Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Observer Design for Polytopic Systems: Application to Chaotic System Reconstruction

  • Conference paper
Nostradamus: Modern Methods of Prediction, Modeling and Analysis of Nonlinear Systems

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 192))

  • 1373 Accesses

Abstract

Many studies concerning design of controllers and observers for a class of nonlinear systems described in polytopic representation are carried out. Such representation includes Takagi-Sugeno models, LPV models, switching models, PLDI... Particularly, T-S models are obtained by interpolation of M local LTI (linear time invariant) models throughout convex functions. The choice of the number of local models may be intuitively chosen by considering some operating regimes. Each LTI model can be obtained by using a direct linearization of an a priori nonlinear model around operating points, or alternatively by using an identification procedure. Based on the Lyapunov method and Linear Matrix Inequalities (LMI) formulation, sufficient conditions have been derived for controllers and observers design. Recently, systems subject to unknown inputs are considered for measurable and immeasurable decision variables. Unknown inputs can result either from model uncertainty, faults or due to the presence of unknown external excitation. These different results have been widely applied in the field of fault diagnosis (FDI), fault tolerance (FTC) and also for secure communications. Indeed, the increasing need of secure communications leads to the development of many techniques which make difficult the detecting of transmitted message. Based on unknown inputs observer design, many works have been carried out on secure communication and chaotic system reconstruction problem. In this framework, unknown inputs Takagi-Sugeno fuzzy observer has been exstensively used. The design of such observers is considered based on LMI and Lyapunov methods. The pole placement in an LMI region is also considered to improve the observer performances. Examples are given to illustrate a chaotic cryptosystem procedure where the plaintext (message) is encrypted using chaotic signals at the drive system side and the plaintext is retrieved via the designed unknown input observer.

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 PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Boyd, S., et al.: Linear matrix inequalities in systems and control theory. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  2. Murray-Smith, R., Johansen, T.: T-S model approaches to modelling and control. Taylor & Francis (1997)

    Google Scholar 

  3. Chadli, M., Akhenak, A., Ragot, J., Maquin, D.: State and Unknown Input Estimation for Discrete Time T-S Model. Journal of Franklin Institute (2009) (in press), doi:10.1016/j.jfranklin.2009.02.011

    Google Scholar 

  4. Chadli, M., Maquin, D., Ragot, J.: An LMI formulation for output feedback stabilisation in T-S model approach. In: IEEE 41st Conference on Decision Control, USA, December 10-13 (2002)

    Google Scholar 

  5. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its application to modelling and control. IEEE Trans. on Systems, Man, Cybernetics. 15(1), 116–132 (1985)

    Article  MATH  Google Scholar 

  6. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A linear Matrix Inequality Approach. John Wiley & Sons, Inc. (2001)

    Google Scholar 

  7. Chadli, M., Akhenak, A., Ragot, J., Maquin, D.: On the Design of Observer for Unknown Iinputs Fuzzy Models. International Journal of Automation and Control 2(1), 113–125 (2008)

    Article  Google Scholar 

  8. Kim, E., Lee, H.: New approaches to relaxed quadratic stability condition of fuzzy control systems. IEEE Trans. on Fuzzy Sets 8(5), 523–534 (2000)

    Article  Google Scholar 

  9. Xiaodiong, L., Qingling, Z.: New approach to H  ∞  controller designs based on observers for T-S fuzzy systems via LMI. Automatica 39, 1571–1582 (2003)

    Article  Google Scholar 

  10. Guan, Y., Saif, M.: A novel approach to the design of unknown input observers. IEEE Transactions on Automatic Control 36(5), 632–635 (1991)

    Article  Google Scholar 

  11. Floquet, T., Barbot, J.P.: A sliding mode approach of unknown input observers for linear systems. In: IEEE Conference on Desicion and Control, pp. 1724–1729 (2004)

