An effective analytical criterion for stability testing of fractional-delay systems

Shi, Min; Wang, Zaihua

doi:10.1016/j.automatica.2011.05.018

Cited by 41 publications

(17 citation statements)

References 17 publications

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“…From (15) and (16) we obtain, respectively, | arg λ i | > π/2, which is equivalent to u i < 0, and h|λ i | < |ω 0i |. From (14) for v i ≥ 0 it follows that arg λ i = arctan(|v i |/|u i |) and by (17) …”

Section: Remarkmentioning

confidence: 93%

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Stability conditions for linear continuous-time fractional-order state-delayed systems

Busłowicz

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.

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Section: Remarkmentioning

confidence: 93%

“…In last years the stability problem of fractional systems with delays has been considered in [13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Stability conditions for linear continuous-time fractional-order state-delayed systems

Busłowicz

2016

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…The value set with respect to (21) becomes a line with two vertices F 1 ðωÞ ¼ ω 0:8 e 0:4πj þ 2ω 0:4 e 0:2πj þ 1 þ 0:5e À 0:1ωj and F 2 ðωÞ ¼ ω 0:8 e 0:4πj þ 2ω 0:4 e 0:2πj þ 1 À 0:5e À 0:1ωj . For this example, the test frequency interval for DðωÞ is ω A ½0:0113; 22:9177 with R min ¼ 0:0113 and R max ¼ 22:9177 via Theorem 3.…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…The condition that all of the poles must locate on the left half of the complex plane is required to be satisfied to ensure the stability of the corresponding interval fractional-order system. And several analytical and graphical stability criteria of fractional-order time-delay systems were investigated in [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Robust stability criterion for fractional-order systems with interval uncertain coefficients and a time-delay

Gao

2015

ISA Transactions

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“…For example, Deng et al [30] studied the stability of n-dimensional linear fractional differential equation with time delays. Shi and Wang [31] presented the BIBO stability criterion of a fractional-order delayed system. Babakhani et al [9] studied the existence of solutions at the neighborhood of equilibrium for fractional-order delayed differential equations and the Hopf bifurcations.…”

Section: Introductionmentioning

confidence: 99%

Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

Chen

Tang

et al. 2017

Shock and Vibration

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The primary resonance of van der Pol oscillator under fractional-order delayed negative feedback and forced excitation is studied. Firstly, the approximate analytical solution is obtained based on the averaging method, and it could be found that the fractionalorder delayed feedback has not only the property of delayed velocity feedback but also that of delayed displacement feedback. Moreover, the amplitude-frequency equation for the steady-state solution is established, and its stability conditions are also obtained. Then, the results of the approximate analytical solution and numerical integration are compared and analyzed. The agreement between the two methods is very high, so that the correctness and accuracy of the approximate analytical solution are verified. Finally, the effects of all the parameters in the fractional-order delayed feedback on the amplitude-frequency curves are analyzed. It could be concluded that fractional-order delayed feedback has important influences on the dynamical behavior of van der Pol oscillator, which is very significant to the optimization and control of a similar system.

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An effective analytical criterion for stability testing of fractional-delay systems

Cited by 41 publications

References 17 publications

Stability conditions for linear continuous-time fractional-order state-delayed systems

Stability conditions for linear continuous-time fractional-order state-delayed systems

Robust stability criterion for fractional-order systems with interval uncertain coefficients and a time-delay

Primary Resonance of van der Pol Oscillator under Fractional-Order Delayed Feedback and Forced Excitation

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