Skip to main content
SpringerLink
Log in

Stability analysis and \(L_{2}\)-gain control for positive fuzzy systems by applying a membership-function-dependent Lyapunov function

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Stability and \(L_2\)-gain control for positive Takagi–Sugeno (T–S) fuzzy systems are further studied in this brief paper. First, considering that the system states are positive, some sufficient conditions of exponential asymptotic stability are obtained by applying a copositive Lyapunov function with membership-function-dependent (MFD) Lyapunov matrices. Based on a preset switching rule, the conditions are expressed as linear matrix inequalities by eliminating the nonconvex factors due to the time derivative of MFD Lyapunov matrices. Then, stability is extended to stabilization by designing a switching controller with time-varying controller gains such that the \(L_{2}\)-gain performance requirements are satisfied. In addition, a quadratic switching strategy is established to further reduce conservativeness. Finally, the applicability and validity of the theoretical results are validated 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

Similar content being viewed by others

Data availability

The simulation data of the current study are available from the corresponding author on reasonable request.

References

  1. Ahmadi, E., Zarei, J., Razavi-Far, R., Saif, M.: A dual approach for positive T–S fuzzy controller design and its application to cancer treatment under immunotherapy and chemotherapy. Biomed. Signal Process. Control 58, 101822 (2020)

    Article  Google Scholar 

  2. Zhang, Q., Liu, C., Zhang, X.: Complexity Analysis and Control of singular Biological Systems. Springer, London (2012)

    Book  Google Scholar 

  3. Benzaouia, A., Mesquine, F., Benhayoun, M., Schulte, H., Georg, S.: Stabilization of positive constrained T–S fuzzy systems: application to a Buck converter. J. Frankl. Inst. 351(8), 4111–4123 (2014)

    Article  MathSciNet  Google Scholar 

  4. Zhao, X., Liu, X., Yin, S., Li, H.: Improved results on stability of continuous-time switched positive linear systems. Automatica 50(2), 614–621 (2014)

    Article  MathSciNet  Google Scholar 

  5. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man. Cybern. SMC–15(1), 116–132 (1985)

    Article  Google Scholar 

  6. Ahmadi, E., Zarei, J., Razavi-Far, R.: Robust \(\ell _{1}\)-controller design for discrete-time positive T–S fuzzy systems using dual approach. IEEE Trans. Syst. Man Cybern. Syst. 52(2), 706–715 (2022)

    Article  Google Scholar 

  7. Zhu, B., Suo, M., Chen, L., Li, S.: Stability and \(L_{1}\)-gain analysis for positive Takagi–Sugeno fuzzy systems with impulse. IEEE Trans. Fuzzy Syst. 26(6), 3893–3901 (2018)

    Article  Google Scholar 

  8. Zheng, X., Wang, X., Yin, Y., Hu, L.: Stability analysis and constrained fuzzy tracking control of positive nonlinear systems. Nonlinear Dyn. 83(4), 2509–2522 (2015)

    Article  MathSciNet  Google Scholar 

  9. Pan, J.-T., Guerra, T.M., Fei, S.-M., Jaadari, A.: Nonquadratic stabilization of continuous T–S fuzzy models: LMI solution for a local approach. IEEE Trans. Fuzzy Syst. 20(3), 594–602 (2012)

    Article  Google Scholar 

  10. Zhang, H., Xie, X.: Relaxed stability conditions for continuous-time T–S fuzzy-control systems via augmented multi-indexed matrix approach. IEEE Trans. Fuzzy Syst. 19(3), 478–492 (2011)

    Article  Google Scholar 

  11. Rhee, B.-J., Won, S.: A new fuzzy Lyapunov function approach for a Takagi–Sugeno fuzzy control system design. Fuzzy Sets Syst. 157(9), 1211–1228 (2006)

    Article  MathSciNet  Google Scholar 

  12. Chen, X., Lam, J., Lam, H.-K.: Positive filtering for positive Takagi–Sugeno fuzzy systems under \(\ell _{1}\) performance. Inf. Sci. 299, 32–41 (2015)

    Article  MathSciNet  Google Scholar 

  13. Wang, L., Lam, H.-K.: A new approach to stability and stabilization analysis for continuous-time Takagi–Sugeno fuzzy systems with time delay. IEEE Trans. Fuzzy Syst. 26(4), 2460–2465 (2018)

    Article  Google Scholar 

  14. Palanimuthu, K., Kim, H.S., Joo, Y.H.: T–S fuzzy sliding mode control for double-fed induction generator-based wind energy system with a membership function-dependent \(H_{\infty }\)-approach. Inf. Sci. 596, 73–92 (2022)

    Article  Google Scholar 

  15. Du, S., Qiao, J., Zhao, X., Wang, R.: Stability and \(L_{1}\)-gain analysis for switched positive T–S fuzzy systems under asynchronous switching. J. Frankl. Inst. 355(13), 5912–5927 (2018)

    Article  Google Scholar 

  16. Li, S., Xiang, Z.: Exponential stability analysis and \(L_{2}\)-gain control synthesis for positive switched T–S fuzzy systems. Nonlinear Anal. Hybrid Syst. 27, 77–91 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 62173123, in part by the Natural Science Foundation of Hebei Province under Grant F2021202045.

Author information

Authors and Affiliations

  1. School of Artificial Intelligence, Hebei University of Technology, Tianjin, 300401, China

    Bo Zheng & Likui Wang

  2. Institute of Advanced Technology Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

    Xiangpeng Xie

  3. Department of Engineering, King’s College London, Strand, London, WC2R 2LS, UK

    Hak-Keung Lam

Authors
  1. Bo Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Likui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiangpeng Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hak-Keung Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. The main idea and method were proposed by [LW]. The first draft of the manuscript was written by [BZ]. [XX] and [H-KL] commented on previous versions of the manuscript. All authors read and approved the final manuscript

Corresponding author

Correspondence to Likui Wang.

Ethics declarations

Conflict of interest

The authors have no relevant financial interests to disclose.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, B., Wang, L., Xie, X. et al. Stability analysis and \(L_{2}\)-gain control for positive fuzzy systems by applying a membership-function-dependent Lyapunov function. Nonlinear Dyn 111, 22255–22265 (2023). https://doi.org/10.1007/s11071-023-09002-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09002-8

Keywords

Search

Navigation