Skip to main content
SpringerLink
Log in

Finite-Time Stability and Stabilization of Fractional Order Positive Switched Systems

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

Abstract

This paper is concerned with finite-time stability analysis and control synthesis of fractional order positive switched systems. By using the linear copositive Lyapunov function integrated with average dwell time switching technique, the finite-time stability of fractional order positive switched systems is first addressed. Then, the finite-time boundedness of fractional order positive switched systems with exogenous input is discussed. Finally, the stabilization of the considered systems is proposed, where a control strategy based on linear programming is designed. Several implemental algorithms are provided to reduce the conservativeness of results. Two numerical examples are given to show the effectiveness of the findings of theory.

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

Similar content being viewed by others

References

  1. F. Amato, M. Ariola, C. Cosentino, Finite-time stability of linear time-varying systems: analysis and controller design. IEEE Trans. Automa. Control 55(4), 1003–1008 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, Finite-time stability of linear time-varying systems with jumps. Automatica 45, 1354–1358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Boroujeni, H. Momeni, Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems. Signal Process. 92, 2365–2370 (2012)

    Article  Google Scholar 

  5. R. Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Theol. 27(3), 201–210 (1983)

    MATH  Google Scholar 

  6. A. Benzaouia, A. Hmamed, F. Mesquine, M. Benhayoun, F. Tadeo, Stabilization of continuous-time fractional positive systems by using a Lyapunov function. IEEE Trans. Autom. Control 59(8), 2203–2208 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Chen, Ubiquitous fractional order controls? In Proceedings of the Second IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, (19–21 July 2006), pp. 481–492

  8. L. Chen, Y. Chai, R. Wu, J. Yang, Stability and stabilization of a class of nonlinear fractional-order systems with Caputo derivative. IEEE Trans. Circuits Syst. II Exp. Briefs 59(9), 602–606 (2012)

    Article  Google Scholar 

  9. L. Chen, Y. He, R. Wu, Y. Chai, L. Yin, Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations. Int. J. Control Autom. Syst. 12(3), 697–702 (2014)

    Article  Google Scholar 

  10. G. Chen, Y. Yang, Finite-time stability of switched positive linear systems. Int. J. Robust Nonlinear Control 24(1), 179–190 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Y. Chen, H. Zou, R. Lu, A. Xue, Finite-time stability and dynamic output feedback stabilization of stochastic systems. Circuits Syst. Signal Process. 33, 53–69 (2014)

    Article  MathSciNet  Google Scholar 

  12. H. Delavari, D. Baleanu, J. Sadati, Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67, 2433–2439 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. H. Dong, Z. Wang, S. Ding, H. Gao, Finite-horizon reliable control with randomly occurring uncertainties and nonlinearities subject to output quantization. Automatica 52, 355–362 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Dong, Z. Wang, S. Ding, H. Gao, Finite-horizon estimation of randomly occurring faults for a class of nonlinear time-varying systems. Automatica 50, 3182–3189 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Dong, Z. Wang, H. Gao, Distributed \(H_{\infty }\) filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks. IEEE Trans. Ind. Electron. 60(10), 4655–4672 (2013)

    Article  Google Scholar 

  16. P. Dorato, Short time stability in linear time-varying systems, in Proceedings of the IRE International Convention Record Part 4, New York, USA (1961), pp. 83–87

  17. H. Du, X. Lin, S. Li, Finite-time stability and stabilization of switched linear systems. Joint 48th IEEE conference on decision and control and 28th Chinese control conference Shanghai, China, (16–18 Dec 2009), pp. 1938–1943

  18. E. Fornasini, M. Valcher, Stability and stabilizability criteria for discrete-time positive switched systems. IEEE Trans. Autom. Control 57(5), 1208–1221 (2012)

    Article  MathSciNet  Google Scholar 

  19. G. Garcia, S. Tarbouriech, J. Bernussou, Finite-time stabilization of linear time-varying continuous systems. IEEE Trans. Autom. Control 54(2), 364–369 (2009)

