Stability of DCESs Under the Hyper-Sampling Mode

  • Arben Çela
  • Mongi Ben Gaid
  • Xu-Guang Li
  • Silviu-Iulian Niculescu
Part of the Communications and Control Engineering book series (CCE)


The main objective of this chapter is to understand the way in which periodic scheduling or hyper-sampling period and DCES-induced delays affect the stability of the controlled plant. Several scenarios are considered, leading to three stability problems. First, a delay-sweeping method is given in the case of constant parameters (hyper-sampling periods and DCES-induced delays). Next, two problems concerning the case when time-varying uncertain parameters are considered. For a system with time-varying uncertain parameters, a sufficient stability condition is given in terms of the existence of an appropriate Lyapunov matrix. The second problem concerns the stability of a real-time system including a constant hyper-sampling period and time-varying uncertain DCES-induced delays. In this case, sufficient conditions expressed in terms of feasibility of some appropriate linear matrix inequalities (LMIs) are proposed. Finally, the third problem concerns the case without DCES-induced delays but subject to time-varying uncertain hyper-sampling periods. The problem is handled by establishing an appropriate connection between the single-sampling and hyper-sampling cases. In this way, the hyper-sampling case appears as a direct application of the results derived in the single-sampling case. Next, a parameter-sweeping method is employed to detect the whole stability region in the corresponding parameter-space. Different examples (including also the case of two inverted pendulums) are given to illustrate the proposed results. It is worth mentioning that each system can be viewed as a switched system composed of “n” sub-systems if the hyper-sampling period has “n” sub-sampling periods. The derived stability regions include, in some cases, sub-regions where some sub-systems are not necessarily stable. This fact helps us to enlarge the stability ranges of the parameters by taking more advantage of the effect induced by the hyper-sampling period on the stability of the overall system. To the best of the authors’ knowledge, such an angle was not sufficiently addressed and exploited for real-time applications.


