Skip to main content
SpringerLink
Log in

Non-fragile Finite-Time Guaranteed Cost Control for a Class of Singular Caputo Fractional-Order Systems with Uncertainties

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

Abstract

In the present paper, for the first time, we consider the problem of non-fragile finite-time guaranteed cost control for a class of fractional-order singular systems with norm-bounded uncertainties. Firstly, we establish a linear quadratic cost function based on the integral of integer order, which serves practical purposes and acts as a performance measure index for the closed-loop system. Then, leveraging finite-time boundedness theory, inequality techniques, and fractional-order calculus properties, we propose a non-fragile state feedback controller that ensures regularity, impulse-free, and finite-time boundedness in the closed-loop system. Furthermore, we establish an upper limit for the performance index. The resulting findings are reliant on the order of the considered fractional-order systems and are obtained through the use of strict linear matrix inequalities (LMIs), devoid of any equality restrictions. Finally, two simulation examples are provided to validate the effectiveness of the proposed method.

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

All data generated or analyzed during this study are included in this article.

References

  1. F. Amato, R. Ambrosino, M. Ariola, G. De Tommasi, A. Pironti, On the finite-time boundedness of linear systems. Automatica 107, 454–466 (2019)

    MathSciNet  Google Scholar 

  2. A. Boukhouima, K. Hattaf, E.M. Lotfi, M. Mahrouf, D.F. Torres, N. Yousfi, Lyapunov functions for fractional-order systems in biology: methods and applications. Chaos, Solitons & Fractals 140, 110224 (2020)

    MathSciNet  Google Scholar 

  3. S. Boy, E. Ghaoui, F. Feron, V. Balakrisshnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadenphia, 1994)

    Google Scholar 

  4. L. Chen, H. Yin, R. Wu, L. Yin, Y. Chen, Robust dissipativity and dissipation of a class of fractional-order uncertain linear systems. IET Control Theory Appl. 13(10), 1454–1465 (2019)

    MathSciNet  Google Scholar 

  5. S.B. Chen, S. Soradi-Zeid, H. Jahanshahi, R. Alcaraz, J.F. Gómez-Aguilar, S. Bekiros, Y.M. Chu, Optimal control of time-delay fractional equations via a joint application of radial basis functions and collocation method. Entropy 22(11), 1213 (2020)

    MathSciNet  Google Scholar 

  6. L. Dai, Singular Control Systems (Springer, Berlin, 1989)

    Google Scholar 

  7. X. Ding, J. Cao, F.E. Alsaadi, Pinning synchronization of fractional-order complex networks with adaptive coupling weights. Int. J. Adapt. Control Signal Process. 33(10), 1478–1490 (2019)

    MathSciNet  Google Scholar 

  8. M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22(1–3), 650–659 (2015)

    MathSciNet  Google Scholar 

  9. T. Feng, B. Wu, L. Liu, Y.E. Wang, Finite-time stability and stabilization of fractional-order switched singular continuous-time systems. Circuits Syst. Signal Process. 38, 5528–5548 (2019)

    Google Scholar 

  10. T. Feng, B. Wu, Y.E. Wang, L. Liu, Input–output finite-time stability of fractional-order switched singular continuous-time systems. Asian J. Control 23(2), 1052–1061 (2021)

    MathSciNet  Google Scholar 

  11. V. Gokulakrishnan, R. Srinivasan, M. Syed Ali, G. Rajchakit, Finite-time guaranteed cost control for stochastic nonlinear switched systems with time-varying delays and reaction–diffusion. Int. J. Comput. Math. 100(5), 1031–1051 (2023)

    MathSciNet  Google Scholar 

  12. D.T. Hong, N.H. Sau, M.V. Thuan, Output feedback finite-time dissipative control for uncertain nonlinear fractional-order systems. Asian J. Control 24(5), 2284–2293 (2022)

