scholar 

  36. Hou, M., Zítek, P., Patton, R.J.: An observer design for linear time-delay systems. IEEE Trans. Autom. Control 47(1), 121–125 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wei, Y.Q., Liu, D.Y., Driss, B., Chen, Y.M.: An improved pseudo-state estimator for a class of commensurate fractional order linear systems based on fractional order modulating functions. Syst. Control Lett. 118, 29–34 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Fliess, M.: Analyse non standard du bruit. C.R. Math. 342(10), 797–802 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Maisel, H.: A first course in numerical analysis. McGraw-Hill, Noida (1978)

    Google Scholar 

  40. Knuth, D.E., Bendix, P.B.: Computational problems in abstract algebra. Pergamon Press, Oxford (1970)

    MATH  Google Scholar 

  41. Liu, C., Liu, D.-Y., Boutat, D., Wang, Y., Wu, Z.-H.: Non-asymptotic and robust estimation for a class of nonlinear fractional-order systems. Commun. Nonlinear Sci Numer Simulat 115, 106752 (2022)

  42. Wang, J.-C., Liu, D.-Y., Boutat, D., Wang, Y.: An innovative modulating functions method for pseudo-state estimation of fractional order system. ISA Trans. (2022)

  43. Wei, Y.-Q., Liu, D.-Y., Boutat, D.: Innovative fractional derivative estimation of the pseudo-state for a class of fractional order linear systems. Automatica 99, 157–166 (2019)

  44. Liu, D.-Y., Gibaru, O., Perruquetti, W., Laleg-Kirati, T.-M.: Fractional order differentiation by integration and error analysis in noisy environment. IEEE Trans. Automatic Control 60(11), 2945–2960 (2015)

Download references

Acknowledgements

The authors are grateful for the financial support of the China Scholarship Council (CSC) via the project National construction High level University government-sponsored graduate student.

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. LISPEN Arts et Métiers Institute of Technology, HESAM University, Lille, F-59000, France

    Lei Wang & Olivier Gibaru

  2. PRISME EA 4229, INSA Centre Val de Loire, Université d’Orléans, Bourges Cedex, 18022, France

    Da-Yan Liu

Authors
  1. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Da-Yan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olivier Gibaru
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Lei Wang wrote the manuscript and performed analyses; Da-Yan Liu and Olivier Gibaru gave constructive advices on the structure and language of the article and guided the revision of the paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Da-Yan Liu.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Liu, DY. & Gibaru, O. A new modulating functions-based non-asymptotic state estimation method for fractional-order systems with MIMO. Nonlinear Dyn 111, 5533–5546 (2023). https://doi.org/10.1007/s11071-022-08128-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-08128-5

Keywords

Search

Navigation