Skip to main content
SpringerLink
Log in

Stability of DCESs Under the Hyper-Sampling Mode

  • Chapter
Optimal Design of Distributed Control and Embedded Systems

Part of the book series: Communications and Control Engineering ((CCE))

  • 1367 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We have multiple lines to derive the stability condition. We may also consider “The asymptotic stability can be ensured if for any odd k and z(0)≠0 such that V(k+2)−V(k)<0”.

References

  1. 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)

    Article  Google Scholar 

  2. 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. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (SIAM, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  4. 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. J. Chen, H.A. Latchman, Frequency sweeping tests for stability independent of delay. IEEE Transactions on Automatic Control 40(9), 1640–1645 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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. 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. 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)

    Article  MathSciNet  Google Scholar 

  9. E. Fridman, A. Seuret, J.-P. Richard, Robust sampled-data stabilization of linear systems: an input delay approach. Automatica 40(8), 1441–1446 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MathSciNet  Google Scholar 

  12. H. Gao, T. Chen, J. Lam, A new delay system approach to network-based control. Automatica 44(1), 39–52 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.P. Hespanha, P. Naghshtabrizi, Y. Xu, A survey of recent results in networked control systems. Proc. IEEE 95(1), 138–162 (2007)

    Article  Google Scholar 

  16. 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. D. Hristu-Varsakelis, W.S. Levine, Handbook of Networked and Embedded Control Systems (Birkhäuser, Boston, 2005)

    Book  MATH  Google Scholar 

  18. 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)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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. F.L. Lian, Analysis, design, modeling, and control of networked control systems. PhD thesis, Univ. Michigan, 2001

    Google Scholar 

  22. R. Luck, A. Ray, An observer-based compensator for distributed delays. Automatica 26(5), 903–908 (1990)

    Article  MATH  Google Scholar 

  23. W. Michiels, S.-I. Niculescu, Stability and Stabilization of Time-Delay Systems. An Eigenvalue-Based Approach (SIAM, Philadelphia, 2007)

    Book  MATH  Google Scholar 

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. 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)

    Article  MathSciNet  Google Scholar 

  26. 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)

    Article  MathSciNet  Google Scholar 

  27. A.S. Poznyak, Advanced Mathematical Tools for Automatic Control Engineers. Deterministic Techniques, vol. 1 (Elsevier, Amsterdam, 2008)

    Google Scholar 

  28. 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. Y.S. Suh, Stability and stabilization of nonuniform sampling systems. Automatica 44(12), 3222–3226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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. 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)

    Article  Google Scholar 

  32. S. Zampieri, Trends in networked control systems, in 17th IFAC World Congress, Seoul, Korea, July 2008

    Google Scholar 

  33. 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. W. Zhang, M.S. Branicky, S.M. Phillips, Stability of networked control systems. IEEE Control Syst. Mag. 21(6), 84–99 (2001)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Telecommunication, Université Paris-Est, ESIEE Paris, Noisy-le-Grand, France

    Arben Çela

  2. Electronic and Real-Time Systems Department, IFP New Energy, Rueil-Malmaison, France

    Mongi Ben Gaid

  3. School of Information Science and Engineering, Northeastern University, Shenyang, People’s Republic of China

    Xu-Guang Li

  4. L2S—Laboratoire des signaux et systèmes, Supélec, Gif-sur-Yvette, France

    Silviu-Iulian Niculescu

Authors
  1. Arben Çela
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mongi Ben Gaid
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xu-Guang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Silviu-Iulian Niculescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Çela, A., Ben Gaid, M., Li, XG., Niculescu, SI. (2014). Stability of DCESs Under the Hyper-Sampling Mode. In: Optimal Design of Distributed Control and Embedded Systems. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-02729-6_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-02729-6_10

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02728-9

  • Online ISBN: 978-3-319-02729-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation