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Mixed \(H_\infty\) and Passive Filtering for A Class of Nonlinear Switched Systems with Unstable Subsystems

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Abstract

This paper investigates the mixed \(H_\infty\) and passive filtering problem for a class of nonlinear switched systems with unstable subsystems. The mixed weighted \(H_\infty\) and passivity performance index is proposed for switched systems. This new performance index covers the weighted \(H_\infty\) performance index and the passivity performance index as special cases. Therefore, based on this new performance index, the weighted \(H_\infty\) filtering problem and the passive filtering problem for nonlinear switched systems with unstable subsystems are solved in a unified framework. The Takagi–Sugeno (T–S) fuzzy model is an effective tool in approximating most complex nonlinear systems, which utilizes local linear system description for each rule. Using the T–S fuzzy model to approximate each nonlinear subsystem, the nonlinear switched systems are modeled as the switched T–S fuzzy systems. The states of the filtering error system constructed by the augmentation technique will be divergent when unstable subsystems are activated. To overcome this problem, a set of mode-dependent filters of a Luenberger-like observer type are constructed in this paper. The multiple Lyapunov functions approach and the average dwell time technique are employed to solve the filtering problem. New sufficient conditions for the existence of mixed \(H_\infty\) and passive filters are developed, which ensure the filtering error system to be asymptotically stable with a prescribed mixed \(H_\infty\) and passivity performance index. Moreover, the desired mixed \(H_\infty\) and passive filters can be constructed by solving a set of linear matrix inequalities. Finally, two numerical examples are given to demonstrate the effectiveness and advantage of the obtained results.

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References

  1. Liberzon, D.: Switching in Systems and Control. Birkhauser, Berlin (2003)

    Book  MATH  Google Scholar 

  2. Hespanha, J.P., Morse, A.S.: Stability of switched systems with average dwell time. In: Proceedings of 38th IEEE Conference on Decision and Control, Phoenix, AZ, pp. 2655–2660 (1999)

  3. Branicky, M.S.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43(4), 475–482 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Zhang, L., Gao, H.: Asynchronously switched control of switched linear systems with average dwell time. Automatica 46(5), 953–958 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zamani, I., Shafiee, M.: On the stability issues of switched singular time-delay systems with slow switching based on average dwell-time. Int. J. Robust Nonlinear Control 24(4), 595–624 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Yuan, C., Wu, F.: Hybrid control for switched linear systems with average dwell time. IEEE Trans. Autom. Control 60(1), 240–245 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybern. SMC–15(1), 116–132 (1985)

    Article  MATH  Google Scholar 

  8. Qiu, J., Feng, G., Gao, H.: Static-output-feedback \(H_\infty\) control of continuous-time T–S fuzzy affine systems via piecewise Lyapunov functions. IEEE Trans. Fuzzy Syst. 21(2), 245–261 (2013)

    Article  Google Scholar 

  9. Qiu, J., Tian, H., Lu, Q., Gao, H.: Nonsynchronized robust filtering design for continuous-time T–S fuzzy affine dynamic systems based on piecewise Lyapunov functions. IEEE Trans. Cybern. 43(6), 1755–1766 (2013)

    Article  Google Scholar 

  10. Cao, K., Gao, X.Z., Lam, H.K., Vasilakos, A.V., Pedrycz, W.: A new relaxed stability condition for Takagi–Sugeno fuzzy control systems using quadratic fuzzy Lyapunov functions and staircase membership functions. Int. J. Fuzzy Syst. 16(3), 327–337 (2014)

    MathSciNet  Google Scholar 

  11. Li, H., Yin, S., Pan, Y., Lam, H.K.: Model reduction for interval type-2 Takagi–Sugeno fuzzy systems. Automatica 61, 308–314 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xie, X., Yue, D., Zhang, H., Xue, Y.: Control synthesis of discrete-time T–S fuzzy systems via a multi-instant homogenous polynomial approach. IEEE Trans. Cybern. 46(3), 630–640 (2016)

