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PART 2: Stability of Fractional Differential Equations and Systems
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PART 2: Stability of Fractional Differential Equations and Systems
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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5 Fractional Model-based Control of the Diffusive RC Line
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6 Stability of Linear FDEs Using the Nyquist Criterion
PART 2
Stability of Fractional Differential Equations and Systems
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