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Synthesis of a Minimum Functional State Observer Approach for Unperturbed/Perturbed Dynamical Systems

  • Ridha Aloui
  • Naceur Benhadj Braiek
Regular Papers Control Theory and Applications
  • 16 Downloads

Abstract

In this paper, we investigate the synthesis of minimum functional state observer for unperturbed and perturbed linear time invariant systems. The principal contribution is the design of a minimum functional state observer, which can estimate directly the state feedback control law. Hence, for the linear time invariant systems, the existence conditions of a minimum functional state observer are obtained by verification of a special dimension condition on system matrices. As a matter of fact, the exact solution of the proposed approach is determined, and the minimum functional state observer that has the same dimension as the control vector is derived by solving a set of linear matrix inequality (LMI) constraints. Whereas, for perturbed linear systems, the proposed minimum functional state observer scheme is developed to ensure the robust quadratic stability of the augmented system. The robustness issue is given via the reconstructed control law designed using an LMI based H method; so that the desired design matrices are derived through the resolution of an optimum LMI system. The effectiveness and usefulness of the proposed approach are validated through a flexible link robot example.

Keywords

Functional observer design LMI optimization problem minimum order observer state feedback control unperturbed/perturbed linear time invariant systems 

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References

  1. [1]
    D.G. Luenberger, “Observers for multivariable systems,” IEEE. Trans. Automat. Contr., vol. 11, pp. 190–197, 1966.CrossRefGoogle Scholar
  2. [2]
    Y. Xu, R. Lu, P. Shi, J. Tao, and S. Xie, “Robust estimation for neural networks with randomly occurring distributed delays and Markovian jump coupling,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 4, pp. 845–855, 2018.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Y. Xu, R. Lu, P. Shi, H. Li, and S. Xie, “Finite-time distributed state estimation over sensor networks with round-robin protocol and fading channels,” IEEE Transactions on Cybernetics, vol. 48, no. 1, pp. 336–345, 2018.CrossRefGoogle Scholar
  4. [4]
    Y. Wu, X. Meng, L. Xie, R. Lu, H. Su, and Z. Wu, “An input-based triggering approach to leaderfollowing problems,” Automatica, vol. 75, pp. 221–228, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Y. Wu, R. Lu, P. Shi, H. Su, and Z. Wu, “Adaptive output synchronization of heterogeneous network with an uncertain leader,” Automatica, vol. 76, pp. 183–192, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    N. Benhadj Braiek and F. Rotella, “State observer design for a class of nonlinear system,” J. Syst. Analysis, Modelling and Simulation, vol. 17, pp. 211–227, 1995.zbMATHGoogle Scholar
  7. [7]
    R. Aloui and N. Benhadj Braiek, “On the determination of an optimal state observer gain for multivariable systems: Application to induction motors,” J. of Automation and Systems Engineering, vol. 2, pp. 206–218, 2008.Google Scholar
  8. [8]
    W. Y. Leong, H. Trinh, and T. Fernando, “A practical functional observer scheme for interconnected time-delay systems,” Int. J. of Control, vol. 88, pp. 1963–1973, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Y. Wu, H Su, P. Shi, Z. Shu, and Z. G. Wu, “Consensus of multiagent systems using aperiodic sampled-data control,” IEEE Transactions on Cybernetics, vol. 46, pp. 2132–2143, 2016.CrossRefGoogle Scholar
  10. [10]
    J. Y. Ng, C. P. Tan, H. Trinh and K.Y. Ng, “A common functional observer scheme for three systems with unknown inputs,” J. of the Franklin Institute, vol. 353, no. 10, pp. 2237–2257, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    H. Hassani, J. Zarei, M. Chadli, and J. Qiu, “Unknown input observer design for interval type-2 TS fuzzy systems with immeasurable premise variables,” IEEE Transactions on Cybernetics, vol. 47, no. 9, pp. 2639–2650, 2017.CrossRefGoogle Scholar
  12. [12]
    Y. Xu, R. Lu, H. Peng, K. Xie, and A. Xue, “Asynchronous dissipative state estimation for stochastic complex networks with quantized jumping coupling and uncertain measurements,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, pp. 268–277, 2017.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Y. Wu, H. Su, P. Shi, R. Lu, and Z. Wu, “Output synchronization of nonidentical linear multiagent systems,” IEEE Transactions on Cybernetics, vol. 47, pp. 130–141, 2017.CrossRefGoogle Scholar
  14. [14]
