- •LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA
- •CONTENTS
- •FOREWORD
- •PREFACE
- •Abstract
- •1. Introduction
- •2. A Nice Equation for an Heuristic Power
- •3. SWOT Method, Non Integer Diff-Integral and Co-Dimension
- •4. The Generalization of the Exponential Concept
- •5. Diffusion Under Field
- •6. Riemann Zeta Function and Non-Integer Differentiation
- •7. Auto Organization and Emergence
- •Conclusion
- •Acknowledgment
- •References
- •Abstract
- •1. Introduction
- •2. Preliminaries
- •3. The Model
- •4. Numerical Simulations
- •5. Synchronization
- •6. Conclusion
- •Acknowledgments
- •References
- •Abstract
- •1. Introduction: A Short Literature Review
- •2. The Injection System
- •3. The Control Strategy: Switching of Fractional Order Controllers by Gain Scheduling
- •4. Fractional Order Control Design
- •5. Simulation Results
- •6. Conclusion
- •Acknowledgment
- •References
- •Abstract
- •Introduction
- •1. Basic Definitions and Preliminaries
- •Conclusion
- •Acknowledgments
- •References
- •Abstract
- •1. Context and Problematic
- •2. Parameters and Definitions
- •3. Semi-Infinite Plane
- •4. Responses in the Semi-Infinite Plane
- •5. Finite Plane
- •6. Responses in Finite Plane
- •7. Simulink Responses
- •Conclusion
- •References
- •Abstract
- •1. Introduction
- •2. Modelling
- •3. Temperature Control
- •4. Conclusion
- •References
- •Abstract
- •1. Introduction
- •2. Preliminaries
- •3. Second Order Sliding Mode Control Strategy
- •4. Adaptation Law Synthesis
- •5. Numerical Studies
- •Conclusion
- •References
- •Abstract
- •1. Introduction
- •2. Rabotnov’s Fractional Operators and Main Formulas of Algebra of Fractional Operators
- •4. Calculation of the Main Viscoelastic Operators
- •5. Relationship of Rabotnov Fractional Operators with Other Fractional Operators
- •8. Application of Rabotnov’s Operators in Problems of Impact Response of Thin Structures
- •9. Conclusion
- •Acknowledgments
- •References
- •Abstract
- •1. Introduction
- •3. Theory of Diffusive Stresses
- •4. Diffusive Stresses
- •5. Conclusion
- •References
- •Abstract
- •Introduction
- •Methods
- •Conclusion
- •Acknowledgment
- •Abstract
- •1. Introduction
- •2. Basics of Fractional PID Controllers
- •3. Tuning Methodology for Fuzzy Fractional PID Controllers
- •4. Optimal Fuzzy Fractional PID Controllers
- •5. Conclusion
- •References
- •INDEX
106 |
Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al. |
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Note that, as long as the frequency of variation of the temperature is low relatively to |
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the diffusivity , the temperature variation is uniform and harmonic. |
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When the frequency of variation of the temperature |
increases, the diffusion depth of |
the temperature decreases in the semi-infinite medium. |
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5. Finite Plane
We consider a finite, homogeneous, isotropic, one-dimensional plane medium of
thickness , conductivity |
, diffusivity , and of initial temperature zero all over the medium |
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(Figure 9). A heat flux |
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there are no losses at this point. Consequently, a temperature gradient ( |
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measurement point, with |
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5.1. System Definition
The one-dimensional heat conduction equation (Özişik, 1985) is given by a partial differential equation:
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Since the initial temperature is zero all over the medium:
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The Fourier law states the boundary conditions (Özişik, 1985) where the flux is applied, as well as at the second end:
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5.2. System Resolution
The system of (Eq. 47-50) represents the relations that model the diffusive interface knowing that the input signal is the flux ( ) whereas the output signal is the temperature
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The initial condition of the temperature being zero, Laplace transform |
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Complimentary Contributor Copy
From the Formal Concept Analysis to the Numerical Simulation … |
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The solution of (Eq. 51) is similar to the previous case, hence:
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Resolving (Eq. 52) for the boundary conditions (Eq. 49-50) gives the values of |
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Taking into consideration the definitions of the characteristic parameters and the outcome of (Eq. 53), thus:
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Under the assumption of an initial condition of the temperature being zero, the transfer
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then, introducing the hyperbolic functions |
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or rewritten as: |
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Complimentary Contributor Copy
108 |
Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al. |
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To clarify, the different temperatures found in the system (Eq. 59) are as follows: |
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functions represented in (Figure 10). |
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The function |
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̅( ) and the fractional derivative of order |
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( ) is a fractional integrator of order |
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The third block is the transfer function ( |
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Complimentary Contributor Copy
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From the Formal Concept Analysis to the Numerical Simulation … |
109 |
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The fourth part is the transfer function |
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the temperature ̅( |
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H(x, s, L) |
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(0, s, L) |
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s0.5T |
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I 0.5(s) |
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Figure 10. Block diagram illustrating the transfer function |
( |
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6. Responses in Finite Plane |
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6.1. Asymptotic Behavior Analysis at |
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At |
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thus |
the temperature ̅( |
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the finite |
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medium of length |
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is given by: |
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̅( |
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( ) |
( |
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(63) |
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The transfer function ( |
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character |
on the |
temperature |
at |
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such, let the change of variable |
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√ ⁄ . |
Considering the definitions in (Para 2.) and (Eq. 60, 61), as a start the following is verified: if then , as ( ) , thus ( ) , accordingly it is found that:
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(64) |
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which is a result similar to what was found in a semi-infinite medium at |
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Now consider when |
then |
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√ ⁄ |
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(65) |
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√ ⁄
Once again, considering the definitions in (Para 2.) and (Eq. 60, 61), the (Eq. 65) is rewritten as:
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(66) |
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Complimentary Contributor Copy
110 |
Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al. |
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Thus for high instances, corresponding to low frequencies (i.e. ), the asymptotic behavior of a finite medium is of capacitive type characterized by an integrator of order . It is different than the behavior in a semi-infinite medium characterized by a fractional
integrator of order |
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For low |
instances, corresponding to |
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frequencies, when |
then |
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(67) |
Accordingly an asymptotic fractional order behavior appears when |
between the |
input flux and the temperature at the boundary where the flux is applied; in fact, it is the same result that appears in the semi-infinite medium.
