Nonlinear Dynamics

, Volume 85, Issue 1, pp 573–582 | Cite as

H \(\infty \) observer-based sliding mode control for singularly perturbed systems with input nonlinearity

  • Wei Liu
  • Yanyan Wang
  • Zhiming Wang
Original Paper


This paper considers the problem of \(H \infty \) observer-based sliding mode control for singularly perturbed systems with input nonlinearities. First, a proper observer is designed such that the observer error system with disturbance attenuation level is asymptotically stable. Then, an observer-based sliding surface is constructed under which a criterion for the input-to-state stability (ISS) of the sliding mode dynamics with respect to the observer error is obtained via linear matrix inequality. The criterion presented is independent of the small parameter, and the upper bound for ISS can be obtained efficiently. In addition, a sliding mode control law is synthesized to guarantee the reachability of the sliding surface in the state estimation space. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed theoretical results.


Singularly perturbed systems Linear matrix inequality (LMI) Sliding mode control (SMC) Input-to-state stability (ISS) 



This paper is supported by the National Natural Science Foundation of China (11171113, 11471118), the Research Foundation of the Henan Higher Education Institutions of China (16A110006) and the Science and Technology Planning Project of Henan Province of China (162102310619).


  1. 1.
    Petersen, I.R., Tempo, R.: Robust control of uncertain systems: classical results and recent developments. Automatica 50, 1315–1335 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Utkin, V.I.: Variable structure systems with sliding modes. IEEE Trans. Autom. Control 22, 212–222 (1977)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Utkin, V.I.: Sliding Modes in Control and Optimization. Springer, Berlin (1992)CrossRefMATHGoogle Scholar
  4. 4.
    Xu, B., Yang, C.G., Shi, Z.K.: Reinforcement learning output feedback NN control using deterministic learning technique. IEEE Trans. Neural Netw. Learn. Syst. 25, 635–641 (2014)CrossRefGoogle Scholar
  5. 5.
    Xu, B., Shi, Z.K., Yang, C.G., Sun, F.C.: Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form. IEEE Trans. Cybern. 44, 2626–2634 (2014)CrossRefGoogle Scholar
  6. 6.
    Hsu, K.C.: Variable structure control design for uncertain dynamic systems with sector nonlinearities. Automatica 34, 505–508 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Huang, J., Sun, L.N., Han, Z.Z., Liu, L.P.: Adaptive terminal sliding mode control for nonlinear differential inclusion systems with disturbance. Nonlinear Dyn. 72, 221–228 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Levant, A.: Transient adjustment of high-order sliding modes. In: Proceedings of the 8th international workshop on variable structure systems, pp. 1–6. Spain (2004)Google Scholar
  9. 9.
    Niu, Y., Daniel, W.C.: Ho: robust observer design for Itô stochastic time-delay systems via sliding mode control. Syst. Control Lett. 55, 781–793 (2006)CrossRefMATHGoogle Scholar
  10. 10.
    Castaños, F., Fridman, L.: Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans. Autom. Control 51, 853–858 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bandyopadhyay, B., Deepak, F., Park, Y.J.: A robust algorithm against actuator saturation using integral sliding mode and composite nonlinear feedback. In: Proceedings of the 17th World Congress the International Federation of Automatic Control Seoul, Korea, pp. 14174–14179 (2008)Google Scholar
  12. 12.
    Wu, L., Zheng, W.X.: Passivity-based sliding mode control of uncertain singular time-delay systems. Automatica 45, 2120–2127 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Wu, L., Daniel, W.C., Ho, : Sliding mode control of singular stochastic hybrid systems. Automatica 46, 779–783 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gao, Y., Sun, B., Lu, G.: Passivity-based integral sliding-mode control of uncertain singularly perturbed systems. IEEE Trans. Circuits Syst. II Express Briefs 58, 386–390 (2011)CrossRefGoogle Scholar
  15. 15.
    Kokotovic, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, London (1986)Google Scholar
  16. 16.
    Naidu, D.S.: Singular perturbations and time scales in control theory and applications: an overview. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 9, 233–278 (2002)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Shao, Z.H.: Robust stability of two-time-scale systems with nonlinear uncertainties. IEEE Trans. Autom. Control 49, 258–261 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shi, P., Shue, S.P., Agarwal, R.K.: Robust disturbance attenuation with stability for a class of uncertain singularly perturbed systems. Int. J. Control 70, 873–891 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Chen, S.J., Lin, J.L.: Maximal stability bounds of singularly perturbed systems. J. Frankl. Inst. 336, 1209–1218 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wang, Z.M., Liu, W., Dai, H.H. Naidu, D.S.: Robust stabilization of model-based uncertain singularly perturbed systems with networked time-delay. In: Proceedings of the 48th IEEE conference on decision and control conference, Shanghai, P.R. China, pp. 7917–7922 (2009)Google Scholar
  21. 21.
    Yang, C.Y., Zhang, Q.L., Sun, J., Chai, T.Y.: Lur’e Lyapunov function and absolute stability criterion for Lur’e singularly perturbed systems. IEEE Trans. Autom. Control 56, 2666–2671 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tuan, H.D., Hosoe, S.: Multivariable circle criteria for multiparameter singularly perturbed systems. IEEE Trans. Autom. Control 45, 720–725 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Saksena, V.R., Kokotovic, P.V.: Singular perturbation of the Popov–Kalman–Yakubovich lemma. Syst. Control Lett. 1, 65–68 (1981)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Liu, H.P., Sun, F.C., Sun, Z.Q.: Stability analysis and synthesis of fuzzy singularly perturbed systems. IEEE Trans. Fuzzy Syst. 13, 273–284 (2005)Google Scholar
  25. 25.
    Moraal, P.E., Grizzle, J.W.: Observer design for nonlinear systems with discrete-time measurements. IEEE Trans. Autom. Control 40, 396–404 (1995)Google Scholar
  26. 26.
    Lu, R.Q., Xu, Y., Xue, A.K.: \(H \infty \) filtering for singular systems with communication delays. Signal Process. 90, 1240–1248 (2010)CrossRefMATHGoogle Scholar
  27. 27.
    Ibrir, S., Xie, W.F., Su, C.Y.: Observer-based control of discrete-time Lipschitzian nonlinear systems: application to one-link flexible joint robot. Int. J. Control 78, 385–395 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Atassi, A.N., Khalil, H.K.: A separation principle for the stabilization of a class of nonlinear systems. IEEE Trans. Autom. Control 44, 1672–1687 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jiang, Z.P., Hill, D.J., Guo, Y.: Semiglobal output feedback stabilization for the nonlinear benchmark example. In: Proceedings of the European Control Conference, Brussels, Belgium (1997)Google Scholar
  30. 30.
    Liberzon, D.: Observer-based quantized output feedback control of nonlinear systems. In: Proceedings of the 17th IFAC World Congress, pp. 8039–8043 (2008)Google Scholar
  31. 31.
    Wang, Z.M., Liu, W.: Output feedback networked control of singular perturbation. In: Proceedings of the 8th World Congress on Intelligent Control and Automation, Taipei, Taiwan, pp. 645–650 (2011)Google Scholar
  32. 32.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River, NJ (2000)MATHGoogle Scholar
  33. 33.
    Boyd, S., Ghaoui, L.E.L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA (1994)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhoukou Normal UniversityZhoukouChina
  2. 2.Center for Applied and Multidisciplinary Mathematics, Department of MathematicsEast China Normal UniversityShanghaiChina

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