Robust Finite-time H Control of Linear Time-varying Delay Systems with Bounded Control via Riccati Equations

  • Piyapong Niamsup
  • Vu N. Phat
Research Article


In this paper, we will present new results on robust finite-time H control for linear time-varying systems with both time-varying delay and bounded control. Delay-dependent sufficient conditions for robust finite-time stabilization and H control are first established to guarantee finite-time stability of the closed-loop system via solving Riccati differential equations. Applications to finite-time H control to a class of linear autonomous time-delay systems with bounded control are also discussed in this paper. Numerical examples are given to illustrate the effectiveness of the proposed method.


Finite-time stability H control bounded control time-varying delay Riccati equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank anonymous associate editor and reviewers for their valuable comments and suggestions, which allowed us to improve the paper.


  1. [1]
    F. Amato, R. Ambrosino, M. Ariola, C. Cosentino, G. De Tommas. Finite-time Stability and Control, New York, USA: Springer, 2014.CrossRefzbMATHGoogle Scholar
  2. [2]
    Y. Q. Wu, C. L. Zhu, Z. C. Zhang. Finite-time stabilization of a general class of nonholonomic dynamic systems via terminal sliding mode. International Journal of Automation and Computing, vol. 13, no. 6, pp. 585–595, 2016.CrossRefGoogle Scholar
  3. [3]
    E. Moulay, M. Dambrine, N. Yeganefar, W. Perruquetti. Finite-time stability and stabilization of time-delay systems. Systems & Control Letters, vol. 57, no. 7, pp. 561–566, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    H. Liu, P. Shi, H. R. Karimi, M. Chadli. Finite-time stability and stabilisation for a class of nonlinear systems with time-varying delay. International Journal of Systems Science, vol. 47, no. 6, pp. 1433–1444, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Y. N. Sun, S. L. Liu, Z. R. Xiang. Robust finite-time H control for uncertain switched neutral systems with mixed delays. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 226, no. 5, pp. 638–650, 2012.Google Scholar
  6. [6]
    L. Van Hien, D. T. Son. Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Applied Mathematics and Computation, vol. 251, pp. 14–23, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    P. Dorato P. Short time stability in linear time-varying systems. In Proceedings of IRE International Convention Record, IEEE, New York, USA, vol. 4, pp. 83–87, 1961.Google Scholar
  8. [8]
    K. M. Zhou, P. P. Khargonekar. An algebraic Riccati equation approach to H optimization. Systems & Control Letters, vol. 11, no. 2, pp. 85–91, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Z. R. Xiang, Y. N. Sun, M. S. Mahmoud. Robust finitetime H control for a class of uncertain switched neutral systems. Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1766–1778, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    W. M. Xiang, J. Xiao. H finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance. Journal of the Franklin Institute, vol. 348, no. 2, pp. 331–352, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    P. Wang, W. W. Sun. Adaptive H control for nonlinear Hamiltonian systems with time delay and parametric uncertainties. International Journal of Automation and Computing, vol. 11, no. 4, 368–376, 2014.CrossRefGoogle Scholar
  12. [12]
    T. Ba¸sar, P. Bernhard. H -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, 2nd ed., Basel, USA: Birkhauser, 1995.Google Scholar
  13. [13]
    P. Balasubramaniam, T. Senthilkumar. Delay-dependent robust stabilization and H control for uncertain stochastic T-S fuzzy systems with discrete interval and distributed time-varying delays. International Journal of Automation and Computing, vol. 10, no. 1, pp. 18–31, 2013.CrossRefGoogle Scholar
  14. [14]
