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Finite-Time Stability and Control of 2D Continuous–Discrete Systems in Roesser Model

  • Jingbo Gao
  • Weiqun Wang
  • Guangchen Zhang
Article

Abstract

This study is conducted to investigate stability and control problems within a finite-time interval for a 2D continuous–discrete system in Roesser model. The concepts of finite-time stability (FTS) and finite-time boundedness (FTB) are naturally extended to the 2D continuous–discrete system. Recursive relations between system states are first obtained, then sufficient conditions for FTS and FTB in the system are derived, and a finite-time controller is supplied to the system. Sufficient conditions for finite-time stabilization are also provided for the linear repetitive process. Examples of metal rolling operation are presented to illustrate the proposed method.

Keywords

2D continuous–discrete system Finite-time stability Finite-time boundedness Linear repetitive process 

Notes

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China under Grant 61573007 and Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20133219110040 for financial support.

References

  1. 1.
    F. Amato, M. Ariola, C. Cosentino, Finite-time control of discrete-time linear systems: analysis and design conditions. Automatica 46(5), 919–924 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37, 1459–1463 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    J. Bai, R. Lu, A. Xue, Q. She, Z. Shi, Finite-time stability analysis of discrete-time fuzzy Hopfield neural network. Neurocomputing 159(2), 263–267 (2015)CrossRefGoogle Scholar
  4. 4.
    M. Buslowicz, Robust stability of the new general 2D model of a class of continuous–discrete linear systems. Bull. Polish Acad. Sci. Tech. Sci. 58(4), 561–565 (2010)zbMATHGoogle Scholar
  5. 5.
    M. Buslowicz, Stability and robust stability conditions for a general model of scalar continuous–discrete linear systems. Pomiary, Automatyka, Kontrola 56, 133–135 (2010)Google Scholar
  6. 6.
    M. Buslowicz, A. Ruszewski, Computer methods for stability analysis of the Roesser type model of 2D continuous–discrete linear systems. Int. J. Appl. Math. Comput. Sci. 22(2), 401–408 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Y. Chen, Q. Liu, R. Lu et al., Finite-time control of switched stochastic delayed systems. Neurocomputing 191, 374 (2016)CrossRefGoogle Scholar
  8. 8.
    P. Dabkowski, K. Galkowski, E. Rogers, Strong practical stability and stabilization of differential linear repetitive processes. Syst. Control Lett. 59(10), 639–644 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. Dandrea, G.E. Dullerud, Distributed control design for spatially interconnected systems. IEEE Trans. Autom. Control 48(9), 1478–1495 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    P. Dorato, C. Abdallah, D. Famularo, Robust finite-time stability design via linear matrix inequalities, in IEEE Conference on Decision and Control. Institute of Electrical Engineers INC (IEE). (1997)Google Scholar
  11. 11.
    K. Galkowski, J. Wood, Multidimensional Signals, Circuits and Systems, Systems and Control Book Series (Taylor and Francis, London, England, 2001)Google Scholar
  12. 12.
    G. Garcia, S. Tarbouriech, J. Bemussou, Finite-time stabilization of linear time-varying continuous systems. IEEE Trans. Autom. Control 54(2), 364–369 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    S. He, F. Liu, On robust controllability with respect to the finite-time interval of uncertain nonlinear jump systems. Trans. Inst. Meas. Control 34(7), 841–849 (2012)CrossRefGoogle Scholar
  14. 14.
    S. He, F. Liu, Optimal finite-time passive controller design for uncertain nonlinear Markovian jumping systems. J. Frankl. Inst. Eng. Appl. Math. 351(7), 3782–3796 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    T. Kaczorek, Positive fractional 2D continuous–discrete linear systems. Bull. Polish Acad. Sci. Tech. Sci. 59(4), 575–579 (2011)zbMATHGoogle Scholar
  16. 16.
    T. Kaczorek, Stability of continuous–discrete linear systems described by the general model. Bull. Polish Acad. Sci. Tech. Sci. 59(2), 189–193 (2011)zbMATHGoogle Scholar
  17. 17.
    G. Kamenkov, On stability of motion over a finite interval of time. J. Appl. Math. Mech. (PMM) 17, 529–540 (1953)MathSciNetGoogle Scholar
  18. 18.
    S. Knorn, A two-dimensional systems stability analysis of vehicle platoons (National University, Maynooth, 2013)Google Scholar
  19. 19.