    Google Scholar 

  12. Yang, F., Wilde, R.W.: Observers for linear systems with unknown inputs. IEEE Trans. Automatic Control 33, 677–681 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Darouach, M., Zasadzinski, M., Xu, S.J.: Full-order observers for linear systems with unknown inputs. IEEE Trans. Automatic Control 39, 606–609 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Syrmos, V.L.: Computational observer design techniques for linear systems with unknown inputs using the concept of transmission zeros. IEEE Transactions on Automatic Control 38(5), 790–794 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin, S.F., Wang, A.P.: Unknown input observers for singular systems designed by eigenstructure assignment. Journal of the Franklin Institute 340(1), 43–61 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koenig, D.: Unknown input proportional T-S-integral observer design for descriptor systems: application to state and fault estimation. IEEE Transactions on Automatique Control 5(2), 213–217 (2005)

    MathSciNet  Google Scholar 

  17. Ha, Q.P., Trinh, H.: State and input simultaneous estimation for a class of nonlinear systems. Automatica 40(10), 1779–1785 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pikovsky, A., Rosemblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press (2001) ISBN 0-521-53352-X

    Google Scholar 

  19. Gonzalez-Miranda, J.M.: Synchronization and Control of Chaos. An introduction for scientists and engineers. Imperial College Press (2004) ISBN 1-86094-488-4

    Google Scholar 

  20. Controlling chaos. In: Schuster, H.G. (ed.) Handbook of Chaos Control. Wiley-VCH, New York

    Google Scholar 

  21. Sushchik, M.M., Rulkov, N.F., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. In: Proceedings of 1995 Intl. Symp. on Nonlinear Theory and Appl., vol. 2, pp. 949–952. IEEE (1995)

    Google Scholar 

  22. Brown, R., Rulkov, N.F., Tracy, E.R.: Modeling and synchronization chaotic system from time-series data. Phys. Rev. E 49, 3784 (1994)

    Article  Google Scholar 

  23. Nolle, L., Goodyear, A., Hopgood, A.A., Picton, P.D., Braithwaite, N.: On Step Width Adaptation in Simulated Annealing for Continuous Parameter Optimisation. In: Reusch, B. (ed.) Fuzzy Days 2001. LNCS, vol. 2206, pp. 589–598. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  24. Nolle, L., Zelinka, I., Hopgood, A.A., Goodyear, A.: Comparison of an self organizing migration algorithm with simulated annealing and differential evolution for automated waveform tuning. Advances in Engineering Software 36(10), 645–653 (2005)

    Article  Google Scholar 

  25. Zelinka, I., Nolle, L.: Plasma reactor optimizing using differential evolution. In: Price, K.V., Lampinen, J., Storn, R. (eds.) Differential Evolution: A Practical Approach to Global Optimization, pp. 499–512. Springer, New York (2006)

    Google Scholar 

  26. Zelinka, I.: Investigation on Evolutionary Deterministic Chaos Control. IFAC, Prague (2005)

    Google Scholar 

  27. Ivan, Z.: Investigation on Evolutionary Deterministic Chaos Control – Extended Study. In: 19th International Conference on Simulation and Modeling (ECMS 2005), Riga, Latvia, June 1-4 (2005b)

    Google Scholar 

  28. Zelinka, I., Senkerik, R., Navratil, E.: Investigation on Evolutionary Optimitazion of Chaos Control. Chaos, Solitons, Fractals (2007), doi:10.1016/j.chaos.2007.07.045

    Google Scholar 

  29. Zelinka, I., Celikovsky, S., Richter, H., Chen, G.: Evolutionary Algorithms and Chaotic Systems. Springer, Germany (2010)

    Book  MATH  Google Scholar 

  30. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis: A linear Matrix Inequality Approach. John Wiley & Sons, Inc. (2001)

    Google Scholar 

  31. Chadli, M., Maquin, D., Ragot, J.: Stability analysis and design for continuous-time Takagi-Sugeno control systems. International Journal of Fuzzy Systems 7(3), 101–109 (2005)

