Skip to main content
SpringerLink
Log in

Finite-Time Interval Observers’ Design for Switched Systems

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

Abstract

In this study, a finite-time interval observers’ design method is developed for switched systems suffering from disturbance. First, the interval observer frames for the systems are constructed. Then, sufficient conditions are derived to guarantee that the upper and lower error systems are both positive and finite-time bound. Unlike the current studies, all the conditions proposed in this paper are formulated in the form of linear programming. Finally, two numerical examples are provided to show the efficiency of designed observers.

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.

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

Abbreviations

\(R^n\) :

n-dimensional Euclidean space

\(R^{n\times m}\) :

The set of \(n{\times }m\) real matrices

\(x>(\ge )0\) :

Its components are positive (nonnegative), i.e., \(x_{i}>(\ge )0\)

\(A>(\ge )0\) :

Its components are positive (nonnegative), i.e., \(A_{ij}>(\ge )0\)

\(E^+\) :

\(\max \{E,0\}\)

\(E^-\) :

\(E^+-E\)

\(||x||_1\) :

The 1-norm of the vector x

\(\overline{\lambda }(v)\) :

The maximum value of the elements of the vector v

\(\underline{\lambda }(v)\) :

The minimum value of the elements of the vector v

\(\mathbf 1 _n\) :

The vector whose entries equal to 1

References

  1. A. Agresti, B. Coull, Approximate is better than exact for interval estimation of binomial proportions. Am. Stat. 52(2), 119–126 (1998)

    MathSciNet  Google Scholar 

  2. F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)

    Article  Google Scholar 

  3. M. Arcak, P. Kokotovic, Nonlinear observers: a circle criterion design and robustness analysis. Automatica 37(12), 1923–1930 (2001)

    Article  MathSciNet  Google Scholar 

  4. C. Briat, M. Khammash, Simple interval observers for linear impulsive systems with applications to sampled-data and switched systems, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)

    Article  Google Scholar 

  5. S. Chebotarev, D. Efimov, A. Zolghadri, Interval observers for continuous-time LPV systems with \(L_1\)/\(L_2\) performance. Automatica 58, 82–89 (2015)

    Article  MathSciNet  Google Scholar 

  6. J. Cheng, H. Zhu, S. Zhong, F. Zheng, Y. Zeng, Finite-time filtering for switched linear systems with a mode-dependent average dwell time. Nonlinear Anal. Hybrid Syst. 15(2), 145–156 (2015)

    Article  MathSciNet  Google Scholar 

  7. K. Degue, D. Efimov, J. Ny, Interval observer approach to output stabilization of linear impulsive systems, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)

  8. H. Du, X. Lin, S. Li, Finite-time stability and stabilization of switched linear systems, in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta (2010)

  9. D. Efimov, T. Raissi, S. Chebotarev, A. Zolghadri, Interval state observer for nonlinear time varying systems. Automatica 49(1), 200–205 (2013)

    Article  MathSciNet  Google Scholar 

  10. H. Ethabet, T. Raissi, M. Amairi, M. Aoun, Interval observers design for continuous-time linear switched systems, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)

    Article  Google Scholar 

  11. L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)

    Book  Google Scholar 

  12. J. Gouze, A. Rapaport, Z. Hadj-Sadok, Interval observers for uncertain biological systems. Ecol. Model. 133(1), 45–56 (2000)

    Article  Google Scholar 

  13. S. Guo, F. Zhu, Interval observer design for discrete-time switched system, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)

  14. Z. He, W. Xie, Control of non-linear switched systems with average dwell time: interval observer-based framework. IET Contr. Theory Appl. 10(1), 10–16 (2016)

    Article  MathSciNet  Google Scholar 

  15. R. Horn, C. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)

    Book  Google Scholar 

  16. B. Hu, G. Zhai, A. Michel, Common quadratic Lyapunov-like functions with associated switching regions for two unstable second-order LTI systems. Int. J. Control 75(14), 1127–1135 (2002)

