Skip to main content
SpringerLink
Log in

\(L_2\)-Gain Analysis and Design of Uncertain Switched Saturated Systems

  • Published:
Journal of Control, Automation and Electrical Systems Aims and scope Submit manuscript

Abstract

This paper studies the problem on \(L_2\)-gain analysis and anti-windup design of uncertain discrete-time switched saturated systems by the multiple Lyapunov functions approach. Firstly, we obtain a sufficient condition of tolerable disturbances, under which the state trajectory starting from the origin will remain inside a bounded set. Then, the upper bound of the restricted \(L_2\)-gain is obtained. Furthermore, the anti-windup compensators and the switched rule, aiming to determine the maximum disturbance tolerance capability and the minimum upper bound of the restricted \(L_2\)-gain, are presented by solving a constraints optimization problem. Finally, we give a numerical example to show 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

  • Benzaouiaa, A., Akhrifb, O., & Saydyc, L. (2010). Stabilisation and control synthesis of switching systems subject to actuator saturation. International Journal of Systems Science, 41(4), 397–409.

    Article  MathSciNet  Google Scholar 

  • Branicky, M. S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43(4), 475–482.

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng, D. (2004). Stabilization of planar switched systems. Systems & Control Letters, 51(2), 79–88.

    Article  MathSciNet  MATH  Google Scholar 

  • Cheng, D. Z., Guo, L., & Huang, J. (2003). On quadratic Lyapunov functions. IEEE Transactions on Automatic Control, 48(5), 885–890.

    Article  MathSciNet  MATH  Google Scholar 

  • da Silva, Gomes, Ghiggi I, J. M., & Tarbouriech, S. (2008). Non-rational dynamic output feedback for time-delay systems with saturating inputs. International Journal of Control, 81(4), 557–570.

    Article  MathSciNet  MATH  Google Scholar 

  • Fang, H., Lin, Z., & Hu, T. (2004). Analysis of linear systems in the presence of actuator saturation and \(L_2\)-disturbances. Automatica, 40(7), 1229–1238.

    Article  MathSciNet  MATH  Google Scholar 

  • Gomes da Silva, J. M., Limon, D., Alamo, T., & Camacho, E. F. (2008). Dynamic output feedback for discrete-time systems under amplitude and rate actuator constraints. IEEE Transactions on Automatic Control, 53(10), 2367–2372.

    Article  MathSciNet  MATH  Google Scholar 

  • Gomes da Silva, J. M., & Tarbouriech, S. (2006). Anti-windup design with guaranteed regions of stability for discrete-time linear systems. Systems & Control Letters, 55(3), 184–192.

    Article  MathSciNet  MATH  Google Scholar 

  • Hespanha, J. P., & Morse, A. S. (1999). Stability of switched systems with average dwell-time. In Proceedings of the 38th IEEE conference on decision and control (pp. 2655–2660). Phoenix, AZ, USA: IEEE.

  • Hu, T., Lin, Z., & Chen, B. M. (2002). Analysis and design for discrete-time linear systems subject to actuator saturation. Systems & Control Letters, 45(2), 97–112.

    Article  MathSciNet  MATH  Google Scholar 

  • Liberzon, D., & Morse, A. S. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19(5), 59–70.

    Article  MATH  Google Scholar 

  • Lin, H., & Antsaklis, P. J. (2006). 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 (pp. 2673–2678). San Diego, CA, USA: IEEE.

  • Lin, H., & Antsaklis, P. J. (2009). Stability and stabilizability of switched linear systems: A survey of recent results. IEEE Transactions on Automatic Control, 54(2), 308–322.

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, L., & Lin, Z. (2008). Design of switched linear systems in the presence of actuator saturation. IEEE Transactions on Automatic Control, 53(6), 1536–1542.

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, L., & Lin, Z. (2010). A switching anti-windup design using multiple Lyapunov functions. IEEE Transactions on Automatic Control, 55(1), 142–148.

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, L., Lin, Z., & Fang, H. (2009). \(L_2\) gain analysis for a class of switched systems. Automatica, 45(4), 965–972.

