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Improved LMI Conditions for Unknown Input Observer Design of Discrete-time LPV Systems


This paper presents a novel unknown input observer (UIO) design for discrete-time linear parameter-varying (LPV) systems. One feature of the proposed approach is the ability of handling LPV systems with variation in both states and outputs matrices. The design problem has been formulated using a less conservative discrete-time stability condition which allows to obtain proportional UIO and proportional-integral UIO structures. Existence conditions for both structures are provided. These poly-quadratic conditions rely on the use of parameter-dependent Lyapunov functions defined in terms of Linear Matrix Inequalities. Furthermore, an extended formulation using the performance index is also derived. Finally, the effectiveness of the proposed design method is illustrated through numerical examples.

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  1. [1]

    J. Daafouz, G. Millerioux, and L. Rosier, “Observer design with guaranteed bound for LPV systems,” IFAC Proceedings Volumes, vol. 38, no. 1, pp. 107–112, 2005.

    Article  Google Scholar 

  2. [2]

    D. Ichalal and S. Mammar, “On unknown input observers for LPV systems,” IEEE Transactions on Industrial Electronics, vol. 62, no. 9, pp. 5870–5880, September 2015.

    Article  Google Scholar 

  3. [3]

    W. P. M. Heemels, J. Daafouz, and G. Millerioux, “Observer-based control of discrete-time LPV systems with uncertain parameters,” IEEE Transactions on Automatic Control, vol. 55, no. 9, pp. 2130–2135, September 2010.

    MathSciNet  Article  Google Scholar 

  4. [4]

    D. Rotondo, F. R. López-Estrada, F. Nejjari, J. C. Ponsart, D. Theilliol, and V. Puig, “Actuator multiplicative fault estimation in discrete-time LPV systems using switched observers,” Journal of the Franklin Institute, vol. 353, no. 13, pp. 3176–3191, September 2016.

    MathSciNet  Article  Google Scholar 

  5. [5]

    S. Ijaz, M. T. Hamayun, H. Anwaar, L. Yan, and M. K. Li, “LPV modeling and tracking control of dissimilar redundant actuation system for civil aircraft,” International Journal of Control, Automation and Systems, vol. 17, no. 3, pp. 705–715, March 2019.

    Article  Google Scholar 

  6. [6]

    D. Rotondo, F. Nejjari, A. Torren, and V. Puig, “Fault tolerant control design for polytopic uncertain LPV systems: application to a quadrotor,” Proc. of Conference on Control and Fault-Tolerant Systems, pp. 643–648, October 2013.

  7. [7]

    P. Apkarian, P. Gahinet, and G. Becker, “Self-scheduled control of linear parameter-varying systems: a design example,” Automatica, vol. 31, no. 9, pp. 1251–1261, September 1995.

    MathSciNet  Article  Google Scholar 

  8. [8]

    A. J. Pérez-Estrada, G. L. Osorio-Gordillo, M. Darouach, and V. H. Olivares-Peregrino, “H generalized dynamic unknown inputs observer design for discrete LPV systems. Application to wind turbine,” European Journal of Control, vol. 44, pp. 40–49, November 2018.

    MathSciNet  Article  Google Scholar 

  9. [9]

    V. F. Montagner, R. C. L. F. Oliveira, V. J. S. Leite, and P. L. D. Peres, “Gain scheduled state feedback control of discrete-time systems with time-varying uncertainties: an LMI approach,” Proceedings of the 44th IEEE Conference on Decision and Control, pp. 4305–4310, December 2005.

  10. [10]

    J. Li, Z. Wang, Y. Shen, and Y. Liu, “Unknown input observer design for Takagi-Sugeno systems with fuzzy output equation,” International Journal of Control, Automation and Systems, vol. 17, no. 1, pp. 267–272, January 2019.

    Article  Google Scholar 

  11. [11]

    S.-K. Chang, You, W.-T., and P.-L. Hsu, “Design of general structured observers for linear systems with unknown inputs,” Journal of the Franklin Institute, vol. 334, no. 2, pp. 213–232, March 1997.

    MathSciNet  Article  Google Scholar 

  12. [12]

    J. J. Martinez, N. Loukkas, and N. Meslem, “H-infinity set-membership observer design for discrete-time LPV systems,” International Journal of Control, pp. 1–25, December 2018.

  13. [13]

    O. Sename, P. Gáspár, and J. Bokor, Robust Control and Linear Parameter Varying Approaches: Application to Vehicle Dynamics, vol. 437, Springer, 2013.

  14. [14]

    K. Watanabe and D. M. Himmelblau, “Instrument fault detection in systems with uncertainties,” International Journal of Systems Science, vol. 13, no. 2, pp. 137–158, May 1982.

    Article  Google Scholar 

  15. [15]

    H. Chen, D. Du, D. Zhu, and Y. Yang, “UIO-based fault estimation and accommodation for nonlinear switched systems,” International Journal of Control, Automation and Systems, vol. 17, no. 2, pp. 435–444, January 2019.

    Article  Google Scholar 

  16. [16]

    F. Xu, J. Tan, X. Wang, V. Puig, B. Liang, B. Yuan, and H. Liu, “Generalized set-theoretic unknown input observer for LPV systems with application to state estimation and robust fault detection,” International Journal of Robust and Nonlinear Control, vol. 27, no. 17, pp. 3812–3832, February 2017.

    MathSciNet  MATH  Google Scholar 

  17. [17]

    A. P. Pandey and M. C. de Oliveira, “On the necessity of LMI-based design conditions for discrete time LPV filters,” IEEE Transactions on Automatic Control, vol. 63, no. 9, pp. 3187–3188, January 2018.

    MathSciNet  Article  Google Scholar 

  18. [18]

    S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1994.

    Book  Google Scholar 

  19. [19]

    J. Lofberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” Proc. of IEEE International Conference on Robotics and Automation, pp. 284–289, September 2004.

  20. [20]

    J. F. Sturm, “Using SeDuMi 1.02, A Matlab toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11, no. 1–4, pp. 625–653, January 1999.

    MathSciNet  Article  Google Scholar 

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Corresponding author

Correspondence to Matheus Senna de Oliveira.

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Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Recommended by Associate Editor Nam H. Jo under the direction of Editor Jessie (Ju H.) Park. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior — Brasil (CAPES) — Finance Code 001.

Matheus Senna de Oliveira received his B.Eng. degree in Control and Automation and an M.S in Electrical Engineering from the Pontifical Catholic University of Rio Grande do Sul, in 2015 and 2018, respectively. He is currently working toward a Ph.D. degree in Electronic Engineering at the Instituto Tecnológico de Aeronáutica. His main research interests are robust control, LPV systems, performance criteria and LMIs.

Renan Lima Pereira received his B.Sc. degree in electrical engineering from Federal University of Maranhäo in 2009, and his M.Sc and Ph.D. degrees in electronic engineering from Instituto Tecnológico de Aeronáutica, in 2011 and 2014, respectively, where he has been a professor since 2018. His research interests include robust control, gain-scheduled control and optimization.

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de Oliveira, M.S., Pereira, R.L. Improved LMI Conditions for Unknown Input Observer Design of Discrete-time LPV Systems. Int. J. Control Autom. Syst. 18, 2543–2551 (2020).

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  • Discrete-time LPV systems
  • linear matrix inequalities
  • poly-quadratic conditions
  • unknown input observers