Skip to main content
SpringerLink
Log in

L 2-gain analysis and anti-windup design of discrete-time switched systems with actuator saturation

  • Regular Papers
  • Published:
International Journal of Automation and Computing Aims and scope Submit manuscript

Abstract

This paper investigates L 2-gain analysis and anti-windup compensation gains design for a class of discrete-time switched systems with saturating actuators and L 2 bounded disturbances by using the switched Lyapunov function approach. For a given set of anti-windup compensation gains, we firstly give a sufficient condition on tolerable disturbances under which the state trajectory starting from the origin will remain inside a bounded set for the corresponding closed-loop switched system subject to L 2 bounded disturbances. Then, the upper bound on the restricted L 2-gain is obtained over the set of tolerable disturbances. Furthermore, the antiwindup compensation gains aiming to determine the largest disturbance tolerance level and the smallest upper bound of the restricted L 2-gain are presented by solving a convex optimization problem with linear matrix inequality (LMI) constraints. A numerical example is given to illustrate the effectiveness of the proposed design 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

Similar content being viewed by others

References

  1. D. Liberzon, A. S. Morse. Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999.

    Article  Google Scholar 

  2. H. Lin, P. J. Antsaklis. Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 308–322, 2009.

    Article  MathSciNet  Google Scholar 

  3. Z. D. Sun, S. S. Ge, T. H. Lee. Controllability and reachability criteria for switched linear systems. Automatica, vol. 38, no. 5, pp. 775–786, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. P. Varaiya. Smart cars on smart roads: Problems of control. IEEE Transactions on Automatic Control, vol. 38, no. 2, pp. 195–207, 1993.

    Article  MathSciNet  Google Scholar 

  5. J. Zhao, M. W. Spong. Hybrid control for global stabilization of the cart-pendulum system. Automatica, vol. 37, no. 12, pp. 1941–1951, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  6. X. M. Sun, G. P. Liu, D. Rees, W. Wang. Stability of systems with controller failure and time-varying delay. IEEE Transactions on Automatic Control, vol. 53, no. 10, pp. 2391–2396, 2008.

    Article  MathSciNet  Google Scholar 

  7. J. Zhao, G. M. Dimirovski. Quadratic stability of a class of switched nonlinear systems. IEEE Transactions on Automatic Control, vol. 49, no. 4, pp. 574–578, 2004.

    Article  MathSciNet  Google Scholar 

  8. D. Cheng. Stabilization of planar switched systems. Systems & Control Letters, vol. 51, no. 2, pp. 79–88, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  9. Y. H. Zhu, D. Z. Cheng. Stability and stabilization of blockcascading switched linear systems. International Journal of Automation and Computing, vol. 3, no. 4, pp. 404–413, 2006.

    Article  MathSciNet  Google Scholar 

  10. M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, vol. 43, no. 4, pp. 475–482, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. P. Hespanha, A. S. Morse. Stability of switched systems with average dwell-time. In Proceedings of the 38th IEEE Conference on Decision and Control, IEEE, Phoenix, USA, vol. 3, pp. 2655–2660, 1999.

    Google Scholar 

  12. Z. D. Sun, S. S. Ge. Analysis and synthesis of switched linear control systems. Automatica, vol. 41, no. 2, pp. 181–195, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Zhai, B. Hu, K. Yasuda, A. N. Michel. Disturbance attenuation properties of time-controlled switched systems. Journal of the Franklin Institute, vol. 338, no. 7, pp. 765–779, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Zhao, D. J. Hill. On stability, and L 2-gain and H ∞ control for switched systems. Automatica, vol. 44, no. 5, pp. 1220–1232, 2008.

    Article  MathSciNet  Google Scholar 

  15. D. G. Roberson, D. J. Stilwell. L 2 gain performance analysis of linear switched systems: Fast switching behavior. In Proceedings of the American Control Conference, IEEE, New York, USA, pp. 2084–2089, 2007.

    Chapter  Google Scholar 

  16. D. Xie, L. Wang, F. Hao, G. Xie. LMI approach to L 2-gain analysis and control synthesis of uncertain switched systems. IEE Proceedings — Control Theory and Applications, vol. 151, no. 1, pp. 21–28, 2004.

    Article  Google Scholar 

  17. H. Lin, P. J. Antsaklis. Switching stabilization and l 2 gain performance controller synthesis for discrete-time switched linear systems. In Proceedings of the 45th IEEE Conference on Decision and Control, IEEE, San Diego, USA, pp. 2673–2678, 2006.

