Observer synthesis for linear discrete-time systems with different fractional orders

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wyślij Rafał Kociszewski Faculty of Electrical Engineering, Białystok University of Technology, Poland

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Abstract

The paper is devoted to observer synthesis for linear discrete-time positive fractional systems with different fractional orders. The problem of finding a nonnegative gain matrix of the observer such that the observer is positive and asymptotically stable is formulated and solved by the use of linear programming (LP) and linear matrix inequality (LMI) methods. The proposed approach to the observer synthesis is illustrated by theoretical example. Numerical calculations and simulations have been performed in the MATLAB/Simulink program environment.

Keywords

discrete-time, fractional order, linear matrix inequality, linear programming, observer, positive, system

Synteza obserwatora układów dyskretnych o różnych niecałkowitych rzędach

Streszczenie

W pracy rozpatrzono problem syntezy obserwatorów dla dodatnich układów dyskretnych różnych niecałkowitych rzędów w równaniu stanu. Wykorzystując podejście oparte na typowym zadaniu programowania liniowego (LP) oraz zadaniu sformułowanym w ramach liniowych nierówności macierzowych (LMI) pokazano, że jest możliwe uzyskanie dodatniego asymptotycznie stabilnego obserwatora. Są to warunki dostateczne, alternatywne w stosunku do podanych w [5, 18] dla układów niedodatnich. Zaprojektowany obserwator poprawnie estymuje (odtwarza) zmienne stanu przyjętego do rozważań dyskretnego układu niecałkowitego rzędu. Wyniki obliczeniowe uzyskano w środowisku programowym MATLAB z wykorzystaniem biblioteki Optimization oraz pakietów SeDuMi [20] i YALMIP [14]. Rezultaty symulacyjne uzyskano przy wykorzystaniu dodatkowej biblioteki Fractional States Space Toolkit [18].

Słowa kluczowe

dodatni, liniowa nierówność macierzowa, obserwator, programowanie liniowe, rząd niecałkowity, układ dyskretny

Bibliografia

  1. Luenberger D.G., Introduction to dynamic systems: Theory, Models, and applications. John Wiley & Sons, New York 1979.
  2. Farina L., Rinaldi S.: Positive linear systems. Theory and applications, Willey, New York 2000.
  3. Debnath L., Recent applications of fractional calculus to science and engineering, “Int. Journal of Mathematics and Mathematical Sciences”, Vol. 54, 2003, 3413-3442 [on-line: www.ijmms.hindawi.com].
  4. Dzieliński A., Sierociuk D., Sarwas G., Some applications of fractional order calculus, “Bull. of the Polish Acad. Of Sciences, Technical Sciences”, Vol. 58, No. 4, 2010, 583-592.
  5. Kilbas A.A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, Elsevier, Amsterdam 2006.
  6. Sabatier J., Agraval O. P., Machado, Advances in fractional calculus. Theoretical developments and applications in physics and engineering, Springer, London 2007.
  7. Dzieliński A, Sierociuk D.: Observer for discrete fractional order state-space systems, 2nd IFAC Workshop on Fractional Diffrentation and its Applications, IFAC FDA’06, Portugal, 2006, 524-529.
  8. Sierociuk D., Estimation and control of discrete-time dybnamical fractional systems described in state space, Ph.D. thesis, Warsaw University of Technology, Warsaw 2007.
  9. Kociszewski R., Controllability and observability of linear time-invariant positive discrete-time systems with delays. Ph.D. thesis, Białystok University of Technology, Białystok 2008.
  10. Ait Rami M., Tadeo F., Positive observation problem for linear discrete positive systems, [in:] Proc. of the 45th IEEE Conf. on Decision and Control, USA 2006, 4729-4733.
  11. Huang Q., Observer design for discrete-time positive systems with delays, [in:] Proc. of Int. Conf. on Intelligent Computation Technology, 2008, 655-659.
  12. Busłowicz M., Simple analytic conditions for stability of fractional discrete-time linear systems with diagonal state matrix, “Bull. of the Polish Acad. of Sciences, Technical Sciences” (in press).
  13. Kaczorek T., Selected problems of fractional systems theory, Springer, Berlin 2011.
  14. Ferris M.C., Mangasarian O.L., Wright S.J., Linear programming with MATLAB, SIAM 2007.
  15. Kaczorek T., Control theory and systems, PWN, Warsaw 1996.
  16. Ogata K., Discrete-time control systems, Prentice Hall, New Jersey 1987.
  17. Boyd S., ElGhaoui L., Feron E., Balakrishnan V., Linear matrix inequalities in system and control theory, SIAM 1994.
  18. Sturm J.F., Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, “Optimization Methods and Software”, 1999, 625-653.
  19. Löfberg J., YALMIP: A toolbox for modeling and optimization in MATLAB [on-line: www.control.ee.ethz.ch/~joloef/yalmip.php].
  20. Sierociuk D., Fractional order discrete state-space system Simulink toolkit user guide, [on-line: www. ee.pw.edu.pl/~sieroci/]