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.

scholarly journals Sliding mode learning control of uncertain nonlinear systems with Lyapunov stability analysis

2018 ◽  
Vol 41 (6) ◽  
pp. 1750-1760
Author(s):  
Erkan Kayacan
Keyword(s):  
Stability Analysis ◽  
Nonlinear Systems ◽  
Lyapunov Stability ◽  
Sliding Mode ◽  
Learning Control ◽  
System Stability ◽  
Uncertain Nonlinear Systems ◽  
System Behaviour ◽  
Lyapunov Stability Analysis ◽  
The Stability

This paper addresses the Sliding Mode Learning Control (SMLC) of uncertain nonlinear systems with Lyapunov stability analysis. In the control scheme, a conventional control term is used to provide the system stability in compact space while a type-2 neuro-fuzzy controller (T2NFC) learns system behaviour so that the T2NFC completely takes over overall control of the system in a very short time period. The stability of the sliding mode learning algorithm has been proven in the literature; however, it is restrictive for systems without overall system stability. To address this shortcoming, a novel control structure with a novel sliding surface is proposed in this paper, and the stability of the overall system is proven for nth-order uncertain nonlinear systems. To investigate the capability and effectiveness of the proposed learning and control algorithms, the simulation studies have been carried out under noisy conditions. The simulation results confirm that the developed SMLC algorithm can learn the system behaviour in the absence of any mathematical model knowledge and exhibit robust control performance against external disturbances.

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.

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.

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.

Synchronization and Anti-Synchronization of the Chaotic Modified Chua's Circuits via a Same Controller

2012 ◽  
Vol 605-607 ◽  
pp. 1972-1975
Author(s):  
Jian Cai Leng ◽  
Rong Wei Guo
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Stability ◽  
Stability Theorem ◽  
Lyapunov Stability Theorem ◽  
Global Synchronization ◽  
Single Input ◽  
Theoretical Results

Based on the Lyapunov stability theorem, a same controller in the form is designed to achieve the global synchronization and anti-synchronization of the chaotic modified Chua's circuits. The controller obtained in this paper is simpler than those obtained in the existing results, and it is a linear single input controller. Numerical simulations verify the correctness and the effectiveness of the proposed theoretical results

Sensor-Based Filtering of Stochastic Distributed Parameter Systems

2015 ◽  
Vol 740 ◽  
pp. 229-233
Author(s):  
Wen Ying Mu ◽  
Bao Tong Cui ◽  
Bin Qi
Keyword(s):  
Numerical Simulation ◽  
State Estimation ◽  
Real Time ◽  
Lyapunov Stability ◽  
Adaptive Filters ◽  
Distributed Parameter Systems ◽  
Stability Theorem ◽  
Distributed Parameter ◽  
Evolution System ◽  
Lyapunov Stability Theorem

This Paper Proposes a Scheme for Filtering of Stochastic Distributed Parameter Systems. it is Assumed that a Real-Time Environment Consists of m Groups of Sensors, each of which Provides Necessarily State Spatially Measurements from Sensing Devices. Base on Lyapunov Stability Theorem and Itô formula, a Class of Distributed Adaptive Filters with Penalty Terms Result in the State Errors Forming a Stable Evolution System and Asymptotically Converge to Stochastic Distributed Parameter Systems, and then the Preferable State Estimation is Derived. Numerical Simulation Demonstrates the Effectiveness of the Proposed Method.

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