Multi-Group Consensus of Heterogeneous Fractional-Order Nonlinear Agents via Pinning Control

2011 ◽  
Author(s):  
Wei Sun ◽  
YangQuan Chen ◽  
Changpin Li
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Pinning Control ◽  
Graph Laplacian ◽  
Control Law ◽  
Group Consensus ◽  
Necessary And Sufficient Condition ◽  
Zero Eigenvalue ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient

The present work concerns the multi-group consensus behavior of directed complex networks. The network consists of agents with heterogeneous fractional-order non-linear dynamics. It can be divided into several groups due to their dynamics or equilibriums. Each group will be stabilized at an equilibrium and different groups may have different steady state values. A necessary and sufficient condition is provided for the proposed pinning control law to be locally Mittag-Leffler stable. The conclusion turns to guarantee the exponential stable for integer-order systems. The collection of heterogeneous equilibriums is determined by the geometric multiplicity of the zero eigenvalue respect to the graph Laplacian. Simulations on fractional-order chaotic systems demonstrated the conclusions.

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

scholarly journals Adaptive Projective Synchronization between Two Different Fractional-Order Chaotic Systems with Fully Unknown Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Liping Chen ◽  
Shanbi Wei ◽  
Yi Chai ◽  
Ranchao Wu
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Projective Synchronization ◽  
Control Law ◽  
Unknown Parameters ◽  
Chen System ◽  
Fractional Order Differential Equations ◽  
Response Systems ◽  
The Stability ◽  
Adaptive Control Law

Projective synchronization between two different fractional-order chaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the stability theory of fractional-order differential equations, a suitable and effective adaptive control law and a parameter update rule for unknown parameters are designed, such that projective synchronization between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown parameters is achieved. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed method.

scholarly journals Fractional-order LCL filters: principle, frequency characteristics, and their analysis

2021 ◽  
Author(s):  
Junhua Xu ◽  
Xiaocong Li ◽  
Xueli Luo ◽  
Liliang Hou ◽  
Jianbo Qin ◽  
...  
Keyword(s):  
Fractional Order ◽  
Theoretical Basis ◽  
Digital Simulation ◽  
Frequency Characteristics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Operating Characteristics ◽  
Theoretical Derivation ◽  
Integer Order ◽  
Necessary And Sufficient

Abstract In this paper, a fractional-order LCL (FOLCL) filter is constructed by introducing fractional-order inductors (FOIs) and fractional-order capacitors (FOCs) to replace the inductors and capacitors in a traditional integer-order LCL (IOLCL) filter, respectively. The principle and frequency characteristics of an FOLCL filter are systematically studied, and five important properties are derived and demonstrated in-depth. One of the most important achievements is that we discover the necessary and sufficient condition for the existence of resonance for an FOLCL filter, that is, the sum of the order of the FOIs and the FOC is equal to 2, which provides a theoretical basis for avoiding the resonance of an FOLCL filter effectively in design. The correctness of the theoretical derivation and analysis are verified by digital simulation. Compared with an IOLCL filter, an FOLCL filter presents more flexible and diverse operating characteristics and has a broader application prospect.

Representation of holomorphic functions by schlömilch’s series

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Peter Rusev
Keyword(s):  
Bessel Function ◽  
Fractional Order ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Zero Index

AbstractA necessary and sufficient condition is given for holomorphic functions to be represented by series of the kind $\sum\limits_{n = 0}^\infty {a_n J_0 (nz),z,a_n \in \mathbb{C},} $ where J 0(z) is the Bessel function of first kind with zero index. To derive the result, we use an Erdélyi-Kober operator of fractional order.

scholarly journals Partial Anti-Synchronization of the Fractional-Order Chaotic Systems through Dynamic Feedback Control

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 718
Author(s):  
Runlong Peng ◽  
Cuimei Jiang ◽  
Rongwei Guo
Keyword(s):  
Feedback Control ◽  
Fractional Order ◽  
Chaotic Systems ◽  
Control Method ◽  
Dynamic Feedback ◽  
Synchronization Problem ◽  
Feedback Control Method ◽  
Single Input ◽  
Dynamic Feedback Control ◽  
Necessary And Sufficient

This paper investigates the partial anti-synchronization problem of fractional-order chaotic systems through the dynamic feedback control method. Firstly, a necessary and sufficient condition is proposed, by which the existence of the partial anti-synchronization problem is proved. Then, an algorithm is given and used to obtain all solutions of this problem. Moreover, the partial anti-synchronization problem of the fractional-order chaotic systems is realized through the dynamic feedback control method. It is noted that the designed controllers are single-input controllers. Finally, two illustrative examples with numerical simulations are used to verify the correctness and effectiveness of the proposed results.

scholarly journals Generalized convergence analysis of the fractional order systems

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 404-411
Author(s):  
Ahmad Ruzitalab ◽  
Mohammad Hadi Farahi ◽  
Gholamhossien Erjaee
Keyword(s):  
Fractional Order ◽  
Fractional Order Systems ◽  
Necessary And Sufficient Condition ◽  
Lyapunov Matrix Equation ◽  
Discrete Time Systems ◽  
Lyapunov Matrix ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Contraction Theory ◽  
Time Systems

Abstract The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

Necessary and Sufficient Condition on Controllability and Reachability of Fractional-Order Linear Switched System

2014 ◽  
Vol 541-542 ◽  
pp. 1319-1326
Author(s):  
Cheng Deng ◽  
Wei Zhu
Keyword(s):  
Analytical Solution ◽  
Hybrid Systems ◽  
Fractional Order ◽  
Switched System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Linear ◽  
Arbitrary Switching ◽  
Necessary And Sufficient ◽  
Important System

Fractional-order linear switched system (FLSS) is an important system of hybrid systems. In this paper, by using the analytical solution of FLSS, the necessary and sufficient condition on controllability and reachability of FLSS is given, respectively. The condition shows that if every subsystem is controllable (reachable), then the whole system is also controllable (reachable) for arbitrary switching rules. And if the whole system is controllable (reachable) for arbitrary switching rules, then every subsystem is controllable (reachable).

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