Stability of interval positive fractional discrete-time linear systems

• Autorzy: Kaczorek T.

Identyfikatory

DOI

10.2478/amcs-2018-0034

• Języki publikacji: EN

Abstrakty

EN

The aim of this work is to show that interval positive fractional discrete-time linear systems are asymptotically stable if and only if the respective lower and upper bound systems are asymptotically stable. The classical Kharitonov theorem is extended to interval positive fractional linear systems.

Słowa kluczowe

EN

interval system; positive system; fractional system; linear discrete-time system; stability

PL

system przedziałowy; układ dodatni; układ ułamkowy; układ liniowy; układ dyskretno-czasowy

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2018
• Tom: Vol. 28, no. 3
• Strony: 451--456
• Opis fizyczny: Bibliogr. 33 poz.

Twórcy

autor

Kaczorek T.

• Faculty of Electrical Engineering, Białystok Technical University, Wiejska 45D, 15-351 Białystok, Poland

Bibliografia

• [1] Berman, A. and Plemmons R.J. (1994). Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA.
• [2] 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 Sciences 56(4): 319–324.
• [3] 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 Sciences 60(2): 279–284.
• [4] 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 Science 19(2): 263–269, DOI: 10.2478/v10006-009-0022-6.
• [5] Farina, L. and Rinaldi, S. (2000). Positive Linear Systems: Theory and Applications, Wiley, New York, NY.
• [6] Kaczorek, T. (1997). Positive singular discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 45(4): 619–631.
• [7] Kaczorek, T. (2002). Positive 1D and 2D Systems, Springer, London.
• [8] Kaczorek, T. (2008). Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science 18(2): 223–228, DOI: 10.2478/v10006-008-0020-0.
• [9] Kaczorek, T. (2010). Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences 58(3): 453–458.
• [10] Kaczorek, T. (2011). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems 58(7): 1203–1210.
• [11] Kaczorek, T. (2012a). Selected Problems of Fractional Systems Theory, Springer, Berlin.
• [12] Kaczorek, T. (2012b). Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(1): 9–12.
• [13] 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 Science 23(1): 29–33, DOI: 10.2478/amcs-2013-0003.
• [14] Kaczorek, T. (2014). Descriptor positive discrete-time and continuous-time nonlinear systems, in R.S. Romaniuk (Ed.), Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, Proceedings of SPIE, Vol. 9290, SPIE, Cardiff, pp. 1–11.
• [15] Kaczorek, T. (2015a). Positivity and stability of discrete-time nonlinear systems, IEEE 2nd International Conference on Cybernetics, Gdynia, Poland, pp. 156–159.
• [16] Kaczorek, T. (2015b). Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems, Computational Problems of Electrical Engineering 5(1): 11–16.
• [17] Kaczorek, T. (2015c). Stability of fractional positive nonlinear systems, Archives of Control Sciences 25(4): 491–496.
• [18] Kaczorek, T. (2016). Analysis of positivity and stability of fractional discrete-time nonlinear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 64(3): 491–494.
• [19] Kaczorek, T. (2018a). Positivity and stability of standard and fractional descriptor continuous-time linear and nonlinear systems, International Journal of Nonlinear Sciences and Numerical Simulation, (in press).
• [20] Kaczorek, T. (2018b). Stability of interval positive continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 66(1): 31–35.
• [21] Kaczorek, T. and Rogowski, K. (2015). Fractional Linear Systems and Electrical Circuits, Springer, Cham.
• [22] Kharitonov, V.L. (1978). Asymptotic stability of an equilibrium position of a family of systems of differential equations, Differentsial’nye uravneniya 14(11): 2086–2088.
• [23] Ortigueira, M. D. (2011). Fractional Calculus for Scientists and Engineers, Springer, Dordrecht.
• [24] Oldham, K.B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY.
• [25] Ostalczyk, P. (2008). Epitome of the Fractional Calculus: Theory and Its Applications in Automatics, Publishing House of the Łódź University of Technology, Łódź, (in Polish).
• [26] Ostalczyk, P. (2016). Discrete Fractional Calculus, World Scientific, River Edge, NJ.
• [27] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.
• [28] Radwan, A.G., Soliman, A.M., Elwakil, A.S. and Sedeek A. (2009). On the stability of linear systems with fractional-order elements. Chaos, Solitones and Fractals 40(5): 2317–2328.
• [29] 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, Springer, Cham, pp. 37–50.
• [30] Sajewski, Ł. (2016b). Descriptor fractional discrete-time linear system with two different fractional orders and its solution, Bulletin of the Polish Academy of Sciences: Technical Sciences 64(1): 15–20.
• [31] Solteiro Pires, E.J., Tenreiro Machado, J.A. and de Moura Oliveira, P.B. (2006). Fractional dynamics in genetic algorithms, IFAC Proceedings Volumes 39(11): 414–419.
• [32] Vinagre, B.M., Monje, C.A. and Calderon A.J. (2002). Fractional order systems and fractional order control actions, Proceedings of the Conference on Decision and Control, Las Vegas, NV, USA, pp. 2550–2554.
• [33] Xiang-Jun, W., Zheng-Mao, W. and Jun-Guo, L. (2008). Stability analysis of a class of nonlinear fractional-order systems, IEEE Transactions on Circuits and Systems II: Express Briefs 55(11): 1178–1182.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).