TIME-DELAYED FEEDBACK CONTROL OF TIME-DELAY CHAOTIC SYSTEMS

2003 ◽  
Vol 13 (01) ◽  
pp. 193-205 ◽  
Author(s):  
XINPING GUAN ◽  
CAILIAN CHEN ◽  
HAIPENG PENG ◽  
ZHENGPING FAN
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
Error Control ◽  
Chaotic Systems ◽  
Controller Design ◽  
Sufficient Conditions ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Control Technique ◽  
Time Delayed Feedback

This paper addresses time-delayed feedback control (DFC) of time-delay chaotic systems. To extend the DFC approach to time-delay chaotic system, alter having been successfully used in chaotic systems without time-delays, the standard feedback control (SFC) method is firstly employed to show the main control technique in this paper based on one error control system. Then sufficient conditions for stabilization and tracking problems via DFC are derived from the results based on SFC. Also, the systematic and analytic controller design method can be obtained to stabilize the system to an unstable fixed point and to tracking an unstable periodic orbit, respectively. Some numerical examples are provided to demonstrate the effectiveness of the presented method.

MASTER-SLAVE SYNCHRONIZATION OF NONAUTONOMOUS CHAOTIC SYSTEMS AND ITS APPLICATION TO ROTATING PENDULUMS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250147 ◽  
Author(s):  
KE DING ◽  
QING-LONG HAN
Keyword(s):  
Chaotic Systems ◽  
Design Method ◽  
Sufficient Conditions ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Phase Locked Loops ◽  
Linear Matrix ◽  
Delayed Feedback Controller ◽  
Delay Dependent ◽  
Time Delayed Feedback

Some mathematical models in engineering and physics, such as rotating pendulums, governors and phase locked loops in circuits, can be described as nonautonomous systems in which there exist chaotic attractors. This paper investigates master-slave synchronization for two nonautonomous chaotic systems by using time-delayed feedback control. Firstly, three delay-dependent synchronization criteria, which are formulated in the form of linear matrix inequalities (LMIs), are established for complete synchronization, lag synchronization and anticipating synchronization, respectively. Secondly, sufficient conditions on the existence of a time-delayed feedback controller are derived by employing these newly-obtained synchronization criteria. The controller gain can be obtained by solving a set of LMIs. Finally, the synchronization criteria and the design method are applied to master-slave synchronization for rotating pendulum systems.

BIFURCATION, CHAOS AND THEIR CONTROL IN A TIME-DELAY DIGITAL TANLOCK LOOP

2013 ◽  
Vol 23 (08) ◽  
pp. 1330029 ◽  
Author(s):  
TANMOY BANERJEE ◽  
BISHWAJIT PAUL ◽  
B. C. SARKAR
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
Parameter Space ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Convergence Time ◽  
Stable Phase ◽  
Bifurcation And Chaos ◽  
Global Parameter ◽  
Time Delayed Feedback

This paper reports the detailed parameter space study of the nonlinear dynamical behaviors and their control in a time-delay digital tanlock loop (TDTL). At first, we explore the nonlinear dynamics of the TDTL in parameter space and show that beyond a certain value of loop gain parameter the system manifests bifurcation and chaos. Next, we consider two variants of the delayed feedback control (DFC) technique, namely, the time-delayed feedback control (TDFC) technique, and its modified version, the extended time-delayed feedback control (ETDFC) technique. Stability analyses are carried out to find out the stable phase-locked zone of the system for both the controlled cases. We employ two-parameter bifurcation diagrams and the Lyapunov exponent spectrum to explore the dynamics of the system in the global parameter space. We establish that the control techniques can extend the stable phase-locked region of operation by controlling the occurrence of bifurcation and chaos. We also derive an estimate of the optimum parameter values for which the controlled system has the fastest convergence time even for a larger acquisition range. The present study provides a necessary detailed parameter space study that will enable one to design an improved TDTL system.

Stochastic Stability of Mathieu-Van Der Pol System with Delayed Feedback Control

2011 ◽  
Vol 105-107 ◽  
pp. 132-138
Author(s):  
Chang Shui Feng ◽  
Shuang Lin Chen
Keyword(s):  
Time Delay ◽  
Lyapunov Exponent ◽  
Feedback Control ◽  
Lyapunov Stability ◽  
Original System ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Control Force ◽  
Van Der Pol ◽  
Time Delayed Feedback

The asymptotic Lyapunov stability with probability one of Mathieu-Van der Pol system with time-delayed feedback control under wide-band noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.

DELAY-DEPENDENT APPROACH TO STABILIZATION OF TIME-DELAY CHAOTIC SYSTEMS VIA STANDARD AND DELAYED FEEDBACK CONTROLLERS

2005 ◽  
Vol 15 (04) ◽  
pp. 1455-1465 ◽  
Author(s):  
SHENGYUAN XU ◽  
JAMES LAM ◽  
YUN ZOU
Keyword(s):  
Time Delay ◽  
Feedback Control ◽  
State Feedback ◽  
Chaotic Systems ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Feedback Controllers ◽  
Problem Of Time ◽  
Unstable Equilibrium Point ◽  
Delay Dependent

In this paper, the stabilization problem of time-delay chaotic systems via standard feedback control (SFC) and delayed feedback control (DFC) methods is investigated. First, new delay-dependent conditions for stabilization via SFC method are obtained in terms of matrix inequalities. The proposed feedback controller stabilizes an unstable equilibrium point in the chaotic system. Then, based on this, delay-dependent conditions for stabilization via DFC method are derived. In both the SFC and DFC cases, algorithms on the design of desired state feedback controllers are proposed. Examples are provided to show the effectiveness of the proposed methods.

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