TY  - BOOK
AU  - Trigeassou,Jean-Claude
AU  - Maamri,Nezha
TI  - Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach
SN  - 9781119648840
AV  - QA314 .T754 2019 
PY  - 2019///
CY  - Newark
PB  - John Wiley & Sons, Incorporated
KW  - Fractional differential equations
KW  - Electronic books
N1  - Cover -- Half-Title Page -- Dedication -- Title Page -- Copyright Page -- Contents -- Foreword -- Preface -- PART 1: Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs) -- 1. The Fractional Integrator -- 1.1. Introduction -- 1.2. Simulation and modeling of integer order ordinary differential equations -- 1.2.1. Simulation with analog computers -- 1.2.2. Simulation with digital computers -- 1.2.3. Initial conditions -- 1.2.4. State space representation and simulation diagram -- 1.2.5. Concluding remarks -- 1.3. Origin of fractional integration: repeated integration -- 1.4. Riemann-Liouville integration -- 1.4.1. Definition -- 1.4.2. Laplace transform of the Riemann-Liouville integral -- 1.4.3. Fractional integration operator -- 1.4.4. Fractional differentiation -- 1.5. Simulation of FDEs with a fractional integrator -- 1.5.1. Simulation of a one-derivative FDE -- 1.5.2. FDE -- 1.5.3. Simulation of the general linear FDE -- A.1. Appendix -- A.1.1. Lord Kelvin's principle -- A.1.2. A brief history of analog computing -- A.1.3. Interpretation of the RK2 algorithm -- A.1.4. The gamma function -- 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 -- 2.3.1. Objective -- 2.3.2. Direct method -- 2.3.3. Indirect method -- 2.3.4. Frequency synthesis of 1/Sn -- 2.4. State space representation of I nd (S) -- 2.5. Modal representation of I nd (S) -- 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 -- A.2.1. Definition of Gn -- A.2.2. Definition of α and η -- 3. Comparison of Two Simulation Techniques -- 3.1. Introduction; 3.2. Simulation with the Grünwald-Letnikov approach -- 3.2.1. Euler's technique -- 3.2.2. The Grünwald-Letnikov fractional derivative -- 3.2.3. Numerical simulation with the Grünwald-Letnikov integrator -- 3.2.4. Some specificities of the Grünwald-Letnikov integrator -- 3.2.5. Short memory principle -- 3.3. Simulation with infinite state approach -- 3.4. Caputo's initialization -- 3.5. Numerical simulations -- 3.5.1. Introduction -- 3.5.2. Comparison of discrete impulse responses (DIRs) -- 3.5.3. Simulation accuracy -- 3.5.4. Static error caused by the short memory principle -- 3.5.5. Caputo's initialization -- 3.5.6. Conclusion -- A.3. Appendix: Mittag-Leffler function -- A.3.1. Definition -- A.3.2. Laplace transform -- A.3.3. Unit step response of 1/sn + a -- A.3.4. Caputo's initialization -- 4. Fractional Modeling of the Diffusive Interface -- 4.1. Introduction -- 4.2. Heat transfer and diffusive model of the plane wall -- 4.2.1. Heat transfer -- 4.2.2. Physical model of the diffusive interface -- 4.2.3. Frequency analysis of the diffusive phenomenon -- 4.2.4. Time analysis of the diffusive phenomenon -- 4.2.5. Conclusion -- 4.3. Fractional commensurate order models -- 4.3.1. Physical origin -- 4.3.2. Analysis of physical commensurate order models -- 4.4. Optimization of the fractional commensurate order model -- 4.4.1. The proposed frequency approach -- 4.4.2. Conclusion -- 4.5. Fractional non-commensurate order models -- 4.5.1. Justification -- 4.5.2. Parameter estimation of Hn1,n2(s) -- 4.5.3. Numerical examples -- 4.5.4. Conclusion -- 4.6. Conclusion -- A.4. Appendix: estimation of frequency responses - the least-squares approach -- A.4.1. Identification of the commensurate order model HN-1,N(jω) -- A.4.2. Parameter estimation of the non-commensurate model Hn1,n2 (jω); 5. Modeling of Physical Systems with Fractional Models: an Illustrative Example -- 5.1. Introduction -- 5.2. Modeling with mathematical models: some basic principles -- 5.3. Modeling of the induction motor -- 5.3.1. Construction of the induction motor -- 5.3.2. Principle of operation -- 5.3.3. Induction motor knowledge model -- 5.3.4. Park's model -- 5.3.5. Fractional Park's model -- 5.4. Identification of fractional Park's model -- 5.4.1. Simplified model -- 5.4.2. Identification algorithm -- 5.4.3. Nonlinear optimization -- 5.4.4. Simulation of ŷk and σk -- 5.4.5. Comments -- 5.4.6. Application to the identification of fractional Park's model -- 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 -- 6.6.1. Reconstruction