    Article  Google Scholar 

  13. Zhou, Q., Yao, D., Wang, J., Wu, C.: Robust control of uncertain semi-Markovian jump systems using sliding mode control method. Appl. Math. Comput. 286, 72–87 (2016)

    MathSciNet  Google Scholar 

  14. Xie, X., Yue, D., Zhang, H., Peng, C.: Control synthesis of discrete-time T-S fuzzy systems: reducing the conservatism whilst alleviating the computational burden. IEEE Trans. Cybern. (2016). doi:10.1109/TCYB.2016.2582747

    Google Scholar 

  15. Zhou, Q., Wu, C., Shi, P.: Observer-based adaptive fuzzy tracking control of nonlinear systems with time delay and input saturation. Fuzzy Sets Syst. 319, 49–68 (2017)

    Article  MathSciNet  Google Scholar 

  16. Liu, D., Wu, C., Zhou, Q., Lam, H.K.: Fuzzy guaranteed cost output tracking control for fuzzy discrete-time systems with different premise variables. Complexity 21(5), 265–276 (2016)

    Article  MathSciNet  Google Scholar 

  17. Chiou, J.S., Wang, C.J., Cheng, C.M., Wang, C.C.: Analysis and synthesis of switched nonlinear systems using the T–S fuzzy model. Appl. Math. Model. 34(6), 1467–1481 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zheng, Q., Zhang, H.: Asynchronous \(H_\infty\) fuzzy control for a class of switched nonlinear systems via switching fuzzy Lyapunov function approach. Neurocomputing 182, 178–186 (2016)

    Article  Google Scholar 

  19. Zhao, X., Yin, Y., Zhang, L., Yang, H.: Control of switched nonlinear systems via T–S fuzzy modeling. IEEE Trans. Fuzzy Syst. 24(1), 235–241 (2016)

    Article  Google Scholar 

  20. Zheng, Q., Zhang, H.: Exponential stability and asynchronous stabilization of nonlinear impulsive switched systems via switching fuzzy Lyapunov function approach. Int. J. Fuzzy Syst. 19(1), 257–271 (2017)

    Article  MathSciNet  Google Scholar 

  21. Yoneyama, J.: \(H_\infty\) filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach. Fuzzy Sets Syst. 160(12), 1738–1748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xia, Z., Li, J.: Switching fuzzy filtering for nonlinear stochastic delay systems using piecewise Lyapunov-Krasovskii function. Int. J. Fuzzy Syst. 14(4), 530–539 (2012)

    MathSciNet  Google Scholar 

  23. Li, H., Pan, Y., Zhou, Q.: Filter design for interval type-2 fuzzy systems with \(D\) stability constraints under a unified frame. IEEE Trans. Fuzzy Syst. 23(3), 719–725 (2015)

    Article  Google Scholar 

  24. Chang, X., Yang, G.: Nonfragile \(H_\infty\) filter design for T–S fuzzy systems in standard form. IEEE Trans. Ind. Electron. 61(7), 3448–3458 (2014)

    Article  Google Scholar 

  25. Nagpal, K.M., Khargonekar, P.P.: Filtering and smoothing in an \(H_\infty\) setting. IEEE Trans. Autom. Control 36(2), 152–166 (1991)

    Article  MATH  Google Scholar 

  26. Zhang, L., Cui, N., Liu, M., Zhao, Y.: Asynchronous filtering of discrete-time switched linear systems with average dwell time. IEEE Trans. Circuits Syst. I 58(5), 1109–1118 (2011)

