Characteristic Equations of the Standard and Descriptor Linear Electrical Circuits of Integer and Fractional Orders

Kaczorek, Tadeusz

doi:10.2478/ama-2018-0032

Abstract

The problem of calculation of the characteristic equations of the standard and descriptor linear electrical circuits of integer and fractional orders is addressed. It is shown that the characteristic equations of standard and descriptor linear electrical circuits are independent of the method used in their analysis: the state space method, the mesh method and the node method. The considerations are illustrated by examples of standard and fractional linear electrical circuits.

References

1. Antsaklis P.J., Michel A.N. (2006), Linear Systems, Birkhauser, Boston.
Search in Google Scholar Back to article
2. Cholewicki T. (1967), Theoretical Electrotechnics, WNT, Warszawa (in Polish).
Search in Google Scholar Back to article
3. Dzieliński A., Sierociuk D., Sarwas G. (2009), Ultracapacitor parameters identification based on fractional order model, Proc. ECC’09, Budapest.10.23919/ECC.2009.7074403
Search in Google Scholar Back to article
4. Farina L., Rinaldi S. (2000), Positive Linear Systems; Theory and Applications, J. Wiley, New York.10.1002/9781118033029
Search in Google Scholar Back to article
5. Kaczorek T. (1992), Linear Control Systems vol. 1, Research Studies Press, J. Wiley, New York.
Search in Google Scholar Back to article
6. Kaczorek T. (2002), Positive 1D and 2D Systems, Springer-Verlag, London.10.1007/978-1-4471-0221-2
Search in Google Scholar Back to article
7. Kaczorek T. (2008a), Fractional positive continuous-time systems and their reachability, Int. J. Appl. Math. Comput. Sci., 18(2), 223–228.10.2478/v10006-008-0020-0
Search in Google Scholar Back to article
8. Kaczorek T. (2008b), Practical stability of positive fractional discrete-time linear systems, Bull. Pol. Acad. Sci. Tech., 56(4), 313–317.
Search in Google Scholar Back to article
9. Kaczorek T. (2009), Asymptotic stability of positive fractional 2D linear systems, Bull. Pol. Acad. Sci. Tech., 57(3), 289–292.10.2478/v10175-010-0131-2
Search in Google Scholar Back to article
10. Kaczorek T. (2010), Positive linear systems with different fractional orders, Bull. Pol. Acad. Sci. Techn., 58(3), 453–458.10.2478/v10175-010-0043-1
Search in Google Scholar Back to article
11. Kaczorek T. (2011a), Positive linear systems consisting of n subsystems with different fractional orders, IEEE Trans. Circuits and Systems, 58(6), 1203–1210.10.1109/TCSI.2010.2096111
Search in Google Scholar Back to article
12. Kaczorek T. (2011b), Positivity and reachability of fractional electrical circuits, Acta Mechanica et Automatica, 5(2), 42–51.
Search in Google Scholar Back to article
13. Kaczorek T. (2011c), Positive electrical circuits and their reachability, Archives of Electrical Engineering, 60(3), 283–301.10.2478/v10171-011-0026-3
Search in Google Scholar Back to article
14. Kaczorek T. (2011d), Controllability and observability of linear electrical circuits, Electrical Review, 87(9a), 248–254.
Search in Google Scholar Back to article
15. Kaczorek T. (2012a), Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar Back to article
16. Kaczorek T. (2012b), Positive unstable electrical circuits, Electrical Review, 88(5a), 187–192.
Search in Google Scholar Back to article
17. Kaczorek T. (2013a), Positive fractional linear electrical circuits. Proceedings of SPIE, 8903, Bellingham WA, USA, Art. No 3903–35.10.1117/12.2034370
Search in Google Scholar Back to article
18. Kaczorek T. (2013b), Constructability and observability of standard and positive electrical circuits, Electrical Review, 89(7), 132–136.
Search in Google Scholar Back to article
19. Kaczorek T. (2013c), Zeroing of state variables in descriptor electrical circuits by state-feedbacks, Electrical Review, 89(10), 200–203.
Search in Google Scholar Back to article
20. Kaczorek T. (2014a), Polynomial approach to fractional descriptor electrical circuits, Computational Models for Business and Engineering Domains - ITHEA, Rzeszow, Poland.
Search in Google Scholar Back to article
21. Kaczorek T. (2014b), Decoupling zeros of positive continuous-time linear systems and electrical circuit, Advances in Systems Science. Advances in Intelligent Systems and Computing, 240, 1–15.10.1007/978-3-319-01857-7_1
Search in Google Scholar Back to article
22. Kaczorek T. (2015a), A class of positive and stable time-varying electrical circuits, Electrical Review, 91(5), 121–124.10.15199/48.2015.05.29
Search in Google Scholar Back to article
23. Kaczorek T. (2015b), Normal positive electrical circuits, IET Circuits Theory and Applications, 9(5), 691–699.10.1049/iet-cta.2014.0183
Search in Google Scholar Back to article
24. Kaczorek T. (2016), Minimal-phase positive electrical circuits, Electrical Review, 92(3), 182–189.10.15199/48.2016.03.43
Search in Google Scholar Back to article
25. Kaczorek T., Rogowski K. (2015), Fractional Linear Systems and Electrical Circuits, Studies in Systems, 13, Springer International Publishing.10.1007/978-3-319-11361-6
Search in Google Scholar Back to article
26. Kailath T. (1980), Linear Systems, Prentice-Hall, Englewood Cliffs, New York.
Search in Google Scholar Back to article
27. Oldham K.B., Spanier J. (1974), The Fractional Calculus, Academic Press, New York.
Search in Google Scholar Back to article
28. Ostalczyk P. (2008), Epitome of the fractional calculus: Theory and its Applications in Automatics, Technical University of Łódź Press, Łódź (in Polish).
Search in Google Scholar Back to article
29. Ostalczyk P. (2016) Discrete Fractional Calculus: Selected Applications in Control and Image Processing, Series in Computer Vision, 4.10.1142/9833
Search in Google Scholar Back to article
30. Podlubny I. (1999), Fractional Differential Equations, Academic Press, San Diego.
Search in Google Scholar Back to article
31. Rosenbrock H. (1970), State-Space and Multivariable Theory, J. Wiley, New York.
Search in Google Scholar Back to article
32. Sajewski Ł. (2016), Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders, Multidimensional Systems and Signal Processing, 27(1), 27–41.10.1007/s11045-014-0287-2
Search in Google Scholar Back to article
33. Vinagre B.M., Monje C.A., Calderon A.J. (2002), Fractional order systems and fractional order control actions, Lecture 3 IEEE CDC’02 TW#2: Fractional calculus Applications in Automatic Control and Robotics.
Search in Google Scholar Back to article
34. Wolovich W.A. (1974), Linear Multivariable Systems, Springer-Verlag, New York.10.1007/978-1-4612-6392-0
Search in Google Scholar Back to article
35. Żak S.H. (2003), Systems and Control, Oxford University Press, New York.
Search in Google Scholar Back to article

Characteristic Equations of the Standard and Descriptor Linear Electrical Circuits of Integer and Fractional Orders

Abstract

Paradigm

My account