Razumikhin-type stability theorems for functional fractional-order differential systems and applications

Chen, Boshan; Chen, Jie‐Jie

doi:10.1016/j.amc.2014.12.010

Cited by 80 publications

(43 citation statements)

References 21 publications

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“…Moreover, many systems modelled with the help of fractional calculus display rich fractional dynamical behavior, such as viscoelastic systems [10], boundary layer effects in ducts [11], electromagnetic waves [12], fractional kinetics [13,14], and electrode-electrolyte polarization [15,16]. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural networks [31,32], fractional-order switched linear systems [33,34], fractional-order singular systems [35,36], positive fractional-order systems [37,38] and fractional chaotic complex networks systems [39]. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural ne...…”

Section: Introductionmentioning

confidence: 99%

“…As a result, the problem of stability analysis and control of fractional-order systems is an important problem in the theory and applications of fractional calculus. Many stability conditions have been proposed for linear fractional-order systems [17][18][19][20][21][22][23], fractional-order nonlinear systems [24][25][26][27][28][29][30], fractional-order neural networks [31,32], fractional-order switched linear systems [33,34], fractional-order singular systems [35,36], positive fractional-order systems [37,38] and fractional chaotic complex networks systems [39]. Among the reported methods, the Lyapunov direct method provides an effective approach to analyze the stability of fractional nonlinear systems.…”

Section: Introductionmentioning

confidence: 99%

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New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

Thuận¹,

Huong²

2017

Asian Journal of Control

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This paper considers the systematic design of robust stabilizing state feedback controllers for fractional-order nonlinear systems. By using the Lyapunov direct method and a recent result on the Caputo fractional derivative of a quadratic function, stabilizability conditions expressed in terms of linear matrix inequalities are derived. The controllers can then be derived by using existing computationally effective convex algorithms. Two numerical examples with simulation results are provided to demonstrate the effectiveness of our results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

Thuận¹,

Huong²

2017

Asian Journal of Control

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“…The most frequently used definitions for noninteger derivatives are Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions [6,9,[23][24][25][26][27][28][29][30][31][33][34][35][36][37][38]. As the Caputo fractional operator is more consistent than another ones, then this operator will be employed in the rest of this paper.…”

Section: Remarkmentioning

confidence: 99%

“…In [27], sufficient stability conditions for fractional-order epidemic systems have been obtained by Volterra-type Lyapunov functions. In [28,39], stability analysis problem of fractional-order systems with delays have been solved by Lyapunov approaches, Razumikhin-type stability theorems, and Lyapunov-Krasovskii approaches. In [64], simple criteria of Mittag-Leffler stability have been proposed for fractional nonlinear systems.…”

Section: Remark 13mentioning

confidence: 99%

“…However, the Lyapunov-Razumikhin functionals are selected as quadratic functions in order to design the control for fractional-order systems with constant delays, timevarying delays, and distributed time-varying delays [1,4,5,[27][28][29][30]63]. In [27][28][29][30], stability theorems for fractional-order systems have been developed by using Riemann-Liouville derivatives, Caputo derivatives, a Lyapunov function, and the Razumikhin technique.…”

Section: Introductionmentioning

confidence: 99%

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Observer-based adaptive neural network control for a class of MIMO uncertain nonlinear time-delay non-integer-order systems with asymmetric actuator saturation

Zouari

Boulkroune

Ibeas

et al. 2016

Neural Comput & Applic

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This paper focuses on the adaptive neural output feedback control of a class of uncertain multi-input-multioutput nonlinear time-delay non-integer-order systems with unmeasured states, unknown control direction, and unknown asymmetric saturation actuator. Thus, the mean value theorem and a Gaussian error function-based continuous differentiable model are used in the paper to describe the unknown asymmetric saturation actuator and to get an affine model in which the control input appears in a linear fashion, respectively. The design of the controller follows a number of steps. Firstly, based on the semigroup property of fractional-order derivative, the system is transformed into a normalized fractional-order system by means of a state transformation in order to facilitate the control design. Then, a simple linear state observer is constructed to estimate the unmeasured states of the transformed system. A neural network is incorporated to approximate the unknown nonlinear functions while a Nussbaum function is used to deal with the unknown control direction. In addition, the strictly positive real condition, the Razumikhin Lemma, the frequency-distributed model, and the Lyapunov method are utilized to derive the parameter adaptive laws and to perform the stability proof. The main advantages of this work are that:(1) it can handle systems with constant, time-varying, and distributed time-varying delays, (2) the considered class of systems is relatively large, (3) the number of adjustable parameters is reduced, (4) the tracking errors converge asymptotically to zero and all signals of the closed-loop system are bounded. Finally, some simulation examples are provided to demonstrate the validity and effectiveness of the proposed scheme.

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Stabilization of uncertain fractional order system with time‐varying delay using BMI approach

Zhou

Kou

et al. 2019

Asian Journal of Control

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This paper considers the systematic design of robust stabilizing state feedback controllers for fractional order nonlinear systems with time‐varying delay being possibly unbounded. By using the fractional Halanay inequality and the Caputo fractional derivative of a quadratic function, stabilizability conditions expressed in terms of bilinear matrix inequalities are derived. The controllers can then be obtained by computing the gain matrices. In order to derive the gain matrices, two algorithms are proposed by using the existing computationally linear matrix inequality techniques. Two numerical examples with simulation results are provided to demonstrate the effectiveness of the obtained results.

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Razumikhin-type stability theorems for functional fractional-order differential systems and applications

Cited by 80 publications

References 21 publications

New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

New Results on Stabilization of Fractional‐Order Nonlinear Systems via an LMI Approach

Observer-based adaptive neural network control for a class of MIMO uncertain nonlinear time-delay non-integer-order systems with asymmetric actuator saturation

Stabilization of uncertain fractional order system with time‐varying delay using BMI approach

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