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E-grāmata: Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State Approach

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  • Formāts: EPUB+DRM
  • Izdošanas datums: 06-Aug-2019
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9781119648819
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Analysis, Modeling and Stability of Fractional Order Differential Systems 1: The Infinite State ApproachPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 06-Aug-2019
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9781119648819
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Modelling of FDEs and FDSs based on fractional integrators represents the generalization of integer order system theory to fractional systems, with important differences due to the specificities of fractional calculus. Fractional integration is performed using two equivalent techniques: frequency synthesis and a modal distributed approach. This modelling technique provides immediate solutions to different initialization problems: fractional derivatives and FDEs/FDSs. Definition of fractional integrator energy allows FDE stability analysis using Lyapunov technique.
Biraword xiii
Preface xv
Part 1 Simulation and Identification of Fractional Differential Equations (FDEs) and Systems (FDSs)
1(126)
Chapter 1 The Fractional Integrator
3(22)
1.1 Introduction
3(1)
1.2 Simulation and modeling of integer order ordinary differential equations
3(7)
1.2.1 Simulation with analog computers
3(2)
1.2.2 Simulation with digital computers
5(1)
1.2.3 Initial conditions
6(1)
1.2.4 State space representation and simulation diagram
7(2)
1.2.5 Concluding remarks
9(1)
1.3 Origin of fractional integration: repeated integration
10(2)
1.4 Riemann-Liouville integration
12(5)
1.4.1 Definition
12(1)
1.4.2 Laplace transform of the Riemann-Liouville integral
13(1)
1.4.3 Fractional integration operator
14(1)
1.4.4 Fractional differentiation
15(2)
1.5 Simulation of FDEs with a fractional integrator
17(3)
1.5.1 Simulation of a one-derivative FDE
17(1)
1.5.2 FDE
18(1)
1.5.3 Simulation of the general linear FDE
18(2)
A.1 Appendix
20(5)
A.1.1 Lord Kelvin's principle
20(1)
A.1.2 A brief history of analog computing
21(1)
A.1.3 Interpretation of the RK2 algorithm
22(1)
A.1.4 The gamma function
23(2)
Chapter 2 Frequency Approach to the Synthesis of the Fractional Integrator
25(30)
2.1 Introduction
25(1)
2.2 Frequency synthesis of the fractional derivator
26(2)
2.3 Frequency synthesis of the fractional integrator
28(5)
2.3.1 Objective
28(1)
2.3.2 Direct method
29(1)
2.3.3 Indirect method
30(2)
2.3.4 Frequency synthesis of 1/Sn
32(1)
2.4 State space representation of Ind(s)
33(3)
2.5 Modal representation of Ind(s)
36(4)
2.6 Numerical algorithm
40(1)
2.7 Frequency validation
41(3)
2.8 Time validation
44(3)
2.9 Internal state variables
47(2)
A.2 Appendix: design of fractional integrator parameters
49(6)
A.2.1 Definition of Gn
49(2)
A.2.2 Definition of α and η
51(4)
Chapter 3 Comparison of Two Simulation Techniques
55(26)
3.1 Introduction
55(1)
3.2 Simulation with the Grunwald-Letnikov approach
56(10)
3.2.1 Euler's technique
56(2)
3.2.2 The Grunwald-Letnikov fractional derivative
58(2)
3.2.3 Numerical simulation with the Grunwald-Letnikov integrator
60(1)
3.2.4 Some specificities of the Grunwald-Letnikov integrator
61(2)
3.2.5 Short memory principle
63(3)
3.3 Simulation with infinite state approach
66(2)
3.4 Caputo's initialization
68(1)
3.5 Numerical simulations
69(9)
3.5.1 Introduction
69(1)
3.5.2 Comparison of discrete impulse responses (DIRs)
70(2)
3.5.3 Simulation accuracy
72(2)
3.5.4 Static error caused by the short memory principle
74(1)
3.5.5 Caputo's initialization
75(3)
3.5.6 Conclusion
78(1)
A.3 Appendix: Mittag-Leffler function
78(3)
A.3.1 Definition
78(1)
A.3.2 Laplace transform
