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Exact State Estimation for Linear Systems with Unbounded Disturbances

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Abstract

A finite-time observer is designed for linear invariant systems in the presence of unknown inputs or disturbances with unavailable upper bound. The main condition for designing the observer is strongly detectability. Geometric control theory is used to decompose the given system into two parts: strongly observable subsystem and strongly detectable subsystem. Through a series of transformations, the former can be partitioned into two parts: affected and unaffected by unknown inputs (UI-free and UI-dependent), and then the states can be exactly estimated via time-delayed observer in pre-defined time. The nonstrongly observable subsystem can be observed asymptotically. Without the upper bound of disturbances, the observer ensures that the convergence time can be set arbitrarily. A numerical example illustrates the effective of the proposed estimation schemes.

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References

  1. G. Basile, J. Marro, On the observability of linear time-invariant systems with unknown inputs. J. Optim. Theory Appl. 3(6), 410–415 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory (Prentice Hall, Englewood Cliffs, 1992)

    MATH  Google Scholar 

  3. F. Bejarano, L. Fridman, A. Pisano, Global Hierarchical Observer for Linear Systems with Unknown Inputs, in 47th IEEE Conference on Decision and Control. Mexico (2008)

  4. F. Bejarano, L. Fridman, A. Poznyak, Unknown input and state estimation for unobservable systems. SIAM J. Control Optim. 48(2), 1155–1178 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. F. Bejarano, T. Floquet, W. Perruquetti, G. Zheng, Observability and detectability of singular linear systems with unknown inputs. Automatica 49(3), 793–800 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. F. Bejarano, Partial unknown input reconstruction for linear systems. Automatica 47(8), 1751–1756 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Chehimi, S.H. Said, F. Msahli, Unknown inputs observer based output feedback controller for rotor resistance estimation in IMs. Int. J. Sci. Tech. Autom. Control Comput. Eng. 5(1), 1532–1543 (2011)

    Google Scholar 

  8. M.S. Chen, C.C. Chen, Unknown input observer for linear non-minimum phase systems. J. Frankl. Inst. 347(2), 577–588 (2010)

    Article  MATH  Google Scholar 

  9. J. Chen, R.J. Patton, H.Y. Zhang, Design of unknown input observers and robust fault detection filters. Int. J. Control 63(1), 85–105 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Darouach, M. Zasadzinski, S.J. Xu, Full-order observers for linear systems with unknown inputs. IEEE Trans. Autom. Control 39(3), 606–609 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Darouach, Complements to full order observers design for linear systems with unknown inputs. Appl. Math. Lett. 22(7), 1107–1111 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. X. Deng, Y. Shen, Aapproach to design semi-global finite-time observers for a class of nonlinear systems. Sci. China Ser. F. Inf. Sci. 52(10), 1746–1753 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Du, C. Qian, S. Yang, S. Li, Recursive design of finite-time convergent observers for a class of time-varying nonlinear systems. Automatica 49(2), 601–609 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Edwards, S.K. Spurgeon, On the development of discontinuous observers. Int. J. Control 59(5), 1211–1229 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Ekramian, F. Sheikholeslam, S. Hosseinnia, Adaptive state observer for Lipschitz nonlinear systems. Syst. Control Lett. 62(4), 319–323 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Engel, G. Kreisselmeier, A continuous-time observer which converges in finite time. IEEE Trans. Autom. Control 47(7), 1202–1204 (2002)

    Article  MathSciNet  Google Scholar 

  17. L. Fridman, J. Davial, A. Levant, High-order sliding-mode observation for linear systems with unknown inputs. Nonlinear Anal. 5(2), 189–205 (2011)

    MATH  MathSciNet  Google Scholar 

  18. L. Fridman, A. Levant, J. Davila, Observation of linear systems with unknown inputs via high-order sliding-mode. Int. J. Syst. Sci. 38(10), 773–791 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Guidorzi, G. Marro, On Wonham stabilizability condition in the synthesis of observers for unknown input systems. IEEE Trans. Autom. Control 16(5), 499–500 (1971)

    Article  Google Scholar 

  20. M. Hautus, Strong detectability and observers. Linear Algebra Appl. 50, 353–368 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  21. J. He, C. Zhang, Fault reconstruction based on sliding mode observers for nonlinear systems. Math. Probl. Eng. (2012). doi:10.1155/2012/451843

  22. Y. Hong, J. Huang, Y. Xu, On an output feedback-time stabilization problem. IEEE Trans. Autom. Control. 46(2), 305–309 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Hou, P.C. Mller, Design of observers for linear systems with unknown inputs. IEEE Trans. Autom. Control 37(6), 871–875 (1992)

    Article  MATH  Google Scholar 

  24. S. Hui, S.H. Zak, Observer design for systems with unknown inputs. Int. J. Appl. Math. Comput 15(4), 431–446 (2005)

    MATH  MathSciNet  Google Scholar 

  25. B. Kulcsar, J. Bokor, J. Shinar, Unknown input reconstruction for LPV systems. Int. J. Robust Nonlinear Control 20(5), 579–595 (2010)

    MATH  MathSciNet  Google Scholar 

  26. K.S. Lee, T.G. Park, New results on fault reconstruction using a finite-time converging unknown observer. IET Control Theory Appl. 6(9), 1258–1269 (2012)

