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Research Papers

# $H∞$ and Sliding Mode Observers for Linear Time-Invariant Fractional-Order Dynamic Systems With Initial Memory Effect

[+] Author and Article Information
Sang-Chul Lee

School of Mechatronics,
Gwangju Institute of Science
and Technology (GIST),
123 Cheomdan-gwagiro, Buk-gu,
Gwangju 500-712, Korea
e-mail: sclee@gist.ac.kr

Yan Li

School of Control Science and Engineering,
Shandong University,
Jinan Shandong Province 250061, China
e-mail: liyan.sdu@gmail.com

YangQuan Chen

School of Engineering,
University of California Merced,
Merced, CA 95343
e-mail: ychen53@ucmerced.edu

Hyo-Sung Ahn

School of Mechatronics,
Gwangju Institute of Science
and Technology (GIST),
123 Cheomdan-gwagiro, Buk-gu,
Gwangju 500-712, Korea
e-mail: hyosung@gist.ac.kr

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 13, 2013; final manuscript received March 21, 2014; published online July 9, 2014. Assoc. Editor: Hashem Ashrafiuon.

J. Dyn. Sys., Meas., Control 136(5), 051022 (Jul 09, 2014) (13 pages) Paper No: DS-13-1354; doi: 10.1115/1.4027289 History: Received September 13, 2013; Revised March 21, 2014

## Abstract

The $H∞$ and sliding mode observers are important in integer-order dynamic systems. However, these observers are not well explored in the field of fractional-order dynamic systems. In this paper, the $H∞$ filter and the fractional-order sliding mode unknown input observer are developed to estimate state of the linear time-invariant fractional-order dynamic systems with consideration of proper initial memory effect. As the first result, the fractional-order $H∞$ filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the $H∞$ norm. Based on the extended bounded real lemma, the $H∞$ filter design is formulated in a linear matrix inequality form, and it will be seen that numerical methods to solve convex optimization problems are feasible in fractional-order systems (FOSs). As the second result of this paper, not only state but also unknown input disturbance are estimated by fractional-order sliding-mode unknown input observer, simultaneously. In this paper, it is shown that the design and stability analysis of the two estimation techniques are not related with the initial history. Through two numerical examples, the performance of the fractional-order $H∞$ filter and the fractional-order sliding-mode observer is illustrated with consideration of the initialization functions.

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## Figures

Fig. 1

Stable regions for fractional-order systems. (a) 0 < r < 1, (b) r = 1, and (c) 1 < r < 2.

Fig. 2

H∞ filtering setup

Fig. 3

Results of state estimation using FO-H∞

Fig. 4

Energy ratio between the noise and the estimation error

Fig. 5

Results of state estimation using FO-SMUIO

Fig. 6

Unknown input estimation using FO-SMUIO

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