79(1)
A.3.3 Unit step response of 1/(sn + a)
79(1)
A.3.4 Caputo's initialization
80(1)
Chapter 4 Fractional Modeling of the Diffusive Interface
81(26)
4.1 Introduction
81(1)
4.2 Heat transfer and diffusive model of the plane wall
82(6)
4.2.1 Heat transfer
82(1)
4.2.2 Physical model of the diffusive interface
83(2)
4.2.3 Frequency analysis of the diffusive phenomenon
85(1)
4.2.4 Time analysis of the diffusive phenomenon
86(1)
4.2.5 Conclusion
87(1)
4.3 Fractional commensurate order models
88(3)
4.3.1 Physical origin
88(1)
4.3.2 Analysis of physical commensurate order models
89(2)
4.4 Optimization of the fractional commensurate order model
91(6)
4.4.1 The proposed frequency approach
91(5)
4.4.2 Conclusion
96(1)
4.5 Fractional non-commensurate order models
97(5)
4.5.1 Justification
97(1)
4.5.2 Parameter estimation of Hn1,2(s)
97(1)
4.5.3 Numerical examples
98(3)
4.5.4 Conclusion
101(1)
4.6 Conclusion
102(1)
A.4 Appendix: estimation of frequency responses -- the least-squares approach
102(5)
A.4.1 Identification of the commensurate order model HN-1,N(jω)
103(1)
A.4.2 Parameter estimation of the non-commensurate model Hn1,n2(jω)
104(3)
Chapter 5 Modeling of Physical Systems with Fractional Models: an Illustrative Example
107(20)
5.1 Introduction
107(1)
5.2 Modeling with mathematical models: some basic principles
108(1)
5.3 Modeling of the induction motor
109(8)
5.3.1 Construction of the induction motor
109(1)
5.3.2 Principle of operation
109(1)
5.3.3 Induction motor knowledge model
110(2)
5.3.4 Park's model
112(2)
5.3.5 Fractional Park's model
114(3)
5.4 Identification of fractional Park's model
117(10)
5.4.1 Simplified model
117(1)
5.4.2 Identification algorithm
118(1)
5.4.3 Nonlinear optimization
119(3)
5.4.4 Simulation of yk and σk
122(1)
5.4.5 Comments
122(1)
5.4.6 Application to the identification of fractional Park's model
123(4)
Part 2 The Infinite State Approach
127(142)
Chapter 6 The Distributed Model of the Fractional Integrator
129(30)
6.1 Introduction
129(1)
6.2 Origin of the frequency distributed model
130(3)
6.3 Frequency distributed model
133(1)
6.4 Finite dimension approximation of the fractional integrator
134(2)
6.5 Frequency synthesis and distributed model
136(2)
6.6 Numerical validation of the distributed model
138(4)
6.6.1 Reconstruction of the weighting function
138(2)
6.6.2 Reconstruction of the impulse response
140(2)
6.7 Riemann-Liouville integration and convolution
142(5)
6.7.1 Conclusion
147(1)
6.8 Physical interpretation of the frequency distributed model
147(9)
6.8.1 The infinite RC transmission line
147(2)
6.8.2 RC line and spatial Fourier transform
149(2)
6.8.3 Impulse response of the RC line
151(2)
6.8.4 General solution
153(2)
6.8.5 Initialization in the time and spatial domains
155(1)
A.6 Appendix: inverse Laplace transform of the fractional integrator
156(3)
Chapter 7 Modeling of FDEs and FDSs
159(34)
7.1 Introduction
159(1)
7.2 Closed-loop modeling of an elementary FDS
160(2)
7.3 Closed-loop modeling of an FDS
162(6)
7.3.1 Modeling of an N-derivative FDS
162(3)
7.3.2 Distributed state
165(3)
7.4 Transients of the one-derivative FDS
168(5)
7.4.1 Numerical simulation
168(1)
7.4.2 Initialization at t = t1
169(2)
7.4.3 Initialization at different instants
171(2)
7.5 Transients of a two-derivative FDS
173(2)
7.6 External or open-loop modeling of commensurate fractional order FDSs
175(7)
7.6.1 Introduction
175(1)
7.6.2 External model of an elementary FDE
176(3)
7.6.3 External representation of a two-derivative FDE
179(1)
7.6.4 External representation of an N-derivative FDE
180(2)
7.7 External and internal representations of an FDS
182(1)
7.8 Computation of the Mittag-Leffler function
183(6)
7.8.1 Introduction
183(1)
7.8.2 Divergence of direct computation
184(1)
7.8.3 Step response approach
185(1)
7.8.4 Improved step response approach
186(3)
