Bifurcation analysis in active control system with time delay feedback

Jian, Peng; Wang, Lianhua; Zhao, Yanpu; Zhao, Yaobing

doi:10.1016/j.amc.2013.04.014

Cited by 16 publications

(14 citation statements)

References 20 publications

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“…Kalmar-Nagy, Stepan, and Moon [41] studied the existence of a subcritical Hopf bifurcation in the delay-differential equation model of the so-called regenerative machine tool vibration. Peng et al [42] investigated the stability and bifurcation of an SDOFsystem with time-delayed feedback. Li et al [43] investigated the nonlinear dynamics of a Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay.…”

Section: Introductionmentioning

confidence: 99%

Time-Delayed Feedback Control of Piezoelectric Elastic Beams under Superharmonic and Subharmonic Excitations

Jian

Xiang

et al. 2019

Applied Sciences

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The time-delayed displacement feedback control is provided to restrain the superharmonic and subharmonic response of the elastic support beams. The nonlinear equations of the controlled elastic beam are obtained with the help of the Euler–Bernoulli beam principle and time-delayed feedback control strategy. Based on Galerkin method, the discrete nonlinear time-delayed equations are derived. Using the multiscale method, the first-order approximate solutions and stability conditions of three superharmonic and 1/3 subharmonic resonance response on controlled beams are derived. The influence of time-delayed parameters and control gain are obtained. The results show that the time-delayed displacement feedback control can effectively suppress the superharmonic and subharmonic resonance response. Selecting reasonably the time-delayed quantity and control gain can avoid the resonance region and unstable multi-solutions and improve the efficiency of the vibration control. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.

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Section: Introductionmentioning

confidence: 99%

Time-Delayed Feedback Control of Piezoelectric Elastic Beams under Superharmonic and Subharmonic Excitations

Jian

Xiang

et al. 2019

Applied Sciences

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“…then Σ(t, (ψ, α)) ∈ W c loc which by (51) implies that Σ(t, ψ) ∈ W c loc (α). We note that if ψ ∈ W c loc (α) then by (50)…”

Section: Parameter-dependent Local Center Manifoldsmentioning

confidence: 99%

“…which is used to control the response of structures to internal or external excitation, see [50]. The function f is substituted by βx 3 (t − 1) and the parameters g u = 0.1, g v = 0.52, β = 0.1, We conclude that this Hopf-Hopf bifurcation is of 'difficult' type, since (Re g 2100 )(Re g 0021 ) = −0.0166 < 0, see [43].…”

Section: S211 Simulation With Pydelaymentioning

confidence: 99%

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Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bosschaert¹,

Janssens²,

Kuznetsov³

2020

SIAM J. Appl. Dyn. Syst.

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In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

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“…Pyragas [19] proposed a method for chaotic control using delayed feedback signals. A lot of researches have been carried out on time-delay control [20,21,22,23,24,25]. Compared with other methods, time-delayed control is a simple and effective method to control chaos in economic models.…”

Section: Introductionmentioning

confidence: 99%

Chaos Control of a Resource-Economic-Pollution Dynamic System

Yin

Huang

Fan

2019

JERR

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How to control chaos in the economic system has aroused the interest of researchers. We research the chaos control in a new Resource-Economic-Pollution system by time-delayed feedback control. By determining the appropriate range of time delay τ and feedback strength k, the chaotic phenomena of the system are controlled. We verify the linear stability and the existence of Hopf bifurcation of the system. Numerical simulations show that chaos control can eliminate the chaotic behavior of the system and stabilize the system at the equilibrium point. When the time lag term is in a certain interval, the chaotic phenomenon of the system will disappear and the system will be controlled in a stable state. In practice, due to capacity and financial constraints, the firm or the government often restrains output through many methods to confine the range of fluctuations in these variables. This shows that the government or corporate decision makers have often used this approach consciously or unconsciously to promote steady economic growth.

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Bifurcation analysis in active control system with time delay feedback

Cited by 16 publications

References 20 publications

Time-Delayed Feedback Control of Piezoelectric Elastic Beams under Superharmonic and Subharmonic Excitations

Time-Delayed Feedback Control of Piezoelectric Elastic Beams under Superharmonic and Subharmonic Excitations

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Chaos Control of a Resource-Economic-Pollution Dynamic System

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