    MathSciNet  Google Scholar 

  32. Johansson, M., Rantzer, A., Arzn, K.: Piecewise quadratic stability of fuzzy systems. IEEE Trans. on Fuzzy Systems 7(6), 713–722 (1999)

    Article  Google Scholar 

  33. Tanaka, K., Hori, T., Wang, H.O.: A multiple Lyapunov function approach to stabilization of fuzzy control systems. IEEE Transactions on Fuzzy Systems 11(4), 582–589 (2003)

    Article  Google Scholar 

  34. Chadli, M.: An LMI approach to design observer for unknown inputs Takagi-Sugeno fuzzy models. Asian Journal of Control 12(4), 524–530 (2010)

    MathSciNet  Google Scholar 

  35. Chilali, M., Gahinet, P.: H  ∞  Design with pole placement constraints: an LMI approch. IEEE Transactions on Automatic Control 41(3), 358–367 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. Li, C., Liao, X., Wong, K.: Lag synchronization of hyperchaos with application to secure communications. Chaos, Solitons and Fractals 23, 183–193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Chen, M., Zhou, D., Shang, Y.: A new observer-based synchronization scheme for private communication. Chaos, Solitons and Fractals 24, 1025–1030 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Boutayeb, M., Darouach, M., Rafaralahy, H.: Generalized State-Space Observers for Chaotic Synchronization and Secure Communication. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 49(3), 345–349 (2002)

    Article  MathSciNet  Google Scholar 

  39. Alvares, G., Montoya, F., Romera, M., Pastor, G.: Breaking parameter modulated chaotic secure communication system. Chaos, Solitons & Fractals 21(4), 783–787 (2004)

    Article  Google Scholar 

  40. Akhenak, A., Chadli, M., Ragot, J., Maquin, D.: Unknown input multiple observer based approach: application to secure communication. In: 1st IFAC Conference on Analysis and Control of Chaotic Systems, Reims, France, June 28-30 (2006)

    Google Scholar 

  41. Edwards, C., Spurgeon, S.K., Patton, R.J.: Sliding mode observers for fault detection and isolation. Automatica 36(4), 541–553 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  43. Chadli, M., Karimi, H.R.: Robust Observer Design for Unknown Inputs Takagi-Sugeno Models. IEEE Trans. on Fuzzy Systems (2012), doi:10.1109/TFUZZ.2012.2197215

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Picardie Jules Verne, 7, Rue du Moulin Neuf, 80000, Amiens, France

    Mohammed Chadli

Authors
  1. Mohammed Chadli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed Chadli .

Editor information

Editors and Affiliations

  1. Faculty of Electrical Engineering and Co, Department of Computer Science, VÅ B-Technical University of Ostrava, 17. listopadu 15, Ostrava-Poruba, 70833, Czech Republic

    Ivan Zelinka

  2. Institute of Physical and, Theoretical Chemistry, University of Tuebingen, Auf der Morgenstelle 8, Tuebingen, 72076, Germany

    Otto E. Rössler

  3. , Faculty of Electrical Engineering, Department of Computer Science, 17. listopadu 15, Ostrava-Poruba, 708 33, Czech Republic

    Václav Snášel

  4. (MIR Labs), Scientific Network for Innovation and, Machine Intelligence Research Labs, MIR Labs Campus, Auburn, 98071, Washington, USA

    Ajith Abraham

  5. , Dept. de Informática y Automática, University of Salamanca, Plaza de la Merced, s/n, Salamanca, 37008, Spain

    Emilio S. Corchado

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chadli, M. (2013). Observer Design for Polytopic Systems: Application to Chaotic System Reconstruction. In: Zelinka, I., Rössler, O., Snášel, V., Abraham, A., Corchado, E. (eds) Nostradamus: Modern Methods of Prediction, Modeling and Analysis of Nonlinear Systems. Advances in Intelligent Systems and Computing, vol 192. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33227-2_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33227-2_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33226-5

  • Online ISBN: 978-3-642-33227-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation