Stabilization of Chaotic System with Uncertain Parameters

2011 ◽  
Vol 66-68 ◽  
pp. 217-219
Author(s):  
Wei Ming Sun ◽  
Xin Yu Wang ◽  
Jun Wei Lei
Keyword(s):  
Lyapunov Stability ◽  
Chaotic Systems ◽  
Adaptive Strategy ◽  
Stability Theorem ◽  
Convergence Time ◽  
Uncertain Parameters ◽  
Lyapunov Stability Theorem ◽  
Unknown Parameters ◽  
Dynamic Information ◽  
Adaptive Weight

Based on Lyapunov stability theorem, a common adaptive strategy was designed for a common class of chaotic systems with uncertain parameters. And the discuss on the sign of unknown parameters were avoided because of the adopting of a novel kind of adaptive weight turning law. At last, the convergence time was analyzed to provide a dynamic information for system users

ANTI-SYNCHRONIZATION OF A CLASS OF COUPLED CHAOTIC SYSTEMS VIA LINEAR FEEDBACK CONTROL

2006 ◽  
Vol 16 (04) ◽  
pp. 1041-1047 ◽  
Author(s):  
CHUANDONG LI ◽  
XIAOFENG LIAO
Keyword(s):  
Lyapunov Stability ◽  
Chaotic Systems ◽  
Sufficient Conditions ◽  
Stability Theorem ◽  
Linear Feedback ◽  
Generalized Synchronization ◽  
Lyapunov Stability Theorem ◽  
Linear Feedback Control ◽  
Relevant Variables ◽  
Special Case

As a special case of generalized synchronization, chaos anti-synchronization can be characterized by the vanishing of the sum of relevant variables. In this paper, based on Lyapunov stability theorem for ordinary differential equations, several sufficient conditions for guaranteeing the existence of anti-synchronization in a class of coupled identical chaotic systems via linear feedback or adaptive linear feedback methods are derived. Chua's circuit is presented as an example to demonstrate the effectiveness of the proposed approach by computer simulations.

Anti-Synchronization for a Modified Chen-Qi-Like Chaotic System

2012 ◽  
Vol 220-223 ◽  
pp. 2113-2116
Author(s):  
Su Hai Huang
Keyword(s):  
Numerical Simulation ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Stability Theorem ◽  
Control Law ◽  
Lyapunov Stability Theorem ◽  
Unknown Parameters ◽  
Dynamical Characteristics ◽  
Control Scheme ◽  
Adaptive Control Scheme

A modified Chen-Qi-like chaotic system is presented. Some basic dynamical characteristics of this system are studied by calculating the Lyapunov exponent and phase figure. Based on the Lyapunov stability theorem, adaptive control scheme and parameters update law are presented for the anti-synchronization of new chaotic systems with fully unknown parameters. Finally, the numerical simulation verify that the control law and parameter changing are correct.

scholarly journals Model Reference Control of Hyperchaotic Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Pengfei Zhao ◽  
Cai Liu ◽  
Xuan Feng
Keyword(s):  
Lyapunov Stability ◽  
Reference System ◽  
Stability Theorem ◽  
Hyperchaotic System ◽  
Uncertain Parameters ◽  
General Description ◽  
Lyapunov Stability Theorem ◽  
Model Reference ◽  
Model Reference Control ◽  
Hyperchaotic Systems

We have applied a famous engineering method, called model reference control, to control hyperchaos. We have proposed a general description of the hyperchaotic system and its reference system. By using the Lyapunov stability theorem, we have obtained the expression of the controller. Four examples for the both certain case and the uncertain case show that our method is very effective for controlling hyperchaotic systems with both certain parameters and uncertain parameters.

scholarly journals Adaptive Integral Observer-Based Synchronization for Chaotic Systems with Unknown Parameters and Disturbances

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiuchun Li ◽  
Jianhua Gu ◽  
Wei Xu
Keyword(s):  
Neural Network ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Original System ◽  
Stability Theorem ◽  
Lyapunov Stability Theorem ◽  
Unknown Parameters ◽  
External Perturbations ◽  
Response Systems ◽  
Barbalat Lemma

Considering the effects of external perturbations on the state vector and the output of the original system, this paper proposes a new adaptive integral observer method to deal with chaos synchronization between the drive and response systems with unknown parameters. The analysis and proof are given by means of the Lyapunov stability theorem and Barbalat lemma. This approach has fewer constraints because many parameters related to chaotic system can be unknown, as shown in the paper. Numerical simulations are performed in the end and the results show that the proposed method is not only suitable to the representative chaotic systems but also applied to some neural network chaotic systems.