    MathSciNet  Google Scholar 

  13. Y. Ji, J. Qiu, Stabilization of fractional-order singular uncertain systems. ISA Trans. 56, 53–64 (2015)

    Google Scholar 

  14. T. Kaczorek, Singular fractional linear systems and electrical circuits. Int. J. Appl. Math. Comput. Sci. 21(2), 379–384 (2011)

    MathSciNet  Google Scholar 

  15. S. Kang, H. Wang, M. Chen, P.X. Liu, C. Li, Event-triggered adaptive backstepping tracking control for a class of nonlinear fractional order systems. Int. J. Adapt. Control Signal Process. 35(4), 442–458 (2021)

    MathSciNet  Google Scholar 

  16. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Application of Fractional Differential Equations (Elsevier, New York, 2006)

    Google Scholar 

  17. K.A. Lazopoulos, D. Karaoulanis, A.K. Lazopoulos, On fractional modelling of viscoelastic mechanical systems. Mech. Res. Commun. 78, 1–5 (2016)

    Google Scholar 

  18. J.F. Li, H. Jahanshahi, S. Kacar, Y.M. Chu, J.F. Gómez-Aguilar, N.D. Alotaibi, K.H. Alharbi, On the variable-order fractional memristor oscillator: data security applications and synchronization using a type-2 fuzzy disturbance observer-based robust control. Chaos, Solitons & Fractals 145, 110681 (2021)

    MathSciNet  Google Scholar 

  19. L. Liu, X. Cao, Z. Fu, S. 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)

    MathSciNet  Google Scholar 

  20. X. Liu, Q. Liu, Y. Li, Finite-time guaranteed cost control for uncertain mean-field stochastic systems. J. Frankl. Inst. 357(5), 2813–2829 (2020)

    MathSciNet  Google Scholar 

  21. L. Liu, Y. Di, Y. Shang, Z. Fu, B. Fan, Guaranteed cost and finite-time non-fragile control of fractional-order positive switched systems with asynchronous switching and impulsive moments. Circuits Syst. Signal Process. 40, 3143–3160 (2021)

    Google Scholar 

  22. Z. Liu, H. Jahanshahi, J.F. Gómez-Aguilar, G. Fernandez-Anaya, J. Torres-Jiménez, A.A. Aly, A.M. Aljuaid, Fuzzy adaptive control technique for a new fractional-order supply chain system. Phys. Scr. 96(12), 124017 (2021)

    Google Scholar 

  23. S. Marir, M. Chadli, M.V. Basin, Bounded real lemma for singular linear continuous-time fractional-order systems. Automatica 135, 109962 (2022)

    MathSciNet  Google Scholar 

  24. V. Martynyuk, M. Ortigueira, Fractional model of an electrochemical capacitor. Signal Process. 107, 355–360 (2015)

    Google Scholar 

  25. K. Mathiyalagan, K. Balachandran, Finite-time stability of fractional-order stochastic singular systems with time delay and white noise. Complexity 21(S2), 370–379 (2016)

    MathSciNet  Google Scholar 

  26. A. Narang, S.L. Shah, T. Chen, Continuous-time model identification of fractional-order models with time delays. IET Control Theory Appl. 5(7), 900–912 (2022)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  28. P. Niamsup, K. Ratchagit, V.N. Phat, Novel criteria for finite-time stabilization and guaranteed cost control of delayed neural networks. Neurocomputing 160, 281–286 (2015)

    Google Scholar 

  29. P. Niamsup, N.T. Thanh, V.N. Phat, Finite-time \(H_{\infty }\) control of linear singular fractional differential equations with time-varying delay. IMA J. Math. Control. Inf. 39(2), 773–788 (2022)

    MathSciNet  Google Scholar 

  30. N.T. Phuong, N.H. Sau, M.V. Thuan, Finite-time dissipative control design for one-sided Lipschitz nonlinear singular Caputo fractional order systems. Int. J. Syst. Sci. 54(8), 1694–1712 (2023)

