Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

q-State Space Least Mean Family of Algorithms

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

Abstract

It is generally known that model-based estimation algorithms (such as Kalman filter and its family) perform better than the non-model-based algorithms [such as least mean square (LMS), recursive least squares] due to extra information available in terms of system dynamics (which can be used to provide state space model of the system). However, the computational complexity of the model based algorithms is very high. On the other hand, the convergence performance of model based least mean type algorithms [such as state space least mean (SSLM) algorithms] is slower and highly dependent on the step-size choice. Thus, the larger step size can provide faster convergence but gives poor steady-state excess mean square error (EMSE). To meet this conflicting demand, we propose to employ the q-calculus to minimize the generalized least mean cost function. The main advantage of using the q-calculus is that it can provide a nonlinear correction term in the adaptation of the state estimate vector. Consequently, this results in an intelligent adaptation by providing both faster convergence in the initial phase of adaptation and a lower steady-state EMSE in the final phase. The developed algorithms are termed as q-state space least mean (q-SSLM) algorithms. The performance of the proposed q-state space least mean square (q-SSLMS) algorithm is also investigated both in terms of convergence in the mean and the mean square sense. The supremacy of the proposed algorithm is validated by performing several simulations and it is also contrasted with the performance of the well-known Kalman filter. Finally, the theoretical convergence analysis is also validated via simulations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 7

Similar content being viewed by others

Notes

  1. Accuracy of the algorithm depends mainly on the dynamics of the problem.

  2. The SSLM algorithm is contrasted with the KF in terms of optimality and computational complexity in the Remarks presented at the end of this section.

  3. The choice of matrix \(\mathbf{G}\) is discussed in the Remarks.

  4. The analysis for higher values of L is quite involved which can be a future arena to explore.

  5. For higher noise variance, the system simulation time resolution had to be increased for stable simulation of the system.

  6. Similar to example 1, for higher noise variance the system simulation time resolution had to be increased for stable simulation of the system.

  7. The full algorithm and its variables can be referred to in [7, 12].

References

  1. A. Ahmed, M. Moinuddin, U.M. Al-Saggaf, State space least mean fourth algorithm for dynamic state estimation in power systems. Arab. J. Sci. Eng

  2. A. Ahmed, M. Moinuddin, U.M. Al-Saggaf, State space least mean fourth algorithm, in International Conference on Electrical and Computer Engineering

  3. A. Ahmed, M. Moinuddin, U.M. Al-Saggaf, State space least mean square for state estimation of synchronous motor, in International Conference on Electrical and Computer Engineering

  4. A. Ahmed, U.M. Al-Saggaf, M. Moinuddin, State space least mean fourth algorithm for state estimation of synchronous motor. Asian J. Eng. Sci. Technol. 4, 9–12 (2014)

    Google Scholar 

  5. U.M. Al-Saggaf, M. Moinuddin, M. Arif, A. Zerguine, The q-least mean squares algorithm. Sig. Process. 111, 50–60 (2014)

    Article  Google Scholar 

  6. I. Arasaratnam, S. Haykin, R.J. Elliott, Discrete-time nonlinear filtering algorithms using Gauss–Hermite quadrature. IEEE Trans. Electromagn. Compat. 95, 953–977 (2007)

    Google Scholar 

  7. C.K. Chui, G. Chen, Kalman Filtering with Real-Time Applications, 4th edn. (Springer, Berlin, 2009)

    MATH  Google Scholar 

  8. P. Diniz, Adaptive Filtering: Algorithms and Practical Implementation, (Springer, Berlin, 2010). https://books.google.co.nz/books?id=DYancQAACAAJ

  9. T. Ernst, The history of q-calculus and a new method. U.U.D.M. Report 2000:16. Department of Mathematics, Uppsala University, Sweden (2000)

  10. G. Evensen, The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53(4), 343–367 (2003). doi:10.1007/s10236-003-0036-9

    Article  Google Scholar 

  11. B. Farhang-Boroujeny, Adaptive Filters: Theory and Applications (Wiley, London, 1999). https://books.google.co.nz/books?id=jxd-QgAACAAJ

