Research Papers

Lyapunov Stability of Commensurate Fractional Order Systems: A Physical Interpretation

[+] Author and Article Information
Jean-Claude Trigeassou

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

Nezha Maamri

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

Alain Oustaloup

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 December 16, 2015; published online February 3, 2016. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 11(5), 051007 (Feb 03, 2016) (8 pages) Paper No: CND-15-1184; doi: 10.1115/1.4032387 History: Received June 24, 2015; Revised December 16, 2015

Lyapunov stability of linear commensurate order fractional systems is revisited with the energy balance principle. This 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. Previous stability results are interpreted, thanks to an equivalent fictitious fractional RLC circuit. Energy balance is used to analyze the usual Lyapunov function and to provide a physical interpretation to the weighting positive matrix. Moreover, the classical linear matrix inequality (LMI) condition is interpreted in terms of internal and external Joule losses.

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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]
Chen, D. , Zhang, R. , Lin, X. , and Ma, X. , 2014, “ Fractional Order Lyapunov Stability Theorem and Its Application in Synchronization of Complex Dynamical Networks,” Comun. 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]
Wang, X. Y. , and Song, J. M. , 2009, “ Synchronization of the Fractional Order Hyperchaos Lorentz Systems With Activation Feedback Control,” Commun. Nonlinear Sci. Num. Simul., 14(8), pp. 3351–3357. [CrossRef]
Da, L. , and Wang, X. Y. , 2010, “ Observer Based Decentralized Fuzzy Neuronal Sliding Mode Control for Interconnected Unknown Chaotic Systems Via Network Structure Adaptation,” Fuzzy Sets Syst., 161(15), pp. 2066–2080. [CrossRef]
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.
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]
Li, Y. , Chen, Y. Q. , and Podlubny, I. , 2010, “ Stability of Fractional Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–Leffler Stability,” Comput. Math. Appl., 59(5), pp. 1810–1821. [CrossRef]
Fahd, J. , Thabet, A. , Emrah, G. , and Dumitru, B. , 2011, “ On the Mittag–Leffler Stability of Q-Fractional Nonlinear Dynamical Systems,” Proc. Rom. Acad., Ser. A, 12(4/2011), pp. 309–314.
Sadati, S. J. , Baleanu, D. , Ranjbar, A. , Ghaderi, R. , and Abdeljawad (Maraaba), T. , 2010, “ Mittag–Leffler Stability Theorem for Fractional Nonlinear Systems With Delay,” Abstr. Appl. Anal., 2010, p. 108651. [CrossRef]
Boyd, S. , El Ghaoui , L., Feron , E., and Balakrishnan , V. , 1994, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA.
Sabatier, J. , Moze, M. , and Farges, C. , 2010, “ LMI Stability Conditions for Fractional Order Systems,” Comput. Math. Appl., 59(5), 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.
Ortega, R. , Loria, A. , Nicklasson, P. J. , and Sira-Ramirez, H. , 1998, Passivity Based Control of Euler Lagrange Systems, Springer-Verlag, Berlin.
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2015, “ Lyapunov Stability of Non Commensurate Fractional Order Systems: An Energy Balance Approach,” ASME J. Comput. Nonlinear Dyn., 11(4), p. 041007. [CrossRef]
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]
Heleschewitz, D. , and Matignon, D. , 1998, “ Diffusive Realizations of Fractional Integro-Differential Operators: Structural Analysis Under Approximation,” Conference IFAC, System, Structure and Control, Nantes, France, Vol. 2, pp. 243–248.
Montseny, G. , 1998, “ Diffusive Representation of Pseudo Differential Time Operators,” ESSAIM, Vol. 5, pp. 159–175. [CrossRef]
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]
Hartley, T. T. , and Lorenzo, C. F. , 2015, “ Realizations for Determining the Energy Stored in Fractional Order Operators,” ASME IDETC-CIE Conference, Boston, MA.
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,” ESSAIM, 5, pp. 145–158. [CrossRef]
Trigeassou, J. C. , Maamri, N. , and Oustaloup, A. , 2010, “ The Pseudo State Space Model of Linear Fractional Differential Systems,” FDA’2010 Conference, Badajoz, Spain.
Chen, C. T. , 1984, Linear System Theory and Design, Holt, Rinehart and Winston, New York.
Niu, Y. , and Wang, X. Y. , 2011, “ An Anonymous Key Agreement Protocol Based on Chaotic Maps,” Commun. Nonlinear Sci. Numer. Simul., 16(4), pp. 1986–1992. [CrossRef]


Grahic Jump Location
Fig. 1

{R(ω) , L(ω)} elementary series circuit

Grahic Jump Location
Fig. 2

Series RL*C* circuit

Grahic Jump Location
Fig. 3

Simulation of a two-derivative FDE




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