Mittag–Leffler stability of fractional order nonlinear dynamic systems

Li, Yan; Chen, YangQuan; Podlubný, Igor

doi:10.1016/j.automatica.2009.04.003

Cited by 1,356 publications

(646 citation statements)

References 11 publications

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“…The researches on the stability of NNs, FOS, and stochastic systems have attracted the attention of a large number of researchers, and many achievements have been made [29][30][31][32][33][34][35][36][37][38][39][40][41]. The sliding mode control (SMC) is a very popular strategy for a general class of nonlinear uncertain systems, with a very large frame of applications fields.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

Meng

Wang

2018

Mathematical Problems in Engineering

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Adaptive synchronization for a class of uncertain delayed fractional-order Hopfield neural networks (FOHNNs) with external disturbances is addressed in this paper. For the unknown parameters and external disturbances of the delayed FOHNNs, some adaptive estimations are designed. Firstly, a fractional-order switched sliding surface is proposed for the delayed FOHNNs. Then, according to the fractional-order extension of the Lyapunov stability criterion, a fractional-order sliding mode controller is constructed to guarantee that the synchronization error of the two uncertain delayed FOHNNs converges to an arbitrary small region of the origin. Finally, a numerical example of two-dimensional uncertain delayed FOHNNs is given to verify the effectiveness of the proposed method.

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Section: Introductionmentioning

confidence: 99%

Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

Meng

Wang

2018

Mathematical Problems in Engineering

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“…Applications of fractionalorder differential equations to different areas are investigated by many researchers, and some basic results have been achieved. 9,21,22 Another significant problem associated with synchronizing FOCSs is to enable the synchronization error to satisfy some PP, which indicates that the errors between the two FOCSs converge to an arbitrary small region of zero with the convergence rate being grater than a performance function given in advance. This kind of controller had been firstly introduced in Ref.…”

Section: Introductionmentioning

confidence: 99%

Prescribed performance synchronization controller design of fractional-order chaotic systems: An adaptive neural network control approach

Jiao

2017

AIP Advances

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In this study, an adaptive neural network synchronization (NNS) approach, capable of guaranteeing prescribed performance (PP), is designed for non-identical fractional-order chaotic systems (FOCSs). For PP synchronization, we mean that the synchronization error converges to an arbitrary small region of the origin with convergence rate greater than some function given in advance. Neural networks are utilized to estimate unknown nonlinear functions in the closed-loop system. Based on the integer-order Lyapunov stability theorem, a fractional-order adaptive NNS controller is designed, and the PP can be guaranteed. Finally, simulation results are presented to confirm our results.

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“…Matignon [8] studied linear time invariant FOSs and laid the theoretical foundation of the stability analysis in 1998. Further, it was revealed that linear or nonlinear FOSs have more general stability type called Mittag-Leffler stability in contrast to the classical exponential stability [9] . Many linear matrix inequality (LMI) criteria are available for the stability and robustness of certain or uncertain FOSs [10−15] .…”

Section: Introductionmentioning

confidence: 99%

Bounded real lemmas for fractional order systems

Liang

Wei

Pan

et al. 2015

Int. J. Autom. Comput.

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This paper derives the bounded real lemmas corresponding to L∞ norm and H∞ norm (L-BR and H-BR) of fractional order systems. The lemmas reduce the original computations of norms into linear matrix inequality (LMI) problems, which can be performed in a computationally efficient fashion. This convex relaxation is enlightened from the generalized Kalman-YakubovichPopov (KYP) lemma and brings no conservatism to the L-BR. Meanwhile, an H-BR is developed similarly but with some conservatism. However, it can test the system stability automatically in addition to the norm computation, which is of fundamental importance for system analysis. From this advantage, we further address the synthesis problem of H∞ control for fractional order systems in the form of LMI. Three illustrative examples are given to show the effectiveness of our methods.

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Mittag–Leffler stability of fractional order nonlinear dynamic systems

Cited by 1,356 publications

References 11 publications

Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

Prescribed performance synchronization controller design of fractional-order chaotic systems: An adaptive neural network control approach

Bounded real lemmas for fractional order systems

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