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.

Initialization Energy in Fractional-Order Systems

2015 ◽  
Author(s):  
Tom T. Hartley ◽  
Jean-Claude Trigeassou ◽  
Carl F. Lorenzo ◽  
Nezha Maamri
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Fractional Order System ◽  
Order System ◽  
Time Varying ◽  
Fractional Order Systems ◽  
History Of ◽  
Infinite Energy ◽  
Order Element

This paper seeks a deeper understanding of the need for time-varying initialization of fractional-order systems. Specifically, the paper determines the energy stored in a fractional-order element based on the history of the element, and shows how this initialization energy is manifest into the future as an initialization function. Further, it is shown that infinite energy is required to initialize a fractional-order system when using the Caputo derivative Laplace transform.

Further Remarks on the Existence of Periodic Solutions of Linear Time Varying Periodic Fractional Order Systems

2017 ◽  
Author(s):  
XueFeng Zhang ◽  
YangQuan Chen
Keyword(s):  
Laplace Transform ◽  
Periodic Solutions ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Linear Time ◽  
Periodic Systems ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Order Linear ◽  
Existence Of Periodic Solutions

Existence of periodic solutions of fractional order dynamic systems is an important and difficult issue in fractional order systems field. In this paper, the non existence of completely periodic solutions and existence of partly periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic systems are discussed. A new property of Laplace transform of periodic function is derived. The non existences of completely periodic solutions of fractional order linear time varying periodic systems and fractional order nonlinear time varying periodic fractional order systems are presented by Laplace transform method and contradiction approach. The existence of partly periodic solutions of fractional order dynamic systems are proved by constructing numerical examples and considering Laplace transform property approaches. The examples and state figures are given to illustrate the effectiveness of conclusion presented.

scholarly journals Stability analysis of fractional-order systems with randomly time-varying parameters

2021 ◽  
Vol 26 (3) ◽  
pp. 440-460
Author(s):  
Dehua Wang ◽  
Xiao-Li Ding ◽  
Juan J. Nieto
Keyword(s):  
Fractional Order ◽  
Stability Conditions ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Alternative Approach ◽  
Special Cases ◽  
Time Varying Parameters ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Linear And Nonlinear Systems

This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results.

scholarly journals Optimal Fault-Tolerant Control against Descriptor Time-Varying Systems with Nonlinear Input

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Xiaoming Su ◽  
Jingyao Wang ◽  
Hongyan Shi
Keyword(s):  
State Feedback ◽  
Fault Tolerant ◽  
Sufficient Conditions ◽  
Fault Tolerant Control ◽  
Time Varying ◽  
Quadratic Performance Index ◽  
Nonlinear Input ◽  
Time Varying Systems ◽  
Quadratic Performance ◽  
The Stability

The problem of optimal fault-tolerant control for a class of descriptor time-varying systems with nonlinear input is considered. Based on the Lyapunov stability theorem, the sufficient conditions of the stability are obtained when the system is normal and ineffective. Furthermore, the fault-tolerant control of the systems is carried out in two cases, and the state feedback fault-tolerant controller is obtained to satisfy the quadratic performance index and reach the minimum value in order to achieve the optimal fault-tolerant control. Finally, the validity of the proposed approach is illuminated by a numerical example.

scholarly journals Robust Stability Analysis of Nonlinear Fractional-Order Time-Variant Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Chen Caixue ◽  
Xie Yunxiang
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Robust Stability ◽  
State Feedback ◽  
Stability Theorem ◽  
Feedback Controller ◽  
Sufficient Condition ◽  
Fractional Order Systems ◽  
Numerical Example ◽  
Robust Stability Analysis

This paper presents a stability theorem for a class of nonlinear fractional-order time-variant systems with fractional orderα  (1<α<1)by using the Gronwall-Bellman lemma. Based on this theorem, a sufficient condition for designing a state feedback controller to stabilize such fractional-order systems is also obtained. Finally, a numerical example demonstrates the validity of this approach.

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.

Robust Stability Analysis of Uncertain Linear Fractional-Order Systems With Time-Varying Uncertainty for 0 < α < 2

2018 ◽  
Vol 141 (3) ◽  
Author(s):  
Mohammad Tavazoei ◽  
Mohammad Hassan Asemani
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Robust Stability ◽  
Stability Condition ◽  
Upper Bound ◽  
Order System ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Robust Stability Analysis ◽  
The Stability

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

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.

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.

scholarly journals Robust finite-time stability of uncertain neutral nonhomogeneous fractional-order systems with time-varying delays

2020 ◽  
pp. 16-16
Author(s):  
Mihailo Lazarevic ◽  
Darko Radojevic ◽  
Stjepko Pisl ◽  
Guido Maione
Keyword(s):  
Fractional Order ◽  
Finite Time ◽  
Test Procedure ◽  
Sufficient Condition ◽  
Time Varying ◽  
Fractional Order Systems ◽  
Time Stability ◽  
Extended Form ◽  
Finite Time Stability ◽  
Numerical Example

This article addresses the problem of finite-time stability for uncertain neutral nonhomogeneous fractional-order systems with time-varying delays where a stability test procedure is suggested. Based on the extended form of the generalized Gr?nwall inequality, a new sufficient condition for robust finite-time stability of such systems is established. Finally, a numerical example is given to show the effectiveness of the obtained result.

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