Skip to main content
SpringerLink
Log in

Kalman Filter Reinforced by Least Mean Square for Systems with Unknown Inputs

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper addresses the state estimation problem of linear discrete-time time-varying stochastic systems with unknown inputs (UIs). It is shown that the globally optimal unbiased minimum-variance filters may not satisfy the minimum-variance property, and hence they cannot eliminate noises appropriately. If this is the case, the well-known Kalman filter may give a better solution, which however may also not be the best one due to that the imbedded unknown input model may not be practical. To remedy the filtering degradation problem, a robust filter named as the KFLMS, which has good noise rejection property for such systems, is developed in this paper, where the UI estimates are obtained by using least mean square algorithm and the state estimation is achieved via the previous proposed two-stage Kalman filtering approach. Numerical examples are provided to show the effectiveness of the proposed results. Specifically, simulation results illustrate the goodness of the new method in the sense of lower root mean square error and better noise rejection property.

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
Fig. 6

Similar content being viewed by others

References

  1. J. Chen, R.J. Patton, Optimal filtering and robust fault diagnosis of stochastic systems with unknown disturbances. IEE Proc. Control Theory Appl. 143(1), 31–36 (1996)

    Article  Google Scholar 

  2. J. Chen, Y. Liu, X. Wang, Recursive least squares algorithm for nonlinear dual-rate systems using missing-output estimation model. Circuits Syst. Signal Process. 36(4), 1406–1425 (2017)

    Article  MathSciNet  Google Scholar 

  3. J. Chen, J. Ma, Y. Liu, F. Ding, Identification methods for time-delay systems based on the redundant rules. Signal Process. 137, 192–198 (2017)

    Article  Google Scholar 

  4. Y. Cheng, H. Ye, Y. Wang, D. Zhou, Unbiased minimum-variance state estimation for linear systems with unknown input. Automatica 45(2), 485–491 (2009)

    Article  MathSciNet  Google Scholar 

  5. M. Darouach, M. Zasadzinski, Unbiased minimum variance estimation for systems with unknown exogenous inputs. Automatica 33(4), 717–719 (1997)

    Article  MathSciNet  Google Scholar 

  6. T. Floquet, J.-P. Barbot, State and unknown input estimation for linear discrete-time systems. Automatica 42(11), 1883–1889 (2006)

    Article  MathSciNet  Google Scholar 

  7. S. Gillijns, B. De Moor, Unbiased minimum-variance input and state estimation for linear discrete-time systems. Automatica 43(1), 111–116 (2007)

    Article  MathSciNet  Google Scholar 

  8. S. Gillijns, B. De Moor, Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica 43(5), 934–937 (2007)

    Article  MathSciNet  Google Scholar 

  9. M.S. Grewal, Kalman filtering, in International Encyclopedia of Statistical Science (Springer, Berlin, 2011), pp. 705–708

    Chapter  Google Scholar 

  10. S. Haykin, B. Widrow, Least-Mean-square Adaptive Filters, vol. 31 (Wiley, New York, 2003)

    Book  Google Scholar 

  11. M. Hou, R.J. Patton, Optimal filtering for systems with unknown inputs. IEEE Trans. Autom. Control 43(3), 445–449 (1998)

    Article  MathSciNet  Google Scholar 

  12. C.-S. Hsieh, Robust two-stage Kalman filters for systems with unknown inputs. IEEE Trans. Autom. Control 45(12), 2374–2378 (2000)

    Article  MathSciNet  Google Scholar 

  13. C.-S. Hsieh, Optimal minimum-variance filtering for systems with unknown inputs, in The Sixth World Congress on Intelligent Control and Automation, 2006 (WCICA 2006), volume 1 (IEEE, 2006), pp. 1870–1874

  14. C.-S. Hsieh, On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs. Automatica 46(4), 708–715 (2010)

    Article  MathSciNet  Google Scholar 

  15. C.-S. Hsieh, On the optimality of two-stage Kalman filtering for systems with unknown inputs. Asian J. Control 12(4), 510–523 (2010)

    MathSciNet  Google Scholar 

  16. C.-S. Hsieh, F.-C. Chen, Optimal solution of the two-stage Kalman estimator. IEEE Trans. Autom. Control 44(1), 194–199 (1999)

    Article  MathSciNet  Google Scholar 

  17. C.-S. Hsieh, M.A. Majidi, Implementation issues of unbiased minimum-variance state estimation for systems with unknown inputs, in 2014 CACS International Automatic Control Conference (CACS) (IEEE, 2014), pp. 323–328

  18. K.P. Kitanidis, Unbiased minimum-variance linear state estimation. Automatica 23(6), 775–778 (1987)

    Article  Google Scholar 

  19. H.A. Sayed, T. Kailath, A state-space approach to adaptive RLS filtering. IEEE Signal Process. Mag. 11(3), 18–60 (1994)

    Article  Google Scholar 

  20. S. Sundaram, C.N. Hadjicostis, Delayed observers for linear systems with unknown inputs. IEEE Trans. Autom. Control 52(2), 334–339 (2007)

    Article  MathSciNet  Google Scholar 

  21. D. Wang, Y. Gao, Recursive maximum likelihood identification method for a multivariable controlled autoregressive moving average system. IMA J. Math. Control Inf. 33(4), 1015–1031 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Center of Excellence on Soft Computing and Intelligent Information Processing, Ferdowsi University of Mashhad, Mashhad, 9177948974, Iran

    Mohammad Ali Majidi & Hadi Sadoghi Yazdi

  2. Department of Electrical and Electronics Engineering, Ta Hwa University of Science and Technology, Qionglin, Taiwan, ROC

    Chien-Shu Hsieh

Authors
  1. Mohammad Ali Majidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chien-Shu Hsieh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hadi Sadoghi Yazdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hadi Sadoghi Yazdi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Majidi, M.A., Hsieh, CS. & Yazdi, H.S. Kalman Filter Reinforced by Least Mean Square for Systems with Unknown Inputs. Circuits Syst Signal Process 37, 4955–4972 (2018). https://doi.org/10.1007/s00034-018-0792-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-018-0792-x

Keywords

Search

Navigation