    Article  MathSciNet  Google Scholar 

  20. R. Hilfer, Application of Fractional Calculus in Physics (World Science Publishing, Singapore, 2000)

    Book  MATH  Google Scholar 

  21. T. Kaczorek, Positive linear systems consisting of subsystems with different fractional orders. IEEE Trans. Circuits Syst. I Reg. Papers 58(6), 1203–1210 (2011)

    Article  MathSciNet  Google Scholar 

  22. T. Kaczorek, Selected Problems of Fractional Systems Theory (Springer, Berlin, 2012)

    MATH  Google Scholar 

  23. T. Kaczorek, Minimum energy control of fractional positive continuous-time linear systems. Bull. Pol. Acad.Sci. Tech. Sci. 61(4), 803–807 (2013)

    Google Scholar 

  24. T. Kaczorek, An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs. Bull. Pol. Acad. Sci. Tech. Sci. 62(2), 227–231 (2014)

    Google Scholar 

  25. M. Lazarević, D. Debeljković, Finite time stability analysis of linear autonomous fractional order systems with delayed state. Asian J. Control 7(4), 440–447 (2005)

    Article  MathSciNet  Google Scholar 

  26. Y. Li, Y. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. X. Li, H. Gao, A heuristic approach to static output-feedback controller synthesis with restricted frequency-domain specifications. IEEE Trans. Autom. Control 59(4), 2008–2014 (2014)

    Article  MathSciNet  Google Scholar 

  28. X. Li, H. Gao, Robust frequency-domain constrained feedback design via a two-stage heuristic approach. IEEE Trans. Cybern. 45(10), 2065–2075 (2015)

    Article  Google Scholar 

  29. H. Li, C. Wu, P. Sh, Y. Gao, Control of nonlinear networked systems with packet dropouts: interval type-2 fuzzy model-based approach. IEEE Trans. Cybern. 45(11), 2378–2389 (2015)

    Article  Google Scholar 

  30. H. Li, C. Wu, L. Wu, H.K. Lam, Y. Gao. Filtering of interval type-2 fuzzy systems with intermittent measurements. IEEE Trans. Cybern. doi:10.1109/TCYB.2015.2413134

  31. J. Liu, J. Lian, Y. Zhuang, Output feedback \(L_{1}\) finite-time control of switched positive delayed systems with MDADT. Nonlinear Anal. Hybrid Syst. 15, 11–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Lu, G. Chen, Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Trans. Autom. Control 54(6), 1294–1299 (2009)

    Article  MathSciNet  Google Scholar 

  33. O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Trans. Autom. Control 52, 1346–1349 (2007)

    Article  MathSciNet  Google Scholar 

  34. M. Ortigueira, J. Machado, Special issue on fractional signal processing and applications. Signal Process. 83(11), 2285–2480 (2003)

    Article  Google Scholar 

  35. I. Podlubny, Fractional Differential Equations (Academic Press, Waltham, 1999)

    MATH  Google Scholar 

  36. J. Qiu, G. Feng, H. Gao, Fuzzy-model-based piecewise \(H_{\infty }\) static-output-feedback controller design for networked nonlinear systems. IEEE Trans. Fuzzy Syst. 18(5), 919–933 (2010)

    Article  Google Scholar 

  37. J. Qiu, G. Feng, H. Gao, Observer-based piecewise affine output feedback controller synthesis of continuous-time T-S fuzzy affine dynamic systems using quantized measurements. IEEE Trans. Fuzzy Syst. 20(6), 1046–1062 (2012)

    Article  Google Scholar 

  38. J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust \(H_{\infty }\) filtering for discrete-time T-S fuzzy systems with time-varying delay. IEEE Trans. Fuzzy Syst. 17(5), 1044–1058 (2009)

