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by Jean-Claude Trigeassou, Nezha Maamri
Analysis, Modeling and Stability of Fractional Order Differential Systems 2
Cover
Foreword
Preface
PART 1: Initialization, State Observation and Control
1 Initialization of Fractional Order Systems
1.1. Introduction
1.2. Initialization of an integer order differential system
1.3. Initialization of a fractional differential equation
1.4. Initialization of a fractional differential system
1.5. Some initialization examples
2 Observability and Controllability of FDEs/FDSs
2.1. Introduction
2.2. A survey of classical approaches to the observability and controllability of fractional differential systems
2.3. Pseudo-observability and pseudo-controllability of an FDS
2.4. Observability and controllability of the distributed state
2.5. Conclusion
3 Improved Initialization of Fractional Order Systems
3.1. Introduction
3.2. Initialization: problem statement
3.3. Initialization with a fractional observer
3.4. Improved initialization
A.3. Appendix
4 State Control of Fractional Differential Systems
4.1. Introduction
4.2. Pseudo-state control of an FDS
4.3. State control of the elementary FDE
4.4. State control of an FDS
4.5. Conclusion
5 Fractional Model-based Control of the Diffusive RC Line
5.1. Introduction
5.2. Identification of the RC line using a fractional model
5.3. Reset of the RC line
PART 2: Stability of Fractional Differential Equations and Systems
6 Stability of Linear FDEs Using the Nyquist Criterion
6.1. Introduction
6.2. Simulation and stability of fractional differential equations
6.3. Stability of ordinary differential equations
6.4. Stability analysis of FDEs
6.5. Stability analysis of ODEs with time delays
6.6. Stability analysis of FDEs with time delays
7 Fractional Energy
7.1. Introduction
7.2. Pseudo-energy stored in a fractional integrator
7.3. Energy stored and dissipated in a fractional integrator
7.4. Closed-loop and open-loop fractional energies
8 Lyapunov Stability of Commensurate Order Fractional Systems
8.1. Introduction
8.2. Lyapunov stability of a one-derivative FDE
8.3. Lyapunov stability of an N-derivative FDE
8.4. Lyapunov stability of a two-derivative commensurate order FDE
8.5. Lyapunov stability of an N-derivative FDE (N > 2) > 2)
A.8. Appendix
9 Lyapunov Stability of Non-commensurate Order Fractional Systems
9.1. Introduction
9.2. Stored energy, dissipation and energy balance in fractional electrical devices
9.3. The usual series RLC circuit
9.4. The series RLC* fractional circuit
9.5. The series RLL*C* circuit
9.6. The series RL*C* fractional circuit
9.7. Stability of a commensurate order FDE: energy balance approach
9.8. Stability of a commensurate order FDE: physical interpretation of the usual approach
A.9. Appendix
10 An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems
10.1. Introduction
10.2. Indirect Lyapunov method
10.3. Lyapunov direct method
10.4. The Van der Pol oscillator
10.5. Analysis of local stability
10.6. Large signal analysis
References
Index
End User License Agreement
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Dedication
Table of Contents
Cover
Foreword
Preface
PART 1: Initialization, State Observation and Control
1 Initialization of Fractional Order Systems
1.1. Introduction
1.2. Initialization of an integer order differential system
1.3. Initialization of a fractional differential equation
1.4. Initialization of a fractional differential system
1.5. Some initialization examples
2 Observability and Controllability of FDEs/FDSs
2.1. Introduction
2.2. A survey of classical approaches to the observability and controllability of fractional differential systems
2.3. Pseudo-observability and pseudo-controllability of an FDS
2.4. Observability and controllability of the distributed state
2.5. Conclusion
3 Improved Initialization of Fractional Order Systems
3.1. Introduction
3.2. Initialization: problem statement
3.3. Initialization with a fractional observer
3.4. Improved initialization
A.3. Appendix
4 State Control of Fractional Differential Systems
4.1. Introduction
4.2. Pseudo-state control of an FDS
4.3. State control of the elementary FDE
