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Delay-Dependent Stability and \(H_{\infty }\) Control for 2-D Markovian Jump Delay Systems with Missing Measurements and Sensor Nonlinearities

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Abstract

This paper investigates the problem of stability and \({H}_\infty \) control for 2-D time-delayed Markovian jump systems with missing measurements and sensor nonlinearities. The measured signals transmitted between the system and the controller are supposed to be imperfect, which involves missing measurements and sensor nonlinearities. The data missing is described by a random variable that obeys the Bernoulli binary distribution, and the sensor nonlinearities are assumed to satisfy the sector conditions. The problem addressed is the design of an output feedback controller such that the resulting closed-loop system is mean-square asymptotically stable and has a prescribed \({H}_\infty \) performance level. By using the Lyapunov method and the discrete Jensen inequality, stability criteria and \({H}_\infty \) performance level are established, and then, the controller design problem is cast into optimization problems via two methods. Finally, a numerical example is exploited to illustrate the effectiveness of the proposed design method.

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References

  1. Y.-Y. Cao, Z.L. Lin, B.M. Chen, An output feedback H\(\infty \) controller design for lin-ear systems subject to sensor nonlinearities. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50(7), 914–921 (2003)

    Article  Google Scholar 

  2. X.-H. Chang, G.-H. Yang, New results on output feedback H\(\infty \) control for linear discrete-time systems. IEEE Trans. Autom. Control 59(5), 1355–1359 (2014)

    Article  MathSciNet  Google Scholar 

  3. O.L. Costa, M.D. Fragoso, Stability results for discrete-time linear systems with Markovian jumping parameters. J. Math. Anal. Appl. 179(1), 154–178 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. O.L. Costa, R.P. Marques, Mixed H 2/H\(\infty \)-control of discrete-time Markovi-an jump linear systems. IEEE Trans. Autom. Control 43(1), 95–100 (1998)

    Article  Google Scholar 

  5. J.T. Dai, G. Guo, H\(\infty \) control of 2-D Markovian jump delay systems with missing measurements and sensor nonlinearities. https://www.researchgate.net/publication/29483052

  6. Y.C. Ding, H. Zhu, S.M. Zhong, Y.P. Zhang, Y. Zeng, H\(\infty \) filtering for stochastic systems with Markovian switching and partly unknown transition probabilities. Circuits Syst. Signal Process. 32(2), 559–583 (2013)

    Article  MathSciNet  Google Scholar 

  7. C.L. Du, L.H. Xie, C.S. Zhang, H\(\infty \) control and robust stabilization of two dimensional systems in Roesser models. Automatica 37(2), 205–211 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. C.L. Du, L.H. Xie, Stability analysis and stabilization of uncertain two dimensional discrete systems: an LMI approach. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 46(11), 1371–1374 (1999)

    Article  MATH  Google Scholar 

  9. L. El Ghaoui, F. Oustry, M. AitRami, A cone complementarity linearization algorithm for static output-feedback and related problems. IEEE Trans. Autom. Control 42(8), 1171–1176 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Z.-Y. Feng, L. Xu, M. Wu, Y. He, Delay-dependent robust stability and stabilisation of uncerta-in two-dimensional discrete systems with time-varying delays. IET Control Theory Appl. 4(10), 1959–1971 (2010)

    Article  MathSciNet  Google Scholar 

  11. H.J. Gao, J. Lam, S.Y. Xu, C.H. Wang, Stabilization and H\(\infty \) control of two-dimensional Markovian jump systems. IMA J. Math. Control Inform. 21(4), 377–392 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. He, M. Wu, G.-P. Liu, J.-H. She, Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Trans. Autom. Control 53(10), 2372–2377 (2008)

    Article  MathSciNet  Google Scholar 

  13. T. Hinamoto, The Fornasini–Marchesini model with no overflow oscillations and its application to 2-D digital filter design, in Proceedings of the International Symposium on Circuits and Systems, Portland, Oregon, pp. 1680–1683 (1989)

  14. T. Hinamoto, Stability of 2-D discrete systems described by the Fornasini Marc-hesini second model. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44(3), 254–257 (1997)

