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Research Papers

A New Look at the Fractional Initial Value Problem: The Aberration Phenomenon

[+] Author and Article Information
Yanting Zhao

Department of Automation,
University of Science and Technology of China,
Hefei 230027, China
e-mail: zyt9lsb@mail.ustc.edu.cn

Yiheng Wei

Department of Automation,
University of Science and Technology of China,
Hefei 230027, China
e-mail: neudawei@ustc.edu.cn

Yuquan Chen

Department of Automation,
University of Science and Technology of China,
Hefei 230027, China
e-mail: cyq@mail.ustc.edu.cn

Yong Wang

Department of Automation,
University of Science and Technology of China,
Hefei 230027, China
e-mail: yongwang@ustc.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 3, 2018; final manuscript received September 24, 2018; published online October 19, 2018. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(12), 121004 (Oct 19, 2018) (8 pages) Paper No: CND-18-1253; doi: 10.1115/1.4041621 History: Received June 03, 2018; Revised September 24, 2018

A typical phenomenon of the fractional order system is presented to describe the initial value problem from a brand-new perspective in this paper. Several simulation examples are given to introduce the named aberration phenomenon, which reflects the complexity and the importance of the initial value problem. Then, generalizations on the infinite dimensional property and the long memory property are proposed to reveal the nature of the phenomenon. As a result, the relationship between the pseudo state-space model and the infinite dimensional exact state-space model is demonstrated. It shows the inborn defects of the initial values of the fractional order system. Afterward, the pre-initial process and the initialization function are studied. Finally, specific methods to estimate exact state-space models and fit initialization functions are proposed.

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Figures

Grahic Jump Location
Fig. 1

The exact state-space model of the fractional order system

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Fig. 2

Input of convex functions and concave functions

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Fig. 3

The aberration phenomenon

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Fig. 4

System outputs with controller (16)

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Fig. 5

System outputs with controller (17)

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Fig. 6

Sliding manifolds with controller (17)

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Fig. 7

Fractional order state observer

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