    C. Liu, H. Lam, T. Fernando, and H. H. Iu, “Design of fuzzy functional observer-controller via higher order derivatives of Lyapunov function for nonlinear systems,” IEEE Transactions on Cybernetics, vol. 47, pp. 1630–1640, 2017.CrossRefGoogle Scholar
  15. [15]
    L. Li, M. Chadli, S. X. Ding, J. Qiu, and Y. Yang, “Diagnostic observer design for TS fuzzy systems: application to realTime weighted fault detection approach,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 2, pp. 805–816, 2018.CrossRefGoogle Scholar
  16. [16]
    A. S. Tlili, “Robustness design of a dynamic outputfeedback decentralized controller using H¥ synthesis and LMI paradigm,” International Journal of Control, Automation, and Systems, vol. 15, no. 4, pp. 1544–1552, 2017.CrossRefGoogle Scholar
  17. [17]
    J. R. Roman and T. E. Bullock, “Design of minimal order stable observers for linear functions of the state via realization theory,” IEEE Transactions on Automatic Control, vol. 20, pp. 613–622, 1975.CrossRefzbMATHGoogle Scholar
  18. [18]
    M. Darouach, “Functional observers for linear descriptor systems,” Proc. of 17th Mediterranean Conference on Control and Automation, 2009.Google Scholar
  19. [19]
    H. Trinh and T. Fernando, Functional Observers for Dynamical Systems, Springer, 2012.CrossRefzbMATHGoogle Scholar
  20. [20]
    R. Aloui, R. Ouhibi, and N. Benhadj Braiek, “Minimal functional observer for linear time invariant systems,” Proc. of Int. Conf. on Control, Engineering and Information Technology (CEIT’13), vol. 4, pp. 165–168, 2013.Google Scholar
  21. [21]
    R. Mohajerpoor, H. Abdi, and S. Nahavandi, “Minimal unknown-input functional observers for multi-input multioutput LTI systems,” J. Process Control, vol. 35, pp. 143–153, 2015.CrossRefGoogle Scholar
  22. [22]
    N. Eskandari, Z. J. Wang, and G. A. Dumont, “On the existence and design of functional observers for LTI systems, with application to user modeling,” Asian Journal of Control, vol. 18, no. 1, pp. 192–205, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    F. Rotella and I. Zambettakis, “A direct design procedure for linear state functional observers,” Automatica, vol. 70, pp. 211–216, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Darouach, “Existence and design of functional observers for linear systems,” IEEE Transactions on Automatic Control, vol. 45, pp. 940–943, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    T. E. Fortmann and D. Williamson, “Design of low-order observers for linear feedback control laws,” IEEE Transactions on Automatic Control, vol. 17, pp. 301–308, 1972.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Hou, A. C. Pugh, and P. C. Muller, “Disturbance decoupled functional observers,” IEEE Transactions on Automatic Control, vol. 44, pp. 382–386, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    P. Murdoch, “Observer design for a linear functional of the state vector,” IEEE Transactions on Automatic Control, vol. 18, pp. 308–310, 1973.MathSciNetCrossRefGoogle Scholar
  28. [28]
    D. W. Russell and T. E. Bullock, “A frequency domain approach to minimal-order observer design for several linear functions of the state,” IEEE Transactions on Automatic Control, vol. 22, pp. 600–604, 1977.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    T. Fernando, K. Emami, S. Yu, H. H. Iu, and K. P. Wong, “A Novel quasi-decentralized functional observer approach to LFC of interconnected power systems,” Accepted by IEEE Transactions on Pouwer Systems, 2016.Google Scholar
  30. [30]
    C. C. Tsui, “What is the minimum function observer order,” Proc. 3rd Word Congress Intell. Control Autom., China, pp. 2811–2816, 2000.Google Scholar
  31. [31]
    T. L. Fernando, H. M. Trinh, and L. Jennings, “Functional observability and the design of minimum order linear functional observers,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1268–1273, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    C. Y. Kee, C. P. Tan, K. Y. Ng, and H. Trinh, “New results in robust functional state estimation using two sliding mode observers in cascade,” Int. J. Robust Nonlinear Control, vol. 24, pp. 2079–2097, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    A. Z. Asanov and D. N. Demyanov, “Analytical synthesis of functional low-order observers,” Journal of Computer and Systems Sciences International, vol. 54, no. 4, pp. 505–513, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    C. Tsui, “Observer design-a survey,” Int. J. Automation and Computing, vol. 12, no. 1, pp. 50–61, 2015.CrossRefGoogle Scholar
  35. [35]
    R. Mohajerpoor, L. Shanmugam, H. Abdi, S. Nahavandi, and J. H. Park, “Delay-dependent functional observer design for linear systems with unknown timevarying state delays,” IEEE Transactions on Cybernetics, DOI:10.1109/TCYB.2017.2726106, 2017.Google Scholar
