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A Multiple Models Method for the Interval Uncertain Nonlinear System State Estimation

Received: 1 September 2023    Accepted: 22 September 2023    Published: 27 September 2023
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Abstract

The following paper proposes a novel Multiple Models Method for observer design to solve the problem of state and parameter estimation of uncertain nonlinear time-varying parameters systems with unknown but bounded disturbance. Classically speaking, an interval observer is a special class of observers that generates a bounded interval vector for the real state vector in a guaranteed way under the assumption that the uncertainties are unknown but bounded; it gives an upper and lower estimate for the system states at each time instant (determining a certain interval for the estimated state variations). Several approaches have been developed and adapted to different kinds of models (linear, nonlinear, fuzzy, etc.). However, in the proposed approach, the objective is not to design an interval observer, but rather a classical Luemberger observer, based on an interval multiple model of the nonlinear system model. The novelty introduced in the paper is about proposing a new interval Multiple Models representation of the uncertain nonlinear system. The observer’s gains are developed based on the Lyapunov stability theory proving that the state and parameter estimation errors are stable and converge to an origin-centred ball of a given radius to be minimized. The design conditions are formulated into linear matrix inequalities constraints, which can be efficiently solved. A numerical example is given to illustrate the design and validate the performance of the interval observers.

Published in International Journal of Industrial and Manufacturing Systems Engineering (Volume 8, Issue 2)
DOI 10.11648/j.ijimse.20230802.11
Page(s) 17-29
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Uncertain Systems, Sector Nonlinearity Approcha, Interval Multiple Models

References
[1] Pourasghar, M., Nguyen, A-T. and Guerra, T-M. (2021) Robust Zonotopic Observer Design: Avoiding Unmeasured Premise Variables for Takagi-Sugeno Fuzzy Systems. IFAC-PapersOnLine, 54 (4): 68-73.
[2] Wang, Z., Lim, C-C., and Shen, Y. (2018) Interval observer design for uncertain discrete-time linear systems. Journal of Systems & Control Letters, 116: 3077-41-46.
[3] Huang, J. , Zhang, H. and Ra˜Assi, T. (2023). Distributed Interval Estimation Methods for Multiagent Systems, in IEEE Systems Journal, 17 (2): 1843-1852.
[4] Bezzaoucha Rebai, S. “an Interval Multiple Models Approach for Uncertain Nonlinear Systems Estimation,” IECON, IEEE Industrial Electronics Society Conference, Brussels, Belgium, October 2022.
[5] Le, V. T. H., Stoica, C., Alamo, T., Camacho, E.F. and Dumur, D., Zonotopes: from guaranteed state estimation to Control, Hoboken: NJ, USA, Wiley, 2013.
[6] Scott, K. , Raimondo, D. M., Marseglia, G. R. and Braatz, R. D. (2016). Constrained zonotpes: a new tool for set- based estimation and fault detection. Automatica 69: pp 126-136.
[7] Blanchini, F. and Miani, S. Set-theoritic Methods in Control, Springer Nature switzerland AG, 2008.
[8] Rego, B. S. and Raffo, G. V. (2019). Suspended load path trackingcontrolusingatilt-rotorUAVbasedonzonotopic state-estimation. Journal of The Franklin Inst. vol 356 (4): 1695-1729.
[9] Lin, H., Zhai, G. and Antsalkis, P. “Set-valued observers for a class of discrete-time uncertain systems with persistent disturbances”, in American Control conference, Denver, Colorado, USA, 2003.
[10] Raissi, T., Efimov, D., and Zolghadri, A. (2012). Interval state estimation for a class of nonlinear systems. IEEE Transactions on Automatic Control, 51 (1): 260-265.
[11] Khan, A., Xie, W., Zhang, B. and Liu, L-W. (2021) A survey of interval observers design methods and implementation for uncertain systems Author links open overlay panel. Journal of the Franklin Institute, 358 (6): 3077-3126.
[12] Bezzaoucha, S., Marx, B., Maquin, D. and Ragot,J. (2013). Nonlinear joint state and parameter estimation: Application to a wastewater treatment plant. Control Engineering Practice, 21 (10): 1377-1385.
[13] Bezzaoucha, S., Voos, H. and Darouach, M. Book Chapter Part IV-Chapter 12: A survey on the polytopic Takagi-Sugeno approach: application to the inverted pendulum. Book title: The Inverted Pendulum: From Theory to New Innovations in Control and Robotics. The Institution of Engineering and Technology IET- publishing: 283-308, 2017.
[14] Rabehi, D., Meslem, N., Ramdani, N. An LMI approach to design interval observers for discrete-time linear switched systems. hal-02517130, Preprint submitted on 24 Mar 2020.
[15] Marouani, G.,Dinh, T. N.,Raissi, T. and Messaoud, H. “Interval observers design for discrete-time linear switched systems”. 2018 European Control Conference (ECC) June 12-15. Limassol, Cyprus, 2018.
[16] Pati, T. , Khajenejad, M., Daddala, S. P., and Yong, S. Z. “L1-Robust Interval Observer Design for Uncertain Nonlinear Dynamical Systems”. EEE Control Systems Letters paper presented at 2022 IEEE Conference on Decision and Control (CDC) December 6-9. Cancun, Mexico, 2022.
[17] Pourasghar, M., Puig, V. and Ocampo-Martinez, C. “Robust Zonotopic Observer Design: Interval Observer versus Set-membership Approaches,” On the 4th Conference on Control and Fault Tolerant Systems (SysTol), Casablanca, Morocco, 2019.
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  • APA Style