  12. S. Haykin, Adaptive Filter Theory, 3rd edn. (Prentice Hall, Prentice, 1996)

    MATH  Google Scholar 

  13. S. Haykin, B. Widrow, Least-Mean-Square Adaptive Filters, Adaptive and Cognitive Dynamic Systems: Signal Processing, Learning, Communications and Control (Wiley, London, 2003). https://books.google.co.nz/books?id=U8X3mJtawUkC

  14. F.H. Jackson, On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)

    Article  Google Scholar 

  15. T. Kailath, Linear Systems, Information and System Sciences Series (Prentice-Hall, Prentice, 1980). https://books.google.co.nz/books?id=ggYqAQAAMAAJ

  16. H.K. Khalil, Nonlinear Systems, 3rd edn. (Prentice Hall, Prentice, 2001)

    Google Scholar 

  17. J. Koekoev, A note on the \(q\)-derivative operator. J. Math. Anal. Appl. 176, 627–634 (1993)

    Article  MathSciNet  Google Scholar 

  18. B. Lathi, Linear Systems and Signals, Oxford Series in Electrical and Computer Engineering (Oxford University Press, Oxford, 2005). https://books.google.co.nz/books?id=7resQgAACAAJ

  19. M.B. Malik, M. Salman, State-space least mean square. Digit. Signal Proc. 18, 334–345 (2008)

    Article  Google Scholar 

  20. R.J. Meinhold, N.D. Singpurwalla, Understanding the Kalman filter. Am. Stat. 37(2), 123–127 (1983)

    Google Scholar 

  21. M. Moinuddin, U.M. Al-Saggaf, A. Ahmed, Family of state space least mean power of two-based algorithms. EURASIP J. Adv. Signal Process. (2015). doi:10.1186/s13634-015-0219-9

  22. R. Olfati-Saber, Distributed Kalman filter with embedded consensus filters, in Proceedings of the 44th IEEE Conference on Decision and Control, 2005, pp. 8179–8184

  23. A.H. Sayed, Fundamentals of Adaptive Filtering (Wiley, London, 2003)

    Google Scholar 

  24. D. Sierociuk, P. Ziubinski, Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise. Circuits Syst. Signal Process. 33(12), 3861–3882 (2014). doi:10.1007/s00034-014-9835-0

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Sierociuk, M. Macias, W. Malesza, G. Sarwas, Dual estimation of fractional variable order based on the unscented fractional order Kalman filter for direct and networked measurements. Circuits Syst. Signal Process. 35(6), 2055–2082 (2016). doi:10.1007/s00034-016-0255-1

    Article  MathSciNet  MATH  Google Scholar 

  26. A.C. Soterroni, R.L. Galski, F.M. Ramos, The \(q\)-gradient method for global optimization. arXiv: 1209.20484v2 [Math.OC]

  27. E.A. Wan, R.V.D. Merwe, The unscented Kalman filter for nonlinear estimation, in Adaptive Systems for Signal Processing, Communications, and Control Symposium, 2000, pp. 153–158

Download references

Acknowledgements

The authors acknowledge the support provided by King Abdulaziz University and the Centre of Excellence in Intelligent Engineering Systems to carry out this work.

Author information

Authors and Affiliations

  1. Centre of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Jeddah, Saudi Arabia

    Arif Ahmed, Muhammad Moinuddin & Ubaid M. Al-Saggaf

  2. Electrical And Computer Engineering Department, King Abdulaziz University, Jeddah, Saudi Arabia

    Arif Ahmed, Muhammad Moinuddin & Ubaid M. Al-Saggaf

  3. School of Engineering and Computer Science, Victoria University of Wellington, Wellington, New Zealand

    Arif Ahmed

Authors
  1. Arif Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad Moinuddin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ubaid M. Al-Saggaf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arif Ahmed.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmed, A., Moinuddin, M. & Al-Saggaf, U.M. q-State Space Least Mean Family of Algorithms. Circuits Syst Signal Process 37, 729–751 (2018). https://doi.org/10.1007/s00034-017-0569-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-017-0569-7

Keywords

Search

Navigation