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Event-triggered \(H_{\infty }\) controller design for uncertain fractional-order systems with time-varying delays

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Abstract

The primary focus of this paper is to investigate the \(H_{\infty }\) state feedback control problem within uncertain fractional-order systems characterized by time-varying delays. Our approach centers on the development of an event-triggered \(H_{\infty }\) control strategy, facilitated by the refined fractional-order Razumikhin theorem. This strategy is aimed at ensuring the uniformly asymptotic stability of the controlled system while adhering to a predefined \(H_{\infty }\) performance index. The central challenge lies in the memory characteristics of the fractional-order calculus operator, particularly in the context of delayed fractional-order systems, where preventing the occurrence of the Zeno phenomenon is paramount. To address this challenge, we introduce a novel theoretical framework and establish a new condition to prevent Zeno behaviors. This condition is derived using inequality techniques and leverages several essential properties of fractional-order calculus. To verify the effectiveness and feasibility of our proposed method, we present two illustrative examples.

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References

  1. He, B.B., Zhou, H.C., Chen, Y., Kou, C.H.: Asymptotical stability of fractional order systems with time delay via an integral inequality. IET Control Theory Appl. 12(12), 1748–1754 (2018)

    Article  MathSciNet  Google Scholar 

  2. He, B.B., Zhou, H.C., Kou, C.H., Chen, Y.: New integral inequalities and asymptotic stability of fractional-order systems with unbounded time delay. Nonlinear Dyn. 94, 1523–1534 (2018)

    Article  Google Scholar 

  3. He, B.B., Zhou, H.C., Kou, C.H., Chen, Y.: Stabilization of uncertain fractional order system with time-varying delay using BMI approach. Asian J. Control 23(1), 582–590 (2021)

    Article  MathSciNet  Google Scholar 

  4. Vijay Aravind, R., Balasubramaniam, P.: Global asymptotic stability of delayed fractional-order complex-valued fuzzy cellular neural networks with impulsive disturbances. J. Appl. Math. Comput. 68, 4713–4731 (2022)

    Article  MathSciNet  Google Scholar 

  5. Boukal, Y., Zasadzinski, M., Darouach, M., Radhy, N.E.: Fractional order time-varying-delay systems: a delay-dependent stability criterion by using diffusive representation. In: Mathematical Techniques of Fractional Order Systems, pp. 133–158 (2018)

  6. Jin, X.C., Lu, J.G.: Delay-dependent criteria for robust stability and stabilization of fractional-order time-varying delay systems. Eur. J. Control 67, 100704 (2022)

    Article  MathSciNet  Google Scholar 

  7. Jin, X.C., Lu, J.G., Zhang, Q.H.: Delay-dependent and order-dependent conditions for stability and stabilization of fractional-order memristive neural networks with time-varying delays. Neurocomputing 522, 53–63 (2023)

    Article  Google Scholar 

  8. Jin, X.C., Lu, J.G., Zhang, Q.H.: Delay-dependent and order-dependent asymptotic stability conditions for Riemann–Liouville fractional-order systems with time delays. Comput. Appl. Math. 42(3), 116 (2023)

    Article  MathSciNet  Google Scholar 

  9. Xu, S., Lam, J., Mao, X.: Delay-dependent \(H_{\infty }\) control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans. Circuits Syst. I Regul. Pap. 54(9), 2070–2077 (2007)

    Article  MathSciNet  Google Scholar 

  10. Zhou, J., Lai, H., Men, B.: \(H_{\infty }\) control for Lur’e singular systems with time delays. Circuits Syst. Signal Process. 41(3), 1367–1388 (2022)

    Article  Google Scholar 

  11. Gao, H., Zhao, Y., Chen, T.: \( H_ {\infty } \) fuzzy control of nonlinear systems under unreliable communication links. IEEE Trans. Fuzzy Syst. 17(2), 265–278 (2008)

    Google Scholar 

  12. Long, L., Zhao, J.: \( H_{\infty }\) control of switched nonlinear systems in \(p\)-normal form using multiple Lyapunov functions. IEEE Trans. Autom. Control 57(5), 1285–1291 (2012)

    Article  MathSciNet  Google Scholar 

  13. Lu, J.G., Zhao, Y.A.: Decentralised robust \(H_{\infty }\) control of fractional-order interconnected systems with uncertainties. Int. J. Control 90(6), 1221–1229 (2017)

    Article  MathSciNet  Google Scholar 

  14. Zhang, Q.H., Lu, J.G.: \(H_{\infty }\) control for singular fractional-order interval systems: the \(0<\alpha <1\) case. ISA Trans. 110, 105–116 (2021)

    Article  PubMed  Google Scholar 

  15. Shen, J., Lam, J.: \(H_{\infty }\) model reduction for positive fractional order systems. Asian J. Control 16(2), 441–450 (2014)

    Article  MathSciNet  Google Scholar 

  16. Shen, J., Lam, J.: State feedback \(H_{\infty }\) control of commensurate fractional-order systems. Int. J. Syst. Sci. 45(3), 363–372 (2014)

    Article  MathSciNet  Google Scholar 

  17. Zhang, X.M., Han, Q.L.: Event-triggered \(H_{\infty }\) control for a class of nonlinear networked control systems using novel integral inequalities. Int. J. Robust Nonlinear Control 27(4), 679–700 (2017)

    Article  MathSciNet  Google Scholar 

  18. Zha, L., Tian, E., Xie, X., Gu, Z., Cao, J.: Decentralized event-triggered \(H_{\infty }\) control for neural networks subject to cyber-attacks. Inf. Sci. 457, 141–155 (2018)

