Realizations for Determining the Energy Stored in Fractional-Order Operators

2015 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
Energy Storage ◽  
Fractional Order ◽  
Transient Behavior ◽  
Fractional Order Systems ◽  
State Variables ◽  
Electrical Circuits ◽  
Infinite Dimensional

The purpose of this paper is to determine physical electrical circuits, in both impedance and admittance forms, that match fractional-order integrators and differentiators, namely 1/sq and sq. Then, using these idealized infinite-dimensional circuits, the energy storage and loss expressions for them are determined, carefully relating the associated infinite-dimensional state variables to physically meaningful quantities. The resulting realizations and energy expressions allow a variety of implementations for understanding the transient behavior of fractional-order systems.

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.

Energy Storage and Loss in Fractional-Order Systems

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Tom T. Hartley ◽  
Jean-Claude Trigeassou ◽  
Carl F. Lorenzo ◽  
Nezha Maamri
Keyword(s):  
Energy Storage ◽  
Fractional Order ◽  
Approximation Methods ◽  
Fractional Order Systems ◽  
State Representation ◽  
Fractional Order Integrator

As fractional-order systems are becoming more widely accepted and their usage is increasing, it is important to understand their energy storage and loss properties. Fractional-order operators can be implemented using a distributed state representation, which has been shown to be equivalent to the Riemann–Liouville representation. In this paper, the distributed state for a fractional-order integrator is represented using an infinite resistor–capacitor network such that the energy storage and loss properties can be readily determined. This derivation is repeated for fractional-order derivatives using an infinite resistor–inductor network. An analytical example is included to verify the results for a half-order integrator. Approximation methods are included.

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.

State-space analysis of linear fractional-order systems

2008 ◽  
Vol 42 (6-8) ◽  
pp. 825-838 ◽  
Author(s):  
Saïd Guermah ◽  
Saïd Djennoune ◽  
Maâmar Bettayeb
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Fractional Order Systems ◽  
Space Analysis ◽  
State Space Analysis

scholarly journals A predator–prey model involving variable-order fractional differential equations with Mittag-Leffler kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aziz Khan ◽  
Hashim M. Alshehri ◽  
J. F. Gómez-Aguilar ◽  
Zareen A. Khan ◽  
G. Fernández-Anaya
Keyword(s):  
Fractional Order ◽  
Existence And Uniqueness ◽  
Variable Order ◽  
Fractional Order Systems ◽  
New Approach ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Fractional Differential ◽  
The Fixed Point Theorem

AbstractThis paper is about to formulate a design of predator–prey model with constant and time fractional variable order. The predator and prey act as agents in an ecosystem in this simulation. We focus on a time fractional order Atangana–Baleanu operator in the sense of Liouville–Caputo. Due to the nonlocality of the method, the predator–prey model is generated by using another FO derivative developed as a kernel based on the generalized Mittag-Leffler function. Two fractional-order systems are assumed, with and without delay. For the numerical solution of the models, we not only employ the Adams–Bashforth–Moulton method but also explore the existence and uniqueness of these schemes. We use the fixed point theorem which is useful in describing the existence of a new approach with a particular set of solutions. For the illustration, several numerical examples are added to the paper to show the effectiveness of the numerical 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.

Sign in / Sign up

Export Citation Format

Share Document