Skip to main content
SpringerLink
Log in

Physical Significance Variable Control for a Class of Fractional-Order Systems

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

Abstract

This paper studies a class of fractional-order systems (FOSs) and proposes control laws based on physical significance variables. Lyapunov techniques and the methods that derive from Yakubovici–Kalman–Popov Lemma are used, and the frequency criterions that ensure asymptotic stability of the physical significance variable closed-loop system are inferred. The asymptotic stability of the observer system is studied for a sector control law where the output is defined by the physical significance variables. Frequency criterions and conditions for asymptotic stability are determined. The control techniques are extended to a class of linear delay fractional-order systems and nonlinear FOS. Numerical simulations of a class of systems described by fractional-order models show the method efficiency.

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

Similar content being viewed by others

Data Availability

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

References

  1. N. Aguila-Camacho, M. Duarte-Mermoud, J. Callegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)

    Article  MathSciNet  Google Scholar 

  2. R. Argarwal, S. Hristova, D. O’Regan, Lyapunov functions and strict stability of Caputo fractional differential equations. Adv. Differ. Equ. 346, 2–20 (2015)

    MathSciNet  Google Scholar 

  3. R. Agarwal, S. Hristova, D. O’Reagan, Remarks on Lyapunov functions to Caputo fractional neural networks. Ann. Acad. Rom. Sci. 10(1), 169–176 (2018)

    MathSciNet  Google Scholar 

  4. U.M. Al-Saggaf, I.M. Mehedi, Rotary flexible joint control by fractional order controllers. Int. J. Control Autom. Syst. 15(6), 2561–2569 (2017)

    Article  Google Scholar 

  5. S. Dadras, H. Malek, Y. Chen, A note on the Lyapunov stability of fractional order nonlinear systems, in Proc. of the ASME 2017, Cleveland, Orlando, 2017, pp. 123–129

  6. K. Diethelm, The Analysis of Fractional Differential Equations (Springer, London, 2004)

    Google Scholar 

  7. M.M. Garcia, T. Gordon, L. Shu, Extended crossover model for human-control of fractional order plants. IEEE Access 5, 27622–27635 (2017)

    Article  Google Scholar 

  8. N. Heymann, I. Podlubni, Physical interpretation of initial conditions for fractional differential with Riemann-Liouville fractional derivatives. Rheol. Acta 7(23), 45–63 (2014)

    Google Scholar 

  9. K. Hirata, T. Murakami, Stability analysis of disturbance observer based controllers for two-wheel wheelchair systems. Adv. Robot. 28(7), 467–477 (2014)

    Article  Google Scholar 

  10. H. Khalil, Nonlinear Systems (Prentice Hall, New Jersey, 2002)

    MATH  Google Scholar 

  11. D. Khimani, M. Patil, High performance super-twisting control for state delay systems. Int. J. Control Autom. Syst. 16(5), 2063–2073 (2018)

    Article  Google Scholar 

  12. M.S. Koo, H.L. Choi, Fast regulation control of a class of input-delayed linear systems with pre-feedback. Int. J. Control Autom. Syst. 16(1), 141–149 (2018)

    Article  Google Scholar 

  13. M. Lazarevic, A.M. Spasic, Finite-time stability analysis of fractional order time delay systems; Gronwall’s approach. Math. Comput. Model. 49, 475–481 (2009)

    Article  MathSciNet  Google Scholar 

  14. Y. Li, Y.Q. Chen, I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    Article  MathSciNet  Google Scholar 

  15. Y. Li, Y.Q. Chen, I. Podlubny, Stability of fractional order nonlinear systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math Appl. 59, 1810–1821 (2010)

    Article  MathSciNet  Google Scholar 

  16. C. Maharajan, R. Raja, J. Cao, G. Rajchakit, Fractional delay segments method on time-delayed recurrent neural networks with impulsive and stochastic effects: an exponential stability approach. Neurocomputing 323, 277–298 (2019)

    Article  Google Scholar 

  17. C. Monje, Y.Q. Chen, B. Vinagre, D. Hue, V. Feliu, Fractional-order systems and controls (Springer, London, 2010)

    Book  Google Scholar 

  18. M.D. Paola, F.P. Pinnola, M. Zingales, Fractional differential equations and related exact mechanical models. Comput. Math Appl. 66(5), 608–620 (2013)

    Article  MathSciNet  Google Scholar 

  19. I. Petras, Fractional-Order Nonlinear Systems, Modeling, Analysis and Simulation (Higher Education Press, Springer, Berlin, 2011)

    Book  Google Scholar 

  20. N. Popescu, D. Popescu, M. Ivanescu, D. Popescu, C. Vladu, I. Vladu, Force observer-based control for a rehabilitation hand exoskeleton system, in Proc. of Asian Control Conference (ASCC2013), Istanbul, 2013, pp. 1–6