    Article  MathSciNet  Google Scholar 

  19. Wan, Z., Ma, X., Zhang, Y., Jiang, T., Zhou, J.: Event-triggered \(H_{\infty }\) controller design for Lurie systems with switching exponential time-varying gains. Comput. Appl. Math. 42(6), 281 (2023)

    Article  MathSciNet  Google Scholar 

  20. Zhang, L., Sun, M.: Dynamic event-triggered \(H_{\infty }\) control for Markov jump systems with input saturation. Eur. J. Control 70, 100770 (2023)

    Article  Google Scholar 

  21. Li, Q., Shen, B., Wang, Z., Alsaadi, F.E.: An event-triggered approach to distributed \(H_{\infty }\) state estimation for state-saturated systems with randomly occurring mixed delays. J. Frankl. Inst. 355(6), 3104–3121 (2018)

    Article  MathSciNet  Google Scholar 

  22. Yang, C., Xia, J., Park, J.H., Shen, H., Wang, J.: Sliding mode control for uncertain active vehicle suspension systems: an event-triggered \(H_{\infty }\) control scheme. Nonlinear Dyn. 103, 3209–3221 (2021)

    Article  Google Scholar 

  23. Wang, C., Zhou, X., Shi, X., Jin, Y.: Delay-dependent and order-dependent LMI-based sliding mode \(H_{\infty }\) control for variable fractional order uncertain differential systems with time-varying delay and external disturbance. J. Frankl. Inst. 359(15), 7893–7912 (2022)

    Article  Google Scholar 

  24. Xiao, P., Gu, Z.: Adaptive event-triggered consensus of fractional-order nonlinear multi-agent systems. IEEE Access 10, 213–220 (2021)

    Article  Google Scholar 

  25. Xu, B., Li, B.: Event-triggered state estimation for fractional-order neural networks. Mathematics 10(3), 325 (2022)

    Article  CAS  Google Scholar 

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

    Google Scholar 

  27. Baleanu, D., Balas, V.E., Agarwal, P.: Fractional Order Systems and Applications in Engineering. Academic Press, Cambridge (2022)

    Google Scholar 

  28. Kaczorek, T., Rogowski, K.: Fractional Linear Systems and Electrical Circuits. Springer, Cham (2015)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Chen, L., Li, T., Wu, R., Chen, Y., Liu, Z.: Non-fragile control for a class of fractional-order uncertain linear systems with time-delay. IET Control Theory Appl. 14(12), 1575–1589 (2020)

    Article  MathSciNet  Google Scholar 

  31. Parvizian, M., Khandani, K., Majd, V.J.: An \(H_{\infty }\) non-fragile observer-based adaptive sliding mode controller design for uncertain fractional-order nonlinear systems with time delay and input nonlinearity. Asian J. Control 23(1), 423–431 (2021)

    Article  MathSciNet  Google Scholar 

  32. Hui, M., Wei, C., Zhang, J., Iu, H.H.C., Yao, R., Bai, L.: Finite-time synchronization of fractional-order memristive neural networks via feedback and periodically intermittent control. Commun. Nonlinear Sci. Numer. Simul. 116, 106822 (2023)

    Article  MathSciNet  Google Scholar 

  33. Liu, P., Zeng, Z., Wang, J.: Asymptotic and finite-time cluster synchronization of coupled fractional-order neural networks with time delay. IEEE Trans. Neural Netw. Learn. Syst. 31(11), 4956–4967 (2020)

    Article  MathSciNet  PubMed  Google Scholar 

  34. Xing, X., Wu, H., Cao, J.: Event-triggered impulsive control for synchronization in finite time of fractional-order reaction–diffusion complex networks. Neurocomputing 557, 126703 (2023)

    Article  Google Scholar 

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

    Google Scholar 

  36. Feng, T., Wang, Y.E., Liu, L., Wu, B.: Observer-based event-triggered control for uncertain fractional-order systems. J. Frankl. Inst. 357(14), 9423–9441 (2020)

    Article  MathSciNet  Google Scholar 

  37. Meng, X., Gao, C., Jiang, B., Karimi, H.R.: An event-triggered sliding mode control mechanism to exponential consensus of fractional-order descriptor nonlinear multi-agent systems. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. (2023). https://doi.org/10.1177/0959651823119139

    Article  Google Scholar 

  38. Chen, X., Chen, Y., Zhang, B., Qiu, D.: A modeling and analysis method for fractional-order dc–dc converters. IEEE Trans. Power Electron. 32(9), 7034–7044 (2017)

    Article  ADS  Google Scholar 

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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 is funded by the Ministry of Education and Training of Vietnam under Grant Number B2023-TNA-15.

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Authors and Affiliations

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

    Nguyen Huu Sau

  2. Department of Basic Sciences, Thai Nguyen University of Economics and Business Administration, Thái Nguyên, Vietnam

    Tran Nguyen Binh

  3. School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 01 Dai Co Viet Str., Hai Ba Trung District, Hanoi, Viet Nam

    Nguyen Truong Thanh

  4. Department of Mathematics and Informatics, TNU–University of Sciences, Thái Nguyên, Vietnam

    Mai Viet Thuan

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  1. Nguyen Huu Sau
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  2. Tran Nguyen Binh
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Correspondence to Mai Viet Thuan.

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Sau, N.H., Binh, T.N., Thanh, N.T. et al. Event-triggered \(H_{\infty }\) controller design for uncertain fractional-order systems with time-varying delays. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02031-5

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  • DOI: https://doi.org/10.1007/s12190-024-02031-5

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