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On Stability Analysis of a Class of Nonlinear Systems with a Focus on Composite Nonlinear Feedback Approach

Abstract

Obtaining a desired transient performance of closed-loop system is the most important issues in impractical and industrial applications, in general. It is to note that the small overshoot and acceptable settling time of the system response are almost two typical characteristics in the transient performance. There are a number of contradictions between these performances; and therefore, a tradeoff between these ones should deeply be considered. In a word, a so-called composite nonlinear feedback (CNF) method is an efficient and simple technique which is employed to overcome the contradiction of simultaneous achievement of the mentioned transient performances. CNF based sliding mode control (SMC) for nonlinear systems with known upper bound uncertainty are considered in this paper. Unlike the existing results for the linear systems, the proposed CNF based SMC comprises two nonlinear parts. The first term assures the stability of the closed-loop nonlinear system and provides a fast convergence response. The second term reduces the overshoot of the response. Finally, to show the merits of the proposed approach, it is applied to two nonlinear Genesio’s chaotic system and a nonlinear helicopter system. The investigated results verify the effectiveness of the proposed approach.

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References

  1. 1.

    Wei, Qiang, Wang, Xing-yuan, & Xiao-peng, Hu. (2014). Feedback chaotic synchronization of a complex chaotic system with disturbances. Journal of Vibration and Control, 21(15), 3067–3071. https://doi.org/10.1177/1077546314521441.

  2. 2.

    Wang, X.-Y., & Wang, M.-J. (2007). Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos: An Interdisciplinary Journal of Nonlinear Science, 17(3), 033106. https://doi.org/10.1063/1.2755420.

  3. 3.

    Lin, Z., Pachter, M., & Banda, S. (1998). Toward improvement of tracking performance nonlinear feedback for linear systems. International Journal of Control, 70(1), 1–11. https://doi.org/10.1080/002071798222433.

  4. 4.

    Chen, B. M., Lee, T. H., Peng, K., & Venkataramanan, V. (2003). Composite nonlinear feedback control for linear systems with input saturation: Theory and an application. IEEE Transactions on Automatic Control, 48(3), 427–439. https://doi.org/10.1109/tac.2003.809148.

  5. 5.

    He, Y., Chen, B. M., & Wu, C. (2005). Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation. Systems & Control Letters, 54(5), 455–469. https://doi.org/10.1016/j.sysconle.2004.09.010.

  6. 6.

    Xiao, Yu., Lin, Xianwu, & Lan, Weiyao. (2014). Composite nonlinear feedback controller design for an overhead crane servo system. Transactions of the Institute of Measurement and Control, 36(5), 662–672. https://doi.org/10.1177/0142331213518578.

  7. 7.

    Chuan, Hu, Wang, Rongrong, Yan, Fengjun, & Chen, Nan. (2016). Robust composite nonlinear feedback path following control for underactuated surface vessels with desired-heading amendment. IEEE Transactions on Industrial Electronics, 63(10), 6386–6394. https://doi.org/10.1109/TIE.2016.2573240.

  8. 8.

    Mobayen, S., & Tchier, F. (2018). Composite nonlinear feedback integral sliding mode tracker design for uncertain switched systems with input saturation. Communications in Nonlinear Science and Numerical Simulation, 65, 173–184. https://doi.org/10.1016/j.cnsns.2018.05.019.

  9. 9.

    Cheng, G., Peng, K., Chen, B. M., & Lee, T. H. (2007). Improving transient performance in tracking general references using composite nonlinear feedback control and its application to high-speed XY-table positioning mechanism. IEEE Transactions on Industrial Electronics, 54(2), 1039–1051. https://doi.org/10.1109/TIE.2007.892635.

  10. 10.

    Wei, L., Fang, F., & Shi, Y. (2014). Adaptive back-stepping-based composite nonlinear feedback water level control for the nuclear U-tube steam generator. IEEE Transactions on Control Systems Technology, 22(1), 369–377. https://doi.org/10.1109/TCST.2013.2250504.

  11. 11.

    Lin, D., Lan, W., & Li, M. (2011). Composite nonlinear feedback control for linear singular systems with input saturation. Systems & Control Letters, 60(10), 825–831. https://doi.org/10.1016/j.sysconle.2011.06.006.

  12. 12.

    Lin, D., & Lan, W. (2015). Output feedback composite nonlinear feedback control for singular systems with input saturation. Journal of Franklin Institute, 352(1), 384–398. https://doi.org/10.1016/j.jfranklin.2014.10.018.

  13. 13.

