Stability of Interval Positive Fractional Discrete–Time Linear Systems

Kaczorek, Tadeusz

doi:10.2478/amcs-2018-0034

Abstract

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.

References

Berman, A. and Plemmons R.J. (1994). Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA.10.1137/1.9781611971262
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 Sciences 56(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 Sciences 60(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 Science 19(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, Wiley, New York, NY.10.1002/9781118033029
Search in Google Scholar Back to article
Kaczorek, T. (1997). Positive singular discrete-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 45(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 Science 18(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 Sciences 58(3): 453-458.10.2478/v10175-010-0043-1
Search in Google Scholar Back to article
Kaczorek, T. (2011). Positive linear systems consisting of n subsystems with different fractional orders, IEEE Transactions on Circuits and Systems 58(7): 1203-1210.10.1109/TCSI.2010.2096111
Search in Google Scholar Back to article
Kaczorek, T. (2012a). Selected Problems of Fractional Systems Theory, Springer, Berlin.10.1007/978-3-642-20502-6
Search in Google Scholar Back to article
Kaczorek, T. (2012b). Positive fractional continuous-time linear systems with singular pencils, Bulletin of the Polish Academy of Sciences: Technical Sciences 60(1): 9-12.10.2478/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 Science 23(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, in R.S. Romaniuk (Ed.), Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, Proceedings of SPIE, Vol. 9290, SPIE, Cardiff, pp. 1-11.10.1117/12.2074558
Search in Google Scholar Back to article
Kaczorek, T. (2015a). 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. (2015b). Analysis of positivity and stability of discrete-time and continuous-time nonlinear systems, Computational Problems of Electrical Engineering 5(1): 11-16.10.1109/CPEE.2015.7333337
Search in Google Scholar Back to article
Kaczorek, T. (2015c). Stability of fractional positive nonlinear systems, Archives of Control Sciences 25(4): 491-496.10.1515/acsc-2015-0031
Search in Google Scholar Back to article
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): 491-494.10.1515/bpasts-2016-0054
Search in Google Scholar Back to article
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).10.1515/ijnsns-2017-0049
Search in Google Scholar Back to article
Kaczorek, T. (2018b). Stability of interval positive continuous-time linear systems, Bulletin of the Polish Academy of Sciences: Technical Sciences 66(1): 31-35.
Search in Google Scholar Back to article
Kaczorek, T. and Rogowski, K. (2015). Fractional Linear Systems and Electrical Circuits, Springer, Cham.10.1007/978-3-319-11361-6
Search in Google Scholar Back to article
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.
Search in Google Scholar Back to article
Ortigueira, M. D. (2011). Fractional Calculus for Scientists and Engineers, Springer, Dordrecht.10.1007/978-94-007-0747-4
Search in Google Scholar Back to article
Oldham, K.B. and Spanier, J. (1974). The Fractional Calculus, Academic Press, New York, NY.
Search in Google Scholar Back to article
Ostalczyk, P. (2008). Epitome of the Fractional Calculus: Theory and Its Applications in Automatics, Publishing House of the Łód´z University of Technology, Łód´z, (in Polish).
Search in Google Scholar Back to article
Ostalczyk, P. (2016). Discrete Fractional Calculus, World Scientific, River Edge, NJ.10.1142/9833
Search in Google Scholar Back to article
Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego, CA.
Search in Google Scholar Back to article
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): 2317-2328.10.1016/j.chaos.2007.10.033
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, Springer, Cham, pp. 37-50.10.1007/978-3-319-29357-8_4
Search in Google Scholar Back to article
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): 15-20.10.1515/bpasts-2016-0003
Search in Google Scholar Back to article
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): 414-419.10.3182/20060719-3-PT-4902.00070
Search in Google Scholar Back to article
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.
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 on Circuits and Systems II: Express Briefs 55(11): 1178-1182.10.1109/TCSII.2008.2002571
Search in Google Scholar Back to article

Stability of Interval Positive Fractional Discrete–Time Linear Systems

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