Positive electrical circuits with the chain structure and cyclic Metzler state matrices

Journal title

Bulletin of the Polish Academy of Sciences Technical Sciences








Kaczorek, Tadeusz : Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D, 15-351 Białystok, Poland



cyclic matrix ; positive ; electrical circuit ; fractional ; Metzler matrix

Divisions of PAS

Nauki Techniczne




  1.  A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Philadelphia: SIAM, 1994.
  2.  L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications. New York: J. Wiley, 2000.
  3.  T. Kaczorek, Positive 1D and 2D Systems. London: Springer-Verlag, 2002.
  4.  T. Kaczorek, “Positive linear systems with different fractional orders,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 58, no. 3, pp. 453–458, 2010.
  5.  T. Kaczorek, “Normal fractional positive linear systems and electrical circuits,” in Proc. Conf. Automation 2019, Warsaw, 2020, pp. 13–26.
  6.  T. Kaczorek, Selected Problems of Fractional Systems Theory. Berlin: Springer, 2011.
  7.  T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits. Cham: Springer, 2015.
  8.  W. Mitkowski, “Dynamical properties of metzler systems,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 54, no. 4, pp. 309–312, 2008.
  9.  W. Mitkowski, Outline of Control Theory. Kraków: Publishing House AGH, 2019.
  10.  P. Ostalczyk, Discrete Fractional Calculus. River Edge, NJ: World Scientific, 2016.
  11.  I. Podlubny, Fractional Differential Equations. San Diego: Academic Press, 1999.
  12.  T. Kaczorek, “Reachability and observability of positive discrete-time linear systems with integer positive and negative powers of the state frobenius matrices,” Arch. Control Sci., vol. 28, no. 1, pp. 5–20, 2018.
  13.  M.D. Ortigueira and J. A. Tenreiro Machado, “New discrete-time fractional derivatives based on the bilinear transformation: definitions and properties,” J. Adv. Res., vol. 25, pp. 1–10, 2020.
  14.  A. Ruszewski, “Stability of discrete-time fractional linear systems with delays,” Arch. Control Sci., vol. 29, no. 3, pp. 549–567, 2019.
  15.  L. Sajewski, “Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 65, no. 5, pp. 709–714, 2017.
  16.  R. Stanisławski, K. Latawiec, and M. Łukaniszyn, “A comparative analysis of laguerre-based approximatiors to the grunwald-letnikov fractional-order difference,” Math. Probl. Eng., vol. 2015, 2015.
  17.  F.G. Gantmacher, The Theory of Matrices. London: Chelsea Pub. Comp., 1959.
  18.  T. Kaczorek and K. Borawski, “Stability of continuoustime and discrete-time linear systems with inverse state matrices,” Meas. Autom. Monit., vol. 62, no. 4, pp. 132–135, 2016.
  19.  T. Kaczorek, Polynomial and Rational Matrices. London: Springer, 2007.






DOI: 10.24425/bpasts.2021.137731