Skip to main content
SpringerLink
Log in

Both States and Unknown Inputs Simultaneous Estimation for Fractional-Order Linear Systems

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

Abstract

This article is devoted to estimating simultaneously both the states and the inputs of linear time-invariant fractional-order systems (LTI-FOSs) with the order \(0<\alpha <2\). Firstly, a necessary and sufficient stability criterion for LTI-FOSs with the order \(0<\alpha <1\) is derived by the linear matrix inequality technique. Secondly, a novel fractional-order observer combined with state vectors and ancillary vectors is given, which can generalize several forms of existing observers. Moreover, the parameter matrices of the desired observer for both the order \(0<\alpha <1\) and \(1<\alpha <2\) are solved on the basis of the stability theorem and the solution to the generalized inverse matrix. Finally, the fractional-order observer design algorithm is proposed and then applied to an illustrated example, in which the simulation results are reported to verify the effectiveness of the proposed approach.

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

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Z. Belkhatir, T.K. Laleg-Kirati, High-order sliding mode observer for fractional commensurate linear systems with unknown input. Automatica 82, 209–217 (2017)

    Article  MathSciNet  Google Scholar 

  2. E.A. Boroujeni, M. Pourgholi, H.R. Momeni, Reduced order linear fractional order observer, in Proceedings of the International Conference on Control Communication and Computing (ICCC), Thiruvananthapuram, India (2013), pp. 1–4

  3. Y. Boukal, M. Darouach, M. Zasadzinski, N.E. Radhy, \(H_\infty \) observer design for linear fractional-order systems in time and frequency domains, in Proceedings of the European Control Conference (ECC), Strasbourg, France (2014), pp. 2975–2980

  4. Y. Boukal, M. Darouach, M. Zasadzinski, N.E. Radhy, Robust \(H_\infty \) observer-based control of fractional-order systems with gain parametrization. IEEE Trans. Autom. Control 62(11), 5710–5723 (2017)

    Article  MathSciNet  Google Scholar 

  5. M. Chilali, P. Gahinet, P. Apkarian, Robust pole placement in LMI regions. IEEE Trans. Autom. Control 44(12), 2257–2270 (1999)

    Article  MathSciNet  Google Scholar 

  6. R.J. Cui, J.G. Lu, \(H\)_/\(H_\infty \) fault detection observer design for fractional-order singular systems in finite frequency domains. ISA Trans. 129, 100–109 (2022)

    Article  Google Scholar 

  7. W.T. Geng, C. Lin, B. Chen, Observer-based stabilizing control for fractional-order systems with input delay. ISA Trans. 100, 103–108 (2020)

    Article  Google Scholar 

  8. P. Gokul, R. Rakkiyappan, New finite-time stability for fractional-order time-varying time-delay linear systems: a Lyapunov approach. J. Frankl. Inst. 359(14), 7620–7631 (2022)

    Article  MathSciNet  Google Scholar 

  9. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific Publishing, Singapore, 2001)

    Google Scholar 

  10. S. Ibrir, M. Bettayeb, Observer-based control of fractional-order linear systems with norm-bounded uncertainties: new convex-optimization conditions, in Proceedings of the 19th World Congress on the International Federation of Automatic Control, Cape Town, South Africa, vol. 47(3) (2014), pp. 2903–2908

  11. S. Ibrir, M. Bettayeb, New sufficient conditions for observer-based control of fractional-order uncertain systems. Automatica 59, 216–223 (2015)

    Article  MathSciNet  Google Scholar 

  12. S. Kanakalakshmi, R. Sakthivel, L.S. Ramya, A. Parivallal, Disturbance estimator based dynamic compensator design for fractional order fuzzy control systems. Iran. J. Fuzzy Syst. 18(4), 79–93 (2021)

    MathSciNet  Google Scholar 

  13. A. Kilbas, H. Srivastava, J. Trujillo, North-Holland Mathematics Studies: Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)

    Google Scholar 

  14. Y. Lan, H. Huang, Y. Zhou, Observer-based robust control of \(\alpha (1\le \alpha <2)\) fractional-order uncertain systems: a linear matrix inequality approach. IET Control Theory Appl. 6(2), 229–234 (2012)

    Article  MathSciNet  Google Scholar 

  15. S.C. Lee, Y. Li, Y.Q. Chen, H.S. Ahn, \(H_\infty \) and sliding mode observers for linear time-invariant fractional-order dynamic systems with initial memory effect. J. Dyn. Syst. Trans. ASME 136(5), 051022 (2014)

    Article  Google Scholar 

  16. C. Li, J.C. Wang, Robust stability and stabilization of fractional order interval systems with coupling relationships: the \(0<\alpha <1\) case. J. Frankl. Inst. 349(7), 2406–2419 (2012)

    Article  MathSciNet  Google Scholar 

  17. Y.B. Liu, W.C. Sun, H.J. Gao, High precision robust control for periodic tasks of linear motor via B-spline wavelet neural network observer. IEEE Trans. Ind. Electron. 69(8), 8255–8263 (2022)

