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Research Papers

Initial Conditions and Initialization of Fractional Systems

[+] Author and Article Information
Massinissa Tari

LIAS,
University of Poitiers,
Poitiers Cedex 8600, France
e-mail: massinissa.tari@univ-poitiers.fr

Nezha Maamri

LIAS,
University of Poitiers,
Poitiers Cedex 8600, France
e-mail: nezha.maamri@univ-poitiers.fr

Jean-Claude Trigeassou

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 15, 2015; final manuscript received January 28, 2016; published online March 16, 2016. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(4), 041014 (Mar 16, 2016) (7 pages) Paper No: CND-15-1098; doi: 10.1115/1.4032695 History: Received April 15, 2015; Revised January 28, 2016

In this paper, the initialization of fractional order systems is analyzed. The objective is to prove that the usual pseudostate variable x(t) is unable to predict the future behavior of the system, whereas the infinite dimensional variable z(ω,t) fulfills the requirements of a true state variable. Two fractional systems, a fractional integrator and a one-derivative fractional system, are analyzed with the help of elementary tests and numerical simulations. It is proved that the dynamic behaviors of these two fractional systems differ completely from that of their integer order counterparts. More specifically, initialization of these systems requires knowledge of z(ω,t0) initial condition.

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Figures

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Fig. 1

Pseudo impulse excitation u(t)

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Fig. 2

Comparison between the theoretical response x(t) and the numerical approximation xnum(t)

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Fig. 3

zj(t) : components of z(ω,t) at T = 5 s

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Fig. 4

Pseudo impulse excitation u¯(t)

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Fig. 5

Comparison between x¯(t) and x(t)

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Fig. 6

Comparison between the components zj(t) and z¯j(t) at T = 5

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Fig. 7

x¯(t) and x(t): responses in the integer order case

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Fig. 8

x¯(t) and x(t): responses in the integer case

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Fig. 9

Comparison between the theoretical response x(t) and the numerical approximation xnum(t)

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Fig. 10

Comparison between x¯(t) and x(t)

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Fig. 11

Comparison between the component zj(t) and z¯j(t) at T = 5 s

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