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.

Initialization of Fractional-Order Systems Using the Hankel Operator

2011 ◽  
Author(s):  
Jay L. Adams ◽  
Robert J. Veillette ◽  
Tom T. Hartley
Keyword(s):  
Fractional Order ◽  
Hankel Operator ◽  
Order System ◽  
Approximation Methods ◽  
Initial Condition ◽  
Fractional Order Systems ◽  
General Characterization ◽  
System A ◽  
General Initial Condition

This paper demonstrates the use of the Hankel operator to characterize the initial-condition response of a fractional-order system. A general initial-condition response can be determined for any input applied before t = 0. Two techniques for approximating the Hankel operator are discussed. These approximation methods are applied to illustrative examples, demonstrating a general characterization of natural responses.

Finite-Time Controllability of Fractional-Order Systems

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Jay L. Adams ◽  
Tom T. Hartley
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Fractional Order ◽  
Finite Time ◽  
Special Consideration ◽  
Fractional Order Systems ◽  
System Output ◽  
Fractional Order Integrator ◽  
Integrator Output

In this paper, the conditions that lead to a system output remaining at zero with zero input are considered. It is shown that the initialization of fractional-order integrators plays a key role in determining whether the integrator output will remain at a zero with zero input. Three examples are given that demonstrate the importance of initialization for integrators of order less than unity, inclusive. Two examples give a concrete illustration of the role that initialization plays in keeping the output of a fractional-order integrator at zero once it has been driven to zero. The implications of these results are considered, with special consideration given to the formulation of the fractional-order optimal control problem.

Finite-Time Controllability of Fractional-Order Systems

2007 ◽  
Author(s):  
Jay L. Adams ◽  
Tom T. Hartley
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Fractional Order ◽  
Finite Time ◽  
Special Consideration ◽  
Fractional Order Systems ◽  
System Output ◽  
Fractional Order Integrator ◽  
Integrator Output

In this paper the conditions that lead to a system output remaining at zero with zero input are considered. It is shown that the initialization of fractional-order integrators plays a key role in determining whether the integrator output will remain at a zero with zero input. Three examples are given that demonstrate the importance of initialization for integrators of order less than unity, inclusive. Two examples give a concrete illustration of the role that initialization plays in keeping the output of a fractional-order integrator at zero once it has been driven to zero. The implications of these results are considered, with special consideration given to the formulation of the fractional-order optimal control problem.

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.

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
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