Advertisement

Design of an Optimal Input Signal for Parameter Estimation of Linear Fractional-Order Systems

  • Wiktor JakowlukEmail author
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 559)

Abstract

The optimal input signal design is a procedure of generating an informative excitation signal to extract the model parameters with maximum accuracy during the estimation process. Non-integer order calculus is a very useful tool, which is often utilized for modeling and control purposes. In the paper, we present a novel optimal input formulation and a numerical scheme for fractional order LTI system identification. The Oustaloup recursive approximation (ORA) method is used to determine the fractional order differentiation in an integer order state-space form. Then, the presented methodology is adopted to obtain an optimal input signal for fractional order system identification from the order interval \(0.5 \le \alpha \le 2.0\). The fundamental step in the presented method was to reformulate the problem into a similar fractional optimal input design problem described by Lagrange formula with the set of constraints. The methodology presented in the paper was verified using a numerical example, and the computational results were discussed.

Keywords

Fractional calculus Optimal inputs Oustaloup filter Parameter identification 

Notes

Acknowledgement

The present study was supported by a grant S/WI/3/18 from the Bialystok University of Technology and funded from the resources for research by the Ministry of Science and Higher Education.

References

  1. 1.
    Monje, C.A., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional Orders Systems and Controls: Fundamentals and Applications. Advances in Industrial Control. Springer, London (2010).  https://doi.org/10.1007/978-1-84996-335-0CrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, Y., Petráš, X.D.: Fractional order control - a tutorial. In: Proceedings of ACC 2009, American Control Conference, pp. 1397–1411 (2009)Google Scholar
  3. 3.
    Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behaviour of real materials. Trans. ASME’84 51(4), 294–298 (1984).  https://doi.org/10.1115/1.3167615CrossRefzbMATHGoogle Scholar
  4. 4.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circ. Syst. Fundam. Theory Appl. 47(1), 25–40 (2000).  https://doi.org/10.1109/81.817385CrossRefGoogle Scholar
  5. 5.
    Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, Hoboken (1993)zbMATHGoogle Scholar
  6. 6.
    Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Danbury (2006)Google Scholar
  7. 7.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).  https://doi.org/10.1142/3779CrossRefzbMATHGoogle Scholar
  8. 8.
    West, B., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003).  https://doi.org/10.1007/978-0-387-21746-8CrossRefGoogle Scholar
  9. 9.
    Petras, I.: Fractional-Order Nonlinear Systems. Springer, New York (2011).  https://doi.org/10.1007/978-3-642-18101-6_4CrossRefzbMATHGoogle Scholar
  10. 10.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999).  https://doi.org/10.1155/2013/802324CrossRefzbMATHGoogle Scholar
  11. 11.
    Valério, D., Costa, J.: An Introduction to Fractional Control. IET, London (2013)zbMATHGoogle Scholar
  12. 12.
    Sheng, H., Chen, Y.D., Qiu, T.S.: Fractional Processes and Fractional-Order Signal Processing. Springer, London (2012).  https://doi.org/10.1007/978-1-4471-2233-3CrossRefzbMATHGoogle Scholar
  13. 13.
    Mozyrska, D., Torres, D.F.M.: Modified optimal energy and initial memory of fractional continuous-time linear systems. Sig. Process. 91(3), Special Issue: SI, 379–385 (2011).  https://doi.org/10.1016/j.sigpro.2010.07.016CrossRefGoogle Scholar
  14. 14.
    Monje, C., Vinagre, B., Feliu, V., Chen, Y.: Tuning and autotuning of fractional order controllers for industry applications. Control Eng. Pract. 16(7), 798–812 (2008).  https://doi.org/10.1016/j.conengprac.2007.08.006CrossRefGoogle Scholar
  15. 15.
    Kalaba, R., Spingarn, K.: Control, Identification, and Input Optimization. Plenum Press, New York (1982)CrossRefGoogle Scholar
  16. 16.
    Ljung, L.: System Identification: Theory for the User. Prentice Hall Inc., Upper Saddle River (1999)zbMATHGoogle Scholar
  17. 17.
