scholarly journals Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation

2013 ◽  
Vol 371 (1999) ◽  
pp. 20120472 ◽  
Author(s):  
Isabelle Schneider
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Non Invasive ◽  
Selective Stabilization ◽  
Spatio Temporal ◽  
The Individual ◽  
Equivariant Hopf Bifurcation

The modest aim of this case study is the non-invasive and pattern-selective stabilization of discrete rotating waves (‘ponies on a merry-go-round’) in a triangle of diffusively coupled Stuart–Landau oscillators. We work in a setting of symmetry-breaking equivariant Hopf bifurcation. Stabilization is achieved by delayed feedback control of Pyragas type, adapted to the selected spatio-temporal symmetry pattern. Pyragas controllability depends on the parameters for the diffusion coupling, the complex control amplitude and phase, the uncontrolled super-/sub-criticality of the individual oscillators and their soft/hard spring characteristics. We mathematically derive explicit conditions for Pyragas control to succeed.

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.

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