  21. A. Pratap, R. Raja, G. Rajchakit, J. Cao, O. Bagdasar, Mittag-Leffler state estimator design and synchronization analysis for fractional-order BAM neural networks with time delays. Int. J. Adapt. Control Signal Process. 33(3), 1–20 (2019)

    MathSciNet  MATH  Google Scholar 

  22. A. Pratap, R. Raja, J. Cao, G. Rajchakit, H.M. Fardoun, Stability and synchronization criteria for fractional order competitive neural networks with time delays: an asymptotic expansion of Mittag Leffler function. J. Frankl. Inst. 356(4), 2212–2239 (2019)

    Article  MathSciNet  Google Scholar 

  23. A. Pratap, R. Raja, J. Cao, R. Grienggrai, C.P. Lim, Global robust synchronization of fractional order complex valued neural networks with mixed time varying delays and impulses. Int. J. Control Autom. Syst. 17(X), 1–12 (2019)

    Google Scholar 

  24. A. Pratap, R. Raja, C. Sowmiya, O. Bagdasar, J. Cao, G. Rajchakit, Robust generalized Mittag–Leffler synchronization of fractional order neural networks with discontinuous activation and impulses. Neural Netw. 103, 128–141 (2018)

    Article  Google Scholar 

  25. L. Rhong, X. Peng, B. Zhang, A reduced-order fault detection filter design for polytopic uncertain continuous-time markovian jump systems with time-varying delays. Int. J. Control Autom. Syst. 16(5), 2021–2032 (2018)

    Article  Google Scholar 

  26. M. Rivero, S. Rogosin, J.T. Machado, Stability of fractional fractional order systems. In: Mathematical Problems in Engineering, vol 2013, ID 356235, pp 286-293 (2013)

  27. J. Sabatier, M. Moze, C. Farges, LMI stability conditions for fractional order systems. Comput. Math Appl. 59, 1594–1609 (2010)

    Article  MathSciNet  Google Scholar 

  28. N. Sene, Lyapunov characterization of the fractionsal nonlinear systems with exogenous input. Fractal Fract. 2(17), 24–29 (2018)

    Google Scholar 

  29. V.B. Sundara, R. Raja, R.P. Agarwal, G. Rajchakit, A novel controllability analysis of impulsive fractional linear time invariant systems with state delay and distributed delays in control. Discontinuity Nonlinearity Complex. 7(3), 275–290 (2018)

    Article  Google Scholar 

  30. H.T. Tuan, H. Trinh, Stability of fractional-order nonlinear systems by Lyapunov direct method. IET Res. J. Inst. Eng. Technol. 12, 1–5 (2015)

    Google Scholar 

  31. P. Wolm, Dynamic stability control of front wheel drive wheelchair using solid state accelerometer and gyroscope. PhD Thesis, University of Canterbury, 2009

  32. B. Wu, C.-L. Wang, Y.-J. Hu, Stability analysis for time-delay systems with nonlinear disturbances via new generalized integral inequalities. Int. J. Control Autom. Syst. 16(6), 2772–2780 (2018)

    Article  Google Scholar 

  33. S. Xiao, L. Xu, H.B. Zeng, Improved stability criteria for discrete-time delay systems via novel summation inequalities. Int. J. Control Autom. Syst. 16(4), 1592–1662 (2018)

    Article  Google Scholar 

  34. X. Zhang, Some results of linear fractional-order time delay system. Appl. Math. Comput. 187, 407–411 (2008)

    MathSciNet  MATH  Google Scholar 

  35. Y. Zhao, Y. Wang, Z. Liu, Lyapunov function method for linear fractional order systems, in Proc of the 34th Chinese Control Conference, 2015, pp. 1457–1463

  36. X.F. Zhou, L.C. Hu, W. Jiang, Stability criterion for a class of nonlinear fractional differential systems. Appl. Math. Lett. 28, 25–29 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechatronics, University of Craiova, 13 Cuza Street, 1100, Craiova, Romania

    Mircea Ivanescu

  2. Department of Computer Science, University Polytehnica of Bucharest, 313 Splaiul Independentei, Bucharest, Romania

    Nirvana Popescu & Decebal Popescu

Authors
  1. Mircea Ivanescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nirvana Popescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Decebal Popescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MI contributed to conceptualization, validation and supervision; MI and DP helped with formal analysis and investigation; NP contributed to methodology, project administration and resources; DP helped with writing—review and editing, and software.

Corresponding author

Correspondence to Mircea Ivanescu.

Ethics declarations

Conflict of interest

The authors declare 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ivanescu, M., Popescu, N. & Popescu, D. Physical Significance Variable Control for a Class of Fractional-Order Systems. Circuits Syst Signal Process 40, 1525–1541 (2021). https://doi.org/10.1007/s00034-020-01531-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-020-01531-6

Keywords

Search

Navigation