Research Papers

Energy Storage and Loss in Fractional-Order Systems

[+] Author and Article Information
Tom T. Hartley

Department of Electrical and Computer Engineering,
University of Akron,
Akron, OH 44325-3904
e-mail: thartley@uakron.edu

Jean-Claude Trigeassou

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

Carl F. Lorenzo

NASA Glenn Research Center,
Cleveland, OH 44135
e-mail: Carl.F.Lorenzo@nasa.gov

Nezha Maamri

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

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 30, 2014; final manuscript received December 27, 2014; published online April 9, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(6), 061006 (Nov 01, 2015) (8 pages) Paper No: CND-14-1113; doi: 10.1115/1.4029511 History: Received April 30, 2014; Revised December 27, 2014; Online April 09, 2015

As fractional-order systems are becoming more widely accepted and their usage is increasing, it is important to understand their energy storage and loss properties. Fractional-order operators can be implemented using a distributed state representation, which has been shown to be equivalent to the Riemann–Liouville representation. In this paper, the distributed state for a fractional-order integrator is represented using an infinite resistor–capacitor network such that the energy storage and loss properties can be readily determined. This derivation is repeated for fractional-order derivatives using an infinite resistor–inductor network. An analytical example is included to verify the results for a half-order integrator. Approximation methods are included.

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Grahic Jump Location
Fig. 1

An infinite series connection of parallel RC elements

Grahic Jump Location
Fig. 2

Lumped infinite-state model structure

Grahic Jump Location
Fig. 3

Lumped RC approximation of infinite-state representation of a fractional-order integrator

Grahic Jump Location
Fig. 4

An infinite series connection of parallel RL elements



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