Advertisement

Kalman Filters for Continuous-time Fractional-order Systems Involving Fractional-order Colored Noises Using Tustin Generating Function

Regular Papers Control Theory and Applications
  • 42 Downloads

Abstract

This study presents fractional-order Kalman filers for linear fractional-order systems with colored noises using Tustin generating function. A continuous-time fractional-order system with the fractional-order colored process noise is discretized by Tustin generating function. The augmented vector consists of the state and the colored noise is offered to construct an augmented system based on the discretized state equation of a fractional-order system and the colored process noise. The Tustin fractional-order Kalman filter is designed based on the augmented system to obtain the state estimation, effectively. Besides, the colored noise involved in the measurement of a continuous-time fractional-order system is also discussed, and the corresponding Tustin fractional-order Kalman filter is provided in this study. Two illustrative examples are given to verify the effectiveness of Tustin fractional-order Kalman filters for the colored process and measurement noises.

Keywords

Colored noise fractional-order systems Kalman filters state estimation Tustin generating function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. O. Efe, “Fractional order systems in industrial automation–a survey,” IEEE Transactions on Industrial Informatics, vol. 7, no. 4, pp. 582–591, 2011. [click]CrossRefGoogle Scholar
  2. [2]
    D. Valerio, J. J. Trujillo, M. Rivero, J. A. T. Machado, and D. Baleanu, “Fractional calculus: a survey of useful formulas,” European Physical Journal Special Topics, vol. 222, no. 8, pp. 1827–1846, 2013. [click]CrossRefGoogle Scholar
  3. [3]
    J. D. Gabano and T. Poinot, “Fractional modelling and identification of thermal systems,” Signal Processing, vol. 91, no. 3, pp. 531–541, 2011.CrossRefzbMATHGoogle Scholar
  4. [4]
    V. Martynyuk and M. Ortigueira, “Fractional model of an electrochemical capacitor,” Signal Processing, vol. 107, pp. 355–360, 2015. [click]CrossRefGoogle Scholar
  5. [5]
    B. Wang, S. E. Li, H. Peng, and Z. Liu, “Fractional-order modeling and parameter identification for lithium-ion batteries,” Journal of Power Sources, vol. 293, pp.151–161, 2015. [click]CrossRefGoogle Scholar
  6. [6]
    P. Shah and S. Agashe, “Review of fractional PID controller,” Mechatronics, vol. 38, pp. 29–41, 2015.CrossRefGoogle Scholar
  7. [7]
    Y. Chen, Y. Wei, H. Zhong, and Y. Wang, “Sliding mode control with a second-order switching law for a class of nonlinear fractional order systems,” Nonlinear Dynamics, vol. 85, no. 1, 633–643, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Y. Li, Y. Q. Chen, H. S. Ahn, and G. Tian, “A survey on fractional-order iterative learning control,” Journal of Optimization Theory & Applications, vol. 156, no. 1, pp. 127–140, 2013. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Hassanzadeh and M. Tabatabaei, “Calculation of controllability and observability matrices for special case of continuous-time multi-order fractional systems,” ISA Transations, DOI:10.1016/j.isatra.2017.03.006, 2017.Google Scholar
  10. [10]
    S. Marir, M. Chadli, and D. Bouagada, “A novel approach of admissibility for singular linear continuous-time fractional-order systems,” International Journal of Control Automation & Systems, vol. 15, no. 2, pp. 959–964, 2017. [click]CrossRefzbMATHGoogle Scholar
  11. [11]
    S. Marir, M. Chadli, and D. Bouagada, “New admissibility conditions for singular linear continuous-time fractionalorder systems,” vol. 354, no.2. pp.752–766, 2017.Google Scholar
  12. [12]
    X. M, Zhang and Q. L, Han, “Network-based H∞ filtering using a logic jumping-like trigger,” Automatica, vol. 49, no. 5, pp. 1428–1435, 2013. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    X. M. Zhang and Q. L. Han, “Event-based H∞ filtering for sampled-data systems,” Automatica, vol. 51, pp. 55–69, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    B. M. Bell and F. W. Cathey, “The iterated Kalman filter update as a Gauss-Newton method,” IEEE Transactions on Automatic Control, vol. 38, no.2, pp. 294–297, 1993. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Sierociuk and A. Dzielinski, “Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation,” International Journal of Applied Mathematics and Computer Science, vol. 16, no. 1, pp. 129–140, 2006.MathSciNetzbMATHGoogle Scholar
  16. [16]
    H. Sadeghian, H. Salarieh, and A. Alasty, “On the general Kalman filter for discrete time stochastic fractional systems,” Mechatronics, vol. 23, no. 7, pp. 764–771, 2013.CrossRefGoogle Scholar
  17. [17]
    L. Ashayeri, M. Shafiee, and M. Menhaj, “Kalman filter for fractional order singular systems,” Journal of American Science, vol. 9 no. 1, pp. 209–216, 2013.Google Scholar
  18. [18]
    H. Torabi, N. Pariz, and A. Karimpour, “Kalman filters for fractional discrete-time stochastic systems along with timedelay in the observation signal,” The European Physical Journal Special Topics, vol. 225, no. 1, pp. 107–118, 2016. [click]CrossRefGoogle Scholar
  19. [19]
    X. Wu, Y. Sun, and Z. Lu, “A modified Kalman filter algorithm for fractional system under Lévy noises,” Journal of the Franklin Institute, vol. 352, no. 5, pp. 1963–1978, 2015.MathSciNetCrossRefGoogle Scholar
  20. [20]
    D. Sierociuk and P. Ziubinski, “Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise,” Circuits, Systems, and Signal Processing, vol. 33, no. 12, pp. 3861–3882, 2014. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D. Sierociuk, “Fractional Kalman filter algorithms for correlated system and measurement noises,” Control & Cybernetics, vol. 42, no. 2, pp. 471–490, 2013.MathSciNetzbMATHGoogle Scholar
  22. [22]
    S. Najar, M. N. Abdelkrim, M. Abdelhamid, et al., “Discrete fractional Kalman filter,” IFAC Proceedings, vol. 42, no. 9, pp. 520–25, 2009.CrossRefGoogle Scholar
  23. [23]
    A. Kiani-B, K. Fallahi, N. Pariz, et al. “A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 863–879, 2009. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    B. M. Vinagre, Y. Q. Chen, and I. Petras, “Two direct Tustin discretization methods for fractional-order differentiator/integrator,” Journal of the Franklin Institute, vol. 340, no. 5, pp. 349–362, 2003.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Light IndustryLiaoning UniversityLiaoningChina

Personalised recommendations