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Finite-time \(H_{\infty }\) Dynamic Output Feedback Control for One-Sided Lipschitz Nonlinear Rectangular Descriptor Markov Jump Systems

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Abstract

This paper considers the finite-time \(H_{\infty }\) dynamic output feedback control for a class of one-sided Lipschitz nonlinear rectangular descriptor Markov jump systems (DMJSs). The differential matrix \(E\in \mathbb {R}^{m\times n}\) is not subject to any constraints, i.e., it includes two cases of \(m\ge n\) and \(m\le n\). For making the closed-loop system as square DMJSs, the rectangular dynamic output feedback controller is proposed. Firstly, the sufficient conditions are given to guarantee that the augmented closed-loop systems are singular stochastic \(H_{\infty }\) finite-time bounded (SS\(H_{\infty }\)FTB) and have a unique solution simultaneously by adopting a mode-dependent Lyapunov functional and implicit function theorem. Then a novel and rigorous strict linear matrix inequality (LMI) sufficient condition for the existence of a rectangular dynamic output feedback controller is given based on the certain matrix decoupling techniques, and the controller is obtained. Ultimately, numerical examples are provided in order to substantiate the soundness of the results.

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Data Availability

The authors declare that the data supporting the findings of this study are available within the article.

References

  1. M. Asadinia, T. Binazadeh, Finite-time stabilization of descriptor time-delay systems with one-sided Lipschitz nonlinearities: application to partial element equivalent circuit. Circuits Syst. Signal Process. 38(12), 5467–5487 (2019)

    Article  Google Scholar 

  2. E.K. Boukas, Control of Singular Systems with Random Abrupt Changes (Springer, London, 2008)

    Google Scholar 

  3. B. Cai, R. Weng, R. Zhang, Y. Liang, L. Zhang, Stabilization of a class of fuzzy stochastic jump systems with partial information on jump and sojourn parameters. Sci. China Technol. Sci. 64(2), 353–363 (2021)

    Article  Google Scholar 

  4. B. Cai, L. Zhang, Y. Shi, Observed-mode-dependent state estimation of hidden semi-Markov jump linear systems. IEEE Trans. Autom. Control 65(1), 442–449 (2020)

    Article  MathSciNet  Google Scholar 

  5. B. Cai, L. Zhang, Y. Shi, Control synthesis of hidden semi-Markov uncertain fuzzy systems via observations of hidden modes. IEEE Trans. Cybern. 50(8), 3709–3718 (2020)

    Article  Google Scholar 

  6. J. Chen, C. Lin, B. Chen, Output feedback control for singular Markovian jump systems with uncertain transition rates. IET Control Theory Appl. 10(16), 2142–2147 (2016)

    Article  MathSciNet  Google Scholar 

  7. J. Chen, C. Lin, B. Chen, Z. Zhang, Observer-based adaptive neural control for a class of nonlinear singular systems. Int. J. Robust Nonlinear Control 30(10), 4043–4058 (2020)

    Article  MathSciNet  Google Scholar 

  8. L. Dai, Singular Control Systems (Springer, New York, 1989)

    Book  Google Scholar 

  9. G. Duan, Analysis and Design of Descriptor Linear Systems (Springer, New York, 2010)

    Book  Google Scholar 

  10. G. Hu, Observers for one-sided Lipschitz nonlinear systems. IMA J. Math. Control. Inf. 23(4), 395–401 (2006)

    Article  Google Scholar 

  11. J.Y. Ishihara, M.H. Terra, Impulse controllability and observability of rectangular descriptor systems. IEEE Trans. Autom. Control 46(6), 991–994 (2001)

    Article  MathSciNet  Google Scholar 

  12. N.K. Kwon, I.S. Park, P. Park, C. Park, Dynamic output-feedback control for singular Markovian jump system: LMI approach. IEEE Trans. Autom. Control 62(10), 5396–5400 (2017)

    Article  MathSciNet  Google Scholar 

  13. C. Lin, J. Chen, B. Chen, L. Guo, Z. Zhang, Fuzzy normalization and stabilization for a class of nonlinear rectangular descriptor systems. Neurocomputing 219, 263–268 (2017)

    Article  Google Scholar 

  14. Y. Ma, X. Jia, D. Liu, Finite-time dissipative control for singular discrete-time Markovian jump systems with actuator saturation and partly unknown transition rates. Appl. Math. Model. 53, 49–70 (2018)

    Article  MathSciNet  Google Scholar 

  15. H. Nguyen, V.N. Phat, P. Niamsup, Criteria for robust finite-time stabilisation of linear singular systems with interval time-varying delay. IET Control Theory Appl. 11(12), 1968–1975 (2017)

    Article  MathSciNet  Google Scholar 

  16. P. Niamsup, V.N. Phat, A new result on finite-time control of singular linear time-delay systems. Appl. Math. Lett. 60, 1–7 (2016)

    Article  MathSciNet  Google Scholar 

  17. C. Park, N.K. Kwon, P. Park, Dynamic output-feedback \(H_{\infty }\) control for continuous-time singular Markovian jump systems. Int. J. Robust Nonlinear Control 28(11), 3521–3531 (2018)

    Article  MathSciNet  Google Scholar 

  18. I.S. Park, N.K. Kwon, P. Park, Dynamic output-feedback control for singular Markovian jump systems with partly unknown transition rates. Nonlinear Dyn. 95(4), 3149–3160 (2019)

    Article  Google Scholar 

  19. J. Ren, L. Feng, J. Fu, T. Zhuang, Admissibility analysis and passive output feedback control for one-sided Lipschitz nonlinear singular Markovian jump systems with uncertainties. Appl. Math. Comput. 409, 126405 (2021)