4.4. State control of an FDS
4.5. Conclusion
5 Fractional Model-based Control of the Diffusive RC Line
5.1. Introduction
5.2. Identification of the RC line using a fractional model
5.3. Reset of the RC line
PART 2: Stability of Fractional Differential Equations and Systems
6 Stability of Linear FDEs Using the Nyquist Criterion
6.1. Introduction
6.2. Simulation and stability of fractional differential equations
6.3. Stability of ordinary differential equations
6.4. Stability analysis of FDEs
6.5. Stability analysis of ODEs with time delays
6.6. Stability analysis of FDEs with time delays
7 Fractional Energy
7.1. Introduction
7.2. Pseudo-energy stored in a fractional integrator
7.3. Energy stored and dissipated in a fractional integrator
7.4. Closed-loop and open-loop fractional energies
8 Lyapunov Stability of Commensurate Order Fractional Systems
8.1. Introduction
8.2. Lyapunov stability of a one-derivative FDE
8.3. Lyapunov stability of an N-derivative FDE
8.4. Lyapunov stability of a two-derivative commensurate order FDE
8.5. Lyapunov stability of an N-derivative FDE (
N
> 2)
A.8. Appendix
9 Lyapunov Stability of Non-commensurate Order Fractional Systems
9.1. Introduction
9.2. Stored energy, dissipation and energy balance in fractional electrical devices
9.3. The usual series RLC circuit
9.4. The series
RLC*
fractional circuit
9.5. The series
RLL*C*
circuit
9.6. The series
RL*C*
fractional circuit
9.7. Stability of a commensurate order FDE: energy balance approach
9.8. Stability of a commensurate order FDE: physical interpretation of the usual approach
A.9. Appendix
10 An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems
10.1. Introduction
10.2. Indirect Lyapunov method
10.3. Lyapunov direct method
10.4. The Van der Pol oscillator
10.5. Analysis of local stability
10.6. Large signal analysis
References
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1. Influence of α
Table 4.2. Influence of T
ce
Table 4.3. Influence of T
ce
Chapter 5
Table 5.1. Identification of H
n
(s) for T
e
=10
–03
s
Table 5.2. Identification of H
n
(s) for T
e
= 5E
–03
s
Table 5.3. Identification of H
n
1
,
n
2
(s) for T
e
=10
–03
s
Table 5.4. Identification of H
n
1
,
n
2
(s) for T
e
= 5E
–03
s
Table 5.5. H
n
(s) with T
e
=10
–3
s
Table 5.6. H
n
1
,
n
2
(s) with T
e
=10
–3
s
Table 5.7. H
n
(s) with T
e
= 5E
–03
s
Table 5.8. H
n
1
,
n
2
(s) with T
e
= 5E
–03
s
Table 5.9. Influence of k
C
List of Illustrations
Chapter 1
Figure 1.1. The initialization problem
Figure 1.2. Step response of a first-order system
Figure 1.3. Free responses with the same initial value. For a color version of t...
Figure 1.4. Comparison of distributions 2 and 3
Figure 1.5. Comparison of initialization functions
Chapter 2
Figure 2.1. Pseudo-controllability of the commensurate order case. For a color v...
Figure 2.2. Pseudo-controllability of the non-commensurate order case
Chapter 3
Figure 3.1. Input u(t) and outputs y(t), ŷ(t) and y
init
(t). For a color version ...
Figure 3.2. Comparison between modes z
j
(t
0
) and ẑ
j
(t
0
) for j = 0,1,2,…, 20
Figure 3.3. u(t), y(t), ŷ(t) and y
init
(t)
Figure 3.4. Comparison between modes z
1, j
(t
0
) and ẑ
1, j
(t
0
) for j = 0,1,2,…, 20
Figure 3.5. Comparison between modes z
2
, j
(t
0
) and ẑ
2, j
(t
0
) for j = 0,1,2,…, 20
Figure 3.6. The different responses of the system on [0, t
0
]
Figure 3.7. Gradient technique estimation of modes ẑ
j
(t
p
) for different sequence...
Figure 3.8. Direct and improved estimation ẑ
j
(t
0
) for j = 0,1,2,…, 20
Figure 3.9. u(t), ŷ(t) (direct), ŷ(t) (improved) and y
init
(t)
Figure 3.10. Comparison of initialization errors
Figure 3.11. The different responses of the system on [0, t
0
]
Figure 3.12. Gradient estimation of the modes ẑ
1,
j(t
p
) for different sequences
Figure 3.13. Gradient estimation of the modes ẑ
2, j
(t
p
) for different sequences
Figure 3.14. Direct and improved estimations ẑ
1, j
(t
0
)
Figure 3.15. Direct and improved estimations ẑ
2, j
(t
0
)
Figure 3.16. u(t), y(t), y(t) (direct) and ŷ(t) (improved)
Figure 3.17. Comparison of initialization errors
Chapter 4
Figure 4.1. ODE state control. For a color version of the figures in this chapte...
