Research Papers

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

[+] 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 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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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

Grahic Jump Location
Fig. 8

Energy of the RLC* circuit, R = −0.1

Grahic Jump Location
Fig. 9

Derivative of the Lyapunov function



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