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Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis

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Book cover Complex System Modelling and Control Through Intelligent Soft Computations

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 319))

Abstract

This present chapter addresses the robust estimation problem for a class of nonlinear systems with unknown inputs and bilinear terms. The considered nonlinear system is represented by Takagi-Sugeno (T-S) Fuzzy Bilinear Model (FBM). Two cases are considered: the first one deals with the study of FBM with measurable decision variables and the second one assumes that these decision variables are unmeasurable. Then, the proposed Fuzzy Bilinear Observer (FBO) design for fuzzy bilinear models subject to unknown inputs is developed to ensure the asymptotic convergence of the error dynamic using the Lyapunov method. Stability analysis and gain matrices determination are performed by resolving a set of Linear Matrices Inequalities (LMIs) for both cases. The design conditions lead to the resolution of linear constraints easy to solve with existing numerical tools. The given observer is then applied for fault detection. This chapter studies also the problem of robust fault diagnosis based on a fuzzy bilinear observer. Sufficient conditions are established in order to guarantee the convergence of the state estimation error. Thus a residual generator is determined on the basis of LMI conditions such that the estimation error is sensitive to fault vector and insensitive to the unknown inputs. These results are provided for measurable and unmeasurable decision variables cases. The performances of the proposed estimation and fault diagnosis method is successfully applied to academic examples.

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References

  • Azar, A. T. (2010a). Adaptive neuro-fuzzy systems. In: A.T Azar (ed.), Fuzzy Systems. IN-TECH, Vienna, Austria.

    Google Scholar 

  • Azar, A. T. (2010b). Fuzzy systems. Vienna: IN-TECH.

    Google Scholar 

  • Azar, A. T. (2012). Overview of Type-2 fuzzy logic systems. International Journal of Fuzzy System Applications IJFSA, 2(4), 1–28.

    Article  MathSciNet  Google Scholar 

  • Bergsten, P., Palm, R., & Driankov, D. (2002). Observers for Takagi-Sugeno fuzzy systems. IEEE Transactions on System, Man, Cybernetics B, Cybernetics, 32(1), 114–121.

    Article  Google Scholar 

  • Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory. Studies in Applied Mathematics (volume 15). Philadelphia, PA: SIAM.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  • Chadli, M. & Borne, P. (2012). Multimodèles en Automatique: Outils Avancés d’Analyses et de Synthèse. Hermès-Lavoisier.

    Google Scholar 

  • Chadli, M. & Borne, P. (2013). Multiple models approach in automation: Takagi-Sugeno fuzzy systems. Hardcover.

    Google Scholar 

  • Chadli, M. & Coppier, H. (2013). Command-control for real-time systems. Hardcover, page 368.

    Google Scholar 

  • Chadli, M., & Guerra, T.-M. (2012). LMI solution for robust static output feedback control of Takagi-Sugeno fuzzy models. IEEE Transaction on Fuzzy Systems, 20(6), 1160–1165.

    Article  Google Scholar 

  • Chadli, M., & Karimi, H. R. (2012). Robust observer design for unknown inputs Takagi-Sugeno models. IEEE Transaction on Fuzzy Systems, 21(1), 158–164.

    Article  Google Scholar 

  • Darouach, M., Zasadzinski, M., & Xu, S. (1994). Full order observer for linear systems with unknown inputs. IEEE Transaction on Automatic Control, 39(3), 606–609.

    Article  MATH  MathSciNet  Google Scholar 

  • Gao, Z., Shi, X., & Ding, S. X. (2008). Fuzzy state/disturbance observer design for T-S fuzzy systems with application to sensor fault estimation. IEEE Transaction on System, Man, Cybernetics, Part B., 38(3), 875–880.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Hou, M., & Muller, P. (1992). Design of observers for linear systems with unknown inputs. IEEE Transaction on Automatic Control, 37(6), 871–874.