    MathSciNet  Google Scholar 

  31. A. Pratap, R. Raja, R.P. Agarwal, J. Cao, Stability analysis and robust synchronization of fractional-order competitive neural networks with different time scales and impulsive perturbations. Int. J. Adapt. Control Signal Process. 33(11), 1635–1660 (2019)

    MathSciNet  Google Scholar 

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

    Google Scholar 

  33. T. Senthilkumar, P. Balasubramaniam, Delay-dependent robust stabilization and \(H_{\infty }\) control for nonlinear stochastic systems with Markovian jump parameters and interval time-varying delays. J. Optim. Theory Appl. 151, 100–120 (2011)

    MathSciNet  Google Scholar 

  34. T. Senthilkumar, P. Balasubramaniam, Non-fragile robust stabilization and \(H_{\infty }\) control for uncertain stochastic time delay systems with Markovian jump parameters and nonlinear disturbances. Int. J. Adapt. Control Signal Process. 28(3–5), 464–478 (2014)

    MathSciNet  Google Scholar 

  35. Y. Shang, L. Liu, Y. Di, Z. Fu, B. Fan, Guaranteed cost and finite-time event-triggered control of fractional-order switched systems. Trans. Inst. Meas. Control. 43(12), 2724–2733 (2021)

    Google Scholar 

  36. J.E. Solís-Pérez, J. Betancourt-Vera, J.F. Gómez-Aguilar, A.A. Alderremy, S. Aly, Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique. Eur. Phys. J. Spec. Top. 231(10), 1931–1968 (2022)

    Google Scholar 

  37. J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández-Pérez, J. Torres-Jiménez, Dynamical analysis of the \(\cal{M} -\mathbb{C} \)omplex Lorenz system and its anti-synchronization via \(\cal{M} -\)Sliding mode control. Res. Phys. 43, 106080 (2022)

    Google Scholar 

  38. H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Google Scholar 

  39. S.P. Talebi, S.J. Godsill, D.P. Mandic, Filtering structures for \(\alpha -\)stable systems. IEEE Control Syst. Lett. 7, 553–558 (2022)

    MathSciNet  Google Scholar 

  40. N.T. Thanh, V.N. Phat, Switching law design for finite-time stability of singular fractional-order systems with delay. IET Control Theory Appl. 13(9), 1367–1373 (2019)

    MathSciNet  Google Scholar 

  41. N.T. Thanh, V.N. Phat, P. Niamsup, New finite-time stability analysis of singular fractional differential equations with time-varying delay. Fract. Calc. Appl. Anal. 23(2), 504–519 (2020)

    MathSciNet  Google Scholar 

  42. N.T. Thanh, M.V. Thuan, T.N. Tuan, New results on finite-time guaranteed cost control of uncertain polytopic fractional-order systems with time-varying delays. Opt. Control Appl. Methods (2023). https://doi.org/10.1002/oca.2989

    Article  MathSciNet  Google Scholar 

  43. M.V. Thuan, T.N. Binh, D.C. Huong, Finite-time guaranteed cost control of Caputo fractional-order neural networks. Asian J. Control 22(2), 696–705 (2020)

    MathSciNet  Google Scholar 

  44. B. Wang, H. Jahanshahi, S. Bekiros, Y.M. Chu, J.F. Gomez-Aguilar, F.E. Alsaadi, M.O. Alassafi, Tracking control and stabilization of a fractional financial risk system using novel active finite-time fault-tolerant controls. Fractals 29(06), 2150155 (2021)

    Google Scholar 

  45. J. Wang, Y. Ji, C. Zhang, Iterative parameter and order identification for fractional-order nonlinear finite impulse response systems using the key term separation. Int. J. Adapt. Control Signal Process. 35(8), 1562–1577 (2021)

