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Lyapunov Stability of Linear Fractional Systems: Part 1 — Definition of Fractional Energy

[+] Author Affiliations
Jean-Claude Trigeassou, Alain Oustaloup

University of Bordeaux, Bordeaux, France

Nezha Maamri

University of Poitiers, Poitiers, France

Paper No. DETC2013-12824, pp. V004T08A025; 10 pages
  • 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
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5591-1
  • Copyright © 2013 by ASME


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.

Copyright © 2013 by ASME
Topics: Stability



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