    Article  MathSciNet  Google Scholar 

  15. S.P. Huang, Z.R. Xiang, Delay-dependent robust H\(\infty \) control for 2-D discrete nonlinear systems with state delays. Multidim. Syst. Sign. Process. 25(4), 775–794 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Kaczorek, Two-Dimensional Linear Systems (Springer, Berlin, 1985)

    MATH  Google Scholar 

  17. Y.S. Lee, Y.S. Moon, W.H. Kwon. Delay-dependent guaranteed cost control for uncertain state-delayed systems, in Proceedings of American Control Conference, Arlington, Virginia, pp. 3376–3381 (2001)

  18. J.L. Liang, Z.D. Wang, X.H. Liu, Robust state estimation for two-dimensional sto-chastic time-delay systems with missing measurements and sensor saturation. Multidimens Syst. Signal. Process. 25(1), 157–177 (2014)

    Article  MATH  Google Scholar 

  19. W.S. Lu, A. Antoniou, Two-Dimensional Digital Filters (Marcel Dekker, New York, 1992)

    MATH  Google Scholar 

  20. W. Paszke, J. Lam, K. Gałkowski, S.Y. Xu, Z.P. Lin, Robust stability and stabilisation of 2D discrete state-delayed systems. Syst. Control Lett. 51(3), 277–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Seiler, R. Sengupta, A bounded real lemma for jump systems. IEEE Trans. Autom. Control 48(9), 1651–1654 (2003)

    Article  MathSciNet  Google Scholar 

  22. T. Senthilkumar, P. Balasubramaniam, Non-fragile robust stabilization and H\(\infty \) control for uncertain stochastic time delay systems with Markovian jump parameters and nonlinear disturbances. Int. J. Adapt. Control Signal Process. 28(3–5), 464–478 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Z.D. Wang, D.W.C. Ho, Y.R. Liu, X.H. Liu, Robust H\(\infty \) control for a class of nonlinear discrete time-delay stochastic systems with missing measurements. Automatica 45(3), 684–691 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. L.G. Wu, P. Shi, H.J. Gao, C.H. Wang, H\(\infty \) filtering for 2D Markovian jump systems. Automatica 44(7), 1849–1858 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. L.G. Wu, X.M. Yao, W.X. Zheng, Generalized H2 fault detection for two dimen-sional Markovian jump systems. Automatica 48(8), 1741–1750 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. J.M. Xu, L. Yu, H\(\infty \) control of 2-D discrete state delay systems. Int. J. Control Autom. Syst. 4(4), 516–523 (2006)

    Google Scholar 

  27. J.M. Xu, Y.R. Nan, G.J. Zhang, L.L. Ou, H.J. Ni, Delay-dependent H\(\infty \) control for uncertain 2-D discrete systems with state delay in the Roesser model. Circuits Syst. Signal Process. 32(3), 1097–1112 (2013)

    Article  Google Scholar 

  28. S. Ye, W.Q. Wang, J. Yao, Stability analysis for 2-D discrete systems with varying delay, in Proceedings of IEEE International Conference on Control Automation Robotics and Vision, Singapore, pp. 67–72 (2010)

  29. X.-L. Zhu, G.-H. Yang, Jensen inequality approach to stability analysis of discrete-time systems with time-varying delay, in Proceedings of American Control Conference, Seattle, Washington, pp. 1644–1649 (2008)

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Authors and Affiliations

  1. School of Control Science and Engineering, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Jiangtao Dai & Ge Guo

  2. Fundamental Science Department, North China Institute of Astronautic Engineering, Langfang, 065000, People’s Republic of China

    Jiangtao Dai

Authors
  1. Jiangtao Dai
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  2. Ge Guo
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Correspondence to Jiangtao Dai.

Additional information

This work was supported by the National Natural Science Foundation of China (61174060, 61273107), the Fundamental Research Funds for the Central Universities (3132013334), the Dalian Leading Talent Project (841252), the Science and Technology Support Program of Langfang (2014011045).

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Dai, J., Guo, G. Delay-Dependent Stability and \(H_{\infty }\) Control for 2-D Markovian Jump Delay Systems with Missing Measurements and Sensor Nonlinearities. Circuits Syst Signal Process 36, 25–48 (2017). https://doi.org/10.1007/s00034-016-0290-y

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  • DOI: https://doi.org/10.1007/s00034-016-0290-y

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