6.2. Asymptotic Behavior Analysis for
For |
high |
instances, |
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frequencies, |
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then |
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The same asymptotic behavior applies as at |
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by an integrator |
of order |
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behavior in a |
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characterized by a fractional integrator of order |
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For |
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√ ⁄ |
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61) |
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relation between |
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then, |
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It is the same asymptotic behavior as it appears in a semi-infinite medium.
6.3. Frequency Response Analysis at
Now that the mathematical model is established, frequency domain analyses ( |
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can be done. The resulting transfer function ( |
) is the concatenation of four cascading |
Complimentary Contributor Copy
From the Formal Concept Analysis to the Numerical Simulation … |
111 |
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blocks as shown on the (Figure 10) and represented in the (Eq. 58-60). When the study is on
the boundary where the flux is applied, |
, the transfer function is expressed by: |
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(71) |
with ( |
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(72) |
Bode diagrams of the classic fractional integrator of order 0.5 in a finite aluminum
medium are represented in Figure 11. |
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The frequency response study of the function ( |
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understanding |
the |
influence of the finite character of the medium on the temperature at |
. Actually Figure |
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12 represents Bode diagrams of this transfer for three different values of |
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in a finite aluminum medium. |
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Gain (dB)
Phase (deg)
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Frequency (rad/s)
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Figure 11. Bode diagrams of |
( |
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Two asymptotic behaviors clearly appear: |
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Complimentary Contributor Copy
112 |
Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al. |
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t the |
high frequencies, a proportional |
behavior prevails. Actually, |
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(74) |
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( ( |
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The transitional zone between the two asymptotic behaviors is four decades wide and it is centered on the transitional frequency .
Gain (dB)
80
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Phase (deg)
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Frequency (rad/s)
Figure 12. Bode diagrams of of ( |
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* |
+ . |
Now it is time to study the influence of each of the three cascading blocks on the global transfer function ( ). For this purpose, for and , Figure 13 visualizes the contribution of each block to the global response in a finite aluminum medium.
Two asymptotic behaviors clearly appear:
At the low frequencies, an integration behavior of order prevails. Actually,
, ( |
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At the high frequencies, a fractional integration behavior of order |
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Complimentary Contributor Copy
From the Formal Concept Analysis to the Numerical Simulation … |
113 |
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{ | ( |
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( ( |
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Here also, the transitional zone between the two asymptotic behaviors is four decades wide and it is centered on the transitional frequency .
Gain (dB)
Phase (deg)
40
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Frequency (rad/s)
H(0,jw,1m)
H0I0.5(0,jw)
F(0,jw,1m)
G(0,jw,1m)
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H(0,jw,1m)
H0I0.5(0,jw)
F(0,jw,1m)
G(0,jw,1m)
100 101
Figure 13. Bode diagrams of ( |
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( ) ( |
) and ( |
). |
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For different values of |
* |
+ |
, (Figure 14 represents the respective Bode |
diagrams in a finite aluminum medium illustrating the translation of the transitional zone towards the high frequencies when decreases.
Gain (dB)
Phase (deg)
40
0
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-80
-120 10-7
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Figure 14. Bode diagrams of ( |
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Complimentary Contributor Copy
114 |
Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al. |
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For different values of * + , Figure 15 represents respectively ( ) in Black-Nichols plane in a finite aluminum medium.
6.4. Frequency Behavior Analysis for
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(Eq. 56-59) reformulate the expression of the third block: |
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( |
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(77) |
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√ ⁄ |
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Figure 16 represents Bode diagrams of the transfer of |
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let |
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Two behaviors clearly appear: |
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At the low frequencies, |
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Gain (dB)
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-270 |
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Phase (deg)
Figure 15. Black-Nichols plots of ( |
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+ |
Complimentary Contributor Copy
From the Formal Concept Analysis to the Numerical Simulation … |
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Figure 16. Bode diagrams of |
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In order to make the analysis easier, a special frequency |
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⁄√ |
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The frequency |
decreases when |
increases. This is why, for a given frequency , the |
gain and the phase are as smaller as is big as it is evident in Figure 16.
Now it is time to study the influence of each of the four cascading blocks on the global transfer function ( ). For this purpose, for and , Figure 17 visualizes the contribution of each block to the global response in a finite aluminum medium.
Three behaviors clearly appear:
At the low frequencies, an integration behavior of order prevails similar to the one
found at |
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116 |
Riad Assaf, Roy Abi Zeid Daou, Xavier Moreau et al. |
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Gain (dB)
Phase (deg)
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Figure 17. Bode diagrams of |
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( |
) and |
( |
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At the medium frequencies, a |
fractional |
integration |
behavior of |
order |
prevails. |
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Actually, |
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Figure 18 represents Bode diagrams of the transfer of |
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) for a finite aluminum |
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medium of length |
and for four different values |
of the temperature sensor position, |
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* |
+ |
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For a finite aluminum medium of length |
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and for four different values of the |
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sensor |
position, |
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* |
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+ |
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Figure 19 |
represents respectively |
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( |
) in Black-Nichols plane. |
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Complimentary Contributor Copy