Skip to main content
SpringerLink
Log in

Integral Sliding Mode Control for Singularly Perturbed Systems with Mismatched Disturbances

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper presents a novel integral sliding mode control (ISMC) for singularly perturbed systems (SPSs) with mismatched disturbances. A singular perturbation parameter \(\varepsilon \)-dependent disturbance observer is constructed to estimate the disturbances. An ISMC which incorporates the disturbance vector estimate is designed, such that the reachability condition is guaranteed. In order to compensate the effect of mismatched components of the disturbances extracted by the projection matrix theory, \({H_\infty }\) control theory is employed to determine the gain of ISMC by solving a set of linear matrix inequalities. As a result, the closed-loop SPSs under the ISMC are internally exponentially stable with the \({H_\infty }\) performance index guaranteed. Finally, the effectiveness and advantages of the proposed method are demonstrated by two examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. A.E. Ahmed, H.M. Schwartz, V.C. Aitken, Sliding mode control for singularly perturbed system, in Proceedings of 5th Asian Control Conference (2004), pp. 1946–1950

  2. B. Barmish, G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching condition. IEEE Trans. Autom. Control 27(1), 153–158 (1982)

    Article  MATH  Google Scholar 

  3. F. Castanos, L. Fridman, Analysis and design of integral sliding manifolds for systems with unmatched perturbations. IEEE Trans. Autom. Control 51(5), 853–858 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. W.H. Chen, Nonlinear disturbance observer-enhanced dynamic inversion control of missiles. J. Guid. Control Dyn. 26(1), 161–166 (2003)

    Article  Google Scholar 

  5. H.H. Choi, LMI-based sliding surface design for integral sliding model control of mismatched uncertain systems. IEEE Trans. Autom. Control 52(4), 736–742 (2007)

    Article  MATH  Google Scholar 

  6. B.L. Cong, Z. Chen, X.D. Liu, On adaptive sliding mode control without switching gain overestimation. Int. J. Robust Nonlinear Control 24(3), 515–531 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Z. Gao, X.Z. Liao, Integral sliding mode control for fractional-order systems with mismatched uncertainties. Nonlinear Dyn. 72(1), 27–35 (2013)

    Article  MathSciNet  Google Scholar 

  8. Y.B. Gao, G.P. Lu, Z.M. Wang, Passivity analysis of uncertain singularly perturbed systems. IEEE Trans. Circuits Syst. II Express Briefs 57(6), 486–490 (2010)

    Article  Google Scholar 

  9. Y.B. Gao, B.H. Sun, G.P. Lu, Passivity-based integral sliding-mode control of uncertain singularly perturbed systems. IEEE Trans. Circuits Syst. II Express Briefs 58(6), 386–390 (2011)

    Article  Google Scholar 

  10. B. Heck, Sliding-mode control for singularly perturbed systems. Int. J. Control 53(4), 985–1001 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Hu, Z.D. Wang, H.J. Gao, L.K. Stergioulas, Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities. IEEE Trans. Ind. Electron. 59(7), 3008–3015 (2012)

    Article  Google Scholar 

  12. M. Innocenti, L. Greco, L. Pollini, Sliding mode control for two-time scale systems: stability issues. Automatica 39(2), 273–280 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. T.H. Li, J.L. Lin, F.C. Kung, Composite sliding-mode control of singular perturbation systems, in Proceedings of American Control Conference (1995), pp. 2248–2249

  14. S.H. Li, J. Yang, W.H. Chen, X.S. Chen, Generalized extended state observer based control for systems with mismatched uncertainties. IEEE Trans. Ind. Electron. 59(12), 4792–4802 (2012)

    Article  Google Scholar 

  15. S.H. Li, K. Zong, H.X. Liu, A composite speed controller based on a second-order model of permanent magnet synchronous motor system. Trans. Inst. Meas. Control 33(5), 522–541 (2011)

    Article  Google Scholar 

  16. H.X. Liu, S.H. Li, Speed control for PMSM servo system using predictive function control and extended state observer. IEEE Trans. Ind. Electron. 59(2), 1171–1183 (2012)

    Article  Google Scholar 

  17. W. Liu, Y.Y. Wang, Z.H. Wang, \({H_\infty }\) observer-based sliding mode control for singularly perturbed systems with input nonlinearity. Nonlinear Dyn. 85(1), 573–582 (2016)