    MathSciNet  Google Scholar 

  20. N. Rong, Z. Wang, Finite-time stabilization of nonlinear systems using an event-triggered controller with exponential gains. Nonlinear Dyn. 98(1), 15–26 (2019)

    Article  Google Scholar 

  21. W. Saad, A. Sellami, G. Garcia, Robust integral sliding mode-\(H_{\infty }\) control of one-sided Lipschitz nonlinear systems. IET Control Theory Appl. 12(17), 2357–2367 (2018)

    Article  MathSciNet  Google Scholar 

  22. B. Sahereh, J. Aliakbar, K.S. Ali, \(H_{\infty }\) filtering for descriptor systems with strict LMI conditions. Automatica 80, 88–94 (2017)

    Article  MathSciNet  Google Scholar 

  23. J. Song, S. He, Robust finite-time \(H_{\infty }\) control for one-sided Lipschitz nonlinear systems via state feedback and output feedback. J. Franklin Inst. 352(8), 3250–3266 (2015)

    Article  MathSciNet  Google Scholar 

  24. X. Song, S. Ma, Indefinite linear quadratic optimal control for continuous-time rectangular descriptor Markov jump systems: infinite-time case. Int. J. Syst. Sci. 53(15), 3302–3321 (2022)

    Article  MathSciNet  Google Scholar 

  25. S. Song, S. Ma, C. Zhang, Stability and robust stabilisation for a class of non-linear uncertain discrete-time descriptor Markov jump systems. IET Control Theory Appl. 6(16), 2518–2527 (2012)

    Article  MathSciNet  Google Scholar 

  26. M. Sun, G. Zhuang, J. Xia, G. Chen, \(H_{\infty }\) dynamic output feedback control for time-varying delay singular Markovian jump systems based on variable elimination technique. Nonlinear Dyn. 108, 239–249 (2022)

    Article  Google Scholar 

  27. J. Tian, S. Ma, Existence of non-impulsive unique solution and stability for continuous-time linear rectangular descriptor Markov jump systems. Automatica 117, 108953 (2020)

    Article  MathSciNet  Google Scholar 

  28. J. Tian, S. Ma, C. Zhang, Unknown input reduced-order observer design for one-sided Lipschitz nonlinear descriptor Markovian jump systems. Asian J. Control 21(2), 952–964 (2019)

    Article  MathSciNet  Google Scholar 

  29. J. Wang, S. Ma, C. Zhang, Finite-time stabilization for nonlinear discrete-time singular Markov jump systems with piecewise-constant transition probabilities subject to average dwell time. J. Franklin Inst. 354(5), 2102–2124 (2017)

    Article  MathSciNet  Google Scholar 

  30. G. Wang, Q. Zhang, Y. Xing, Analysis and Design of Singular Markovian Jump Systems (Springer, New York, 2015)

    Book  Google Scholar 

  31. Y. Xia, E.K. Boukas, P. Shi, J. Zhang, Stability and stabilization of continuous-time singular hybrid systems. Automatica 45(6), 1504–1509 (2009)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  33. G. Zhang, Regularizability, controllability and observability of rectangular descriptor systems by dynamic compensation. In Proceedings of the American Control Conference, pp. 4393–4398 (2006)

  34. J. Zhang, M. Chadli, F. Zhu, Finite-time observer design for singular systems subject to unknown inputs. IET Control Theory Appl. 13(14), 2289–2299 (2019)

    Article  MathSciNet  Google Scholar 

  35. Y. Zhang, C. Liu, X, Mu, Robust finite-time \(H_{\infty }\) control of singular stochastic systems via static output feedback. Appl. Math. Comput. 218(9), 5629–5640 (2012)

    MathSciNet  Google Scholar 

  36. Y. Zhang, P. Shi, S.K. Nguang, Observer-based finite-time \(H_{\infty }\) control for discrete singular stochastic systems. Appl. Math. Lett. 36, 115–121 (2014)

    Article  MathSciNet  Google Scholar 

  37. W. Zhang, H. Su, F. Zhu, S.P. Bhattacharyya, Improved exponential observer design for one-sided Lipschitz nonlinear systems. Int. J. Robust Nonlinear Control 26(18), 3958–3973 (2016)

    Article  MathSciNet  Google Scholar 

  38. X. Zhang, Z. Zhao, Normalization and stabilization for rectangular singular fractional order T-S fuzzy systems. Fuzzy Sets Syst. 381, 140–153 (2020)

    Article  MathSciNet  Google Scholar 

  39. J. Zhu, S. Ma, Z. Cheng, Singular LQ problem for nonregular descriptor systems. IEEE Trans. Autom. Control 47(7), 1128–1133 (2002)

    Article  MathSciNet  Google Scholar 

  40. A. Zulfiqar, M. Rehan, M. Abid, Observer design for one-sided Lipschitz descriptor systems. Appl. Math. Model. 40(3), 2301–2311 (2016)

    Article  MathSciNet  Google Scholar 

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Funding

This work is supported by National Natural Science Foundation of China (61977042) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61821004).

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Authors and Affiliations

  1. School of Science, Shandong Jiaotong University, Jinan, 250357, China

    Xue Song

  2. School of Mathematics, Shandong University, Jinan, 250100, China

    Xue Song & Shuping Ma

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  1. Xue Song
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  2. Shuping Ma
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Correspondence to Shuping Ma.

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Song, X., Ma, S. Finite-time \(H_{\infty }\) Dynamic Output Feedback Control for One-Sided Lipschitz Nonlinear Rectangular Descriptor Markov Jump Systems. Circuits Syst Signal Process 43, 2695–2722 (2024). https://doi.org/10.1007/s00034-023-02595-w

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