    Souad Bezzaoucha Rebai. (2023). A Multiple Models Method for the Interval Uncertain Nonlinear System State Estimation. International Journal of Industrial and Manufacturing Systems Engineering, 8(2), 17-29. https://doi.org/10.11648/j.ijimse.20230802.11

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    ACS Style

    Souad Bezzaoucha Rebai. A Multiple Models Method for the Interval Uncertain Nonlinear System State Estimation. Int. J. Ind. Manuf. Syst. Eng. 2023, 8(2), 17-29. doi: 10.11648/j.ijimse.20230802.11

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    AMA Style

    Souad Bezzaoucha Rebai. A Multiple Models Method for the Interval Uncertain Nonlinear System State Estimation. Int J Ind Manuf Syst Eng. 2023;8(2):17-29. doi: 10.11648/j.ijimse.20230802.11

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  • @article{10.11648/j.ijimse.20230802.11,
      author = {Souad Bezzaoucha Rebai},
      title = {A Multiple Models Method for the Interval Uncertain Nonlinear System State Estimation},
      journal = {International Journal of Industrial and Manufacturing Systems Engineering},
      volume = {8},
      number = {2},
      pages = {17-29},
      doi = {10.11648/j.ijimse.20230802.11},
      url = {https://doi.org/10.11648/j.ijimse.20230802.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijimse.20230802.11},
      abstract = {The following paper proposes a novel Multiple Models Method for observer design to solve the problem of state and parameter estimation of uncertain nonlinear time-varying parameters systems with unknown but bounded disturbance. Classically speaking, an interval observer is a special class of observers that generates a bounded interval vector for the real state vector in a guaranteed way under the assumption that the uncertainties are unknown but bounded; it gives an upper and lower estimate for the system states at each time instant (determining a certain interval for the estimated state variations). Several approaches have been developed and adapted to different kinds of models (linear, nonlinear, fuzzy, etc.). However, in the proposed approach, the objective is not to design an interval observer, but rather a classical Luemberger observer, based on an interval multiple model of the nonlinear system model. The novelty introduced in the paper is about proposing a new interval Multiple Models representation of the uncertain nonlinear system. The observer’s gains are developed based on the Lyapunov stability theory proving that the state and parameter estimation errors are stable and converge to an origin-centred ball of a given radius to be minimized. The design conditions are formulated into linear matrix inequalities constraints, which can be efficiently solved. A numerical example is given to illustrate the design and validate the performance of the interval observers.},
     year = {2023}
    }
    

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    T1  - A Multiple Models Method for the Interval Uncertain Nonlinear System State Estimation
    AU  - Souad Bezzaoucha Rebai
    Y1  - 2023/09/27
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ijimse.20230802.11
    DO  - 10.11648/j.ijimse.20230802.11
    T2  - International Journal of Industrial and Manufacturing Systems Engineering
    JF  - International Journal of Industrial and Manufacturing Systems Engineering
    JO  - International Journal of Industrial and Manufacturing Systems Engineering
    SP  - 17
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    PB  - Science Publishing Group
    SN  - 2575-3142
    UR  - https://doi.org/10.11648/j.ijimse.20230802.11
    AB  - The following paper proposes a novel Multiple Models Method for observer design to solve the problem of state and parameter estimation of uncertain nonlinear time-varying parameters systems with unknown but bounded disturbance. Classically speaking, an interval observer is a special class of observers that generates a bounded interval vector for the real state vector in a guaranteed way under the assumption that the uncertainties are unknown but bounded; it gives an upper and lower estimate for the system states at each time instant (determining a certain interval for the estimated state variations). Several approaches have been developed and adapted to different kinds of models (linear, nonlinear, fuzzy, etc.). However, in the proposed approach, the objective is not to design an interval observer, but rather a classical Luemberger observer, based on an interval multiple model of the nonlinear system model. The novelty introduced in the paper is about proposing a new interval Multiple Models representation of the uncertain nonlinear system. The observer’s gains are developed based on the Lyapunov stability theory proving that the state and parameter estimation errors are stable and converge to an origin-centred ball of a given radius to be minimized. The design conditions are formulated into linear matrix inequalities constraints, which can be efficiently solved. A numerical example is given to illustrate the design and validate the performance of the interval observers.
    VL  - 8
    IS  - 2
    ER  - 

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Author Information
  • Department of Electrical, Computer Engineering and Automation, EIGSI-École d’ingénieurs Généralistes, La Rochelle, France; MIA Mathématiques, Images & Applications Laboratory - La Rochelle University, La Rochelle, France

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