    Article  Google Scholar 

  39. J. Qiu, Y. Wei, H. Karimi, New approach to delay-dependent \(H_{\infty }\) control for continuous-time Markovian jump systems with time-varying delay and deficient transition descriptions. J. Frankl. Inst. 352, 189–215 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Saleh, Tavazoei, M. Haeri, A note on the stability of fractional order systems. Math. Comput. Simul. 79, 1556–1576 (2009)

    MathSciNet  MATH  Google Scholar 

  41. J. Sabatier, M. Moze, C. Farges, LMI stability conditions for fractional order systems. Comput. Math. Appl. 59, 1594–1609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Shen, J. Lam, State-feedback \(H_{\infty }\) control of commensurate fractional-order systems. Int. J. Syst. Sci. 45(3), 363–372 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Trans. Netw. 14(3), 616–629 (2006)

    Article  MathSciNet  Google Scholar 

  44. Y. Tong, L. Zhang, M. Basin, C. Wang, Weighted \(H_{\infty }\) control with D-stability constraint for switched positive linear systems. Int. J. Robust Nonlinear Control 24(4), 758–774 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Wang, Y. Yu, G. Wen, Hybrid projective synchronization of time-delayed fractional order chaotic systems. Nonlinear Anal. Hybrid Syst. 11, 129–138 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. L. Weiss, E. Infante, Finite time stability under perturbing forces and on product spaces. IEEE Trans. Automat. Control 12(1), 54–59 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  47. W. Xiang, J. Xiao, \(H_{\infty }\) finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance. J. Frankl. Inst. 348(2), 331–352 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Y. Yang, G. Chen, Finite-time stability of fractional order impulsive switched systems. Int. J. Robust Nonlinear Control 25(13), 2207–2222 (2015)

    MathSciNet  MATH  Google Scholar 

  49. Y. Yang, J. Li, G. Chen, Finite-time stability and stabilization of nonlinear stochastic hybrid systems. J. Math. Anal. Appl. 356, 338–345 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Zhang, Z. Han, F. Zhu, Finite-time control and \(L_{1}\)-gain analysis for positive switched systems. Optim. Control Appl. Meth. 36(4), 550–565 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Zhang, Z. Han, F. Zhu, J. Huang, Feedback control for switched positive linear systems. IET Control Theory Appl. 7(3), 464–469 (2013)

    Article  MathSciNet  Google Scholar 

  52. J. Zhang, Z. Han, F. Zhu, X. Zhao, Absolute exponential stability and stabilization of switched nonlinear systems. Syst. Control Lett. 66, 51–57 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  54. 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), 1809–1815 (2012)

    Article  MathSciNet  Google Scholar 

  55. X. Zhao, S. Yin, H. Li, B. Niu, Switching stabilization for a class of slowly switched systems. IEEE Trans. Autom. Control 60(1), 221–226 (2015)

    Article  MathSciNet  Google Scholar 

  56. X. Zhao, T. Wu, X. Zheng, R. Li, Discussions on observer design of nonlinear positive systems via T-S fuzzy modeling. Neurocomput 157, 70–75 (2015)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which have improved the quality of this paper. This work was supported by the National Natural Science Foundation of China under Grants Nos. 61503107, 61104120, 61473107, and 61203123 and also by the Zhejiang Provincial Natural Science Foundation of China under Grant LY16F030005.

Author information

Authors and Affiliations

  1. Institute of Information and Control, Hangzhou Dianzi University, Hangzhou, 310018, China

    Junfeng Zhang & Yun Chen

  2. College of Engineering, Bohai University, Jinzhou, 121013, China

    Xudong Zhao

  3. Chongqing SANY Highintelligent Robots Co. Ltd., Chongqing, 401120, People’s Republic of China

    Xudong Zhao

Authors
  1. Junfeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xudong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junfeng Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Zhao, X. & Chen, Y. Finite-Time Stability and Stabilization of Fractional Order Positive Switched Systems. Circuits Syst Signal Process 35, 2450–2470 (2016). https://doi.org/10.1007/s00034-015-0236-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-015-0236-9

Keywords

Search

Navigation