scholarly journals Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation

2009 ◽  
Vol 58 (1-2) ◽  
pp. 319-344 ◽  
Author(s):  
Min Xiao ◽  
Daniel W. C. Ho ◽  
Jinde Cao
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Small World ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Small World Networks ◽  
Time Delayed Feedback

Criticality of Hopf Bifurcation in Precision Motion Stage With PID and Time-Delayed Feedback Controls

2020 ◽  
Author(s):  
Sunit K. Gupta ◽  
Jiamin Wang ◽  
Oumar R. Barry
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Delayed Feedback ◽  
State Feedback Control ◽  
Lumped Parameter Model ◽  
Delayed Feedback Control ◽  
Use Of Time ◽  
Motion Stage ◽  
Precision Motion ◽  
Time Delayed Feedback

Abstract The use of precision motion stages is very popular among advanced manufacturing and machining industries. However, the performance of these motion stages is usually undermined by friction-induced vibration. In this paper, we propose the use of time-delayed feedback control to minimize the undesirable effects of friction-induced vibrations. The use of time-delayed feedback control is well established in the literature; however, the use of time-delayed feedback control in PID controlled motion-stages has not been explored yet. Here, we consider a lumped parameter model of the PID controlled precision motion stage with a linear time-delayed state feedback control. The dynamical friction in the systemis modeled using the LuGre model. Stability and nonlinear analysis of the system are carried out using analytical methods. The stability analysis reveals the existence of multiple stability lobes and codimension-2 Hopf points for a given choice of system parameters. Also, the nature of Hopf bifurcation is determined by using the method of multiple scales. We observe the existence of both subcritical and supercritical Hopf bifurcations in the system, depending on the choice of control parameters. This observation implies that the nonlinearity in the system could both be stabilizing or destabilizing in nature.

BIFURCATION DYNAMICS IN CONTROL SYSTEMS

1999 ◽  
Vol 09 (01) ◽  
pp. 287-293 ◽  
Author(s):  
GUANRONG CHEN ◽  
JIALIANG LU ◽  
BRENT NICHOLAS ◽  
SWATIPRAKASH M. RANGANATHAN
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Control Systems ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Main Concern ◽  
Feedback Strategy ◽  
Feedback Control Systems ◽  
Time Delayed Feedback ◽  
Control Purpose

This paper is to report the observation that when the popular time-delayed feedback strategy is used for control purpose, it may actually create unwanted bifurcations. Hopf bifurcation created by delayed feedback control is the main concern of this article, but some other types of bifurcations are also observed to exist in such delayed-feedback control systems. The observations are illustrated by computer simulations.

DYNAMICS OF A TWO-DIMENSIONAL DELAYED SMALL-WORLD NETWORK UNDER DELAYED FEEDBACK CONTROL

2006 ◽  
Vol 16 (11) ◽  
pp. 3257-3273 ◽  
Author(s):  
XU XU ◽  
HAIYAN HU ◽  
HUAILEI WANG
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Periodic Solutions ◽  
Small World ◽  
Delayed Feedback ◽  
State Feedback Control ◽  
Delayed Feedback Control ◽  
Small World Network ◽  
Two Dimensional ◽  
Normal Form Theory

This paper presents a detailed analysis on the dynamics of a two-dimensional delayed small-world network under delayed state feedback control. On the basis of stability switch criteria, the equilibrium is studied, and the stability conditions are determined. This study shows that with properly chosen delay and gain in the delayed feedback path, the controlled small-world delayed network may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or the multistability solutions via three types of codimension two bifurcations. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are determined by using the normal form theory and center manifold theorem. In addition, the study shows that the controlled model exhibits period-doubling bifurcations which lead eventually to chaos; and the chaos can also directly occur via the bifurcations from the quasi-periodic solutions. The results show that the delayed feedback is an effective approach in order to generate or annihilate complex behaviors in practical applications.

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