×
Balachandran, K.; Govindaraj, V.; Rivero, M.; Machado, J. A. Tenreiro; Trujillo, J. J.

Observability of nonlinear fractional dynamical systems. (English) Zbl 1295.34008

Abstr. Appl. Anal. 2013, Article ID 346041, 7 p. (2013).
Summary: We study the observability of linear and nonlinear fractional differential systems of order \(0<\alpha<1\) by using the Mittag-Leffler matrix function and the application of Banach’s contraction mapping theorem. Several examples illustrate the concepts.

Cited in 7 Documents

MSC:

34A08 Fractional ordinary differential equations
34H05 Control problems involving ordinary differential equations
93B07 Observability
47N20 Applications of operator theory to differential and integral equations

Keywords:

Mittag-Leffler matrix function; Banach’s contraction mapping theorem
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Machado, J. A. T., And i say to myself: what a fractional world!, Fractional Calculus and Applied Analysis, 14, 4, 635-654 (2011) · Zbl 1273.37002 · doi:10.2478/s13540-011-0037-1
[2] Magin, R. L., Fractional Calculus in Bioengineering (2006), Redding, Mass, USA: Begell House, Redding, Mass, USA
[3] Petráš, I., Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (2011), Heidelberg, Germany: Springer, Heidelberg, Germany · Zbl 1228.34002
[4] Tarasov, V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Nonlinear Physical Science, xvi+504 (2010), Heidelberg, Germany: Springer, Heidelberg, Germany · Zbl 1214.81004 · doi:10.1007/978-3-642-14003-7
[5] Valério, D.; da Costa, J. S., An Introduction to Fractional Control (2012), Stevenage, UK: IET, Stevenage, UK
[6] Zaslavsky, G. M., Hamiltonian Chaos and Fractional Dynamics, xiv+421 (2008), Oxford, UK: Oxford University Press, Oxford, UK · Zbl 1152.37001
[7] Bonilla, B.; Rivero, M.; Rodríguez-Germá, L.; Trujillo, J. J., Fractional differential equations as alternative models to nonlinear differential equations, Applied Mathematics and Computation, 187, 1, 79-88 (2007) · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[8] Klimek, M., On Solutions of Linear Fractional Differential Equations of a Variational Type (2009), Czestochowa, Poland: Czestochowa University of Technology, Czestochowa, Poland
[9] Uchaikin, V. V., Fractional Derivatives for Physicists and Engineers, Volume 1. Back-Ground and Theory; Volume II. Applications (2013), Springer Jointly published with Higher Education Press · Zbl 1312.26002
[10] Machado, J. T.; Kiryakova, V.; Mainardi, F., Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 16, 3, 1140-1153 (2011) · Zbl 1221.26002 · doi:10.1016/j.cnsns.2010.05.027
[11] Tenreiro Machado, J.; Galhano, A. M.; Trujillo, J. J., Science metrics on fractional calculus development since 1966, Fractional Calculus and Applied Analysis, 16, 2, 479-500 (2013) · Zbl 1312.26004 · doi:10.2478/s13540-013-0030-y
[12] Bettayeb, M.; Djennoune, S., New results on the controllability and observability of fractional dynamical systems, Journal of Vibration and Control, 14, 9-10, 1531-1541 (2008) · Zbl 1229.93018 · doi:10.1177/1077546307087432
[13] Matignon, D.; d’Andréa-Novel, B., Some results on controllability and observability of finite dimensional fractional differential systems, Proceedings of the IAMCS, IEEE Conference on Systems, Man and Cybernetics
[14] Sabatier, J.; Merveillaut, M.; Fenetau, L.; Oustaloup, A., On observability of fractional order systems, Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC ’09)
[15] Shamardan, A. B.; Moubarak, M. R. A., Controllability and observability for fractional control systems, Journal of Fractional Calculus, 15, 25-34 (1999) · Zbl 0964.93013
[16] Vinagre, B. M.; Monje, C. A.; Calderon, A. J., Fractional order systems and fractional order control actions, Fractional Calculus Applications in Automatic Control and Robotics (2002)
[17] Monje, C. A.; Chen, Y.; Vinagre, B. M.; Xue, D.; Feliu, V., Fractional-Order Systems and Controls, xxvi+414 (2010), London, UK: Springer, London, UK · Zbl 1211.93002 · doi:10.1007/978-1-84996-335-0
[18] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus. Models and Numerical Methods, 3, xxiv+400 (2012), Hackensack, NJ, USA: World Scientific Publishing, Hackensack, NJ, USA · Zbl 1248.26011 · doi:10.1142/9789814355216
[19] Diethelm, K., The Analysis of Fractional Differential Equations, 2004, viii+247 (2010), Berlin, Germany: Springer, Berlin, Germany · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2
[20] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, 204, xvi+523 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[21] Mainardi, F.; Gorenflo, R., On Mittag-Leffler-type functions in fractional evolution processes, Journal of Computational and Applied Mathematics, 118, 1-2, 283-299 (2000) · Zbl 0970.45005 · doi:10.1016/S0377-0427(00)00294-6
[22] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, xvi+366 (1993), New York, NY, USA: John Wiley & Sons Inc., New York, NY, USA · Zbl 0789.26002
[23] Oldham, K. B.; Spanier, J., The Fractional Calculus, xiii+234 (1974), London, UK: Academic Press, London, UK · Zbl 0292.26011
[24] Caputo, M., Linear model of dissipation whose \(Q\) is almost frequency independent, Geophysical Journal of Royal Astronomical Society, 13, 529-539 (1967)
[25] Éidel’man, S. D.; Chikriĭ, A. A., Dynamic game-theoretic approach problems for fractional-order equations, Ukrainian Mathematical Journal, 52, 11, 1787-1806 (2000) · Zbl 1032.91029 · doi:10.1023/A:1010439422856
[26] Kaczorek, T., Selected Problems of Fractional Systems Theory, 411, xviii+344 (2011), Berlin, Germany: Springer, Berlin, Germany · Zbl 1221.93002 · doi:10.1007/978-3-642-20502-6
[27] Wen, X.-J.; Wu, Z.-M.; Lu, J.-G., Stability analysis of a class of nonlinear fractional-order systems, IEEE Transactions on Circuits and Systems II, 55, 11, 1178-1182 (2008) · doi:10.1109/TCSII.2008.2002571
[28] Zabczyk, J., Mathematical Control Theory: An Introduction, x+260 (1992), Berlin, Germany: Birkhäuser, Berlin, Germany · Zbl 1071.93500
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.