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Robust H ∞ Control for a Class of 2-D Nonlinear Discrete Stochastic Systems

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Abstract

This paper is concerned with the problem of stability and robust H ∞ control for 2-D stochastic systems with parameter uncertainties and sector nonlinearities. The class of systems under investigation is described by the 2-D state-space Roesser model. Our attention is focused on the design of a state feedback controller for 2-D stochastic system with sector nonlinearity, such that the closed-loop 2-D stochastic system is asymptotically stable and has a prescribed H ∞ disturbance attenuation performance. First, a sufficient condition is established for the 2-D nonlinear stochastic systems to be asymptotically stable. Then, we extend the bounded real lemma for 2-D systems to 2-D stochastic systems with sector nonlinearities. Based on this lemma, solvability conditions for the H ∞ control of 2-D nonlinear stochastic systems in the form of LMIs (linear matrix inequalities) are derived. A numerical example illustrates the effectiveness of the proposed results.

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References

  1. S. Boyd, L. El Ghaoul, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 (Society for Industrial and Applied Mathematics, Philadelphia, 1987)

    Google Scholar 

  2. C.E. de Souza, L. Xie, On the discrete-time bounded real lemma with application in the characterization of static state feedback H ∞ controllers. Syst. Control Lett. 18(1), 61–71 (1992)

    Article  MATH  Google Scholar 

  3. H. Deng, M. Krstic, R.J. Williams, Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans. Autom. Control 46(8), 1237–1253 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Du, L. Xie, Y.C. Soh, H ∞ filtering of 2-D discrete systems. IEEE Trans. Signal Process. 48(6), 1760–1768 (2000)

    Article  MATH  Google Scholar 

  5. C. Du, L. Xie, H ∞ Control and Filtering of Two-Dimensional Systems, vol. 278 (Springer, Berlin, 2002)

    MATH  Google Scholar 

  6. A. El Bouhtouri, D. Hinrichsen, A.J. Pritchard, H ∞ control for discrete-time stochastic systems. Int. J. Robust Nonlinear Control 9(13), 923–948 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Fornasini, G. Marchesini, State-space realization theory of two-dimensional filters. IEEE Trans. Autom. Control 21(4), 484–492 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Gao, J. Lam, C. Wang, S. Xu, Robust H ∞ filtering for 2D stochastic systems. Circuits Syst. Signal Process. 23(6), 479–505 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Gao, J. Lam, S. Xu, C. Wang, Stability and stabilization of uncertain 2-D discrete systems with stochastic perturbation. Multidimens. Syst. Signal Process. 16(1), 85–106 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. U. Haussmann, Optimal stationary control with state control dependent noise. SIAM J. Control 9(2), 184–198 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. D. Hinrichsen, A. Pritchard, Stochastic H ∞. SIAM J. Control Optim. 36(5), 1504–1538 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. H. Kar, V. Singh, Robust stability of 2-D discrete systems described by the Fornasini-Marchesini second model employing quantization/overflow nonlinearities. IEEE Trans. Circuits Syst. II, Express Briefs 51(11), 598–602 (2004)

    Article  Google Scholar 

  15. H.K. Khalil, Nonlinear Systems, 3rd edn. (Prentice Hall, New Jewsey, 2002)

    MATH  Google Scholar 

  16. W.S. Lu, Two-Dimensional Digital Filters, vol. 80 (CRC Press, Boca Raton, 1992)

    MATH  Google Scholar 

  17. W. Lu, On a Lyapunov approach to stability analysis of 2-D digital filters. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 41(10), 665–669 (1994)

    Article  Google Scholar 

  18. W.M. Wonham, On a matrix Riccati equation of stochastic control. SIAM J. Control 6(4), 681–697 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Xie, C. Du, Y.C. Soh, C. Zhang, H ∞ and robust control of 2-D systems in FM second model. Multidimens. Syst. Signal Process. 13(3), 265–287 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Xie, L. Xie, Decentralized stabilization of a class of interconnected stochastic nonlinear systems. IEEE Trans. Autom. Control 45(1), 132–137 (2000)

    Article  MATH  Google Scholar 

  21. S. Ye, W. Wang, Stability analysis and stabilisation for a class of 2-D nonlinear discrete systems. Int. J. Inf. Syst. Sci. 42(5), 839–851 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Yue, J.a. Fang, S. Won, Delay-dependent robust stability of stochastic uncertain systems with time delay and Markovian jump parameters. Circuits Syst. Signal Process. 22(4), 351–365 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the Associate Editor and anonymous reviewers for their constructive comments and suggestions which led to the considerable improvement of the presentation of this paper.

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Authors and Affiliations

  1. Fundamental Science Department, North China Institute of Astronautic Engineering, Langfang, Hebei, 065000, P.R. China

    Jiangtao Dai & Zhaozhuang Guo

  2. Dept. of Basic Courses, Xuzhou Air Force College, Xuzhou, Jiangsu, 221000, P.R. China

    Shuling Wang

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  1. Jiangtao Dai
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  2. Zhaozhuang Guo
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  3. Shuling Wang
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Correspondence to Jiangtao Dai.

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Dai, J., Guo, Z. & Wang, S. Robust H ∞ Control for a Class of 2-D Nonlinear Discrete Stochastic Systems. Circuits Syst Signal Process 32, 2297–2316 (2013). https://doi.org/10.1007/s00034-013-9573-8

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  • DOI: https://doi.org/10.1007/s00034-013-9573-8

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