This paper addresses the stability of linear commensurate order fractional systems, Dn (X) = AX 0 < n < 1, using the infinite state approach. First, the energy of a fractional integrator is defined, using the distributed energy of its initial state. Compared to the integer order case, this energy is characterized by a long memory decay, which is the characteristic feature of fractional systems. Then, it is applied to define the energy V(t) of a one derivative system. Numerical simulations exhibit the influence of initial conditions on V(t). Thanks to the definition of a dissipation function, a stability condition is derived. Finally, the general case is investigated and a weighted Lyapunov function is derived, using a positive P matrix, related to the eigenvalues of A matrix.
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ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
August 4–7, 2013
Portland, Oregon, USA
Conference Sponsors:
- Design Engineering Division
- Computers and Information in Engineering Division
ISBN:
978-0-7918-5591-1
PROCEEDINGS PAPER
Lyapunov Stability of Linear Fractional Systems: Part 1 — Definition of Fractional Energy
Jean-Claude Trigeassou,
Jean-Claude Trigeassou
University of Bordeaux, Bordeaux, France
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Nezha Maamri,
Nezha Maamri
University of Poitiers, Poitiers, France
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Alain Oustaloup
Alain Oustaloup
University of Bordeaux, Bordeaux, France
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Jean-Claude Trigeassou
University of Bordeaux, Bordeaux, France
Nezha Maamri
University of Poitiers, Poitiers, France
Alain Oustaloup
University of Bordeaux, Bordeaux, France
Paper No:
DETC2013-12824, V004T08A025; 10 pages
Published Online:
February 12, 2014
Citation
Trigeassou, J, Maamri, N, & Oustaloup, A. "Lyapunov Stability of Linear Fractional Systems: Part 1 — Definition of Fractional Energy." Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 4: 18th Design for Manufacturing and the Life Cycle Conference; 2013 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications. Portland, Oregon, USA. August 4–7, 2013. V004T08A025. ASME. https://doi.org/10.1115/DETC2013-12824
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