A New Fractional Order Observer Design for Fractional Order Nonlinear Systems

2011 ◽  
Author(s):  
Sara Dadras ◽  
Hamid Reza Momeni
Keyword(s):  
Fractional Order ◽  
Original System ◽  
Observer Design ◽  
Fractional Order Systems ◽  
State Variables ◽  
Lyapunov Approach ◽  
Error System ◽  
Simulation Results ◽  
System States ◽  
The Stability

In this paper, a class of fractional order systems is considered and simple fractional order observers have been proposed to estimate the system’s state variables. By introducing a fractional calculus into the observer design, the developed fractional order observers guarantee the estimated states reach the original system states. Using the fractional order Lyapunov approach, the stability (zero convergence) of the error system is investigated. Finally, the simulation results demonstrate validity and effectiveness of the proposed scheme.

Functional interval observer design for singular fractional-order systems with disturbances

2020 ◽  
pp. 014233122094489
Author(s):  
Dinh Cong Huong ◽  
Dao Thi Hai Yen
Keyword(s):  
Fractional Order ◽  
Design Method ◽  
Observer Design ◽  
Effective Algorithm ◽  
Fractional Order Systems ◽  
Full State ◽  
Simulation Results ◽  
Interval Observer ◽  
Interval Observers ◽  
State Functions

In this study, we consider the problem of designing functional interval observers for a class of singular fractional-order systems. The goal of this work is to design not a full state interval observer for singular fractional-order systems, but to design an observer for state functions of this kind of systems. Conditions for the existence of such functional interval observers are given and an effective algorithm for computing unknown observer matrices is provided in this study. Two numerical examples and simulation results are provided to illustrate the effectiveness of the proposed design method.

scholarly journals Hybrid Stability Checking Method for Synchronization of Chaotic Fractional-Order Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Seng-Kin Lao ◽  
Lap-Mou Tam ◽  
Hsien-Keng Chen ◽  
Long-Jye Sheu
Keyword(s):  
Fractional Order ◽  
Design Stage ◽  
Fractional Order Systems ◽  
State Variables ◽  
Driving System ◽  
Control Scheme ◽  
The Stability ◽  
Synchronization Scheme ◽  
Hyperchaotic Systems ◽  
Stability Checking

A hybrid stability checking method is proposed to verify the establishment of synchronization between two hyperchaotic systems. During the design stage of a synchronization scheme for chaotic fractional-order systems, a problem is sometimes encountered. In order to ensure the stability of the error signal between two fractional-order systems, the arguments of all eigenvalues of the Jacobian matrix of the erroneous system should be within a region defined in Matignon’s theorem. Sometimes, the arguments depend on the state variables of the driving system, which makes it difficult to prove the stability. We propose a new and efficient hybrid method to verify the stability in this situation. The passivity-based control scheme for synchronization of two hyperchaotic fractional-order Chen-Lee systems is provided as an example. Theoretical analysis of the proposed method is validated by numerical simulation in time domain and examined in frequency domain via electronic circuits.

CHAOS SYNCHRONIZATION OF FRACTIONAL ORDER UNIFIED CHAOTIC SYSTEM VIA NONLINEAR CONTROL

2011 ◽  
Vol 25 (03) ◽  
pp. 407-415 ◽  
Author(s):  
XIANG RONG CHEN ◽  
CHONG XIN LIU
Keyword(s):  
Nonlinear Control ◽  
Fractional Order ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Control Method ◽  
Fractional Order Systems ◽  
Unified Chaotic System ◽  
Simulation Results ◽  
Nonlinear Control Method ◽  
The Stability

Based on the stability theory of fractional order systems, an effective but theoretically rigorous nonlinear control method is proposed to synchronize the fractional order chaotic systems. Using this method, chaos synchronization between two identical fractional order unified systems is studied. Simulation results are shown to illustrate the effectiveness of this method.

scholarly journals Realization of a fractional-order RLC circuit via constant phase element

2021 ◽  
Author(s):  
Riccardo Caponetto ◽  
Salvatore Graziani ◽  
Emanuele Murgano
Keyword(s):  
Experimental Data ◽  
Carbon Black ◽  
Fractional Order ◽  
Constant Phase Element ◽  
Polymeric Matrix ◽  
Fractional Order Systems ◽  
New Device ◽  
Rlc Circuit ◽  
Simulation Results

AbstractIn the paper, a fractional-order RLC circuit is presented. The circuit is realized by using a fractional-order capacitor. This is realized by using carbon black dispersed in a polymeric matrix. Simulation results are compared with the experimental data, confirming the suitability of applying this new device in the circuital implementation of fractional-order systems.

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

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 A Fractional-Order Discrete Noninvertible Map of Cubic Type: Dynamics, Control, and Synchronization

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Xiaojun Liu ◽  
Ling Hong ◽  
Lixin Yang ◽  
Dafeng Tang
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Dimensional Space ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
State Variables ◽  
Discrete Maps ◽  
The Stability ◽  
Control And Synchronization ◽  
Simultaneous Variation

In this paper, a new fractional-order discrete noninvertible map of cubic type is presented. Firstly, the stability of the equilibrium points for the map is examined. Secondly, the dynamics of the map with two different initial conditions is studied by numerical simulation when a parameter or a derivative order is varied. A series of attractors are displayed in various forms of periodic and chaotic ones. Furthermore, bifurcations with the simultaneous variation of both a parameter and the order are also analyzed in the three-dimensional space. Interior crises are found in the map as a parameter or an order varies. Thirdly, based on the stability theory of fractional-order discrete maps, a stabilization controller is proposed to control the chaos of the map and the asymptotic convergence of the state variables is determined. Finally, the synchronization between the proposed map and a fractional-order discrete Loren map is investigated. Numerical simulations are used to verify the effectiveness of the designed synchronization controllers.

scholarly journals Stability of Fractional Order Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Margarita Rivero ◽  
Sergei V. Rogosin ◽  
José A. Tenreiro Machado ◽  
Juan J. Trujillo
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
State Of The Art ◽  
Point Of View ◽  
Fractional Order Systems ◽  
Short Period ◽  
Systems And Control ◽  
Different Types ◽  
The Stability ◽  
And Control

The theory and applications of fractional calculus (FC) had a considerable progress during the last years. Dynamical systems and control are one of the most active areas, and several authors focused on the stability of fractional order systems. Nevertheless, due to the multitude of efforts in a short period of time, contributions are scattered along the literature, and it becomes difficult for researchers to have a complete and systematic picture of the present day knowledge. This paper is an attempt to overcome this situation by reviewing the state of the art and putting this topic in a systematic form. While the problem is formulated with rigour, from the mathematical point of view, the exposition intends to be easy to read by the applied researchers. Different types of systems are considered, namely, linear/nonlinear, positive, with delay, distributed, and continuous/discrete. Several possible routes of future progress that emerge are also tackled.

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