    Article  MathSciNet  Google Scholar 

  17. S. Ibrir, Circle-criterion approach to discrete-time nonlinear observer design. Automatica 43(8), 1432–1441 (2007)

    Article  MathSciNet  Google Scholar 

  18. D. Liberzon, Switching in Systems and Control (Springer, Berlin, 2012)

    MATH  Google Scholar 

  19. X. Luan, F. Liu, P. Shi, Finite-time filtering for non-linear stochastic systems with partially known transition jump rates. IET Contr. Theory Appl. 4(5), 735–745 (2010)

    Article  MathSciNet  Google Scholar 

  20. M. Moisan, O. Bernard, Robust interval observers for global Lipschitz uncertain chaotic systems. Syst. Control Lett. 59(11), 687–694 (2010)

    Article  MathSciNet  Google Scholar 

  21. T. Raissi, D. Efimov, A. Zolghadri, Interval state estimation for a class of nonlinear systems. IEEE Trans. Autom. Control 57(1), 260–265 (2011)

    Article  MathSciNet  Google Scholar 

  22. M. Rami, C. Cheng, C. Prada, Tight robust interval observers: an LP approach, in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun (2008)

  23. Y. Su, J. Huang, Stability of a class of linear switching systems with applications to two consensus problems. IEEE Trans. Autom. Control 57(6), 1420–1430 (2012)

    Article  MathSciNet  Google Scholar 

  24. M. Wang, J. Feng, G. Dimirovski, J. Zhao, Stabilization of switched nonlinear systems using multiple Lyapunov function method, in Proceedings of the 2009 American Control Conference, St. Louis (2009)

  25. Y. Wang, D. Bevly, R. Rajamani, Interval observer design for LPV systems with parametric uncertainty. Automatica 60(10), 79–85 (2015)

    Article  MathSciNet  Google Scholar 

  26. J. Zhang, Z. Han, F. Zhu, Finite-time control and \(L_1\)-gain analysis for positive switched systems. Optim. Control Appl. Methods 36(4), 550–565 (2015)

    Article  Google Scholar 

  27. Y. Zhang, C. Liu, X. Mu, Robust finite-time stabilization of uncertain singular Markovian jump systems. Appl. Math. Model. 36(10), 5109–5121 (2012)

    Article  MathSciNet  Google Scholar 

  28. Y. Zhang, P. Shi, S. Nguang, H. Karimi, Observer-based finite-time fuzzy \(H_{\infty }\) control for discrete-time systems with stochastic jumps and time-delays. Signal Process. 97, 252–261 (2014)

    Article  Google Scholar 

  29. Y. Zhang, Y. Shi, P. Shi, Robust and non-fragile finite-time \(H_{\infty }\) control for uncertain Markovian jump nonlinear systems. Appl. Math. Comput. 279, 125–138 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Y. Zhang, Y. Shi, P. Shi, Resilient and robust finite-time \(H_{\infty }\) control for uncertain discrete-time jump nonlinear systems. Appl. Math. Model. 49, 612–629 (2017)

    Article  MathSciNet  Google Scholar 

  31. X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1915 (2012)

    Article  MathSciNet  Google Scholar 

  32. X. Zhao, L. Zhang, P. Shi, M. Liu, Stability of a class of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132–1137 (2012)

    Article  MathSciNet  Google Scholar 

  33. G. Zheng, D. Efimov, F. Bejarano, W. Perruquetti, H. Wang, Interval observer for a class of uncertain nonlinear singular systems. Automatica 71(9), 159–168 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (61403267), Natural Science Foundation of Jiangsu Province (BK20130322), and China Postdoctoral Science Foundation (2017M611903).

Author information

Authors and Affiliations

  1. School of Mechanical and Electrical Engineering, Soochow University, Suzhou, 215131, China

    Xiang Ma, Jun Huang & Liang Chen

Authors
  1. Xiang Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Huang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, X., Huang, J. & Chen, L. Finite-Time Interval Observers’ Design for Switched Systems. Circuits Syst Signal Process 38, 5304–5322 (2019). https://doi.org/10.1007/s00034-019-01122-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-019-01122-0

Keywords

Search

Navigation