    Article  MATH  MathSciNet  Google Scholar 

  • Ichalal, D., Marx, B., Maquin, D., & Ragot, J. (2012). Observer design and fault tolerant control of Takagi-Sugeno nonlinear systems with unmeasurable premise variables. Fault Diagnosis in Robotic and Industrial Systems, Gerasimos Rigatos ed., iConceptPress.

    Google Scholar 

  • Ichalal, D., Marx, B., Ragot, J., & Maquin, D. (2009). Fault diagnosis for Takagi-Sugeno nonlinear systems. In 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, pages 504–509.

    Google Scholar 

  • Keller, H. (1987). Non-linear observer design by transformation into a generalized observer canonical form. International Journal of Control, 46(6), 1915–1930.

    Article  MATH  MathSciNet  Google Scholar 

  • Khalil, H. K. (1996). Nonlinear systems (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Lendek, Z., Lauber, J., Guerra, T. M., Babuska, R., & De Schutter, B. (2010). Adaptive observers for TS fuzzy systems with unknown polynomial inputs. Fuzzy Sets and Systems, 161(15), 2043–2065.

    Article  MATH  MathSciNet  Google Scholar 

  • Li, T. H. S., & Tsai, S. H. (2007). T-S Fuzzy bilinear model and fuzzy controller design for a class of nonlinear systems. IEEE Transaction on Fuzzy Systems, 15(3), 494–506.

    Article  Google Scholar 

  • Li, T. H. S., Tsai, S. H., Lee, J. Z., Hsiao, M. Y., & Chao, C. H. (2008). Robust H infinity fuzzy control for a class of uncertain discrete fuzzy bilinear systems. IEEE Transaction on System, Man, and Cybernetics, 38(2), 510–527.

    Article  Google Scholar 

  • Liu, X., & Zhang, Q. (2003). New approaches to H ∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI. Automatica, 39(9), 1571–1582.

    Article  MATH  MathSciNet  Google Scholar 

  • Ma, K. M. (2002). Observer design for a class of fuzzy systems. In Proceedings of the First International Conference on Machine Learning and Cybernetics, Vol. 1, pages 46–49.

    Google Scholar 

  • Ma, X. J., & Sun, Z. Q. (2001). Analysis and design of fuzzy reduced-dimensional observer and fuzzy functional observer. Fuzzy Sets and Systems, 120(1), 35–63.

    Article  MATH  MathSciNet  Google Scholar 

  • Marx, B., Koenig, D., & Ragot, J. (2007). Design of observers for Takagi-Sugeno descriptor systems with unknown inputs and application to fault diagnosis. Control Theory and Applications, IET, 1(5), 1487–1495.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Patton, R. J., Chen, J., & Lopez-Toribio, C. J. (1998). Fuzzy observers for nonlinear dynamic systems fault diagnosis. In 37th IEEE Conference on Decision and Control, Vol. 1, pages 84–89.

    Google Scholar 

  • Saoudi, D., Chadli, M., & Braeik, N. B. (2013a). Design of an active fault tolerant control based on the fuzzy bilinear observer for nonlinear systems. In 10th International Multi-Conference on Systems, Signals and Devices SSD’13, IEEE, pages 1–6.

    Google Scholar 

  • Saoudi, D., Chadli, M., & Braeik, N. B. (2013b). State estimation for unknown input fuzzy bilinear systems: application to fault diagnosis. In European Control Conference ECC’13, pages 2465–2470.

    Google Scholar 

  • Saoudi, D., Chadli, M., & Braeik, N. B. (2014). Robust estimation design for fuzzy bilinear systems with unmeasurable premise variables. In International Conference on Control, Engineering and Information Technology CEIT’14, pages 1–6.

    Google Scholar 

  • Saoudi, D., Chadli, M., Mechmeche, C., & Braeik, N. B. (2010). T-S fuzzy bilinear observer for a class of nonlinear systems. In 18th Medit. Conf. Contr. Aut., pages 1395–1400.