    Mobayen, S. (2014). Robust tracking controller for multivariable delayed systems with input saturation via composite nonlinear feedback. Nonlinear Dynamics, 76(1), 827–838. https://doi.org/10.1007/s11071-013-1172-5.

  14. 14.

    Wang, J., & Zhao, J. (2016). On improving transient performance in tracking control for switched systems with input saturation via composite nonlinear feedback. International Journal Robust Nonlinear Control, 26(3), 509–518. https://doi.org/10.1002/rnc.3322.

  15. 15.

    Wang, R., Hu, C., Yan, F., & Chadli, M. (2016). Composite nonlinear feedback control for path following of four-wheel independently actuated autonomous ground vehicles. IEEE Transactions on Intelligent Transportation Systems, 17(7), 2063–2074. https://doi.org/10.1109/TITS.2015.2498172.

  16. 16.

    Mobayen, S., Majd, V. J., & Sojoodi, M. (2012). An LMI-based composite nonlinear feedback terminal sliding-mode controller design for disturbed MIMO systems. Mathematics and Computers in Simulation, 85, 1–10. https://doi.org/10.1016/j.matcom.2012.09.006.

  17. 17.

    Mobayen, S. (2014). Design of CNF-based nonlinear integral sliding surface for matched uncertain linear systems with multiple state-delays. Nonlinear Dynamics, 77(3), 1047–1054. https://doi.org/10.1007/s11071-014-1362-9.

  18. 18.

    Li, Y., Tong, S., & Li, T. (2015). Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with Input saturatio. IEEE Transactions on Cybernetics, 45(10), 2299–2308. https://doi.org/10.1109/TCYB.2014.2370645.

  19. 19.

    Zhang, H., Wang, X.-Y., & Lin, X.-H. (2017). Topology identification and module–phase synchronization of neural network with time delay. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(6), 885–892. https://doi.org/10.1109/tsmc.2016.2523935.

  20. 20.

    Ebrahimi Mollabashi, H., Mazinan, A. H., & Hamidi, H. (2018). Takagi–sugeno fuzzy-based CNF control approach considering a class of constrained nonlinear systems. IETE Journal of Research. https://doi.org/10.1080/03772063.2018.1464969.

  21. 21.

    Lin, D., & Wang, X.-Y. (2010). Observer-based decentralized fuzzy neural sliding mode control for interconnected unknown chaotic systems via network structure. Fuzzy Sets and Systems, 161, 2066–2080. https://doi.org/10.1016/j.fss.2010.03.006.

  22. 22.

    Mobaye, S., & Tchier, F. (2017). Composite nonlinear feedback control technique for master/slave synchronization of nonlinear systems. Nonlinear Dynamics, 87(3), 1731–1747. https://doi.org/10.1007/s11071-016-3148-8.

  23. 23.

    Mollabashi, H. E., & Mazinan, A. H. (2018). Adaptive composite non-linear feedback based sliding mode controller for non-linear systems. Electronics Letters, 54, 973–974. https://doi.org/10.1049/el.2018.0619.

  24. 24.

    Wang, X.-Y., & Song, J.-M. (2009). Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Communications in Nonlinear Science and Numerical Simulation, 14, 3351–3357. https://doi.org/10.1016/j.cnsns.2009.01.010.

  25. 25.

    Wang, T., Wang, X., & Wang, M. (2011). Chaotic control of Hénon map with feedback and non-feedback methods. Communications in Nonlinear Science and Numerical Simulation, 16, 3367–3374. https://doi.org/10.1016/j.cnsns.2010.11.014.

  26. 26.

    Liu, L., Pu, J., Song, X., Fu, Z., & Wang, X. (2014). Adaptive sliding mode control of uncertain chaotic systems with input nonlinearity. Nonlinear Dynamics, 76(4), 1857–1865. https://doi.org/10.1007/s11071-013-1163-6.

  27. 27.

    Samadi, B., & Rodrigues, L. (2014). A sum of squares approach to back-stepping controller synthesis for piecewise affine and polynomial systems. International Journal of Robust and Nonlinear Control, 24(16), 2365–2387. https://doi.org/10.1002/rnc.2994.

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Correspondence to A. H. Mazinan.

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Mollabashi, H.E., Mazinan, A.H. On Stability Analysis of a Class of Nonlinear Systems with a Focus on Composite Nonlinear Feedback Approach. Sens Imaging 19, 37 (2018). https://doi.org/10.1007/s11220-018-0221-z

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Keywords

  • Stability analysis
  • Composite nonlinear feedback
  • Sliding mode control
  • Genesio’s chaotic systems