    MathSciNet  Google Scholar 

  46. Y. Wei, W.T. Peter, B. Du, Y. Wang, An innovative fixed-pole numerical approximation for fractional order systems. ISA Trans. 62, 94–102 (2016)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  48. Z.G. Wu, H. Su, P. Shi, J. Chu, Analysis and Synthesis of Singular Systems with Time-delays (Springer, Berlin, 2013)

    Google Scholar 

  49. S. Xiao, Z. Wang, C. Wang, Passivity analysis of fractional-order neural networks with interval parameter uncertainties via an interval matrix polytope approach. Neurocomputing 477, 96–103 (2022)

    Google Scholar 

  50. S. Xu, J. Lam, Robust Control and Filtering of Singular Systems (Springer, Berlin, 2006)

    Google Scholar 

  51. D. Yang, K.Y. Cai, Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems. Expert Syst. Appl. 37(10), 6963–6967 (2010)

    Google Scholar 

  52. A. Yousefpour, H. Jahanshahi, J.M. Munoz-Pacheco, S. Bekiros, Z. Wei, A fractional-order hyper-chaotic economic system with transient chaos. Chaos, Solitons & Fractals 130, 109400 (2020)

    MathSciNet  Google Scholar 

  53. X. Zhang, Y. Chen, Admissibility and robust stabilization of continuous linear singular fractional order systems with the fractional order \(\alpha \): The \(0 < \alpha < 1\) case. ISA Trans. 82, 42–50 (2018)

    Google Scholar 

  54. Q.H. Zhang, J.G. Lu, Positive real lemmas for singular fractional-order systems: the \(0 < \alpha < 1\) case. IET Control Theory Appl. 14(18), 2805–2813 (2020)

    MathSciNet  Google Scholar 

  55. G. Zhang, Q. Zhu, Finite-time guaranteed cost control for uncertain delayed switched nonlinear stochastic systems. J. Frankl. Inst. 359(16), 8802–8818 (2022)

    MathSciNet  Google Scholar 

  56. L. Zhang, B. Huang, J. Lam, LMI synthesis of \(H_2\) and mixed \(H_2/H_{\infty }\) controllers for singular systems. IEEE Trans. Circuits Syst II Analog Digial Signal Process. 50(9), 615–626 (2003)

    Google Scholar 

  57. Y. Zhou, J. Wang, L. Zhang, Basic Theory of Fractional Differential Equations (World Scientific, Singapore, 2016)

    Google Scholar 

Download references

Acknowledgements

The author would like to thank the editor(s) and anonymous reviewers for their constructive comments which helped to improve the present paper. The research of N.H. Muoi was supported by Vietnam Academy of Science and Technology under Grant No. CSCL01.02/22-23.

Author information

Authors and Affiliations

  1. Faculty of Fundamental and Applied Sciences, Thai Nguyen University of Technology, Thai Nguyen, Vietnam

    Nguyen Thi Phuong

  2. Department of Mathematics and Informatics, TNU–University of Sciences, Thai Nguyen, Vietnam

    Mai Viet Thuan

  3. Faculty of Fundamental Science, Hanoi University of Industry, 298 Cau Dien Street, Bac Tu Liem District, Hanoi, Vietnam

    Nguyen Huu Sau

  4. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

    Nguyen Huyen Muoi

Authors
  1. Nguyen Thi Phuong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mai Viet Thuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Huu Sau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nguyen Huyen Muoi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mai Viet Thuan.

Ethics declarations

Conflict of interest

The authors declare that there is 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

Phuong, N.T., Thuan, M.V., Sau, N.H. et al. Non-fragile Finite-Time Guaranteed Cost Control for a Class of Singular Caputo Fractional-Order Systems with Uncertainties. Circuits Syst Signal Process 43, 795–820 (2024). https://doi.org/10.1007/s00034-023-02513-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-023-02513-0

Keywords

Mathematics Subject Classification

Search

Navigation