Skip to main content
SpringerLink
Log in

Finite frequency \(H_\infty \) control of 2-D continuous systems in Roesser model

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

This paper investigates the finite frequency (FF) \(H_\infty \) control problem of two-dimensional (2-D) continuous systems in Roesser Model. Our attention is focused on designing state feedback controllers guaranteeing the bounded-input-bounded-output stability and FF \(H_\infty \) performance of the corresponding closed-loop system. A generalized 2-D Kalman-Yakubovich-Popov (KYP) lemma is presented for 2-D continuous systems. By the generalized 2-D KYP lemma, the existence conditions of \(H_\infty \) controllers are obtained in terms of linear matrix inequalities. Two examples are given to validate the proposed methods.

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
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Anderson, B., & Vongpanitlerd, S. (1973). Network analysis and aynthesis: A modern systems theory approach. Englewood Cliffs: Prentice-Hall.

    Google Scholar 

  • Bachelier, O., & Mehdi, D. (2006). On the KYP lemma, the hybrid Roesser models and the matrix \(\partial D\)-regularity. LAII-ESIP Research Report No. 20060916OB.

  • Bachelier, O., Paszke, W., & Mehdi, D. (2008). On the Kalman-Yakubovich-Popov lemma and the multidimensional models. Multidimensional Systems and Signal Processing, 19(3), 425–447.

    Article  MathSciNet  MATH  Google Scholar 

  • Bachelier, O., Paszke, W., Yeganefar, N., & Mehdi, D. (2016). LMI stability conditions for 2-D Roesser models. IEEE Transactions on Automatic Control, 61(3), 766–770.

    Article  MathSciNet  MATH  Google Scholar 

  • Bracewell, R. (1995). Two-dimensional imaging. In Series Prentice Hall Signal Processing Series. Upper Saddle River. Englewood Cliffs, NJ: Prentice-Hall.

  • Chen, S., & Fong, I. (2006). Robust filtering for 2-D state-delayed systems with NFT uncertainties. IEEE Transactions on Signal Processing, 54(1), 274–285.

    Article  Google Scholar 

  • Chen, S., & Fong, I. (2007). Delay-dependent robust \({H_\infty }\) filtering for uncertain 2-D state-delayed systems. Signal Processing, 87(11), 2659–2672.

    Article  MATH  Google Scholar 

  • Chesi, G., & Middleton, R. (2015). \({H_\infty }\) and \({H_2 }\) norms of 2D mixed continuous-discrete-time systems via rationally-dependent complex Lyapunov functions. IEEE Transactions on Automatic Control, 60(10), 2614–2625.

    Article  MathSciNet  MATH  Google Scholar 

  • Du, C., & Xie, L. (2002). \({H_\infty }\) control and filtering of two-dimensional systems. Berlin: Springer.

  • Du, C., Xie, L., Guo, G., & Teoh, J. (2007). A generalized KYP lemma based approach for disturbance rejection in data storage systems. Automatica, 43(12), 2112–2118.

    Article  MathSciNet  MATH  Google Scholar 

  • El-Kasri, C., Hmamed, A., Tissir, E., & Tadeo, F. (2013). Robust \({H_\infty }\) filtering for uncertain two-dimensional continuous systems with time-varying delays. Multidimensional Systems and Signal Processing, 24(4), 685–706.

    Article  MathSciNet  MATH  Google Scholar 

  • Fornasini, E., & Marchesini, G. (1978). Doubly-indexed dynamical systems: State-space models and structural properties. Mathematical Systems Theory, 12(1), 59–72.

    Article  MathSciNet  MATH  Google Scholar 

  • Gahinet, P., & Apkarian, P. (1994). A linear matrix inequality approach to \({H_\infty }\) control. International Journal of Robust and Nonlinear Control, 4(4), 421–448.

    Article  MathSciNet  MATH  Google Scholar 

  • Gao, H., & Li, X. (2011). \({H_\infty }\) filtering for discrete-time state-delayed systems with finite frequency specifications. IEEE Transactions on Automatic Control, 56(12), 2935–2941.

    Article  MathSciNet  Google Scholar 

  • Ghous, I., & Xiang, Z. (2016a). Robust state feedback \({H_\infty }\) control for uncertain 2-D continuous state delayed systems in the Roesser model. Multidimensional Systems and Signal Processing, 27(2), 297–319.

    Article  MathSciNet  MATH  Google Scholar 

  • Ghous, I., & Xiang, Z. (2016b). \({H_\infty }\) control of a class of 2-D continuous switched delayed systems via state-dependent switching. International Journal of Systems Science, 47(2), 300–313.

    Article  MathSciNet  MATH  Google Scholar 

  • Iwasaki, T., & Hara, S. (2005). Generalized KYP lemma: Unified frequency domain inequalities with design applications. IEEE Transactions on Automatic Control, 50(1), 41–59.

    Article  MathSciNet  MATH  Google Scholar 

  • Iwasaki, T., & Hara, S. (2007). Feedback control synthesis of multiple frequency domain specifications via generalized KYP lemma. International Journal of Robust and Nonlinear Control, 17(5–6), 415–434.