    Article  MathSciNet  Google Scholar 

  27. J. LI, Y. Fang, S. Shi, Robust output-feedback for hydraulic servo-position system of cold-strip rolling mill. Control Theory Appl. 29(3), 331–336 (2012)

    Google Scholar 

  28. Y. Li, X. Xia, Y. Shen, A high-gain-based global finite-time nonlinear observer, in 9th ICCA Santiago Chile (2011)

  29. D.G. Luenberger, An introduction to observers. IEEE Trans. Autom. Control 16(6), 596–602 (1971)

    Article  MathSciNet  Google Scholar 

  30. M. Lungu, R. Lungu, Reduced order observer for linear time-invariant multivariable systems with unknown inputs. Circuits Syst. Signal Process. 32(6), 2883–2898 (2013)

    Article  MathSciNet  Google Scholar 

  31. M. Lungu, R. Lungu, Full-order observer design for linear systems with unknown inputs. Int. J. Control 85(10), 1602–1615 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Marro F. Morbidi, L. Ntogramatzidis, D. Prattichizzo, Geometric Control Theory for Linear Systems:a Tutorial, in Proceedings of the 19th International Symposium on MTNS Budapest Hungary (2010)

  33. T. Menard, E. Moulay, W. Perruquetti, A global high-gain finite-time observer. IEEE Trans. Autom. Control 55(6), 1500–1506 (2010)

    Article  MathSciNet  Google Scholar 

  34. P.H. Menold, R. Findeisen, F. Allgower, Finite time convergent observers for linear time-varying systems, in Proceedings of the 11th Mediterranean Conference on Control and Automation (MED’03) Rhodos Greece (2003)

  35. H. Michalska, D. Mayne, Moving horizon observers and observer-based control. IEEE Trans. Autom. Control 40(6), 995–1006 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  36. B.P. Molinari, A strong controllability and observability in linear multivariable control. IEEE Trans. Autom. Control 21(5), 761–764 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  37. J. O’Reilly, Observers for Linear Systems (Academic Press, New York, 1983)

    MATH  Google Scholar 

  38. T.G. Park, Estimation strategies for fault isolation of linear systems with disturbances. IET Control Theory Appl. 4(12), 2781–2792 (2010)

    Article  MathSciNet  Google Scholar 

  39. W. Perruquetti, T. Floquet, E. Moulay, Finite-time observers: application to secure communication. IEEE Trans. Autom. Control 53(1), 356–360 (2008)

    Article  MathSciNet  Google Scholar 

  40. F. Plestan, J.W. Grizzle, E.R. Westervelt, G. Abba, Stable walking of a 7-dof biped robot. IEEE Trans. Robot. Autom. 219(4), 653–668 (2003)

    Article  Google Scholar 

  41. F. Sauvage, M. Guay, D. Dochain, Design of a nonlinear finite-time converging observer for a class of nonlinear systems. J. Control Sci. Eng. (2007). doi:10.1155/2007/36954

  42. S. Shao, P.W. Wheeler, J.C. Clare, Fault detection for modular multilevel converters based on sliding mode observer. IEEE Trans. Power Electron. 28(11), 4867–4872 (2013)

    Article  Google Scholar 

  43. Y. Shen, X. Xia, Semi-global finite-time observers for nonlinear systems. Automatica 44(12), 3152–3156 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  44. y Shtessel, C. Edwards, L. Fridman, A. Levant, Sliding Mode Control and Observation (Springer, New York, 2013)

    Google Scholar 

  45. C.P. Tan, X. Yu, Z. Man, Terminal sliding mode observers for a class of nonlinear systems. Automatica 46(8), 1401–1404 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  46. H. Trentelman, A.A. Stoorvogel, M. Hautus, Control Theory for Linear Systems, Communications and Control Engineering (Springer, New York, 2001)

    Book  MATH  Google Scholar 

  47. V.I. Utkin, Sliding Modes in Control and Optimization (Springer, Berlin, 1992)

    Book  MATH  Google Scholar 

  48. B. Walcott, S. Zak, State observation of nonlinear uncertain dynamical systems. IEEE Trans. Autom. Control 32(2), 166–170 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  49. X.G. Yan, C. Edwards, Adaptive sliding-mode-observer-based fault reconstruction for nonlinear ystems with parametric uncertainties. IEEE Trans. Ind. Electron. 55(11), 4029–4036 (2008)

    Article  Google Scholar 

  50. F. Zhu, State estimation and unknown input reconstruction via both reduced-order and high-order mode observers. J. Process. Control 22(1), 296–302 (2012)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Control Engineering, Naval Aeronautical and Astronautical University, Yantai, China

    Fei Li, Guorong Zhao & Jingli Huang

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  1. Fei Li
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  2. Guorong Zhao
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  3. Jingli Huang
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Correspondence to Fei Li.

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Li, F., Zhao, G. & Huang, J. Exact State Estimation for Linear Systems with Unbounded Disturbances. Circuits Syst Signal Process 34, 1519–1533 (2015). https://doi.org/10.1007/s00034-014-9918-y

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  • DOI: https://doi.org/10.1007/s00034-014-9918-y

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