Identification and Adaptive Control of the Uncertain Lorenz System

2005 ◽  
Author(s):  
Hossein Nejat Pishkenari ◽  
Mohammad Shahrokhi
Keyword(s):  
Lorenz System ◽  
Rapid Identification ◽  
Open Loop ◽  
Stability Theorem ◽  
Lyapunov Stability Theorem ◽  
Unknown Parameters ◽  
Identification Method ◽  
System A ◽  
Identification Technique ◽  
The Stability

In this paper an identification method which can estimate the unknown parameters of a general nonlinear system based on three techniques (gradient, least-squares and rapid identification) has been developed. The stability of the proposed schemes has been shown using the Lyapunov stability theorem. The properties of each identification technique have been discussed briefly. Open loop identification of the Lorenz chaotic system is presented to show the effectiveness of the proposed approach. To illustrate the efficiency of the identification method for control purposes, it has been applied for controlling the well-known Lorenz system. By exploiting the property of the system a novel singularity-free controller is proposed. The stability of controller has been shown by a Lyapunov function. The designed controller coupled with the proposed identification technique can stabilize the uncertain Lorenz system. The effectiveness of the approach has been shown through simulation.

Switched Projective Synchronization of the Stochastic Newton-Leipnik System

2013 ◽  
Vol 328 ◽  
pp. 570-574
Author(s):  
Duan Dong ◽  
Shao Juan Ma ◽  
Jie Zheng
Keyword(s):  
Hilbert Spaces ◽  
Chaotic Systems ◽  
Control Method ◽  
Random Parameter ◽  
Adaptive Controller ◽  
Stability Theorem ◽  
Projective Synchronization ◽  
Deterministic System ◽  
Lyapunov Stability Theorem ◽  
Orthogonal Polynomial Expansion

The paper is involved with switched projective synchronization of two identical chaotic systems with random parameter using adaptive control method. Based on the orthogonal polynomial expansion of the Hilbert spaces, the Newton-Leipnik system with random parameter is transformed as the equivalent deterministic system. At last, an adaptive controller can be designed by the Lyapunov stability theorem for achieving switched projective synchronization of the equivalent deterministic system with different initial values. Corresponding numerical simulations are performed to verify the effectiveness of presented schemes for synchronizing the stochastic Newton-Leipnik system.

scholarly journals Lyapunov Stability Analysis of Gradient Descent-Learning Algorithm in Network Training

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Ahmad Banakar
Keyword(s):  
Lyapunov Stability ◽  
Gradient Descent ◽  
Learning Algorithm ◽  
Stability Theorem ◽  
Stability And Convergence ◽  
Lyapunov Stability Theorem ◽  
Network Training ◽  
Learning Rates ◽  
Lyapunov Stability Analysis ◽  
The Stability

The Lyapunov stability theorem is applied to guarantee the convergence and stability of the learning algorithm for several networks. Gradient descent learning algorithm and its developed algorithms are one of the most useful learning algorithms in developing the networks. To guarantee the stability and convergence of the learning process, the upper bound of the learning rates should be investigated. Here, the Lyapunov stability theorem was developed and applied to several networks in order to guaranty the stability of the learning algorithm.

SYNCHRONIZATION BETWEEN TWO DIFFERENT HYPERCHAOTIC SYSTEMS WITH UNCERTAIN PARAMETERS

2013 ◽  
Vol 27 (13) ◽  
pp. 1350044
Author(s):  
XING-YUAN WANG ◽  
YU-HONG YANG ◽  
MING-KU FENG
Keyword(s):  
Lyapunov Stability ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Lyapunov Stability Theory ◽  
Adaptive Controller ◽  
Sufficient Condition ◽  
Uncertain Parameters ◽  
Hyperchaotic Lorenz System ◽  
Hyperchaotic Systems

This paper studies the problem of chaos synchronization between two different hyperchaotic systems with uncertain parameters. Based on the Lyapunov stability theory, we obtain the sufficient condition of synchronization between two different hyperchaotic systems with uncertain parameters. A new adaptive controller with parameter update laws is designed to synchronize these chaotic systems. We proved it in theory with an uncertain hyperchaotic Lorenz system and an uncertain hyperchaotic Rössler system. Numerical results verified the validation of the proposed scheme.

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