    Hussain, M.: Review of the applications of neural networks in chemical process control-simulation and on-line implementation. Artif. Intell. Eng. 13, 55–68 (1999).  https://doi.org/10.1016/S0954-1810(98)00011-9CrossRefGoogle Scholar
  18. 18.
    Gevers, M., Ljung, L.: Optimal experiment designs with respect to the intended model application. Automatica 22(5), 543–554 (1986)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bombois, X., Scorletti, G., Gevers, M., Van den Hof, P.M.J., Hildebrand, R.: Least costly identification experiment for control. Automatica 42(10), 1651–1662 (2006).  https://doi.org/10.1016/j.automatica.2006.05.016CrossRefzbMATHGoogle Scholar
  20. 20.
    Bombois, X., Hjalmarsson, H., Scorletti, G.: Identification for robust deconvolution filtering. Automatica 46(3), 577–584 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rivera, D., Lee, H., Braun, M., Mittelmann, H.: Plant friendly system identification: a challenge for the process industries. In: Proceeding of the SYSID 2003, Rotterdam, The Netherlands, pp. 917–922 (2003)Google Scholar
  22. 22.
    Narasimhan, S., Rengaswamy, R.: Plant friendly input design: convex relaxation and quality. IEEE Trans. Autom. Control 56, 1467–1472 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Potters, M.G., Bombois, X., Forgione, M., Modén, P.E., Lundh, M., Hjalmarsson, H., Van den Hof, P.M.J.: Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification. In: Proceedings of European Control Conference, Strasbourg, France, pp. 726–731 (2014)Google Scholar
  24. 24.
    Larsson, C.A., Rojas, C.R., Bombois, X., Hjalmarsson, H.: Experiment evaluation of model predictive control with excitation (MPC-X) on an industrial depropanizer. J. Process Control 31, 1–16 (2015)CrossRefGoogle Scholar
  25. 25.
    Annergren, M., Larson, C.A., Hjalmarsson, H., Bombois, X., Wahlberg, B.: Application-oriented input design in system identification. Optimal input design for control. IEEE Control Syst. Mag. 37, 31–56 (2017)CrossRefGoogle Scholar
  26. 26.
    Jakowluk, W.: Fractional-order linear systems modeling in time and frequency domains. In: 16th IFIP TC8 International Conference in Computer Information Systems and Industrial Management, pp. 502–513. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-319-59105-6_43CrossRefGoogle Scholar
  27. 27.
    Jakowluk, W.: Optimal input signal design for a second order dynamic system identification subject to D-efficiency constraints. In: 14th IFIP TC8 International Conference in Computer Information Systems and Industrial Management, pp. 351–362. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-319-24369-6_29CrossRefGoogle Scholar
  28. 28.
    Jakowluk, W.: Plant friendly input design for parameter estimation in an inertial system with respect to D-efficiency constraints. Entropy 16(11), 5822–5837 (2014).  https://doi.org/10.3390/e16115822CrossRefGoogle Scholar
  29. 29.
    Jakowluk, W.: Free final time input design problem for robust entropy-like system parameter estimation. Entropy 20(7), 528 (2018).  https://doi.org/10.3390/e20070528CrossRefGoogle Scholar
  30. 30.
    Mozyrska, D.: Multiparameter fractional difference linear control systems. Discret. Dyn. Nat. Soc. 2014, 8 (2014).  https://doi.org/10.1155/2014/183782MathSciNetCrossRefGoogle Scholar
  31. 31.
    Tricaud, C., Chen, Y.: Solving fractional order optimal control problems in Riots\(\_95\) a general-purpose optimal control problem solver. In: 3rd IFAC Workshop on Fractional Differentiation and Its Applications, Ankara, Turkey (2008)Google Scholar
  32. 32.
    Schwartz, A., Polak, E., Chen, Y.: Riots a MATLAB toolbox for solving optimal control problems. Version 1.0 for Windows (1997). http://www.schwartz-home.com/RIOTS/
  33. 33.
    Kaczorek, T.: Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition. Bull. Pol. Acad. Sci. Tech. Sci. 65, 45–51 (2017).  https://doi.org/10.1515/bpasts-2017-0006CrossRefGoogle Scholar
  34. 34.
    Mozyrska, D., Torres, D.F.M.: Modified optimal energy and initial memory of fractional continuous-time linear systems. Sig. Process. 91, Special Issue: SI, 379–385 (2011).  https://doi.org/10.1016/j.sigpro.2010.07.016CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Computer ScienceBialystok University of TechnologyBialystokPoland

Personalised recommendations