Title

Mittag-Leffler stability analysis of a class of homogeneous fractional systems

Journal title

Archives of Control Sciences

Yearbook

2021

Volume

vol. 31

Issue

No 2

Affiliation

Fajraoui, Tarek : University of Gafsa, Tunisia, Faculty of Sciences of Gafsa, Department of Mathematics, University campus Sidi Ahmed Zarroug 2112 Gafsa, Tunisia ; Ghanmi, Boulbaba : University of Gafsa, Tunisia, Faculty of Sciences of Gafsa, Department of Mathematics, University campus Sidi Ahmed Zarroug 2112 Gafsa, Tunisia ; Mabrouk, Fehmi : University of Gafsa, Tunisia, Faculty of Sciences of Gafsa, Department of Mathematics, University campus Sidi Ahmed Zarroug 2112 Gafsa, Tunisia ; Omri, Faouzi : University of Gafsa, Tunisia, Faculty of Sciences of Gafsa, Department of Mathematics, University campus Sidi Ahmed Zarroug 2112 Gafsa, Tunisia

Authors

Fajraoui, Tarek ; Ghanmi, Boulbaba ; Mabrouk, Fehmi ; Omri, Faouzi

Keywords

homogeneous fractional systems ; Lyapunov homogeneous function ; Mittag-Leffler stability

Divisions of PAS

Nauki Techniczne

Coverage

401-415

Publisher

Committee of Automatic Control and Robotics PAS

Bibliography

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[2] A. Bacciotti and L. Rosier: Liapunov Functions and Stability in Control Theory. Lecture Notes in Control and Inform. Sci, 267 (2001), DOI: 10.1007/b139028.
[3] K. Diethelm: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Series on Complexity, Nonlinearity and Chaos, Springer, Heidelberg, 2010.
[4] M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, and R. Castro-Linares: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul., 22(1-3) (2015), 650–659, DOI: 10.1016/j.cnsns. 2014.10.008.
[5] H. Hermes: Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In: Differential equations: stability and control, Proc. Int. Conf., Colorado Springs/CO (USA) 1989, Lect. Notes Pure Appl. Math. 127, 249-260 (1990).
[6] H. Hermes: Nilpotent and high-order approximations of vector field systems. SIAM Rev, 33, (1991), 238–264, DOI: 10.1137/1033050.
[7] Y. Li, Y. Chen, and I. Podlubny: Stability of fractional-order nonlinear dynamic system: Lyapunov direct method and generalized Mittag- Leffler stability. Comput. Math. Appl, 59(5) (2010), 1810–1821, DOI: 10.1016/j.camwa.2009.08.019.
[8] Y. Li, Y. Chen, and I. Podlubny: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45 (2009), 1965–1969, DOI: 10.1140/epjst/e2011-01379-1.
[9] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204 , Elsevier Science B.V., Amsterdam 2006, DOI: 10.1016/s0304- 0208(06)80001-0.
[10] T. Menard, E. Moulay, and W. Perruquetti: Homogeneous approximations and local observer design. ESAIM: Control, Optimization and Calculus of Variations, 19 (2013), 906–929, DOI: 10.1051/cocv/2012038.
[11] K.B. Oldham and J. Spanier: The Fractional Calculus. Academic Press, New-York, 1974.
[12] I. Podlubny: Fractional Differential Equations. Mathematics in Sciences and Engineering. Academic Press, San Diego, 1999.
[13] H. Rios, D. Efmov, L. Fridman, J. Moreno, and W. Perruquetti: Homogeneity based uniform stability analysis for time-varying systems. IEEE Transactions on automatic control, 61(3), (2016), 725–734, DOI: 10.1109/TAC.2015.2446371.
[14] R. Rosier: Homogeneous Lyapunov function for homogeneous continuous vector field. System & Control Letters, 19 (1992), 467–473, DOI: 10.1016/0167-6911(92)90078-7.
[15] H.T. Tuan and H. Trinh: Stability of fractional-order nonlinear systems by Lyapunov direct method. IET Control Theory Appl, 12 (2018), DOI: 10.1049/ict-cta.2018.5233.
[16] F. Zhang, C. Li, and Y.Q. Chen: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ., (2011), 1–12, DOI: 10.1155/2011/635165.

Date

2021.07.01

Type

Article

Identifier

DOI: 10.24425/acs.2021.137424