Fractional descriptor continuous–time linear systems described by the Caputo–Fabrizio derivative

Tadeusz Kaczorek; Kamil Borawski

doi:10.1515/amcs-2016-0037

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Fractional descriptor continuous–time linear systems described by the Caputo–Fabrizio derivative

International Journal of Applied Mathematics and Computer Science

Volume 26 (2016): Issue 3 (September 2016)

By: Tadeusz Kaczorek and Kamil Borawski

Open Access

|Sep 2016

Abstract

The Weierstrass–Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo–Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated with a numerical example.

References

Bru, R., Coll, C., Romero-Vivo, S. and Sanchez, E. (2003). Some problems about structural properties of positive descriptor systems, in L. Benvenuti et al. (Eds.), Positive Systems, Lecture Notes in Control and Information Sciences, Vol. 294, Springer, Berlin, pp. 233–240.10.1007/978-3-540-44928-7_32
Search in Google Scholar Back to article
Bru, R., Coll, C. and Sanchez, E. (2000). About positively discrete-time singular systems, in M.E. Mastorakis (Ed.), System and Control: Theory and Applications, World Scientific and Engineering Society, Athens, pp. 44–48.
Search in Google Scholar Back to article
Campbell, S.L., Meyer, C. and Rose, N. (1976). Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients, SIAM Journal on Applied Mathematics31(3): 411–425.10.1137/0131035
Search in Google Scholar Back to article
Caputo, M. and Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications1(2): 1–13.
Search in Google Scholar Back to article
Dai, L. (1989). Singular Control Systems, Lecture Notes in Control and Information Sciences, Vol. 118, Springer-Verlag, Berlin.
Search in Google Scholar Back to article
Dodig, M. and Stosic, M. (2009). Singular systems state feedbacks problems, Linear Algebra and Its Applications431(8): 1267–1292.10.1016/j.laa.2009.04.024
Search in Google Scholar Back to article
Duan, G.R. (2010). Analysis and Design of Descriptor Linear Systems, Springer, New York, NY.10.1007/978-1-4419-6397-0
Search in Google Scholar Back to article
Fahmy, M.M. and O’Reilly, J. (1989). Matrix pencil of closed-loop descriptor systems: Infinite-eigenvalues assignment, International Journal of Control49(4): 1421–1431.10.1080/00207178908559713
Search in Google Scholar Back to article
Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, J. Wiley, New York, NY.10.1002/9781118033029
Search in Google Scholar Back to article
Gantmacher, F.R. (1959). The Theory of Matrices, Chelsea Publishing Company, London.
Search in Google Scholar Back to article
Kaczorek, T. (1997). Positive singular discrete time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences45(4): 619–631.
Search in Google Scholar Back to article
Kaczorek, T. (1998a). Positive descriptor discrete-time linear systems, Problems of Nonlinear Analysis in Engineering Systems1(7): 38–54.
Search in Google Scholar Back to article
Kaczorek, T. (1998b). Vectors and Matrices in Automation and Electrotechnics, WNT, Warsaw, (in Polish).
Search in Google Scholar Back to article
Kaczorek, T. (2001). Positive 1D and 2D Systems, Springer-Verlag, London.10.1007/978-1-4471-0221-2
Search in Google Scholar Back to article
Kaczorek, T. (2011a). New stability tests of positive standard and fractional linear systems, Circuits and Systems2(4): 261–268.10.4236/cs.2011.24036
Search in Google Scholar Back to article
Kaczorek, T. (2011b). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems58(6): 1203–1210.10.1109/TCSI.2010.2096111
Search in Google Scholar Back to article
Kaczorek, T. (2012). Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar Back to article
Kaczorek, T. (2014a). Descriptor positive discrete-time and continuous-time nonlinear systems, 34 IEEE-SPIE Joint Symposium, Wilga, Poland, DOI:10.1117/12.2074558.10.1117/12.2074558
Search in Google Scholar Back to article
Kaczorek, T. (2014b). Positivity and linearization of a class of nonlinear discrete-time systems by state feedbacks, Logistyka (6): 5078–5083.
Search in Google Scholar Back to article
Kaczorek, T. (2015a). Analysis of positive and stable fractional continuous-time linear systems by the use of Caputo–Fabrizio derivative, 12th International Conference on Mechatronic Systems and Materials (MSM’2016), Białystok, Poland.10.7148/2016-0053
Search in Google Scholar Back to article
Kaczorek, T. (2015b). Positivity and stability of discrete-time nonlinear systems, IEEE 2nd International Conference on Cybernetics (CYBCONF), Gdynia, Poland.10.1109/CYBConf.2015.7175924
Search in Google Scholar Back to article
Klamka, J. (2013). Controllability of dynamical systems: A survey, Bulletin of the Polish Academy of Sciences: Technical Sciences61(2): 221–229.10.2478/bpasts-2013-0031
Search in Google Scholar Back to article
Kucera, V. and Zagalak, P. (1988). Fundamental theorem of state feedback for singular systems, Automatica24(5): 653–658.10.1016/0005-1098(88)90112-4
Search in Google Scholar Back to article
Losada, J. and Nieto, J. (2015). Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications1(2): 87–92.
Search in Google Scholar Back to article
Ostalczyk, P. (2016). Discrete Fractional Calculus, World Scientific Publishing Company, River Edge, NJ.10.1142/9833
Search in Google Scholar Back to article
Van Dooren, P. (1979). The computation of Kronecker’s canonical form of a singular pencil, Linear Algebra and Its Applications27: 103–140.10.1016/0024-3795(79)90035-1
Search in Google Scholar Back to article
Virnik, E. (2008). Stability analysis of positive descriptor systems, Linear Algebra and Its Applications429(10): 2640–2659.10.1016/j.laa.2008.03.002
Search in Google Scholar Back to article
Zhang, H., Xie, D., Zhang, H. and Wang, G. (2014a). Stability analysis for discrete-time switched systems with unstable subsystems by a mode-dependent average dwell time approach, ISA Transactions53(4): 1081–1086.10.1016/j.isatra.2014.05.02024925136
Search in Google Scholar Back to article
Zhang, J., Han, Z., Wu, H. and Hung, J. (2014b). Robust stabilization of discrete-time positive switched systems with uncertainties and average dwell time switching, Circuits, Systems and Signal Processing33(1): 71–95.10.1007/s00034-013-9632-1
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/amcs-2016-0037 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

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Language: English

Page range: 533 - 541

Submitted on: Nov 26, 2015

Accepted on: May 25, 2016

Published on: Sep 29, 2016

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

Caputo–Fabrizio derivative

Related subjects:

Mathematics,

Applied mathematics

© 2016 Tadeusz Kaczorek, Kamil Borawski, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 26 (2016): Issue 3 (September 2016)