    Article  MATH  Google Scholar 

  18. M. Liu, L.X. Zhang, P. Shi, H.R. Karimi, Robust control of stochastic systems against bounded disturbances with application to flight control. IEEE Trans. Ind. Electron. 61(3), 1504–1515 (2014)

    Article  Google Scholar 

  19. Z. Liu, L. Zhao, H.M. Xiao, C.C. Gao, Adaptive \({H_\infty }\) integral sliding mode control for uncertain singular time-delay systems based on observer. Circuits Syst. Signal Process. 36(11), 4365–4387 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. X.P. Ma, Q.J. Wang, L.N. Zhou, C.Y. Yang, Controller design and analysis for singularly perturbed switched systems with actuator saturation. Int. J. Robust Nonlinear Control 26(15), 3404–3420 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Morawiec, The adaptive backstepping control of permanent magnet synchronous motor supplied by current source inverter. IEEE Trans. Indus. Inform. 9(2), 1047–1055 (2013)

    Article  Google Scholar 

  22. R.K. Munje, B.B. Musmade, J.G. Parkhe, B.M. Patre, Sliding mode control for three time scale system with matched disturbances, in Proceedings of India Conference (2012), pp. 131–136

  23. R.K. Munje, B.M. Patre, S.R. Shimjith, A.P. Tiwari, Sliding mode control for spatial stabilization of advanced heavy water reactor. IEEE Trans. Nucl. Sci. 60(4), 3040–3050 (2013)

    Article  Google Scholar 

  24. T. Nguyen, W.C. Su, Z. Gajic, Variable structure control for singularly perturbed linear continuous systems with matched disturbances. IEEE Trans. Autom. Control 57(3), 777–783 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Nguyen, W.C. Su, Z. Gajic, Sliding mode control for singularly perturbed linear continuous time systems: composite control approaches, in Proceedings of IEEE International Symposium on Computer-Aided Control System Design (2010), pp. 2011–2016

  26. Z.H. Shao, Robust stability of two-time-scale systems with nonlinear uncertainties. IEEE Trans. Autom. Control 49(2), 258–261 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. W.C. Su, Sliding surface design for singularly perturbed systems. Int. J. Control 72(11), 990–995 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. H.C. Ting, J.L. Chang, Y.P. Chen, Output feedback integral sliding mode controller of time-delay systems with mismatch disturbances. Asian J. Control 14(1), 85–94 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. S.C. Tong, Y.M. Li, Adaptive fuzzy output feedback tracking backstepping control of strict-feedback nonlinear systems with unknown dead zones. IEEE Trans. Fuzzy Syst. 20(1), 168–180 (2012)

    Article  Google Scholar 

  30. S.W. Wang, D.W. Yu, D.L. Yu, Compensation for unmatched uncertainty with adaptive RBF network. Int. J. Eng. Sci. Technol. 3(6), 35–43 (2011)

    Google Scholar 

  31. G.L. Wang, Q.L. Zhang, C.Y. Yang, Exponential stability of stochastic singular delay systems with general Markovian switchings. Int. J. Robust Nonlinear Control 25(17), 3478–3494 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Z.G. Wu, P. Shi, H.Y. Su, J. Chu, Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled data. IEEE Trans. Cybern. 43(6), 1796–1806 (2013)

    Article  Google Scholar 

  33. J. Yang, W.H. Chen, S.H. Li, Non-linear disturbance observer-based robust control for systems with mismatched disturbances/uncertainties. IET Control Theory Appl. 5(18), 2053–2062 (2011)

    Article  MathSciNet  Google Scholar 

  34. J. Yang, S.H. Li, J.Y. Su, X.H. Yu, Continuous nonsingular terminal sliding mode control for systems with mismatched disturbances. Automatica 49(7), 2287–2291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Yang, S.H. Li, C.Y. Sun, L. Guo, Nonlinear-disturbance-observer-based robust flight control for airbreathing hypersonic vehicles. IEEE Trans. Aerosp. Electron. Syst. 49(2), 1263–1275 (2013)

    Article  Google Scholar 

  36. J. Yang, S.H. Li, X.H. Yu, Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans. Ind. Electron. 60(1), 160–169 (2012)

