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Research Papers

# Stability and Stabilization of a Class of Fractional-Order Nonlinear Systems for 1 < α < 2

[+] Author and Article Information
Sunhua Huang

Department of Electrical Engineering,
Northwest A&F University,
Yangling 712100, Shaanxi, China

Bin Wang

Department of Electrical Engineering,
Northwest A&F University,
Yangling 712100, Shaanxi, China
e-mail: binwang@nwsuaf.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 19, 2016; final manuscript received November 6, 2017; published online January 10, 2018. Editor: Bala Balachandran.

J. Comput. Nonlinear Dynam 13(3), 031003 (Jan 10, 2018) (8 pages) Paper No: CND-16-1436; doi: 10.1115/1.4038443 History: Received September 19, 2016; Revised November 06, 2017

## Abstract

This study is interested in the stability and stabilization of a class of fractional-order nonlinear systems with Caputo derivatives. Based on the properties of the Laplace transform, Mittag-Leffler function, Jordan decomposition, and Grönwall's inequality, some sufficient conditions that ensure local stability and stabilization of a class of fractional-order nonlinear systems under the Caputo derivative with $1<α<2$ are presented. Finally, typical instances, including the fractional-order three-dimensional (3D) nonlinear system and the fractional-order four-dimensional (4D) nonlinear hyperchaos, are implemented to demonstrate the feasibility and validity of the proposed method.

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## Figures

Fig. 1

State trajectories of uncontrolled fractional-order nonlinear system (34) with α=1.4

Fig. 2

State trajectories of controlled fractional-order nonlinear system (37) with α=1.4

Fig. 3

State trajectories of controlled fractional-order nonlinear system (37) with α=1.9

Fig. 4

State trajectories of uncontrolled fractional-order nonlinear system (40) with α=1.1

Fig. 5

State trajectories of controlled fractional-order nonlinear system (43) with α=1.1

Fig. 6

State trajectories of controlled fractional-order nonlinear system (43) with α=1.9

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