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Technical Brief

Sliding Mode Control of Vibration in Single-Degree-of-Freedom Fractional Oscillators

[+] Author and Article Information
Jian Yuan

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: yuanjianscar@gmail.com

Youan Zhang

Institute of Technology,
Yantai Nanshan University,
Yantai 265713, China
e-mail: zhangya63@sina.com

Jingmao Liu

Shandong Nanshan International Flight Co., Ltd.,
Yantai 265713, China
e-mail: liujingmao@nanshan.com.cn

Bao Shi

Institute of System Science and Mathematics,
Naval Aeronautical and Astronautical University,
Yantai 264001, China
e-mail: baoshi781@sohu.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 12, 2016; final manuscript received April 2, 2017; published online July 10, 2017. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 139(11), 114503 (Jul 10, 2017) (6 pages) Paper No: DS-16-1183; doi: 10.1115/1.4036665 History: Received April 12, 2016; Revised April 02, 2017

This paper proposes sliding mode control of vibration in three types of single-degree-of-freedom (SDOF) fractional oscillators: the Kelvin–Voigt type, the modified Kelvin–Voigt type, and the Duffing type. The dynamical behaviors are all described by second-order differential equations involving fractional derivatives. By introducing state variables of physical significance, the differential equations of motion are transformed into noncommensurate fractional-order state equations. Fractional sliding mode surfaces are constructed and the stability of the sliding mode dynamics is proved by means of the diffusive representation and Lyapunov stability theory. Then, sliding mode control laws are designed for fractional oscillators, respectively, in cases where the bound of the external exciting force is known or unknown. Furthermore, sliding mode control laws for nonzero initialization case are designed. Finally, numerical simulations are carried out to validate the above control designs.

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Figures

Grahic Jump Location
Fig. 1

Vibration response of the SDOF fractional Kelvin–Voigt oscillators

Grahic Jump Location
Fig. 2

Sliding mode control of vibration in SDOF fractional Kelvin–Voigt oscillators

Grahic Jump Location
Fig. 3

Adaptive sliding mode control of vibration in SDOF fractional Kelvin–Voigt oscillators

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