Reachability and observability of fractional positive electrical circuits

2013;
: pp. 28-36
Authors:
1
Białystok University of Technology

Necessary and sufficient conditions for the reachability and observability of fractional positive continuous-time linear electrical circuits are established. Effectiveness of the proposed conditions is demonstrated on examples of electrical circuits.

  1.  P. Antsaklis, A. Michel, Linear Systems, Birkhauser, Boston, 2006.
  2. M.Busłowicz, “Stability of linear continuous-time fractional order systems with delays of the retarded type”, Bull. Pol. Acad. Sci. Tech., vol. 56, no. 4, pp. 319-324, 2008.
  3. A. Dzieliński, D.Sierociuk, G.Sarwas, “Ultracapacitor parameters identification based on fractional order model”, in Proc. European Control Conference, Budapest, Hungary, 2009.
  4. A.Dzieliński, D.Sierociuk, “Stability of discrete fractional order state-space systems”, Journal of Vibrations and Control, vol. 14, no. 9-10, pp. 1543-1556, 2008.
  5. L.Farina, S.Rinaldi, Positive Linear Systems: Theory and Applications, J. Wiley & Sons, New York, 2000.
  6. T.Kaczorek, “Asymptotic stability of positive fractional 2D linear systems”, Bull. Pol. Acad. Sci. Tech., vol. 57, no. 3, pp. 289-292, 2009.
  7. T.Kaczorek, “Decomposition of the pairs (A,B) and (A,C) of the positive discrete-time linear systems”, Archives of Control Sciences, vol. 20, no. 3, pp. 341-361, 2010.
  8. T.Kaczorek, “Fractional positive continuous-time linear systems and their reachablity”, Int. J. Appl. Math. Comput. Sci., vol. 18, no. 2, pp. 223-228, 2008.
  9. T.Kaczorek, Positive 1D and 2D Systems, London, UK: Springer-Verlag, 2002.
  10. T.Kaczorek, “Controllability and obser­vability of linear electrical circuits”, Electrical Review, vol. 87, no. 9a, pp. 248-254, 2011.
  11. T. Kaczorek, “Decoupling zeros of positive discrete-time linear systems”, Circuits and Systems, vol. 1, pp. 41-48, 2010.
  12. T. Kaczorek, “Positive electrical circuits and their reachability”, Archives of Electrical Engineering, vol. 60, no. 3, pp. 283-301, 2011 and also Selected classes of positive electrical circuits and their reachability, Monograph Computer Application in Electrical Engineering, Poznan University of Technology, Poznan, Poland, 2012.
  13. T. Kaczorek, “Positivity and reachability of fractional electrical circuits”, Acta Mechanica et Automatica, vol. 5, no. 2, pp. 42-51, 2011.
  14. T. Kaczorek, “Positive linear systems consisting of n subsystems with different fractional orders”, IEEE Trans. Circuits and Systems, vol. 58, no. 6, pp. 1203-1210, 2011.
  15. T. Kaczorek, “Practical stability of positive fractional discrete-time linear systems”, Bull. Pol. Acad. Sci. Tech., vol. 56, no. 4, pp. 313-317, 2008.
  16. T. Kaczorek, “Reachability and controllability to zero tests for standard and positive fractional discrete-time systems”, Journal Européen des Systemes Automatisés, JESA, vol. 42, no. 6-8, pp. 769-787, 2008.
  17. Kaczorek, T., “Stability of positive continuous-time systems with delays”, Bull. Pol. Acad. Sci. Tech., vol. 57, no. 4, pp. 395-398, 2009.
  18. T. Kaczorek, Selected Problems of Fractional Systems Theory, Berlin, Germany: Springer-Verlag, 2011.
  19. T. Kaczorek, “Constructability and observability of standard and positive electrical circuits”, Electrical Review, vol. 89, no. 7, pp. 132-136, 2013.
  20. T. Kaczorek, “Reachability and observability of fractional positive continuous-time linear systems”, in Proc. XV Conf. System Modelling and Control, Sept. 23-24, Lodz, Poland, 2013.
  21. T.Kailath, Linear systems, Prentice Hall, Englewood Cliffs, New York, 1980.
  22. R.Kalman, “Mathematical Description of Linear Systems”, SIAM J. Control, vol. 1, no. 2, pp. 152-192, 1963.
  23. R.Kalman, “On the general theory of control systems”, in Proc. First Intern. Congress on Automatic Control, London, UK: Butterworth, pp. 481-493, 1960.
  24. K.Oldham, J.Spanier, The Fractional Calculus: Integrations and Differentiations of Arbitrary Order, New York, USA: Academic Press, 1974.
  25. P.Ostalczyk, Epitome of the Fractional Calculus, Theory and its Applications in Automatics, Lodz, Poland: Technical University of Lodz Press, 2008. (Polish)
  26. I.Podlubny, Fractional Differential Equations, San Diego, USA: Academic Press, 1999.
  27. H. Rosenbrock, State-space and Multivariable Theory, New York, USA: J. Wiley, 1970.
  28. J Tokarzewski, ”Finite zeros of positive linear discrete time systems”, Bull. Pol. Acad. Sci. Tech., vol. 59, no. 3, pp. 287-292, 2011.
  29. J.Tokarzewski, “Finite zeros of positive continuous-time systems”, Bull. Pol. Acad. Sci. Tech., vol. 59, no. 3, pp. 293-298, 2011.
  30. J Tokarzewski, Finite Zeros in Discrete Time Control Systems, Berlin, Germany: Springer-Verlag, 2006.
  31. W. Wolovich, Linear Multivariable Systems, New York, UAS: Springer-Verlag, 1974.