    Article  Google Scholar 

  18. Z.T. Liu, H.J. Gao, X.H. Yu, W.Y. Lin, J.B. Qiu, J.J. Rodríguez-Andina, D.S. Qu, B-spline wavelet neural network-based adaptive control for linear motor-driven systems via a novel gradient descent algorithm. IEEE Trans. Ind. Electron. 71(2), 1896–1905 (2022)

    Article  Google Scholar 

  19. J.G. Lu, Y.Q. Chen, Robust stability and stabilization of fractional-order interval systems with the fractional order \(\alpha \): the \(0 < \alpha < 1\) case. IEEE Trans. Autom. Control 55(1), 152–158 (2010)

    Article  MathSciNet  Google Scholar 

  20. S. Marir, M. Chadli, D. Bouagada, New admissibility conditions for singular linear continuous-time fractional-order systems. J. Frankl. Inst. 354(2), 752–766 (2016)

    Article  MathSciNet  Google Scholar 

  21. D. Matignon, Stability result on fractional differential equations with applications to control processing, in Proceedings of the Computation Engineering in System and Application Multiconference (IMACS, IEEE-SMC), Lille, France (1996), pp. 963–968

  22. D. Melchor-Aguilar, Mikhailov stability criterion for fractional commensurate order systems with delays. J. Frankl. Inst. 359(15), 8395–8408 (2022)

    Article  MathSciNet  Google Scholar 

  23. I. N’Doye, M. Darouach, M. Zasadzinski, N.E. Radhy, Observers design for singular fractional-order systems, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) Orlando, FL, USA (2011), pp. 4017–4022

  24. I. N’Doye, M. Darouach, H. Voos, M. Zasadzinski, Design of unknown input fractional-order observers for fractional-order systems. Int. J. Appl. Math. Comput. Sci. 23(3), 491–500 (2013)

    Article  MathSciNet  Google Scholar 

  25. I. N’Doye, M. Darouach, M. Zasadzinski, N.E. Radhy, Robust stabilization of uncertain descriptor fractional-order systems. Automatica 49(6), 1907–1913 (2013)

    Article  MathSciNet  Google Scholar 

  26. C.C. Peng, W.H. Zhang, Back-stepping stabilization of fractional-order triangular system with applications to chaotic systems. Asian J. Control 23(1), 143–154 (2021)

    Article  MathSciNet  Google Scholar 

  27. C.C. Peng, W.H. Zhang, Linear feedback synchronization and anti-synchronization of a class of fractionalorder chaotic systems based on triangular structure. Eur. Phys. J. Plus 134(6), 292 (2019)

    Article  Google Scholar 

  28. K. Rabah, S. Ladaci, A fractional adaptive sliding mode control configuration for synchronizing disturbed fractional-order chaotic systems. Circuits Syst. Signal Process. 39(3), 1244–1264 (2020)

    Article  Google Scholar 

  29. C. Rao, S. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971)

    Google Scholar 

  30. J. Sabatier, M. Moze, C. Farges, On stability of fractional order systems, in Plenary Lecture VIII on 3rd IFAC Workshop on Fractional Differentiation and Its Applications, Ankara, Turkey (2008)

  31. N.H. Sau, N.T.T. Huyen, Passivity analysis of fractional-order neural networks with time-varying delay based on LMI approach. Circuits Syst. Signal Process. 39(12), 5906–5925 (2020)

    Article  Google Scholar 

  32. P.W. Shi, W.C. Sun, X.B. Yang, I.J. Rudas, H.J. Gao, Master-slave synchronous control of dual-drive gantry stage with cogging force compensation. IEEE Trans. Syst. Man Cybern. Syst. 53(1), 216–225 (2023)

    Article  Google Scholar 

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

    Article  Google Scholar 

  34. M. Wyrwasm, Full-order observers for linear fractional multi-order difference systems. Bull. Pol. Acad. Sci. Tech. 65(6), 891–898 (2017)

    Google Scholar 

  35. M.H. Xiong, Y.S. Tan, D.S. Du, B.Y. Zhang, S.M. Fei, Observer-based event-triggered output feedback control for fractional-order cyber-physical systems subject to stochastic network attacks. ISA Trans. 104, 15–25 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 62203247)

Author information

Authors and Affiliations

  1. School of Information and Control Engineering, Qingdao University of Technology, Qingdao, 266520, China

    Chenchen Peng, Ling Ren & Zihao Zhao

Authors
  1. Chenchen Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ling Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zihao Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chenchen Peng.

Ethics declarations

Conflict of interest

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

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

Peng, C., Ren, L. & Zhao, Z. Both States and Unknown Inputs Simultaneous Estimation for Fractional-Order Linear Systems. Circuits Syst Signal Process 43, 895–915 (2024). https://doi.org/10.1007/s00034-023-02522-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-023-02522-z

Keywords

Search

Navigation