    M.P. Lazarević, A.M. Spasić, Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Math. Comput. Model. 49(3), 475–481 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    A. Lebedev, The problem of stability in a finite interval of time. J. Appl. Math. Mech. (PMM) 18, 75–94 (1954)Google Scholar
  21. 21.
    Y. Li, M. Cantoni, E. Weyer, On water-level error propagation in controlled irrigation channels, in Proceedings of IEEE Conference on Decision Control and European Control Conference, (Seville, Spain, 2005), pp. 2101–2106Google Scholar
  22. 22.
    Y. Ma, B. Wu, Y.E. Wang, Finite-time stability and finite-time boundedness of fractional order linear systems. Neurocomputing. 173, 2076–2082 (2016)CrossRefGoogle Scholar
  23. 23.
    A. Maidi, M. Diaf, J. Corriou, Optimal linear PI fuzzy controller design of a heat exchanger. Chem. Eng. Process. 47(5), 938–945 (2008)CrossRefGoogle Scholar
  24. 24.
    E. Moulay, W. Perruquetti, Finite time stability and stabilization of a class of continuous systems. J. Math. Anal. Appl. 323(2), 1430–1443 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    D.H. Owens, E. Rogers, Stability analysis for a class of 2D continuous–discrete linear systems with dynamic boundary conditions. Syst. Control Lett. 37(1), 55–60 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    W. Paszke, O. Bachelier, New robust stability and stabilization conditions for linear repetitive processes, in International Workshop on Multidimensional, (IEEE, 2009), pp. 1–6 Google Scholar
  27. 27.
    W. Paszke, P. Dabkowski, E. Rogers, K. Galkowski, Stability and robustness of discrete linear repetitive processes in the finite frequency domain using the KYP lemma, in IEEE 52nd Annual Conference on Decision and Control (CDC), 2013, pp. 3421–3426 (2013)Google Scholar
  28. 28.
    W. Paszke, E. Rogers, K. Galkowski, Z. Cai, Robust finite frequency range iterative learning control and experimental verification. Control Eng. Pract. 21(10), 1310–1320 (2013)CrossRefGoogle Scholar
  29. 29.
    R.P. Roesser, A discrete state-space model for linear image processing. IEEE Trans. Autom. Control 20(1), 1–10 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    E. Rogers, K. Galkowski, D.H. Owens, Control Systems Theory and Applications for Linear Repetitive Processes (Springer, Berlin Heidelberg, 2007)zbMATHGoogle Scholar
  31. 31.
    S.B. Stojanović, D.L. Debeljković, N. Dimitrijević, Finite-time stability of discrete-time systems with time-varying delay. Chem. Ind. Chem. Eng. Q./CICEQ 18(4), 525–533 (2012)CrossRefGoogle Scholar
  32. 32.
    T. Tan, B. Zhou, G.R. Duan, Finite-time stabilization of linear time-varying systems by piecewise constant feedback. Automatica 68, 277–285 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    L. Wang, W. Wang, J. Gao, Stability and robust stabilization of 2D continuous–discrete systems in Roesser model based on KYP lemma. Multidim. Syst. Sign. Process. 28(1), 1–14 (2017)CrossRefGoogle Scholar
  34. 34.
    L. Wu, H. Gao, C. Wang, Quasi sliding mode control of differential linear repetitive processes with unknown input disturbance. IEEE Trans. Ind. Electron. 58(7), 3059–3068 (2011)CrossRefGoogle Scholar
  35. 35.
    R. Wu, Y. Lu, L. Chen, Finite-time stability of fractional delayed neural networks. Neurocomputing 149, 700–707 (2015)CrossRefGoogle Scholar
  36. 36.
    L. Wu, P. Shi, H. Gao, H∞ filtering for 2D Markovian jump systems. Automatica 44(7), 1849–1858 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    L. Wu, R. Yang, P. Shi, Stability analysis and stabilization of 2D switched systems under arbitrary and restricted switching. Automatica 59, 206–215 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    W. Zhang, X. An, Finite-time control of linear stochastic systems. Int. J. Innov. Comput. Inf. Control 4(3), 689–696 (2008)Google Scholar
  39. 39.
    G. Zhang, H.L. Trentelman, W. Wang, Input-output finite-region stability and stabilization for discrete 2D Fornasini–Marchesini models. Syst. Control Lett. 99, 9–16 (2017)CrossRefzbMATHGoogle Scholar
  40. 40.
    G. Zhang, W. Wang, Finite-region stability and boundedness for discrete 2D Fornasini–Marchesini second models. Int. J. Syst. Sci. 48(4), 778–787 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    G. Zhang, W. Wang, Finite-region stability and finite-region boundedness for 2D Roesser models. Math. Methods Appl. Sci. 39(18), 5757–5769 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ScienceNanjing University of Science and TechnologyNanjingChina

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