A.7 Appendix: inverse Laplace transform of 1/(sn + a)
189(4)
Chapter 8 Fractional Differentiation
193(26)
8.1 Introduction
193(1)
8.2 Implicit fractional differentiation
194(1)
8.3 Explicit Riemann-Liouville and Caputo fractional derivatives
195(4)
8.3.1 Definitions
195(2)
8.3.2 Theoretical prerequisites
197(1)
8.3.3 Comments
198(1)
8.4 Initial conditions of fractional derivatives
199(6)
8.4.1 Introduction
199(1)
8.4.2 Implicit derivative
200(1)
8.4.3 Caputo derivative
201(2)
8.4.4 Riemann-Liouville derivative
203(1)
8.4.5 Relations between initial conditions
204(1)
8.5 Initial conditions in the general case
205(3)
8.5.1 Introduction
205(1)
8.5.2 Implicit derivatives
205(1)
8.5.3 Caputo derivatives
206(1)
8.5.4 Riemann-Liouville derivatives
207(1)
8.5.5 Relations between initial conditions
207(1)
8.6 Unicity of FDS transients
208(4)
8.6.1 Transients of the elementary FDE
208(1)
8.6.2 Unicity of transients
209(1)
8.6.3 Conclusion
210(2)
8.7 Numerical simulation of Caputo and Riemann-Liouville transients
212(7)
8.7.1 Introduction
212(1)
8.7.2 Simulation of Caputo derivative initialization
212(3)
8.7.3 Simulation of Riemann-Liouville initialization
215(4)
Chapter 9 Analytical Expressions of FDS Transients
219(24)
9.1 Introduction
219(2)
9.2 Mittag-Leffler approach
221(6)
9.2.1 Free response of the elementary FDS
221(2)
9.2.2 Free response of the N-derivative FDS
223(2)
9.2.3 Complete solution of the N-derivative FDS
225(2)
9.3 Distributed exponential approach
227(10)
9.3.1 Introduction
227(1)
9.3.2 Solution of Dn (x(t)) = ax(t) using frequency discretization
227(3)
9.3.3 Solution of Dn (x(t)) = ax(t) using a continuous approach
230(2)
9.3.4 Solution of Dnx(t)) = ax(t) using Picard's method
232(3)
9.3.5 Solution of Dn-(X(t)) = AX(t)
235(2)
9.3.6 Solution of Dn(X(t)) = AX(t) +Bu(t)
237(1)
9.4 Numerical computation of analytical transients
237(6)
9.4.1 Introduction
237(1)
9.4.2 Computation of the forced response
238(2)
9.4.3 Step response of a three-derivative FDS
240(3)
Chapter 10 Infinite State and Fractional Differentiation of Functions
243(26)
10.1 Introduction
243(1)
10.2 Calculation of the Caputo derivative
244(6)
10.2.1 Fractional derivative of the Heaviside function
245(1)
10.2.2 Fractional derivative of the power function
246(2)
10.2.3 Fractional derivative of the exponential function
248(1)
10.2.4 Fractional derivative of the sine function
249(1)
10.3 Initial conditions of the Caputo derivative
250(3)
10.4 Transients of fractional derivatives
253(4)
10.4.1 Introduction
253(1)
10.4.2 Heaviside function
254(1)
10.4.3 Power function
255(1)
10.4.4 Exponential function
256(1)
10.4.5 Sine function
256(1)
10.5 Calculation of fractional derivatives with the implicit derivative
257(5)
10.5.1 Introduction
257(1)
10.5.2 Fractional derivative of the Heaviside function
258(1)
10.5.3 Fractional derivative of the power function
259(1)
10.5.4 Fractional derivative of the exponential function
260(1)
10.5.5 Fractional derivative of the sine function
261(1)
10.5.6 Conclusion
262(1)
10.6 Numerical validation of Caputo derivative transients
262(4)
10.6.1 Introduction
262(2)
10.6.2 Simulation results
264(2)
A.10 Appendix: convolution lemma
266(3)
References 269(16)
Index 285
Jean-Claude Trigeassou is Honorary Professor at Bordeaux University, France, and has been associated with the research activities of its IMS-LAPS lab since 2006. His main research interests include the modeling of fractional order systems, based on the infinite state approach.

Nezha Maamri is Associate Professor at Poitiers University, France. Her research activities concern the method of moments, robust control using integer order and fractional order controllers, plus the modeling, initialization and stability of fractional order systems.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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