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Research Papers

Lyapunov Stability of Noncommensurate Fractional Order Systems: An Energy Balance Approach

[+] Author and Article Information
Jean-Claude Trigeassou

IMS-LAPS,
University of Bordeaux 1,
Talence Cedex 33405, France
e-mail: jean-claude.trigeassou@ims-bordeaux.fr

Nezha Maamri

LIAS ENSIP,
University of Poitiers,
Poitiers Cedex 86000, France
e-mail: nezha.maamri@univ-poitiers.fr

Alain Oustaloup

IMS-LAPS,
University of Bordeaux 1,
Talence Cedex 33405, France
e-mail: alain.oustaloup@ims-bordeaux.fr

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 24, 2015; final manuscript received October 11, 2015; published online November 19, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(4), 041007 (Nov 19, 2015) (9 pages) Paper No: CND-15-1183; doi: 10.1115/1.4031841 History: Received June 24, 2015; Revised October 11, 2015

Lyapunov stability of linear noncommensurate order fractional systems is treated in this paper. The proposed methodology is based on the concept of fractional energy stored in inductor and capacitor components, where natural decrease of the stored energy is caused by internal Joule losses. The Lyapunov function is expressed as the sum of the different reversible fractional energies, whereas its derivative is interpreted in terms of internal and external Joule losses. Stability conditions are derived from the energy balance principle, adapted to the fractional case. Examples are taken from electrical systems, but this methodology applies also directly to mechanical and electromechanical systems.

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References

Khalil, H. K. , 1996, Non Linear Systems, Prentice Hall, Upper Saddle River, NJ.
Aguila-Camacho, N. , Duarte-Mermoud, M. A. , and Gallegos, J. A. , 2014, “ Lyapunov Functions for Fractional Order Systems,” Commun. Nonlinear Sci. Numer. Simul., 19(9), pp. 2951–2957. [CrossRef]
Baleanu, D. , Ranjbar, N. A. , Sadati, R. S. J. , Delavari, H. , Abdeljawad, T. , and Gejji, V. , 2011, “ Lyapunov–Krasovskii Stability Theorem for Fractional Systems With Delay,” Rom. J. Phys., 56(5–6), pp. 636–643.
Chen, D. , Zhang, R. , Lin, X. , and Ma, X. , 2014, “ Fractional Order Lyapunov Stability Theorem and Its Application in Synchronization of Complex Dynamical Networks,” Commun. Nonlinear Sci. Numer. Simul., 19(12), pp. 4105–4121. [CrossRef]
Hu, J. B. , Lu, G. P. , Zhang, S. H. , and Zhao, L. D. , 2015, “ Lyapunov Stability Theorem About Fractional System Without and With Delay,” Commun. Nonlinear Sci. Numer. Simul., 20(3), pp. 905–913. [CrossRef]
Li, Y. , and Chen, Y. Q. , 2014, “ Lyapunov Stability of Fractional Order Non Linear Systems: A Distributed Order Approach,” ICFDA’14, Catania, June 23–25.
Momani, S. , and Hadid, S. , 2004, “ Lyapunov Stability Solutions of Fractional Integrodifferential Equations,” Int. J. Math. Math. Sci., 47, pp. 2503–2507. [CrossRef]
Jarad, F. , Abdeljawad, T. , Gudogdu, D. , and Baleanu, D. , 2011, “ On the Mittag-Leffler Stability of Q-Fractional Nonlinear Dynamical Systems,” Proc. Rom. Acad., Ser. A, 12(4), pp. 309–314.
Li, Y. , Chen, Y. Q. , and Podlubny, I. , 2009, “ Mittag Leffler Stability of Fractional Order Non Linear Dynamic Systems,” Automatica, 45(8), pp. 1965–1969. [CrossRef]
Sadati, S. J. , Baleanu, D. , Ranjbar, A. , Ghaderi, R. , and Abdeljawad, T. , 2010, “ Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems With Delay,” Abstr. Appl. Anal., 2010, p. 108651. [CrossRef]
Sabatier, J. , Moze, M. , and Farges, C. , 2010, “ LMI Stability Conditions for Fractional Order Systems,” Comput. Math. Appl., 9, pp. 1594–1609. [CrossRef]
Trigeassou, J. C. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2011, “ A Lyapunov Approach to the Stability of Fractional Differential Equations,” Signal Process., 91(3), pp. 437–445. [CrossRef]
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2013, “ Lyapunov Stability of Linear Fractional Systems. Part 1: Definition of Fractional Energy and Part 2: Derivation of a Stability Condition,” ASME Paper No. DETC2013-12824.
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2014, “ Lyapunov Stability of Fractional Order Systems: The Two Derivatives Case,” ICFDA’14, Catania, June 23–25.
Yuan, J. , Shi, B. , and Ji, W. , 2013, “ Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems,” Adv. Math. Phys., 2013, p. 576709.
Hartley, T. T. , Trigeassou, J. C. , Lorenzo, C. F. , and Maamri, N. , 2015, “ Energy Storage and Loss in Fractional Order Systems,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061006. [CrossRef]
Ortega, R. , Loria, A. , Nicklasson, P. J. , and Sira-Ramirez, H. , 1998, Passivity Based Control of Euler Lagrange Systems, Springer-Verlag, Berlin, Germany.
Oldham, K. B. , and Spanier, J. , 1974, The Fractional Calculus, Academic Press, New York.
Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Trigeassou, J. C. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2012, “ State Variables and Transients of Fractional Order Differential Systems,” Comput. Math. Appl., 64(10), pp. 3117–3140. [CrossRef]
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2013, “ The Infinite State Approach: Origin and Necessity,” Comput. Math. Appl., 66(5), pp. 892–907. [CrossRef]
Montseny, G. , 1998, “ Diffusive Representation of Pseudo Differential Time Operators,” Proc. ESSAIM, 5, pp. 159–175. [CrossRef]
Maamri, N. , Tari, M. , and Trigeassou, J. C. , 2014, “ Physical Interpretation and Initialization of the Fractional Integrator,” ICFDA’14, Catania, June 23–25.
Retiere, N. , and Ivanes, M. , 1998, “ Modeling of Electrical Machines by Implicit Derivative Half Order Systems,” IEEE Power Eng. Rev., 18(9), pp. 62–64.
Matignon, D. , 1998, “ Stability Properties for Generalized Fractional Differential Systems,” Proc. ESSAIM, 5, pp. 145–158. [CrossRef]
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2015, “ Lyapunov Stability of Commensurate Fractional Order Systems: A Physical Interpretation,” ASME J. Comput. Nonlinear Dyn. (submitted).

Figures

Grahic Jump Location
Fig. 2

Simulation of the elementary cell

Grahic Jump Location
Fig. 3

Fractional RC* circuit

Grahic Jump Location
Fig. 4

Series RLC circuit

Grahic Jump Location
Fig. 5

Energy of the RLC circuit

Grahic Jump Location
Fig. 6

Current and voltage of RLC* circuit

Grahic Jump Location
Fig. 7

Energy of the RLC* circuit, R = 0.1

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Fig. 8

Energy of the RLC* circuit, R = −0.1

Grahic Jump Location
Fig. 9

Derivative of the Lyapunov function

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