    Article  Google Scholar 

  37. C.Y. Yang, T.T. Ma, Z.Y. Che, L.N. Zhou, An adaptive-gain sliding mode observer for sensorless control of permanent magnet linear synchronous. IEEE Access 6(1), 3469–3478 (2018)

    Article  Google Scholar 

  38. C.Y. Yang, L. Ma, X.P. Ma, X.S. Wang, Stability analysis of singularly perturbed control systems with actuator saturation. J. Frankl. Inst. 353(6), 1284–1296 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Yang, J.F. Ni, X. Pan, J.W. Liu, Global robust sliding mode control for time-delay systems with mismatched uncertainties. Circuits Syst. Signal Process. 35(8), 3015–3026 (2016)

    Article  MATH  Google Scholar 

  40. C.Y. Yang, J. Sun, X.P. Ma, Stabilization bound of singularly perturbed systems subject to actuator saturation. Automatica 49(2), 457–462 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. L. Yang, J.Y. Yang, Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems. Int. J. Robust Nonlinear Control 21(16), 1865–1879 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Yang, S. Yang, B. Richard, Modeling and robust discrete-time sliding-mode control design for a fluid power electrohydraulic actuator (EHA) system. IEEE/ASME Trans. Mechatron 18(1), 1–10 (2013)

    Article  Google Scholar 

  43. C.Y. Yang, Q.L. Zhang, Multiobjective control for T–S fuzzy singularly perturbed systems. IEEE Trans. Fuzzy Syst. 17(1), 104–115 (2009)

    Article  Google Scholar 

  44. C.Y. Yang, Q.L. Zhang, J. Sun, T.Y. Chai, Lur’e Lyapunov function and absolute stability criterion for Lur’e singularly perturbed systems. IEEE Trans. Autom. Control 56(11), 2666–2671 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. C.Y. Yang, L.N. Zhou, \({H_\infty }\) control and \(\varepsilon \)-bound estimation of discrete-time singularly perturbed systems. Circuits Syst. Signal Process. 35(7), 2640–2654 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. J. Yang, A. Zolotas, W.H. Chen, K. Michail, S.H. Li, Robust control of nonlinear MAGLEV suspension system with mismatched uncertainties via DOBC approach. ISA Trans. 50(3), 389–396 (2011)

    Article  Google Scholar 

  47. J.Y. Yao, Z.X. Jiao, D.W. Ma, Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans. Ind. Electron. 61(11), 6285–6293 (2014)

    Article  Google Scholar 

  48. X.H. Yu, O. Kaynak, Sliding-mode control with soft computing: a survey. IEEE Trans. Ind. Electron. 56(9), 3275–3285 (2009)

    Article  Google Scholar 

  49. D. Yue, S.F. Xu Sliding mode control of singular perturbation systems, in Proceedings of the IEEE International Conference on Systems (1996), pp. 113–116

  50. J.H. Zhang, G. Feng, Y.Q. Xia, Design of estimator-based sliding-mode output-feedback controllers for discrete-time systems. IEEE Trans. Ind. Electron. 61(5), 2432–2440 (2014)

    Article  Google Scholar 

  51. J.H. Zhang, X.W. Liu, Y.Q. Xia, Z.Q. Zuo, Y.J. Wang, Disturbance observer based integral sliding mode control for systems with mismatched disturbances. IEEE Trans. Ind. Electron. 63(11), 7041–7047 (2016)

    Article  Google Scholar 

  52. J.H. Zhang, P. Shi, Y.Q. Xia, Robust adaptive sliding-mode control for fuzzy systems with mismatched uncertainties. IEEE Trans. Fuzzy Syst. 18(4), 700–711 (2010)

    Article  Google Scholar 

  53. J.H. Zhang, W.X. Zheng, Design of adaptive sliding mode controllers for linear systems via output feedback. IEEE Trans. Ind. Electron. 61(7), 3553–3562 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (2017XKQY055).

Author information

Authors and Affiliations

  1. School of Information and Control Engineering, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Chunyu Yang, Zhiyuan Che & Linna Zhou

Authors
  1. Chunyu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiyuan Che
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Linna Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunyu Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, C., Che, Z. & Zhou, L. Integral Sliding Mode Control for Singularly Perturbed Systems with Mismatched Disturbances. Circuits Syst Signal Process 38, 1561–1582 (2019). https://doi.org/10.1007/s00034-018-0925-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-018-0925-2

Keywords

Search

Navigation