Skip to main content
SpringerLink
Log in

Local stabilization for unstable bilinear systems with input saturation

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

Abstract

This paper is concerned with local stabilization for unstable bilinear systems with input saturation. Given a prespecified polytope \(\mathcal{P}\) of the state space containing the zero equilibrium point, a linear state feedback is designed to guarantee that the closed-loop system is asymptotically stable in \(\mathcal{P}\) and \(\mathcal{P}\) is enclosed in the domain of attraction of the zero equilibrium point. Such sufficient conditions are derived via linear matrix inequalities (LMIs). Finally, an example illustrates the effectiveness of the proposed method.

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. Juang, J.N.: Continuous-time bilinear system identification. Nonlinear Dyn. 39, 79–94 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Souley Ali, H., Zasadzinski, M., Rafaralahy, H., Darouach, M.: Robust H ∞ reduced order filtering for uncertain bilinear systems. Automatica 42, 405–415 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akpan, E.: Robust control for a class of bilinear systems with application to induction motor. Electr. Power Compon. Syst. 31, 261–273 (2003)

    Article  Google Scholar 

  4. Pardalos, P.M., Yatsenko, V.: Optimization and Control of Bilinear Systems. Springer, Berlin (2009)

    Google Scholar 

  5. Mohler, R.R.: Bilinear Control Processes. Academic Press, New York (1973)

    MATH  Google Scholar 

  6. Liu, C.H., Hsiao, M.Y., Tsai, S.H.: Stabilization for a class of time-delay discrete bilinear system. Inf. Technol. J. 10, 457–460 (2011)

    Article  Google Scholar 

  7. Li, T.H.S., Tsai, S.H.: T-S fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems. IEEE Trans. Fuzzy Syst. 15, 494–506 (2007)

    Article  Google Scholar 

  8. Tao, C.W., Wang, W.Y., Chan, M.L.: Design of sliding mode controllers for bilinear systems with time varying uncertainties. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 34, 639–645 (2004)

    Article  Google Scholar 

  9. Chen, M.S., Tsao, S.T.: Exponential stabilization of a class of unstable bilinear systems. IEEE Trans. Autom. Control 45, 989–992 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chiou, J.S., Kung, F.C., Li, T.H.S.: Robust stabilization of a class of singular perturbed discrete bilinear systems. IEEE Trans. Autom. Control 45, 1187–1191 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ho, D.W.C., Lu, G., Zheng, Y.: Global stabilization for bilinear systems with time delay. IEE Proc., Control Theory Appl. 149, 89–94 (2002)

    Article  Google Scholar 

  12. Lu, G., Ho, D.W.C.: Global stabilization controller design for discrete-time bilinear systems with time-delays. In: Proceedings of the 4th World Congress on Intelligent Control and Automation, pp. 10–14 (2002)

    Google Scholar 

  13. Jerbi, H.: Global feedback stabilization of new class of bilinear systems. Syst. Control Lett. 42, 313–320 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lu, G., Ho, D.W.C.: Continuous stabilization controllers for singular bilinear systems: The state feedback case. Automatica 42, 309–314 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mahout, V., Tarbouriech, S., Garcia, G.: Controller design for unstable uncertain bilinear systems. In: Proceedings of the 10th Mediterranean Conference on Control and Automation—MED2002 (2002)

    Google Scholar 

  16. Amato, F., Cosentino, C., Fiorillo, A.S., Merola, A.: Stabilization of bilinear systems via linear state-feedback control. IEEE Trans. Circuits Syst. II, Express Briefs 56, 76–80 (2009)

    Article  Google Scholar 

  17. Hu, T., Lin, Z., Chen, B.: An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38, 351–359 (2002)

    Article  MATH  Google Scholar 

  18. Cai, X., Huang, J., Liu, L.: Stability analysis of linear time-delay differential inclusion systems subject to input saturation. IET Control Theory Appl. 4, 2592–2602 (2010)

    Article  MathSciNet  Google Scholar 

  19. Zhu, Z., Xia, Y., Fu, M.: Adaptive Sliding Mode Control for Attitude Stabilization with Actuator Saturation. IEEE Trans. Ind. Electron. 58, 4898–4907 (2011)

    Article  Google Scholar 

  20. Niculescu, S.I., Tarbouriech, S., Dion, J.M., Dugard, L.: Stability criteria for bilinear systems with delayed state and saturating actuators. In: Proceedings of the 34th Conference on Decision and Control, New Orleans, pp. 2064–2069 (1995)

    Google Scholar 

  21. Lee, C.H.: On the stability of uncertain homogeneous bilinear systems subjected to time-delay and constrained inputs. J. Chin. Inst. Chem. Eng. 31, 529–534 (2008)

    Google Scholar 

  22. Kang, Y., Zhao, P., Ling, Q.: Stability of markovian jump bilinear systems with delayed saturating actuators. In: Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pp. 2929–2934 (2009)

    Google Scholar 

  23. Kang, Y., Shang, W., Xi, H.: Estimating the Delay-time for the Stability of Markovian Jump Bilinear Systems with Saturating Actuators. Acta Autom. Sin. 36, 762–767 (2010)

    Article  MathSciNet  Google Scholar 

  24. Lee, C.H., Chen, C.Y.: On the robust stability of homogeneous perturbed large-scale bilinear systems with time delays and constrained inputs. World Acad. Sci., Eng. Technol. 65, 131–136 (2010)

    Google Scholar 

  25. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

Download references

Acknowledgement

The authors are grateful for the support of the National Natural Science Foundation of China under Grants 61074011 and 60904023.

Author information

Authors and Affiliations

  1. College of Electric and Information Engineering, Henan University of Science and Technology, Luoyang, 471003, China

    Leipo Liu, Xiaona Song & Zhumu Fu

  2. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, China

    Xiushan Cai

Authors
  1. Leipo Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaona Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhumu Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiushan Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leipo Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, L., Song, X., Fu, Z. et al. Local stabilization for unstable bilinear systems with input saturation. Nonlinear Dyn 70, 249–254 (2012). https://doi.org/10.1007/s11071-012-0448-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-012-0448-5

Keywords

Search

Navigation