  36. [36]
    Y. Xiong and M. Saif, “Unknown dusturbance inputs estimation based on a state functional observer design,” Automatica, vol. 39, pp. 1389–1398, 2003.CrossRefzbMATHGoogle Scholar
  37. [37]
    H. Deng and H. X. Li, “Functional observers for linear systems with unknown inputs,” Asian Journal of Control, vol. 6, pp. 462–468, 2004.CrossRefGoogle Scholar
  38. [38]
    T. Fernando and H. Trinh, “Design of redured-order state/unknown input observers based on a descriptor system approach,” Asian Journal of Control, vol. 9, pp. 458–465, 2007.MathSciNetCrossRefGoogle Scholar
  39. [39]
    M. Darouach, “On the functional observersfor linear descriptor systems,” Systems & Control Letters, vol. 61, pp. 427–434, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Sassi, M. Zasadzinski, A. H. Souley, and K. Abderrahim, “A robust reduced order functional H¥ observer for time delay bilinear systems,” Proc. of 25th Mediterranean Conference on Control and Automation (MED), Malta, pp. 637–642, 2017.Google Scholar
  41. [41]
    Q. P. Ha, H. Trinh, H. T. Nguyen, and H. D. Tuan, “Dynamic output feedback sliding mode control using pole placement and linear functional observers,” IEEE Transactions on Industrial Electronics, vol. 50, pp. 1030–1037, 2003.CrossRefGoogle Scholar
  42. [42]
    M. Ezzine, H. Souley Ali, M. Darouach, and H. Messaoud, “A controller design based on a functional H¥ filter for descriptor systems: the time and frequency domain cases,” Automatica, vol. 48, pp. 542–549, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    J. Lan and R. J. Patton, “Robust fault-tolerant control based on a functional observer for linear descriptor systems,” IFAC-PapersOnLine, vol. 48, pp. 138–143, 2015.CrossRefGoogle Scholar
  44. [44]
    C. C. Tsui, “An overview of the applications and solution of a fundamental matrix equation pair,” J. of Franklin Institute, vol. 341, pp. 465–75, 2004.MathSciNetCrossRefGoogle Scholar
  45. [45]
    H. Trinh, S. Nahavandi, and T. D. Tran, “Algorithms for designing reduced-order functional observers of linear systems,” Int. J. Innovative Computing, Information and Control, vol. 4, pp. 321–333, 2008.Google Scholar
  46. [46]
    F. Rotella and I. Zambettakis, “Minimal single linear functional observers for linear systems,” Automatica, vol. 47, pp. 164–169, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    F. Rotella and I. Zambettakis, “On minimum functional observers for linear time-varying systems,” IEEE Transactions on Automatic Control, 2011.Google Scholar
  48. [48]
    H. M. Tran and H. Trinh, “Minimal-order functional observer-based residual generators for fault detection and isolation of dynamical systems,” Mathematical Problems in Engineering, vol. 2016, Article ID 2740645,17 pages, 2016.Google Scholar
  49. [49]
    R. Mohajerpoor, L. Shanmugama, H. Abdia, S. Nahavandia, and M. Saif, “Functional observer design for retarded system with interval time-varying delays,” Int. J. of Systems Science, vol. 48, pp. 1060–1070, 2017.MathSciNetCrossRefGoogle Scholar
  50. [50]
    P. Borne, G. Dauphin-Tanguy, J. P. Richard, F. Rotella, and I. Zambettakis, Commande et Optimisation des Processus, vol. 1, Technip, 1990.zbMATHGoogle Scholar
  51. [51]
    S. D. G. Cumming, “Design of observers of reduced dynamics,” Electron. Lett., vol. 5, pp. 213–214, 1969.CrossRefGoogle Scholar
  52. [52]
    H. R. Sirisena, “Minimal order observers for linear functions of a state vector,” Int. J. Control, vol. 29, pp. 235–54, 1979.MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    M. Darouach, “Linear functional observers for systems with delays in state variables,” IEEE Transactions on Automatic Control, vol. 46, no. 3, pp. 491–496, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    P. C. Muller and M. Hou, “On the observer design for descriptor systems,” IEEE Transactions on Automatic Control, vol. 38, pp. 1666–1671, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    R. Engel and G. Kreisselmeier, “A continuous-time observer which converges in finite time,” IEEE Transactions on Automatic Control, vol. 47, pp. 1202–1204, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    A. J. Koshkouei and A. S. I. Zinober, “Sliding mode state observation for non-linear systems,” Int. J. Control, vol. 77, pp. 118–127, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    R. Aloui and N. Benhadj Braiek, “Optimal state observer design for nonlinear dynamical systems,” Nonlinear Dynamics and Systems Theory, vol. 12, pp. 37–48, 2012.MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Advanced Systems (L.S.A)Polytechnic School of TunisiaLa MarsaTunisia

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