Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order $$\varvec{PI}^{\varvec{\lambda }} \varvec{D}^{\varvec{\mu }}$$ P I λ D μ feedback controller

Chen, Lincong; Zhao, Tianlong; Li, Wei; Zhao, Jun

doi:10.1007/s11071-015-2345-1

Cited by 46 publications

(12 citation statements)

References 38 publications

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“…Compared with integer-order controllers, the fractional-order controllers have better dynamic performances and robustness, and recently, various fractional-order controllers have been developed. 39,40 And we obtained the critical conditions when the system (5) will exhibit stochastic P-bifurcation through the above analysis, which can make the system switch between monostable and bistable states by selecting the corresponding unfolding parameters; this can provide theoretical guidance for the design of fractional-order controllers.…”

Section: The Influence With Variation Of Time-delay S On the Systemmentioning

confidence: 99%

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

Wang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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This paper investigates the stochastic P-bifurcation behavior of bistability in a generalized Van der Pol system with fractional time-delay feedback under additive and multiplicative Gaussian white noise excitations. First, using the minimal mean square error principle, the fractional derivative is found to be equivalent to a linear combination of damping and restoring forces, and the original system is transformed into an equivalent integer-order system. Second, the stationary probability density function of system amplitude is obtained by stochastic averaging, and according to singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the different types of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each region divided by the transition set curves. The consistency between the analytical solutions and the Monte Carlo simulation results verifies the theoretical analysis in this paper. The method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.

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Section: The Influence With Variation Of Time-delay S On the Systemmentioning

confidence: 99%

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

Wang

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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“…Xu et al [20] considered a complex Duffing system subjected to nonstationary random excitation. Chen et al [21][22][23] studied the random jumping and bifurcation behaviors of fractionalorder Duffing oscillators under combined harmonic and white noise excitation, and they designed a controller based on the fractional-order PID.…”

Section: Introductionmentioning

confidence: 99%

Threshold for Chaos of a Duffing Oscillator with Fractional‐Order Derivative

Xing

Chen

Chang

et al. 2019

Shock and Vibration

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In this paper, the necessary condition for the chaotic motion of a Duffing oscillator with the fractional-order derivative under harmonic excitation is investigated. The necessary condition for the chaos in the sense of Smale horseshoes is established based on the Melnikov method, and then the chaotic threshold curve is obtained. The largest Lyapunov exponents are provided, and some other typical numerical simulation results, including the time histories, frequency spectrograms, phase portraits, and Poincare maps, are presented and compared. From the analysis of the numerical simulation results, it could be found that, near the chaotic threshold curve, the system generates chaos via the period-doubling bifurcation, from single periodic motion to period-2 motion and period-4 motion to chaotic motion. The effects of fractional-order parameters, the stiffness coefficient, and the damping coefficient on the threshold value of the chaotic motion are analytically discussed. The results show that the coefficient of the fractional-order derivative has great effect on the threshold value of the chaotic motion, while the order of the fractional-order derivative has less. The analysis results reveal some new phenomena, and it could be useful for designing or controlling dynamic systems with the fractional-order derivative.

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“…Stochastic perturbations are ubiquitous in the real world, so it is necessary to study the dynamical behaviors of the fractionalorder stochastic systems. A lot of methods have been put forward to study the fractional-order stochastic systems, such as the stochastic averaging method [14][15][16][17], multiple scales method [18][19][20], Wiener path integral technique [21], and statistical linearization-based technique [22]. Some recent articles on this topic are as follows.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Bifurcations of a Fractional‐Order Vibro‐Impact System Driven by Additive and Multiplicative Gaussian White Noises

Yang

Sun

2019

Complexity

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Stochastic fractional-order systems or stochastic vibro-impact systems can present rich dynamical behaviors, and lots of studies dealing with stochastic fractional-order systems or stochastic vibro-impact systems are available now, while the discussion on the stochastic systems with both vibro-impact factors and fractional derivative element is rare. This paper is concerned with the stochastic bifurcation of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. Firstly, we can remove the discontinuity of the original system with the help of nonsmooth transformation and obtain the equivalent stochastic system. Then, we adopt the stochastic averaging method to get the approximately analytical solutions. At last, an example is discussed in detail to assess the reliability of the developed approach. We also find that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can induce the stochastic bifurcation.

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Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order $$\varvec{PI}^{\varvec{\lambda }} \varvec{D}^{\varvec{\mu }}$$ P I λ D μ feedback controller

Cited by 46 publications

References 38 publications

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

Stochastic P-bifurcation in a generalized Van der Pol oscillator with fractional delayed feedback excited by combined Gaussian white noise excitations

Threshold for Chaos of a Duffing Oscillator with Fractional‐Order Derivative

Stochastic Bifurcations of a Fractional‐Order Vibro‐Impact System Driven by Additive and Multiplicative Gaussian White Noises

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