Figure 4.2. FDE pseudo-state control
Figure 4.3. Distributions of FDE internal states
Figure 4.4. Initialization procedure
Figure 4.5. Fractional integrator state control
Figure 4.6. Initial and final distributions of the internal state
Figure 4.7. State control of a one-derivative FDE
Figure 4.8. FDE state control
Figure 4.9. Two fractional integrators in series
Figure 4.10. Initialization procedure
Figure 4.11. State control of two fractional integrators in series
Figure 4.12. Distribution of initial and final internal states
Figure 4.13. Two-derivative FDE state control
Chapter 5
Figure 5.1. Prediction and simulation quadratic criteria. For a color version of...
Figure 5.2. The RC line
Figure 5.3. Space discretization
Figure 5.4. Interface at x = 0
Figure 5.5. Improved numerical interface at x = 0
Figure 5.6. Step responses of the RC line
Figure 5.7. Internal voltage distributions
Figure 5.8. Data file:T
e
= 10
–03
s
Figure 5.9. Data file: T
e
= 5E
–03
s
Figure 5.10. Initialization procedure
Figure 5.11. Natural relaxation of the RC line
Figure 5.12. Initial and final distribution of the distributed state
Figure 5.13. Initial and final distribution of the space variable
Figure 5.14. Excitation and response of the RC line
Figure 5.15. Initial and final distribution of the distributed states
Figure 5.16. Excitation and response of the RC line
Figure 5.17. Excitation and response of the RC line
Chapter 6
Figure 6.1. Closed-loop simulation of an FDE with fractional integrators
Figure 6.2. D contour excluding the origin
Figure 6.3. Example of the Nyquist contour for a stable ODE
Figure 6.4. Open-loop graph of a third-order ODE
Figure 6.5. Nichols chart of the third-order ODE
Figure 6.6. Open-loop Nyquist’s graph of the third-order ODE
Figure 6.7. Open-loop approximate graph
Figure 6.8. Open-loop graph of the one-derivative FDE
Figure 6.9. Open-loop Nyquist’s graph of the one-derivative FDE
Figure 6.10. Open-loop graph of the stable FDE
Figure 6.11. Open-loop graph of the conditionally stable FDE
Figure 6.12. Instability caused by n1 = 2
Figure 6.13. Instability caused by n1 > 2
Figure 6.14. Two-derivative FDE with a1 > 0
Figure 6.15. Nichols chart with a0 = 0.7
Figure 6.16. Step response with a0 = 0.7
Figure 6.17. Step response with a0 = 0.4
Figure 6.18. Definition of sectors in the {a, b} plane
Figure 6.19. Stability regions in the {a, b} plane
Figure 6.20. Nyquist’s diagram for different values of K
Chapter 7
Figure 7.1. Fractional integrator input. For a color version of the figures in t...
Figure 7.2. Realization of the input
Figure 7.3. Fractional integrator
Figure 7.4. Pseudo-energy of the fractional integrator
Figure 7.5. Distributed electrical network
Figure 7.6. Infinite length RC line
Figure 7.7. Integer order integrator
Figure 7.8. Equivalent electrical circuit
Figure 7.9. Fractional order integrator
Figure 7.10. Theoretical and simulated energies
Figure 7.11. Fractional energy
Figure 7.12. Closed-loop ODE model
Figure 7.13. Closed-loop and open-loop energies
Chapter 8
Figure 8.1. Comparison of Lyapunov functions with IC1. For a color version of th...
Figure 8.2. Comparison of Lyapunov functions with IC
2
Figure 8.3. Derivative of the Lyapunov function with IC
1
and IC
2
Figure 8.4. Set of acceptable values
Figure 8.5. RLC* circuit
Figure 8.6. Matignon’s stability condition
Chapter 9
Figure 9.1. Capacitor circuit
Figure 9.2. Inductor circuit
Figure 9.3. Infinite length RC line
Figure 9.4. Infinite length GL line
Figure 9.5. Series RLC circuit
Figure 9.6. Energies of the series RLC circuit. For a color version of the figur...
Figure 9.7. Current and voltage of the RLC* circuit
Figure 9.8. Energy of the RLC* circuit, R = 0.1
Figure 9.9. Energy of the RLC* circuit, R = –0.1
Figure 9.10. Derivative of the Lyapunov function
Figure 9.11. Series RLL*C* circuit
Figure 9.12. Series RL*C* circuit
Figure 9.13. Simulation of a two-derivative FDE
Figure 9.14. Elementary cell of the LG line
Chapter 10
Figure 10.1. Comparison of
and
Figure 10.2. System nonlinearity
Figure 10.3. Stability domain
Figure 10.4. Fractional Van der Pol oscillator
Figure 10.5. Influence of the fractional order n. For a color version of the fig...
Figure 10.6. Influence of the parameter α
Figure 10.7. Stability of the Van der Pol system
Figure 10.8. Nonlinear characteristic i=f(v)
Guide
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