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Research Papers

Lyapunov Stability of Commensurate Fractional Order Systems: A Physical Interpretation

[+] 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 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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References

Figures

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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