Skip to main content
SpringerLink
Log in

Unknown input observer design for one-sided Lipschitz nonlinear systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper considers the observer design problem for one-sided Lipschitz nonlinear systems with unknown inputs. The systems under consideration are a larger class of nonlinearities than the well-studied Lipschitz systems and have inherent advantages with respect to conservativeness. For such systems, we first propose a full-order nonlinear unknown input observer (UIO) by using the linear matrix inequality (LMI) approach. Following a similar design procedure and using state transformation, the reduced-order nonlinear UIO is also constructed. Sufficient conditions to guarantee existence of full-order and reduced-order UIOs are established by carefully considering the one-sided Lipschitz condition together with the quadratic inner-bounded condition. Based on the matrix generalized inverse technique, the UIO conditions are formulated in terms of LMIs. Moreover, the proposed observers are applied to a single-link flexible joint robotic system with unknown inputs. Simulation results are finally given to illustrate the effectiveness of the proposed design.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Wang, S., Davison, E., Dorato, P.: Observing the states of systems with unmeasurable disturbance. IEEE Trans. Autom. Control. 20, 716–717 (1975)

    Article  MATH  Google Scholar 

  2. Bhattacharyya, S.P.: Observer design for linear systems with unknown inputs. IEEE Trans. Autom. Control. 23, 483–484 (1978)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Koening, D.: Unknown input proportional multiple-integral observer design for linear descriptor systems: application to state and fault estimation. IEEE Trans. Autom. Control. 50, 212–217 (2005)

    Article  Google Scholar 

  6. Sundaram, S., Hadjicostis, C.N.: Partial state observers for linear systems with unknown inputs. Automatica 44, 3126–3132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Trinh, H., Tran, T.D., Fernando, T.: Disturbance decoupled observers for systems with unknown inputs. IEEE Trans. Autom. Control. 53, 2397–2402 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  9. Ha, Q., Trinh, H.: State and input simultaneous estimation for a class of nonlinear systems. Automatica 40, 1779–1785 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Koening, D., Marx, B., Jacquet, D.: Unknown input observers for switched nonlinear discrete time descriptor system. IEEE Trans. Autom. Control. 53, 373–379 (2008)

    Article  Google Scholar 

  11. Ding, Z.: Reduced-order observer design for nonlinear systems with unknown inputs. In: Proceedings 9th IEEE International Conference Control Automatic. pp. 842–847. Santiago, Chile (2011)

  12. Darouach, M., Boutat-Baddas, L., Zerrougui, M.: \(H_\infty \) observer for a class of nonlinear singular systems. Automatica 47, 2517–2525 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, M., Wu, Q., Jiang, C.: Disturbance-observer-based robust synchronization control of uncertain chaotic systems. Nonlinear Dyn. 70, 2421–2432 (2012)

    Article  MathSciNet  Google Scholar 

  14. Zhang, W., Su, H., Zhu, F., Wang, M.: Observer-based \(H_\infty \) synchronization and unknown input recovery for a class of digital nonlinear systems. Circuit Syst. Signal Process. 32, 2867–2881 (2013)

    Article  MathSciNet  Google Scholar 

  15. Hammouri, H., Tmar, Z.: Unknown input observer for state affine systems: a necessary and sufficient condition. Automatica 46, 271–278 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, H., Duan, Z.: Unknown input observer design for systems with monotone non-linearities. IET Control. Theory Appl. 6, 1941–1947 (2012)

    Article  MathSciNet  Google Scholar 

  17. Yang, J., Zhu, F., Zhang, W.: Sliding-mode observers for nonlinear systems with unknown inputs and measurement noise. Int. J. Control. Autom. Syst. 11, 903–910 (2013)

  18. Gholami, A., Markazi, A.: A new adaptive fuzzy sliding mode observer for a class of MIMO nonlinear systems. Nonlinear Dyn. 70, 2095–2105 (2012)

    Article  MathSciNet  Google Scholar 

  19. Wu, H., Wang, J.: Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics. Nonlinear Dyn. 72, 615–628 (2013)

    Article  MATH  Google Scholar 

  20. Raghavan, S., Hedrick, J.: Observer design for a class of nonlinear systems. Int. J. Control. 59, 515–528 (1994)

  21. Zhu, F., Han, Z.: A note on observers for Lipschitz nonlinear systems. IEEE Trans. Autom. Control. 47, 1751–1754 (2002)

    Article  MathSciNet  Google Scholar 

  22. Chen, W., Saif, M.: Unknown input observer design for a class of nonlinear systems: an LMI approach. In: Proceedings 2006 American Control Conference. pp. 834–838. Minnesota, USA (2006)

