PART 2: The Infinite State Approach
by Nezha Maamri, Jean-Claude Trigeassou
Analysis, Modeling and Stability of Fractional Order Differential Systems 1
- Cover
- Foreword
- Preface
- PART 1: Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs)
- 1 The Fractional Integrator
- 2 Frequency Approach to the Synthesis of the Fractional Integrator
- 2.1. Introduction
- 2.2. Frequency synthesis of the fractional derivator
- 2.3. Frequency synthesis of the fractional integrator
- 2.4. State space representation of
- 2.5. Modal representation of
- 2.6. Numerical algorithm
- 2.7. Frequency validation
- 2.8. Time validation
- 2.9. Internal state variables
- A.2. Appendix: design of fractional integrator parameters
- 3 Comparison of Two Simulation Techniques
- 4 Fractional Modeling of the Diffusive Interface
- 4.1. Introduction
- 4.2. Heat transfer and diffusive model of the plane wall
- 4.3. Fractional commensurate order models
- 4.4. Optimization of the fractional commensurate order model
- 4.5. Fractional non-commensurate order models
- 4.6. Conclusion
- A.4. Appendix: estimation of frequency responses – the least-squares approach
- 5 Modeling of Physical Systems with Fractional Models: an Illustrative Example
- PART 2: The Infinite State Approach
- 6 The Distributed Model of the Fractional Integrator
- 6.1. Introduction
- 6.2. Origin of the frequency distributed model
- 6.3. Frequency distributed model
- 6.4. Finite dimension approximation of the fractional integrator
- 6.5. Frequency synthesis and distributed model
- 6.6. Numerical validation of the distributed model
- 6.7. Riemann–Liouville integration and convolution
- 6.8. Physical interpretation of the frequency distributed model
- A.6. Appendix: inverse Laplace transform of the fractional integrator
- 7 Modeling of FDEs and FDSs
- 7.1. Introduction
- 7.2. Closed-loop modeling of an elementary FDS
- 7.3. Closed-loop modeling of an FDS
- 7.4. Transients of the one-derivative FDS
- 7.5. Transients of a two-derivative FDS
- 7.6. External or open-loop modeling of commensurate fractional order FDSs
- 7.7. External and internal representations of an FDS
- 7.8. Computation of the Mittag-Leffler function
- A.7. Appendix: inverse Laplace transform of
- 8 Fractional Differentiation
- 8.1. Introduction
- 8.2. Implicit fractional differentiation
- 8.3. Explicit Riemann–Liouville and Caputo fractional derivatives
- 8.4. Initial conditions of fractional derivatives
- 8.5. Initial conditions in the general case
- 8.6. Unicity of FDS transients
- 8.7. Numerical simulation of Caputo and Riemann–Liouville transients
- 9 Analytical Expressions of FDS Transients
- 10 Infinite State and Fractional Differentiation of Functions
- 10.1. Introduction
- 10.2. Calculation of the Caputo derivative
- 10.3. Initial conditions of the Caputo derivative
- 10.4. Transients of fractional derivatives
- 10.5. Calculation of fractional derivatives with the implicit derivative
- 10.6. Numerical validation of Caputo derivative transients
- A.10. Appendix: convolution lemma
- 6 The Distributed Model of the Fractional Integrator
- References
- Index
- End User License Agreement
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