  1. 28.
    M.-M. Ben Gaid, A. Çela, Y. Hamam, Optimal real-time scheduling of control tasks with state feedback resource allocation. IEEE Trans. Control Syst. Technol. 17(2), 309–326 (2009) CrossRefGoogle Scholar
  2. 31.
    M.-M. Ben Gaid, R. Kocik, Y. Sorel, R. Hamouche, A methodology for improving software design lifecycle in embedded control systems, in Proceeding of Design, Automation and Test in Europe, DATE ’08, Munich, Germany, 10–14 March 2008 Google Scholar
  3. 40.
    S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994) CrossRefMATHGoogle Scholar
  4. 54.
    A. Cervin, P. Alriksson, Optimal on-line scheduling of multiple control tasks: a case study, in Proceedings of the 18th Euromicro Conference on Real-Time Systems, Dresden, Germany, July 2006 Google Scholar
  5. 61.
    J. Chen, H.A. Latchman, Frequency sweeping tests for stability independent of delay. IEEE Transactions on Automatic Control 40(9), 1640–1645 (1995) MathSciNetCrossRefMATHGoogle Scholar
  6. 65.
    M.B.G. Cloosterman, N. Van de Wouw, W.P.M.H. Heemels, H. Nijmeijer, Robust stability of networked control systems with time-varying network-induced delays, in Proceedings of the 45th Conference on Decision and Control, San Diego, CA, December 2006 Google Scholar
  7. 81.
    T. Estrada, P.J. Antsaklis, Stability of model-based networked control systems with intermittent feedback, in 17th IFAC World Congress, Seoul, Korea, 2008 Google Scholar
  8. 83.
    H. Fazelinia, R. Sipahi, N. Olgac, Stability robustness analysis of multiple time-delayed systems using building block concept. IEEE Transactions on Automatic Control 52(5), 799–810 (2007) MathSciNetCrossRefGoogle Scholar
  9. 88.
    E. Fridman, A. Seuret, J.-P. Richard, Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40(8), 1441–1446 (2004) MathSciNetCrossRefMATHGoogle Scholar
  10. 89.
    H. Fujioka, A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices. IEEE Transactions on Automatic Control 54(10), 2440–2445 (2009) MathSciNetCrossRefGoogle Scholar
  11. 91.
    H. Gao, T. Chen, New results on stability of discrete-time systems with time-varying state delay. IEEE Transactions on Automatic Control 52(2), 328–334 (2007) MathSciNetCrossRefGoogle Scholar
  12. 92.
    H. Gao, T. Chen, J. Lam, A new delay system approach to network-based control. Automatica 44(1), 39–52 (2008) MathSciNetCrossRefMATHGoogle Scholar
  13. 94.
    R. Gielen, S. Olaru, M. Lazăr, On polytopic embeddings as a modeling framework for networked control systems, in Proceedings of the 3rd International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control, Pavia, Italy, 2008 Google Scholar
  14. 105.
    K. Gu, S.-I. Niculescu, J. Chen, On stability of crossing curves for general systems with two delays. J. Math. Anal. Appl. 311(1), 231–253 (2005) MathSciNetCrossRefMATHGoogle Scholar
  15. 119.
    J.P. Hespanha, P. Naghshtabrizi, Y. Xu, A survey of recent results in networked control systems. Proc. IEEE 95(1), 138–162 (2007) CrossRefGoogle Scholar
  16. 120.
    L. Hetel, J. Daafouz, C. Iung, LMI control design for a class of exponential uncertain systems with application to network controlled switched systems, in American Control Conference, New York, New York, USA, 2007, pp. 1401–1406 Google Scholar
  17. 124.
    D. Hristu-Varsakelis, W.S. Levine, Handbook of Networked and Embedded Control Systems (Birkhäuser, Boston, 2005) CrossRefMATHGoogle Scholar
  18. 133.
    X. Jiang, Q.-L. Han, S. Liu, A. Xue, A new H stabilization criterion for networked control systems. IEEE Transactions on Automatic Control 53(4), 1025–1032 (2008) MathSciNetCrossRefGoogle Scholar
  19. 147.
    M.S. Lee, C.S. Hsu, On the τ-decomposition method of stability analysis for retarded dynamical systems. SIAM J. Control 7, 242–259 (1969) MathSciNetCrossRefMATHGoogle Scholar
  20. 148.
    X.-G. Li, A. Çela, S.-I. Niculescu, A. Reama, Some remarks on robust stability of networked control systems, in 18th IEEE International Conference on Control Applications, Saint Petersburg, Russia, July 2009, pp. 19–24 Google Scholar
  21. 152.
    F.L. Lian, Analysis, design, modeling, and control of networked control systems. PhD thesis, Univ. Michigan, 2001 Google Scholar
  22. 162.
    R. Luck, A. Ray, An observer-based compensator for distributed delays. Automatica 26(5), 903–908 (1990) CrossRefMATHGoogle Scholar
  23. 174.
    W. Michiels, S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach (SIAM, Philadelphia, 2007) CrossRefMATHGoogle Scholar
  24. 175.
    L. Mirkin, Some remarks on the use of time-varying delay to model sample-and-hold circuits. IEEE Transactions on Automatic Control 52(6), 1109–1112 (2007) MathSciNetCrossRefGoogle Scholar
  25. 178.
    L.A. Montestruque, P.J. Antsaklis, Stability of model-based networked control systems with time-varying transmission times. IEEE Transactions on Automatic Control 49(9), 1562–1572 (2004) MathSciNetCrossRefGoogle Scholar
  26. 190.
    N. Olgac, R. Sipahi, An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems. IEEE Transactions on Automatic Control 47(5), 793–797 (2002) MathSciNetCrossRefGoogle Scholar
  27. 194.
    A.S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers. Deterministic Techniques, vol. 1 (Elsevier, Amsterdam, 2008) Google Scholar
  28. 209.
    A. Seuret, J.-P. Richard, Control of a remote system over network including delays and packet dropout, in 17th IFAC World Congress, Seoul, Korea, 2008, pp. 6336–6341 Google Scholar
  29. 222.
    Y.S. Suh, Stability and stabilization of nonuniform sampling systems. Automatica 44(12), 3222–3226 (2008) MathSciNetCrossRefMATHGoogle Scholar
  30. 224.
    X.-M. Sun, G.-P. Liu, D. Rees, W. Wang, A novel method of stability analysis for networked control systems, in 17th IFAC World Congress, Seoul, Korea, July 2008, pp. 4852–4856 Google Scholar
  31. 260.
    D. Yue, Q.-L. Han, C. Peng, State feedback controller design of networked control systems. IEEE Transactions on Circuits and Systems II 51(11), 640–644 (2004) CrossRefGoogle Scholar
  32. 261.
    S. Zampieri, Trends in networked control systems, in 17th IFAC World Congress, Seoul, Korea, July 2008 Google Scholar
  33. 263.
    W. Zhang, M.S. Branicky, Stability of networked control systems with time-varying transmission period, in Allerton Conference on Communication Control and Computing, Allerton, 2001 Google Scholar
  34. 264.
    W. Zhang, M.S. Branicky, S.M. Phillips, Stability of networked control systems. IEEE Control Syst. Mag. 21(6), 84–99 (2001) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Arben Çela
    • 1
  • Mongi Ben Gaid
    • 2
  • Xu-Guang Li
    • 3
  • Silviu-Iulian Niculescu
    • 4
  1. 1.Department of Computer Science and TelecommunicationUniversité Paris-Est, ESIEE ParisNoisy-le-GrandFrance
  2. 2.Electronic and Real-Time Systems DepartmentIFP New EnergyRueil-MalmaisonFrance
  3. 3.School of Information Science and EngineeringNortheastern UniversityShenyangPeople’s Republic of China
  4. 4.L2S—Laboratoire des signaux et systèmesSupélecGif-sur-YvetteFrance

Personalised recommendations