    P. Shi, Y. Q. Zhang, M. Chadli, R. K. Agarwal. Mixed H and passive filtering for discrete fuzzy neural networks with stochastic jumps and time delays. IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 4, pp. 903–909, 2016.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Z. R. Xiang, C. H. Qiao, M. S. Mahmoud. Finite-time analysis and H control for switched stochastic systems. Journal of the Franklin Institute, vol. 349, no. 3, pp. 915–927, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Z. Xiang, S. Liu, M. S. Mahmoud. Robust H reliable control for uncertain switched neutral systems with distributed delays. IMA Journal Mathematical Control and Information, vol. 32, pp. 1–19, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H. Liu, Y. Shen, X. D. Zhao. Delay-dependent observerbased H finite-time control for switched systems with time-varying delay. Nonlinear Analysis: Hybrid Systems, vol. 6, no. 3, pp. 885–898, 2012.MathSciNetzbMATHGoogle Scholar
  18. [18]
    H. Bounit, H. Hammouri. Bounded feedback stabilization and global separation principle of distributed parameter systems. IEEE Transactions on Automatic Control, vol. 42, no. 3, pp. 414–419, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    M. Slemrod. Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Mathematics of Control, Signals and Systems, vol. 22, no. 3, pp. 265–285, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    H. J. Sussmann, E. D. Sontag, Y. Yang. A general result on the stabilization of linear systems using bounded controls. IEEE Transactions on Automatic Control, vol. 39, no. 12, pp. 2411–2425, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. L. Corradini, A. Cristofaro, F. Giannoni, G. Orlando. Control Systems with Saturating Inputs: Analysis Tools and Advanced Design, New York, USA: Springer, 2012.CrossRefGoogle Scholar
  22. [22]
    B. Wang, J. Y. Zhai, S. M. Fei. Output feedback tracking control for a class of switched nonlinear systems with time-varying delay. International Journal of Automation and Computing, vol. 11, no. 6, pp. 605–612, 2014.CrossRefGoogle Scholar
  23. [23]
    V. N. Phat, P. Niamsup. Stabilization of linear nonautonomous systems with norm-bounded controls. Journal of Optimization Theory and Applications, vol. 131, no. 1, pp. 135–149, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Y. Guo, Y. Yao, S. C. Wang, B. Q. Yang, K. Liu, X. Zhao. Finite-time control with H constraints of linear time-invariant and time-varying systems. Journal of Control Theory and Applications, vol. 11, no. 2, pp. 165–172, 2013.MathSciNetCrossRefGoogle Scholar
  25. [25]
    G. Garcia, S. Tarbouriech, J. Bernussou. Finite-time stabilization of linear time-varying continuous systems. IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 364–369, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    J. J. Hui, H. X. Zhang, X. Y. Kong. Delay-dependent nonfragile H control for linear systems with interval timevarying delay. International Journal of Automation and Computing, vol. 12, no. 1, pp. 109–116, 2015.CrossRefGoogle Scholar
  27. [27]
    F. Amato, R. Ambrosino, C. Cosentino, G. De Tommasi. Input-output finite time stabilization of linear systems. Automatica, vol. 46, no. 9, pp. 1558–1562, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    H. Du, C. Qian, M. T. Frye, S. Li. Global finite-time stabilisation using bounded feedback for a class of non-linear systems. IET Control Theory & Applications, vol. 6, no. 14, pp. 2326–2336, 2012.MathSciNetCrossRefGoogle Scholar
  29. [29]
    K. Q. Gu, V. L. Kharitonov, J. Chen. Stability of Time- Delay System, Boston, USA: Birkhauser, 2003.CrossRefzbMATHGoogle Scholar
  30. [30]
    A. J. Laub. A Schur method for solving algebraic Riccati equations. IEEE Transactions on Automatic Control, vol. 24, no. 6, pp. 913–921, 1979.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    J. S. Gibson. Linear-quadratic optimal control of hereditary differential systems: infinite dimensional Riccati equations and numerical approximations. SIAM Journal on Control and Optimization, vol. 21, no. 1, pp. 95–139, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    W. T. Reid. Riccati Differential Equations, Volume 86, New York, USA: Academic Press, 1972.zbMATHGoogle Scholar
  33. [33]
    U. Shaked, V. Suplin. A new bounded real lemma representation for the continuous-time case. IEEE Transactions on Automatic Control, vol. 46, no. 9, pp. 1420–1426, 2001.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Automation, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  2. 2.Institute of MathematicsVASTHanoiVietnam

Personalised recommendations