    Article  MathSciNet  MATH  Google Scholar 

  • Pettersson, S. (2003). Synthesis of switched linear systems. In Proceedings of the 42nd IEEE conference on decision and control (pp. 5283–5288). Maui, HI, USA: IEEE.

  • Sun, X. M., Liu, G. P., Rees, D., & Wang, W. (2008). Stability of systems with controller failure and time-varying delay. IEEE Transactions on Automatic Control, 53(10), 2391–2396.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, Z. D., Ge, S. S., & Lee, T. H. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38(5), 775–786.

    Article  MathSciNet  MATH  Google Scholar 

  • Tarbouriech, S., Peres, P. L. D., Garcia, G., & Queinnec, I. (2002). Delay-dependent stabilisation and disturbance tolerance for time-delay systems subject to actuator saturation. IEE Proceedings of the Control Theory & Applications, 149(5), 387–393.

    Article  Google Scholar 

  • Varaiya, P. P. (1993). Smart cars on smart roads: Problems of control. IEEE Transactions on Automatic Control, 38(2), 195–207.

    Article  MathSciNet  Google Scholar 

  • Wada, N., Oomoto, T., & Saeki, M. (2004). \(L_2\)-Gain analysis of discrete-time systems with saturation nonlinearity using parameter dependent Lyapunov function. In Proceedings of the 43rd IEEE conference on decision and control (pp. 1952–1957). Atlantis, Paradise Island, Bahamas: IEEE.

  • Xie, D., Wang, L., Hao, F., & Xie, G. (2004). LMI approach to \(L_2\)-gain analysis and control synthesis of uncertain switched systems. IEE Proceedings of the Control Theory & Applications, 151(1), 21–28.

    Article  Google Scholar 

  • Zhai, G. (2001). Quadratic stabilizability of discrete-time switched systems via state and output feedback. In Proceedings of the 40th IEEE conference on decision and control (pp. 2165–2166).

  • Zhai, G., Hu, B., Yasuda, K., & Michel, A. N. (2001). Disturbance attenuation properties of time-controlled switched systems. Journal of the Franklin Institute, 338(7), 765–779.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, L., Boukas, E., & Haidar, A. (2008). Delay-range-dependent control synthesis for time-delay systems with actuator saturation. Automatica, 44(10), 2691–2695.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, X., Zhao, J., & Dimirovski, G. M. (2011). L2-Gain analysis and control synthesis of uncertain discrete-time switched linear systems with time delay and actuator saturation. International Journal of Control, 84(10), 1746–1758.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, X., Zhao, J., & Dimirovski, G. M. (2012). \(L_2\)-Gain analysis and control synthesis of uncertain switched linear systems subject to actuator saturation. International Journal of Systems Science, 43(4), 731–740.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, J., & Dimirovski, G. M. (2004). Quadratic stability of a class of switched nonlinear systems. IEEE Transactions on Automatic Control, 49(4), 574–578.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, J., & Hill, D. J. (2008). On stability, and \(L_2\)-gain and \(H_{\infty }\) control for switched systems. Automatica, 44(5), 1220–1232.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, J., & Spong, M. W. (2001). Hybrid control for global stabilization of cart-pendulum system. Automatica, 37(12), 1941–1951.

    Article  MathSciNet  MATH  Google Scholar 

  • Zheng, Q., & Wu, F. (2008). Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions. Systems & Control Letters, 57(11), 896–903.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Scientific Research Fund of Education Department of Liaoning Province of China (No. L2014159).

Author information

Authors and Affiliations

  1. School of Information and Control Engineering, Liaoning Shihua University, Fushun, 113001, People’s Republic of China

    Haijun Sun & Xinquan Zhang

  2. National Experimental Teaching Demonstration Center of Petrochemical Process Control, Liaoning Shihua University, Fushun, 113001, People’s Republic of China

    Haijun Sun & Xinquan Zhang

Authors
  1. Haijun Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinquan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinquan Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, H., Zhang, X. \(L_2\)-Gain Analysis and Design of Uncertain Switched Saturated Systems. J Control Autom Electr Syst 29, 399–410 (2018). https://doi.org/10.1007/s40313-018-0392-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40313-018-0392-9

Keywords

Search

Navigation