Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

Zaihua Wang, Haiyan Hu, Qi Xu, G. Stépán

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.

Original languageEnglish
JournalInternational Journal of Non-Linear Mechanics
DOIs
Publication statusAccepted/In press - Jun 20 2016

Fingerprint

Hopf bifurcation
Hopf Bifurcation
Derivatives
Feedback
Derivative
Duffing Oscillator
Bifurcation Control
Active Vibration Control
Characteristic Roots
Nonlinear Oscillator
Vibration control
Control theory
Control Theory
Stability Analysis
Controller
Controllers
Interval

Keywords

  • Acceleration-derivative feedback
  • Duffing oscillator
  • Hopf bifurcation
  • Positive feedback
  • Stable region
  • Time delay

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

Cite this

@article{51621bd9fcec44cd9cc5d850f57c668c,
title = "Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback",
abstract = "This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.",
keywords = "Acceleration-derivative feedback, Duffing oscillator, Hopf bifurcation, Positive feedback, Stable region, Time delay",
author = "Zaihua Wang and Haiyan Hu and Qi Xu and G. St{\'e}p{\'a}n",
year = "2016",
month = "6",
day = "20",
doi = "10.1016/j.ijnonlinmec.2016.10.008",
language = "English",
journal = "International Journal of Non-Linear Mechanics",
issn = "0020-7462",
publisher = "Elsevier Limited",

}

TY - JOUR

T1 - Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

AU - Wang, Zaihua

AU - Hu, Haiyan

AU - Xu, Qi

AU - Stépán, G.

PY - 2016/6/20

Y1 - 2016/6/20

N2 - This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.

AB - This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.

KW - Acceleration-derivative feedback

KW - Duffing oscillator

KW - Hopf bifurcation

KW - Positive feedback

KW - Stable region

KW - Time delay

UR - http://www.scopus.com/inward/record.url?scp=85006100832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85006100832&partnerID=8YFLogxK

U2 - 10.1016/j.ijnonlinmec.2016.10.008

DO - 10.1016/j.ijnonlinmec.2016.10.008

M3 - Article

AN - SCOPUS:85006100832

JO - International Journal of Non-Linear Mechanics

JF - International Journal of Non-Linear Mechanics

SN - 0020-7462

ER -