A Note on the Lyapunov Stability of Fractional-Order Nonlinear Systems

2017 ◽  
Author(s):  
Sara Dadras ◽  
Soodeh Dadras ◽  
Hadi Malek ◽  
YangQuan Chen
Keyword(s):  
Exponential Stability ◽  
Fractional Order ◽  
Lyapunov Stability ◽  
Time Domain ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Sufficient Condition ◽  
Fractional Order Systems ◽  
The Stability ◽  
The Time Domain

In this paper, stability of fractional order (FO) systems is investigated in the sense of the Lyapunov stability theory. A new definition for exponential stability of the fractional order systems is given and sufficient conditions are obtained for the exponential stability of the FO systems using the notion of Lyapunov stability. Besides, a less conservative sufficient condition is derived for asymptotical stability of FO systems. The stability analysis is done in the time domain. Numerical examples are given to show that the obtained conditions are effective and applicable in practice.

Stability of Nonlinear Fractional-Order Time Varying Systems

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Sunhua Huang ◽  
Runfan Zhang ◽  
Diyi Chen
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Local Stability ◽  
State Feedback ◽  
Sufficient Condition ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Time Varying Systems ◽  
The Stability

This paper is concerned with the stability of nonlinear fractional-order time varying systems with Caputo derivative. By using Laplace transform, Mittag-Leffler function, and the Gronwall inequality, the sufficient condition that ensures local stability of fractional-order systems with fractional order α : 0<α≤1 and 1<α<2 is proposed, respectively. Moreover, the condition of the stability of fractional-order systems with a state-feedback controller is been put forward. Finally, a numerical example is presented to show the validity and feasibility of the proposed method.

scholarly journals SYNCHRONIZATION OF CHAOTIC FRACTIONAL-ORDER SYSTEMS VIA LINEAR CONTROL

2010 ◽  
Vol 20 (01) ◽  
pp. 81-97 ◽  
Author(s):  
ZAID M. ODIBAT ◽  
NATHALIE CORSON ◽  
M. A. AZIZ-ALAOUI ◽  
CYRILLE BERTELLE
Keyword(s):  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Sufficient Conditions ◽  
Linear Control ◽  
Fractional Order Systems ◽  
Control Laws ◽  
Response System ◽  
The Stability ◽  
Stability Results

The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractional-order Chen, Rössler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis.

scholarly journals Sufficient Conditions on the Exponential Stability of Neutral Stochastic Differential Equations with Time-Varying Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yanwei Tian ◽  
Baofeng Chen
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Linear Matrix ◽  
Time Varying ◽  
Matrix Inequalities ◽  
Delay Dependent

The exponential stability is investigated for neutral stochastic differential equations with time-varying delays. Based on the Lyapunov stability theory and linear matrix inequalities (LMIs) technique, some delay-dependent criteria are established to guarantee the exponential stability in almost sure sense. Finally a numerical example is provided to illustrate the feasibility of the result.

Synchronization of Hyperchaotic Memristor-Based Chua’s Circuits

2014 ◽  
Vol 644-650 ◽  
pp. 3485-3488
Author(s):  
Hai Long Huang ◽  
Yan Peng ◽  
Jun Jian Huang
Keyword(s):  
Active Control ◽  
Lyapunov Stability ◽  
Stability Theory ◽  
Control Method ◽  
Lyapunov Stability Theory ◽  
Sufficient Condition ◽  
Global Synchronization ◽  
Error System ◽  
The Stability ◽  
Active Control Method

This paper further investigates the problem of synchronization of hyperchaotic memristor-based Chua’s circuits. An active control method is employed to design a controller to achieve the global synchronization of two identical memristor-based systems. Based on Lyapunov stability theory, a sufficient condition is given to guarantee the stability of the synchronization error system.

Robust stability and stabilization of uncertain fractional order systems subject to input saturation

2017 ◽  
Vol 24 (16) ◽  
pp. 3676-3683 ◽  
Author(s):  
Esmat Sadat Alaviyan Shahri ◽  
Alireza Alfi ◽  
JA Tenreiro Machado
Keyword(s):  
Fractional Order ◽  
Robust Stability ◽  
State Feedback ◽  
Input Saturation ◽  
Sufficient Conditions ◽  
Fractional Order Systems ◽  
Numerical Examples ◽  
Stability And Stabilization ◽  
The Stability ◽  
Robust Stability And Stabilization

This paper studies the stability and the stabilization for a class of uncertain fractional order (FO) systems subject to input saturation. The Lipschitz condition and the Gronwall–Bellman lemma are adopted and sufficient conditions are derived to stabilize systems by designing a state feedback controller. Numerical examples demonstrate the effectiveness of the proposed method.

Cluster synchronization in fractional-order network with nondelay and delay coupling

2021 ◽  
pp. 2250006
Author(s):  
Yi Wang ◽  
Zhaoyan Wu
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Cluster Synchronization ◽  
Sufficient Condition ◽  
Fractional Order Systems ◽  
Adaptive Controllers ◽  
Control Scheme ◽  
Control Schemes ◽  
The Stability ◽  
Adaptive Control Scheme

In this paper, cluster synchronization for fractional-order complex network with nondelay and delay coupling is investigated. Based on the stability theory of fractional-order systems and the properties of fractional derivative, both static and adaptive control schemes are adopted to design effective controllers. Sufficient condition for achieving cluster synchronization about static controllers is provided. From the condition, the needed feedback gains can be estimated by simple calculations. Further, adaptive control scheme is introduced to design unified controllers. Noticeably, in the adaptive controllers, the feedback gains need not be calculated in advance and can adjust themselves to the needed values according to updating laws. Finally, numerical simulations are given to demonstrate the correctness of the obtained results.

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