    Chapter  Google Scholar 

  18. J. Daafouz, P. Riedinger, C. Iung. Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1883–1887, 2002.

    Article  MathSciNet  Google Scholar 

  19. J. M. Gomes da Silva, I. Ghiggi, S. Tarbouriech. Nonrational dynamic output feedback for time-delay systems with saturating inputs. International Journal of Control, vol. 81, no. 4, pp. 557–570, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  20. Q. Zheng, F. Wu. Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions. Systems & Control Letters, vol. 57, no. 11, pp. 896–903, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Fang, Z. Lin, T. Hu. Analysis of linear systems in the presence of actuator saturation and L 2-disturbances. Automatica, vol. 40, no. 7, pp. 1229–1238, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. M. Gomes da Silva, D. Limon, T. Alamo, E. F. Camacho. Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints. IEEE Transactions on Automatic Control, vol. 53, no. 10, pp. 2367–2372, 2008.

    Article  MathSciNet  Google Scholar 

  23. F. E. Haoussi, E. H. Tissir. Delay and its time-derivative dependent robust stability of uncertain neutral systems with saturating actuators. International Journal of Automation and Computing, vol. 7, no. 4, pp. 455–462, 2010.

    Article  Google Scholar 

  24. S. Tarbouriech, P. L. D. Peres, G. Garcia, I. Queinnec. Delay-dependent stabilisation and disturbance tolerance for time-delay systems subject to actuator saturation. IEE Proceedings — Control Theory and Applications, vol. 149, no. 5, pp. 387–393, 2002.

    Article  Google Scholar 

  25. L. X. Zhang, E. Boukas, A. Haidar. Delay-range-dependent control synthesis for time-delay systems with actuator saturation. Automatica, vol. 44, no. 10, pp. 2691–2695, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Hu, Z. Lin, B. M. Chen. Analysis and design for discretetime linear systems subject to actuator saturation. Systems & Control Letters, vol. 45, no. 2, pp. 97–112, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. M. Gomes da Silva, S. Tarbouriech. Anti-windup design with guaranteed regions of stability for discrete-time linear systems. Systems & Control Letters, vol. 55, no. 3, pp. 184–192, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  28. X. Zhang, J. Zhao, G. M. Dimirovski. L 2-gain analysis and control synthesis of uncertain switched linear systems subject to actuator saturation. International Journal of Systems Science, vol. 43, no. 4, pp. 731–740, 2012.

    Article  MathSciNet  Google Scholar 

  29. L. Lu, Z. Lin. Design of switched linear systems in the presence of actuator saturation. IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1536–1542, 2008.

    Article  MathSciNet  Google Scholar 

  30. L. Lu, Z. Lin. A switching anti-windup design using multiple Lyapunov functions. IEEE Transactions on Automatic Control, vol. 55, no. 1, pp. 142–148, 2010.

    Article  MathSciNet  Google Scholar 

  31. Y. M. Ma, G. H. Yang. Disturbance rejection of switched discrete-time systems with saturation nonlinearity. In Proceedings of the 46th IEEE Conference on Decision and Control, IEEE, New Orleans, USA, pp. 3170–3175, 2007.

    Google Scholar 

  32. L. Lu, Z. Lin, H. Fang. L 2 gain analysis for a class of switched systems. Automatica, vol. 45, no. 4, pp. 965–972, 2009.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information and Control Engineering, Liaoning Shihua University, Fushun, 113001, China

    Xin-Quan Zhang

  2. State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, China

    Jun Zhao

Authors
  1. Xin-Quan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin-Quan Zhang.

Additional information

This work was supported by National Natural Science Foundation of China (Nos. 61174073 and 90816028).

Xin-Quan Zhang received the B. Sc. degree in automation and M. Sc. degree in control theory and engineering from Liaoning Technical University, China in 2003 and 2007, respectively. Also he received the Ph.D. degree in control theory and control engineering from the Northeastern University, Shenyang, China, in 2012. Currently, he joined the School of Information and Control Engineering at Liaoning Shihua University in China.

His research interests include switched systems, robust control and systems control under constraints.

Jun Zhao received the B. Sc. and M. Sc. degrees in mathematics from Liaoning University, China in 1982 and 1984, respectively. He completed his Ph. D. in control theory and applications in 1991 at Northeastern University, China. From 1992 to 1993, he was a postdoctoral fellow at the same university. Since 1994, as a professor, he has been with College of Information Science and Engineering, Northeastern University, China. From 1998 to 1999, he was a senior visiting scholar at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, USA. From November 2003 to May 2005, he was a research fellow at Department of Electronic Engineering, City University of Hong Kong. From 2007 to 2010, he was a fellow at School of Engineering, Australian National University.

His research interests include switched systems, nonlinear systems and network synchronization.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, XQ., Zhao, J. L 2-gain analysis and anti-windup design of discrete-time switched systems with actuator saturation. Int. J. Autom. Comput. 9, 369–377 (2012). https://doi.org/10.1007/s11633-012-0657-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11633-012-0657-x

Keywords

Search

Navigation