    Article  MathSciNet  Google Scholar 

  27. Wang, D., Shi, P., Wang, J., Wang, W.: Delay-dependent exponential \(H_\infty\) filtering for discrete-time switched delay systems. Int. J. Robust Nonlinear Control 22(13), 1522–1536 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhang, D., Yu, L., Wang, Q., Ong, C.J., Wu, Z.: Exponential \(H_\infty\) filtering for discrete-time switched singular systems with time-varying delays. J. Frankl. Inst. 349(7), 2323–2342 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, L., Dong, X., Qiu, J., Alsaedi, A., Hayat, T.: \(H_\infty\) filtering for a class of discrete-time switched fuzzy systems. Nonlinear Anal. Hybrid Syst. 14, 74–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, B., Zhang, H., Wang, G., Dang, C.: Asynchronous \(H_\infty\) filtering for linear switched systems with average dwell time. Int. J. Syst. Sci. 47(12), 2783–2791 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wu, L., Zheng, W.: Passivity-based sliding mode control of uncertain singular time-delay systems. Automatica 45(9), 2120–2127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kottenstette, N., McCourt, M.J., Xia, M., Gupta, V., Antsaklis, P.J.: On relationships among passivity, positive realness, and dissipativity in linear systems. Automatica 50(4), 1003–1016 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhao, J., Hill, D.J.: Dissipativity theory for switched systems. IEEE Trans. Autom. Control 53(4), 941–953 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wu, Z., Park, J.H., Su, H., Song, B., Chu, J.: Mixed \(H_\infty\) and passive filtering for singular systems with time delays. Signal Process. 93(7), 1705–1711 (2013)

    Article  Google Scholar 

  35. Mathiyalagan, K., Park, J.H., Sakthivel, R., Anthoni, S.M.: Robust mixed \(H_\infty\) and passive filtering for networked Markov jump systems with impulses. Signal Process. 101, 162–173 (2014)

    Article  Google Scholar 

  36. Shi, P., Zhang, Y., Chadli, M., Agarwal, R.K.: Mixed H-infinity and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Trans. Neural Netw. Learn. Syst. 27(4), 903–909 (2016)

    Article  MathSciNet  Google Scholar 

  37. Zhai, G., Hu, B., Yasuda, K., Michel, A.N.: Disturbance attenuation properties of time-controlled switched systems. J. Frankl. Inst. 338(7), 765–779 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lin, H., Antsaklis, P.J.: Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, D., Wang, W., Shi, P., Sun, X.: Controller failure analysis for systems with interval time-varying delay: a switched method. Circuits Syst. Signal Process. 28(3), 389–407 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, Z., Liu, Y., Liu, X.: Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays. IEEE Trans. Autom. Control 55(7), 1656–1662 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhai, G., Hu, B., Yasuda, K., Michel, A.N.: Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach. Int. J. Syst. Sci. 32(8), 1055–1061 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xie, D., Zhang, H., Zhang, H., Wang, B.: Exponential stability of switched systems with unstable subsystems: a mode-dependent average dwell time approach. Circuits Syst. Signal Process. 32(6), 3093–3105 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work was supported by the Natural Science Research of Anhui higher education promotioned program (Grant No. TSKJ2017B25), the Priming Scientific Research Foundation for the Introduction Talent in Anhui Polytechnic University (Grant No. 2017YQQ002), the Anhui Natural Science Foundation (Grant No. 1408085ME105), and the National Natural Science Foundation of China (Grant No. 61374117).

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Authors and Affiliations

  1. College of Electrical Engineering, Anhui Polytechnic University, Wuhu, 241000, Anhui, People’s Republic of China

    Qunxian Zheng & Xingzhong Guo

  2. School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Hongbin Zhang

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  1. Qunxian Zheng
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  2. Xingzhong Guo
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  3. Hongbin Zhang
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Correspondence to Qunxian Zheng.

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Zheng, Q., Guo, X. & Zhang, H. Mixed \(H_\infty\) and Passive Filtering for A Class of Nonlinear Switched Systems with Unstable Subsystems. Int. J. Fuzzy Syst. 20, 769–781 (2018). https://doi.org/10.1007/s40815-017-0340-z

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  • DOI: https://doi.org/10.1007/s40815-017-0340-z

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