    Article  MathSciNet  MATH  Google Scholar 

  • Iwasaki, T., Meinsma, G., & Fu, M. (2000). Generalized S-procedure and finite frequency KYP lemma. Mathematical Problems in Engineering, 6(2–3), 305–320.

    Article  MathSciNet  MATH  Google Scholar 

  • Kaczorek, T. (1985). Two-dimensional linear systems. Berlin: Heidelberg.

    MATH  Google Scholar 

  • Kalman, R. (1963). Lyapunov functions for the problem of Lur’e in automatic control. Proceedings of the National Academy of Sciences of the United States of America, 49(2), 201–205.

    Article  MathSciNet  MATH  Google Scholar 

  • Lam, J., Xu, S., Zou, Y., et al. (2004). Robust output feedback stabilization for two-dimensional continuous systems in Roesser form. Applied Mathematics Letters, 17(12), 1331–1341.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, X., & Gao, H. (2012). Robust finite frequency \({H_\infty }\) filtering for uncertain 2-D Roesser systems. Automatica, 48(6), 1163–1170.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, X., Gao, H., & Wang, C. (2012). Generalized Kalman-Yakubovich-Popov lemma for 2-D FM LSS model. IEEE Transactions on Automatic Control, 57(12), 3090–3103.

    Article  MathSciNet  Google Scholar 

  • Lu, W., & Antoniou, A. (1992). Two-dimensional digital filters. New York: Marcel Dekker.

    MATH  Google Scholar 

  • Paszke, W., & Bachelier, O. (2013). Robust control with finite frequency specification for uncertain discrete linear repetitive processes. Multidimensional Systems and Signal Processing, 24(4), 727–745.

    Article  MathSciNet  MATH  Google Scholar 

  • Paszke, W., Rogers, E., Galkowski, K., & Cai, Z. (2013). Robust finite frequency range iterative learning control design and experimental verification. Control Engineering Practice, 21(10), 1310–1320.

    Article  Google Scholar 

  • Rantzer, A. (1996). On the Kalman-Yakubovich-Popov lemma. Systems and Control Letters, 28(1), 7–10.

    Article  MathSciNet  MATH  Google Scholar 

  • Roesser, R. (1975). A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control, 20(1), 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, W., Khargonekar, P., & Shim, D. (1994). Solution to the positive real control problem for linear time-invariant systems. IEEE Transactions on Automatic Control, 39(10), 2034–2046.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, Z., & Liu, X. (2003). Robust stability of two-dimensional uncertain discrete systems. IEEE Signal Processing Letters, 10(5), 133–136.

    Article  Google Scholar 

  • Wu, L., Shi, P., Gao, H., & Wang, C. (2008). \({H_\infty }\) filtering for 2-D Markovian jump systems. Automatica, 44(7), 1849–1858.

    Article  MathSciNet  MATH  Google Scholar 

  • Wu, L., Wang, Z., Gao, H., & Wang, C. (2007). \({H_\infty }\) and \({l_2} - {l_\infty }\) filtering for two-dimensional linear parameter-varying systems. International Journal of Robust and Nonlinear Control, 17(12), 1129–1154.

    Article  MathSciNet  MATH  Google Scholar 

  • Xie, L., Fu, M., & Li, H. (1998). Passivity analysis and passification for uncertain signal processing systems. IEEE Transactions on Signal Processing, 46(9), 2394–2403.

    Article  Google Scholar 

  • Xu, S., Lam, J., Lin, Z., et al. (2003). Positive real control of two-dimensional systems: Roesser models and linear repetitive processes. International Journal of Control, 76(11), 1047–1058.

    Article  MathSciNet  MATH  Google Scholar 

  • Xu, S., Lam, J., Lin, Z., & Galkowski, K. (2002). Positive real control for uncertain two-dimensional systems. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49(11), 1659–1666.

    Article  MathSciNet  Google Scholar 

  • Xu, S., Lam, J., Zou, Y., Lin, Z., & Galkowski, K. (2008). \(H_\infty \) output feedback control for two-dimensional continuous systems. Dynamics of Continuous Discrete and Impulsive Systems, 1(1), 1–14.

    MathSciNet  MATH  Google Scholar 

  • Xu, S., Lam, J., Zou, Y., Lin, Z., & Paszke, W. (2005). Robust \(H_\infty \) filtering for uncertain 2-D continuous systems. IEEE Transactions on Signal Processing, 53(5), 1731–1738.

    Article  MathSciNet  Google Scholar 

  • Yang, R., Xie, L., & Zhang, C. (2008). Generalized two-dimensional Kalman-Yakubovich-Popov lemma for discrete Roesser model. IEEE Transactions on Circuits and Systems I: Regular Papers, 55(10), 3223–3233.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 61273120.

Author information

Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China

    Zhaoxia Duan & Zhengrong Xiang

Authors
  1. Zhaoxia Duan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhengrong Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhengrong Xiang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duan, Z., Xiang, Z. Finite frequency \(H_\infty \) control of 2-D continuous systems in Roesser model. Multidim Syst Sign Process 28, 1481–1497 (2017). https://doi.org/10.1007/s11045-016-0430-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-016-0430-3

Keywords

Search

Navigation