    Google Scholar 

  • Saoudi, D., Chadli, M., Mechmeche, C., & Braeik, N. B. (2012a). unknown input observer design for fuzzy bilinear systems: an LMI approach. Journal of Mathematical Problems in Engineering, MPE’12, 2012(Special section p1):1–21.

    Google Scholar 

  • Saoudi, D., Mechmeche, C., Chadli, M., & Braeik, N. B. (2012b). design of multimodel bilinear observers for Takagi-Sugeno discrete models. In International Symposium on Security and Safety of Complex Systems 2SCS’12.

    Google Scholar 

  • Saoudi, D., Mechmeche, C., Chadli, M., & Braeik, N. B. (2012c). Robust residual generator design for Takagi-Sugeno fuzzy bilinear systems subject to unknown inputs. In 8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Safeprocess’12, pages 1023–1028.

    Google Scholar 

  • Takagi, T., & Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transaction on System, Man, Cybernetics, SMC, 15(1), 116–132.

    Article  MATH  Google Scholar 

  • Tanaka, K., Ikeda, T., & Wang, H. O. (1998). Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Transaction on Fuzzy System, 6(2), 250–265.

    Article  Google Scholar 

  • Tanaka, K., & Sugeno, M. (1992). Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems, 45(2), 135–156.

    Article  MATH  MathSciNet  Google Scholar 

  • Tanaka, K. & Wang, H. O. (2000). Fuzzy control systems design and analysis: a linear matrix inequality approach. Wiley Inter-Science.

    Google Scholar 

  • Taniguchi, T., Tanaka, K., & Wang, H. O. (2000). Fuzzy descriptor systems and nonlinear model following control. IEEE Transaction on Fuzzy Systems, 8(4), 442–452.

    Article  Google Scholar 

  • Tong, S. & Tang, Y. (2000). Analysis and design of fuzzy robust observer for uncertain nonlinear systems. In Proceedings of 9th IEEE International Conference on Fuzzy System, Vol. 2, pages 993–996.

    Google Scholar 

  • Tsai, S. H., & Li, T. H. S. (2007). robust fuzzy control of a class of fuzzy bilinear systems with time-delay. Chaos, Solitons and Fractals, 39(5), 2028–2040.

    Article  MathSciNet  Google Scholar 

  • Yang, F., & Wilde, R. W. (1988). Observers for linear systems with unknown inputs. IEEE Transaction on Automatic Control, 33(7), 677–681.

    Article  MATH  MathSciNet  Google Scholar 

  • Yoneyama, J. (2009). H ∞ filtering for fuzzy systems with immeasurable premise variables: an uncertain system approach. Fuzzy Sets and Systems, 160(12), 1738–1748.

    Article  MATH  MathSciNet  Google Scholar 

  • Yoneyama, J., Nishikawa, M., Katayama, H., & Ichikawa, A. (2000). Output stabilization of Takagi-Sugeno fuzzy systems. Fuzzy Sets and Systems, 111(2), 253–266.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. LSA—Laboratoire des Systèmes Avancés, Polytechnic High School of Tunisia, University of Carthage, BP 743, 2078, La Marsa, Tunisia

    Dhikra Saoudi & Naceur Benhadj Braeik

  2. UPJV-MIS—University of Picardie Jules Verne, 33, Rue Saint-Leu, 80039, Amiens, France

    Mohammed Chadli

Authors
  1. Dhikra Saoudi
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  2. Mohammed Chadli
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  3. Naceur Benhadj Braeik
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Correspondence to Dhikra Saoudi .

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Editors and Affiliations

  1. Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom

    Quanmin Zhu

  2. Faculty of Computers and Information, Benha University, Benha, Egypt

    Ahmad Taher Azar

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Saoudi, D., Chadli, M., Braeik, N.B. (2015). Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis. In: Zhu, Q., Azar, A. (eds) Complex System Modelling and Control Through Intelligent Soft Computations. Studies in Fuzziness and Soft Computing, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-319-12883-2_23

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  • DOI: https://doi.org/10.1007/978-3-319-12883-2_23

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12882-5

  • Online ISBN: 978-3-319-12883-2

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