Angles between matrices and between polynomials in fractional linear systems and electrical circuits

: pp. 1-10
Białystok University of Technology

The notions of angles between matrices and between polynomials of fractional linear systems and electrical circuits are proposed. In analysis of  angles between state matrices of fractional linear systems the Hadamard product of two matrices  is applied. The angles between matrices and their functions are also addressed. The angles between symmetrical and asymmetrical part of matrices are investigated. The angles between polynomials of transfer matrices of fractional linear systems are analyzed and some new properties are established.

  1. M. Ait Rami and F. Tadeo, “Controller Synthesis for Positive Linear Systems With Bounded Controls”, IEEE Transactions on Circuits and Systems, vol. 54, no. 2, pp. 151-155, 2007.
  2. S. Domek and P. Dworakm Theoretical Developments and Aplications of Non-Integer Order Systems, Springer 2016.
  3. L.Farina and S.Rinaldi, Positive Linear Systems; Theory and Applications. J. Wiley, New York, 2000.
  4. F.R.Gantmacher, The Theory of Matrices. Chelsea 1974.
  5. T. Kaczorek,: “A class of positive and stable time-varying electrical circuits. Electrical Review”, vol. 91, no. 5, 2015, 121-124.
  6. T. Kaczorek, “Angles between matrices and between polynomials in analysis of electrical circuits”, Proceedings of SPIE, Bellingham WA, USA, Art. vol. 8903, no 3903-35.
  7. T. Kaczorek, “Normal positive electrical circuits”, IET Circuits Theory and Applications, vol. 9, no. 5, pp. 691-699, 2015.
  8. T. Kaczorek, Positive 1D and 2D Systems. Springer-Verlag. London, 2002.
  9. T. Kaczorek, “Positive electrical circuits and their reachability”, Archives of Electrical Engineering, vol. 60, no. 3, pp. 283-301, 2011.
  10. T. Kaczorek Positive fractional linear electrical circuits, Proceedings of SPIE, vol. 8903, Bellingham WA, USA, Art. No 3903-35.
  11. T. Kaczorek, Positive and stable electrical circuits with state matrices, Archives of Electrical Engineering,  vol.67, no.3, pp. 563-578, 2018.
  12. T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin,  2011.
  13. T. Kaczorek, Vectors and Matrices in Automation and  Electrotechnics, WNT, Warszawa,1998 (in Polish).
  14. T. Kaczorek and K. Borawski, “Stability of continuous-time and discrete-time linear systems with inverse state matrices”, Measurement Automation Monitoring, vol. 62, no. 4, pp. 132-135, 2016.
  15. T. Kaczorek and K. Rogowski, “Fractional Linear Systems and Electrical Circuits”, Studies in Systems, Decision and Control, vol. 13, Springer, 2015.
  16. T. Kaczorek and L. Sajewski, “Transfer matrices with positive coefficients for standard and fractional positive linear systems”, in Proc.  MMAR, Międzyzdroje 2018, http:/
  17. K.J.L atawiec, M. Lukaniszyn and R. Stanisławski, Advances in Modelling and Control of Non-integer Order Systems, Springer 2015.
  18. A. B. Malinowska, D. Mozyrska and L. Sajewski Advances in Non-Integer Order Calculus and Its Applications, Springer 2020.
  19. P. Ostalczyk, Discrete Fractional Calculus, World Scientific,2016,