A New Interval Observer Design Method with Application to Fault Detection

  • Liliang Li
  • Zhijie Shao
  • Rui Niu
  • Gang Liu
  • Zhenhua WangEmail author
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)


This chapter proposes a novel interval observer for continuous-time linear systems with unknown disturbance. Based on the stability criterion of a Metzler matrix, the interval observer design problem is converted into a series of nonlinear inequalities. To attenuate the effect of unknown disturbance on estimation error, an interval observer design method based on constrained optimization is proposed. The proposed interval observer is able to estimate upper and lower bounds of the states in the general assumption that disturbance is unknown but bounded. Thus it is particularly suitable for fault detection for uncertain linear systems. Therefore, the proposed method is further used to generate dynamic thresholds to achieve fault detection. Finally, a flight control system is simulated to demonstrate the effectiveness of the proposed method.


Interval observer Metzler matrix Constrained optimization Fault detection 


  1. 1.
    Bernard, O., Gouzé, J.L.: Closed loop observers bundle for uncertain biotechnological models. J. Process Control 14(7), 765–774 (2004)CrossRefGoogle Scholar
  2. 2.
    Chambon, E., Apkarian, P., Burlion, L.: Metzler matrix transform determination using a non-smooth optimization technique with an application to interval observers. SIAM Conf. Control Appl. 205–211 (2015)Google Scholar
  3. 3.
    Efimov, D., Fridman, L., Raïssi, T., Zolghadri, A., Seydou, R.: Application of interval observers and HOSM differentiators for fault detection. In: Proceedings 8th International Federation of Automatic Control (IFAC) Symposium, pp. 516–521 (2012)CrossRefGoogle Scholar
  4. 4.
    Efimov, D., Raïssi, T., Zolghadri, A.: Control of nonlinear and LPV systems: interval observer-based framework. IEEE Trans. Autom. Control 58(3), 773–778 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000)CrossRefGoogle Scholar
  6. 6.
    Gouzé, J.L., Rapaport, A., Hadj-Sadok, M.Z.: Interval observers for uncertain biological systems. Ecol. Model. 133(1), 45–56 (2000)CrossRefGoogle Scholar
  7. 7.
    Keel, L.H., Bhattacharyya, S.P., Howze, J.W.: Robust control with structured perturbations. IEEE Trans. Autom. Control 33(1), 68–78 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mazenc, F., Bernard, O.: Interval observers for linear time-invariant systems with disturbances. Automatica 47(1), 140–147 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mazenc, F., Bernard, O.: Asymptotically stable interval observers for planar systems with complex poles. IEEE Trans. Autom. Control 55(2), 523–527 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Moisan, M., Bernard, O., Gouzé, J.L.: Near optimal interval observers bundle for uncertain bioreactors. Automatica 45(1), 291–295 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Narendra, K.S., Shorten, R.: A characterization of the Hurwitz stability of Metzler matrices. In: Proceedings of the American Control Conference, pp. 1833–1837. IEEE Press, New York (2009)Google Scholar
  12. 12.
    Narendra, K.S., Shorten, R.: Hurwitz stability of Metzler matrices. IEEE Trans. Autom. Control 55(6), 1484–1487 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rami, M.A., Cheng, C.H., Prada, C.De.: Tight robust interval observers: an LP approach. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 2967–2972. IEEE Press, New York (2008)Google Scholar
  14. 14.
    Raïssi, T., Efimov, D., Zolghadri, A.: Interval state estimation for a class of nonlinear systems. IEEE Trans. Autom. Control 57(1), 260–265 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Smith, J.L.: Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Math. Surv. Monogr. 41 (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Liliang Li
    • 1
  • Zhijie Shao
    • 1
  • Rui Niu
    • 1
  • Gang Liu
    • 1
  • Zhenhua Wang
    • 2
    Email author
  1. 1.Shanghai Institute of Spaceflight Control TechnologyShanghaiPeople’s Republic of China
  2. 2.School of AstronauticsHarbin Institute of TechnologyHarbinPeople’s Republic of China

Personalised recommendations