of the weighting function -- 6.6.2. Reconstruction of the impulse response -- 6.7. Riemann-Liouville integration and convolution -- 6.7.1. Conclusion -- 6.8. Physical interpretation of the frequency distributed model -- 6.8.1. The infinite RC transmission line -- 6.8.2. RC line and spatial Fourier transform -- 6.8.3. Impulse response of the RC line -- 6.8.4. General solution -- 6.8.5. Initialization in the time and spatial domains -- 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 -- 7.3.1. Modeling of an N-derivative FDS -- 7.3.2. Distributed state -- 7.4. Transients of the one-derivative FDS -- 7.4.1. Numerical simulation; 7.4.2. Initialization at t = t1 -- 7.4.3. Initialization at different instants -- 7.5. Transients of a two-derivative FDS -- 7.6. External or open-loop modeling of commensurate fractional order FDSs -- 7.6.1. Introduction -- 7.6.2. External model of an elementary FDE -- 7.6.3. External representation of a two-derivative FDE -- 7.6.4. External representation of an N-derivative FDE -- 7.7. External and internal representations of an FDS -- 7.8. Computation of the Mittag-Leffler function -- 7.8.1. Introduction -- 7.8.2. Divergence of direct computation -- 7.8.3. Step response approach -- 7.8.4. Improved step response approach -- A.7. Appendix: inverse Laplace transform of 1/Sn + a -- 8. Fractional Differentiation -- 8.1. Introduction -- 8.2. Implicit fractional differentiation -- 8.3. Explicit Riemann-Liouville and Caputo fractional derivatives -- 8.3.1. Definitions -- 8.3.2. Theoretical prerequisites -- 8.3.3. Comments -- 8.4. Initial conditions of fractional derivatives -- 8.4.1. Introduction -- 8.4.2. Implicit derivative -- 8.4.3. Caputo derivative -- 8.4.4. Riemann-Liouville derivative -- 8.4.5. Relations between initial conditions -- 8.5. Initial conditions in the general case -- 8.5.1. Introduction -- 8.5.2. Implicit derivatives -- 8.5.3. Caputo derivatives -- 8.5.4. Riemann-Liouville derivatives -- 8.5.5. Relations between initial conditions -- 8.6. Unicity of FDS transients -- 8.6.1. Transients of the elementary FDE -- 8.6.2. Unicity of transients -- 8.6.3. Conclusion -- 8.7. Numerical simulation of Caputo and Riemann-Liouville transients -- 8.7.1. Introduction -- 8.7.2. Simulation of Caputo derivative initialization -- 8.7.3. Simulation of Riemann-Liouville initialization -- 9. Analytical Expressions of FDS Transients -- 9.1. Introduction -- 9.2. Mittag-Leffler approach -- 9.2.1. Free response of the elementary FDS; 9.2.2. Free response of the N-derivative FDS -- 9.2.3. Complete solution of the N-derivative FDS -- 9.3. Distributed exponential approach -- 9.3.1. Introduction -- 9.3.2. Solution of Dn (x(t)) = ax(t) using frequency discretization -- 9.3.3. Solution of Dn (x(t)) = ax(t) using a continuous approach -- 9.3.4. Solution of Dn (x(t)) = ax(t) using Picard's method -- 9.3.5. Solution of Dn ( X (t)) = AX (t) -- 9.3.6. Solution of Dn (X(t)) = AX(t) + Bu(t) -- 9.4. Numerical computation of analytical transients -- 9.4.1. Introduction -- 9.4.2. Computation of the forced response -- 9.4.3. Step response of a three-derivative FDS -- 10. Infinite State and Fractional Differentiation of Functions -- 10.1. Introduction -- 10.2. Calculation of the Caputo derivative -- 10.2.1. Fractional derivative of the Heaviside function -- 10.2.2. Fractional derivative of the power function -- 10.2.3. Fractional derivative of the exponential function -- 10.2.4. Fractional derivative of the sine function -- 10.3. Initial conditions of the Caputo derivative -- 10.4. Transients of fractional derivatives -- 10.4.1. Introduction -- 10.4.2. Heaviside function -- 10.4.3. Power function -- 10.4.4. Exponential function -- 10.4.5. Sine function -- 10.5. Calculation of fractional derivatives with the implicit derivative -- 10.5.1. Introduction -- 10.5.2. Fractional derivative of the Heaviside function -- 10.5.3. Fractional derivative of the power function -- 10.5.4. Fractional derivative of the exponential function -- 10.5.5. Fractional derivative of the sine function -- 10.5.6. Conclusion -- 10.6. Numerical validation of Caputo derivative transients -- 10.6.1. Introduction -- 10.6.2. Simulation results -- A.10. Appendix: convolution lemma -- References -- Index -- Other titles from iSTE in Systems and Industrial Engineering - Robotics -- EULA
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