Skip to main content

Positivity of Fractional Descriptor Linear Discrete–Time Systems Cover

.blurhash-client-img { display: none !important; }

Positivity of Fractional Descriptor Linear Discrete–Time Systems

International Journal of Applied Mathematics and Computer Science

Volume 29 (2019): Issue 2 (June 2019)

By: Tadeusz Kaczorek

Open Access

|Jul 2019

Abstract

The positivity of fractional descriptor linear discrete-time systems is investigated. The solution to the state equation of the systems is derived. Necessary and sufficient conditions for the positivity of fractional descriptor linear discrete-time systems are established. The discussion is illustrated with numerical examples.

References

Ali, R.M. and Napp, D. (2012). Characterization and stability of autonomous positive descriptor systems, IEEE Transactions on Automatic Control57(10): 2668–2673.10.1109/TAC.2012.2190211
Search in Google Scholar Back to article
Berman, A. and Plemmons, R.J. (1994). Nonnegative Matrices in the Mathematical Sciences, SIAM, New York, NY.10.1137/1.9781611971262
Search in Google Scholar Back to article
Borawski, K. (2018). Analysis of the positivity of descriptor continuous-time linear systems by the use of Drazin inverse matrix method, in R. Szewczyk et al. (Eds.), Automation 2018, Advances in Intelligent Systems and Computing, Vol. 743, Springer, Cham, pp. 172–182.10.1007/978-3-319-77179-3_16
Search in Google Scholar Back to article
Busłowicz, M. (2008). Stability of linear continuous-time fractional order systems with delays of the retarded type, Bulletin of the Polish Academy of Sciences: Technical Sciences56(4): 319–324.
Search in Google Scholar Back to article
Busłowicz, M. (2012). Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences60(2): 279–284.10.2478/v10175-012-0037-2
Search in Google Scholar Back to article
Busłowicz, M. and Kaczorek, T. (2009). Simple conditions for practical stability of positive fractional discrete-time linear systems, International Journal of Applied Mathematics and Computer Science19(2): 263–269, DOI: 10.2478/v10006-009-0022-6.10.2478/v10006-009-0022-6
Open DOI 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
Kaczorek, T. (1993). Theory of Control and Systems, Polish Scientific Publishers, Warsaw, (in Polish).
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. (2002). Positive 1D and 2D Systems, Springer, London.10.1007/978-1-4471-0221-2
Search in Google Scholar Back to article
Kaczorek, T. (2008). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science18(2): 223–228, DOI: 10.2478/v10006-008-0020-0.10.2478/v10006-008-0020-0
Open DOI Search in Google Scholar Back to article
Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences58(3): 453–458.10.2478/v10175-010-0043-1
Search in Google Scholar Back to article
Kaczorek, T. (2011a). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems58(7): 1203–1210.10.1109/TCSI.2010.2096111
Search in Google Scholar Back to article
Kaczorek, T. (2011b). Selected Problems of Fractional Systems Theory, Springer, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar Back to article
Kaczorek, T. (2012). Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences60(1): 9–12.10.2478/v10175-012-0002-0
Search in Google Scholar Back to article
Kaczorek, T. (2013). Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils, International Journal of Applied Mathematics and Computer Science23(1): 29–33, DOI: 10.2478/amcs-2013-0003.10.2478/amcs-2013-0003
Open DOI Search in Google Scholar Back to article
Kaczorek, T. (2014). Descriptor positive discrete-time and continuous-time nonlinear systems, Proceedings of SPIE9290, DOI: 10.1117/12.2074558.10.1117/12.2074558
Open DOI Search in Google Scholar Back to article
Kaczorek, T. (2015a). Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems, Computational Problems of Electrical Engineering5(1): 11–16.10.1109/CPEE.2015.7333337
Search in Google Scholar Back to article
Kaczorek, T. (2015b). Positivity and stability of discrete-time nonlinear systems, IEEE 2nd International Conference on Cybernetics, Gdynia, Poland, pp. 156–159.10.1109/CYBConf.2015.7175924
Search in Google Scholar Back to article
Kaczorek, T. (2015c). Stability of fractional positive nonlinear systems, Archives of Control Sciences25(4): 491–496, DOI: 10.1515/acsc-2015-0031.10.1515/acsc-2015-0031
Open DOI Search in Google Scholar Back to article
Kaczorek, T. (2016a). Analysis of positivity and stability of fractional discrete-time nonlinear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences64(3): 491–494.10.1515/bpasts-2016-0054
Search in Google Scholar Back to article
Kaczorek, T. (2016b). Drazin inverse matrix method for fractional descriptor discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences64(2): 395–399.10.1515/bpasts-2016-0044
Search in Google Scholar Back to article
Kaczorek, T. (2018). Stability of interval positive continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences66(1): 2018.
Search in Google Scholar Back to article
Kaczorek, T. (2019). Positivity and stability of standard and fractional descriptor continuous-time linear and nonlinear systems, International Journal of Nonlinear Sciences and Numerical Simulation19(3–4): 299–307.10.1515/ijnsns-2017-0049
Search in Google Scholar Back to article
Klamka, J. (2002). Positive controllability of positive dynamical systems, Proceedings of the American Control Conference, ACC-2000, Anchorage, AK, USA, pp. 4395–4400.10.1109/ACC.2002.1025385
Search in Google Scholar Back to article
Mitkowski, W. (2000). Remarks on stability of positive linear systems, Control and Cybernetics29(1): 295–304.
Search in Google Scholar Back to article
Mitkowski, W. (2008). Dynamical properties of Metzler systems, Bulletin of the Polish Academy of Sciences, Technical Sciences56(4): 309–312.
Search in Google Scholar Back to article
Sajewski, Ł. (2016a). Descriptor fractional discrete-time linear system and its solution—Comparison of three different methods, in R. Szewczyk et al. (Eds.), Challenges in Automation, Robotics and Measurement Techniques, Advances in Intelligent Systems and Computing, Vol. 440, Springer, Berlin/Heidelberg, pp. 37–50.10.1007/978-3-319-29357-8_4
Search in Google Scholar Back to article
Sajewski, Ł. (2016). Descriptor fractional discrete-time linear system with two different fractional orders and its solution, Bulletin of the Polish Academy of Sciences, Technical Sciences64(1): 15–20.10.1515/bpasts-2016-0003
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(10): 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
Xiang-Jun, W., Zheng-Mao, W. and Jun-Guo, L. (2008). Stability analysis of a class of nonlinear fractional-order systems, IEEE Transactions Circuits and Systems II: Express Briefs55(11): 1178–1182.10.1109/TCSII.2008.2002571
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amcs-2019-0022 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

Journal RSS Feed

Language: English

Page range: 305 - 310

Submitted on: Oct 22, 2018

|

Accepted on: Mar 5, 2019

|

Published on: Jul 4, 2019

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

Related subjects:

Applied mathematics

© 2019 Tadeusz Kaczorek, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 29 (2019): Issue 2 (June 2019)

Previous article