  23. Pertew, A., Marquez, H., Zhao, Q.: \(H_\infty \) synthesis of unknown input observers for a nonlinear Lipschitz systems. Int. J. Control. 78, 1155–1165 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xiong, Y., Saif, M.: Sliding mode observer for nonlinear uncertain systems. IEEE Trans. Autom. Control. 46, 2012–2017 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kalsi, K., Lian, J., Huib, S., Zak, S.: Sliding-mode observers for systems with unknown inputs: a high-gain approach. Automatica 46, 347–353 (2010)

    Article  MATH  Google Scholar 

  26. Abbaszadeh, M., Marquez, H.: Nonlinear observer design for one-sided Lipschitz systems. In: Proceeding 2010 American Control Conference. pp. 5284–5289. Baltimore, USA (2010)

  27. Hu, G.: Observers for one-sided lipschitz non-linear systems. IMA J. Math. Control. Inf. 23, 395–401 (2006)

    Article  MATH  Google Scholar 

  28. Xu, M., Hu, G., Zhao, Y.: Reduced-order observer design for one-sided Lipschitz non-linear systems. IMA J. Math. Control. Inf. 26, 299–317 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao, Y., Tao, J., Shi, N.Z.: A note on observer design for one-sided Lipschitz nonlinear systems. Syst. Control. Lett. 59, 66–71 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fu, M., Hou, M., Duan, G.: Stabilization of quasi-one-sided Lipschitz nonlinear systems. IMA J. Math. Control. Inf. 26, 299–317 (2012)

    Google Scholar 

  31. Zhang, W., Su, H., Liang, Y., Han, Z.: Nonlinear observer design for one-sided Lipschitz nonlinear systems: a linear matrix inequality approach. IET Control. Theory Appl. 6, 1297–1303 (2012)

    Article  MathSciNet  Google Scholar 

  32. Zhang, W., Su, H., Wang, H., Han, Z.: Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations. Commun. Nonlinear Sci. Numer. Simul. 17, 4968–4977 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, W., Su, H., Zhu, F., Yue, D.: A note on observers for discrete-time Lipschitz nonlinear systems. IEEE Trans. Cricuit. Syst. II Exp. Br. 59, 123–127 (2012)

    Google Scholar 

  34. Benallouch, M., Boutayeb, M., Zasadzinski, M.: Observer design for one-sided Lipschitz discrete-time systems. Syst. Control. Lett. 61, 879–886 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Barbata, A., Zasadzinski, M., Ali, H.S., Messaoud, H.: Exponential observer for a class of one-sided Lipschitz stochastic nonlinear systems. IEEE Trans. Autom. Control. (2014). doi:10.1109/TAC.2014.2325391

  36. Cortes, J.: Discontinuous dynamical systems. IEEE Control. Syst. Mag. 28, 36–73 (2008)

    Article  MathSciNet  Google Scholar 

  37. Yu, W., DeLellis, P., Chen, G., Bernardo, M., Kurths, J.: Distributed adaptive control of synchronization in complex networks. IEEE Trans. Autom. Control. 57, 2153–2158 (2012)

    Article  Google Scholar 

  38. Su, H., Chen, G., Wang, X., Lin, Z.: Adaptive second-order consensus of networked mobile agents with nonlinear dynamics. Automatica 47, 368–375 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grant 61473129, the Program for New Century Excellent Talents in University from Chinese Ministry of Education under Grant NCET-12-0215, the Innovation Foundation of Shanghai Municipal Education Commission under Grant 12YZ156, the Fund of SUES under Grant 2012gp45, and Shanghai Municipal Natural Science Foundation under Grant 12ZR1412200.

Author information

Authors and Affiliations

  1. Laboratory of Intelligent Control and Robotics, Shanghai University of Engineering Science, Shanghai, 201620, China

    Wei Zhang

  2. School of Automation, Image Processing and Intelligent Control Key Laboratory of Education Ministry of China, Huazhong University of Science and Technology, Wuhan, 430074, China

    Housheng Su

  3. College of Electronics and Information Engineering, Tongji University, Shanghai, 200092, China

    Fanglai Zhu

  4. Department of Mathematics and Computer Science, Lawrence Technological University, Southfield, MI, 48075, USA

    Ghassan M. Azar

Authors
  1. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Housheng Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fanglai Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ghassan M. Azar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Housheng Su.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, W., Su, H., Zhu, F. et al. Unknown input observer design for one-sided Lipschitz nonlinear systems. Nonlinear Dyn 79, 1469–1479 (2015). https://doi.org/10.1007/s11071-014-1754-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1754-x

Keywords

Search

Navigation