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E-grāmata: Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State Approach

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  • Formāts: EPUB+DRM
  • Izdošanas datums: 19-Dec-2019
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9781119686842
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Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State ApproachPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 19-Dec-2019
  • Izdevniecība: ISTE Ltd and John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9781119686842
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This book introduces an original fractional calculus methodology (?the infinite state approach?) which is applied to the modeling of fractional order differential equations (FDEs) and systems (FDSs). Its modeling is based on the frequency distributed fractional integrator, while the resulting model corresponds to an integer order and infinite dimension state space representation. This original modeling allows the theoretical concepts of integer order systems to be generalized to fractional systems, with a particular emphasis on a convolution formulation. With this approach, fundamental issues such as system state interpretation and system initialization ? long considered to be major theoretical pitfalls ? have been solved easily. Although originally introduced for numerical simulation and identification of FDEs, this approach also provides original solutions to many problems such as the initial conditions of fractional derivatives, the uniqueness of FDS transients, formulation of analytical transients, fractional differentiation of functions, state observation and control, definition of fractional energy, and Lyapunov stability analysis of linear and nonlinear fractional order systems. This second volume focuses on the initialization, observation and control of the distributed state, followed by stability analysis of fractional differential systems.
Foreword xiii
Preface xv
Part 1 Initialization, State Observation and Control
1(166)
Chapter 1 Initialization of Fractional Order Systems
3(32)
1.1 Introduction
3(1)
1.2 Initialization of an integer order differential system
4(6)
1.2.1 Introduction
4(1)
1.2.2 Response of a linear system
4(2)
1.2.3 Input/output solution
6(1)
1.2.4 State space solution
7(1)
1.2.5 First-order system example
8(2)
1.3 Initialization of a fractional differential equation
10(4)
1.3.1 Introduction
10(1)
1.3.2 Free response of a simple FDE
10(4)
1.4 Initialization of a fractional differential system
14(3)
1.4.1 Introduction
14(1)
1.4.2 State space representation
14(1)
1.4.3 Input/output formulation
15(2)
1.5 Some initialization examples
17(18)
1.5.1 Introduction
17(1)
1.5.2 Initialization of the fractional integrator
17(2)
1.5.3 Initialization of the Riemann-Liouville derivative
19(2)
1.5.4 Initialization of an elementary FDS
21(12)
1.5.5 Conclusion
33(2)
Chapter 2 Observability and Controllability of FDEs/FDSs
35(32)
2.1 Introduction
35(2)
2.2 A survey of classical approaches to the observability and controllability of fractional differential systems
37(3)
2.2.1 Introduction
37(1)
2.2.2 Definition of observability and controllability
37(1)
2.2.3 Observability and controllability criteria for a linear integer order system
37(2)
2.2.4 Observability and controllability of FDS
39(1)
2.3 Pseudo-observability and pseudo-controllability of an FDS
40(20)
2.3.1 Introduction
40(1)
2.3.2 Elementary approach
41(4)
2.3.3 Cayley-Hamilton approach
45(4)
2.3.4 Gramian approach
49(3)
2.3.5 Gilbert's approach
52(5)
2.3.6 Conclusion
57(1)
2.3.7 Pseudo-controllability example
58(2)
2.4 Observability and controllability of the distributed state
60(5)
2.4.1 Introduction
60(2)
2.4.2 Observability of the distributed state
62(2)
2.4.3 Controllability of the distributed state
64(1)
2.5 Conclusion
65(2)
Chapter 3 Improved Initialization of Fractional Order Systems
67(32)
3.1 Introduction
67(1)
3.2 Initialization: problem statement
68(3)
3.3 Initialization with a fractional observer
71(10)
3.3.1 Fractional observer definition
71(1)
3.3.2 Stability analysis
72(2)
3.3.3 Convergence analysis
74(2)
3.3.4 Numerical example 1: one-derivative system
76(2)
3.3.5 Numerical example 2: non-commensurate order system
78(3)
3.4 Improved initialization
81(18)
3.4.1 Introduction
81(1)
3.4.2 Non-commensurate order principle
82(2)
3.4.3 Gradient algorithm
84(3)
3.4.4 One-derivative FDE example
87(4)
3.4.5 Two-derivative FDE example
91(4)
A.3 Appendix
95(1)
A.3.1 Convergence of gradient algorithm
95(3)
A.3.2 Stability and limit value of X
98(1)
Chapter 4 State Control of Fractional Differential Systems
99(34)
4.1 Introduction
99(1)
4.2 Pseudo-state control of an FDS
100(3)
4.2.1 Introduction
100(1)
4.2.2 Numerical simulation example
101(2)
4.3 State control of the elementary FDE
103(18)
4.3.1 Introduction
103(1)
4.3.2 State control of a fractional integrator
104(17)
4.4 State control of an FDS
121(10)
4.4.1 Introduction
121(1)
4.4.2 Principle of state control
122(2)
4.4.3 State control of two integrators in series
124(2)
4.4.4 Numerical example
126(3)
4.4.5 State control of a two-derivative FDE
129(1)
4.4.6 Pseudo-state control of the two-derivative FDE
130(1)
4.5 Conclusion
131(2)
Chapter 5 Fractional Model-based Control of the Diffusive RC Line
133(34)
5.1 Introduction
133(1)
5.2 Identification of the RC line using a fractional model
134(20)
5.2.1 Introduction
134(1)
5.2.2 An identification algorithm dedicated to fractional models
134(5)
5.2.3 Simulation of the diffusive RC line
139(10)
5.2.4 Experimental identification
149(5)
5.3 Reset of the RC line
154(13)
5.3.1 Introduction
154(1)
5.3.2 Natural relaxation
155(1)
5.3.3 Principle of the reset technique
156(2)
5.3.4 Proposed reset procedure
158(1)
5.3.5 Experimental results
159(5)
5.3.6 Comments
164(1)
5.3.7 Conclusion
165(2)
Part 2 Stability of Fractional Differential Equations and Systems
167(210)
Chapter 6 Stability of Linear FDEs Using the Nyquist Criterion
169(36)
6.1 Introduction
169(2)
6.2 Simulation and stability of fractional differential equations
171(4)
6.2.1 Simulation of an FDE
171(1)
6.2.2 Stability of the simulation scheme
172(2)
6.2.3 Stability analysis of FDEs using the Nyquist criterion
174(1)
6.3 Stability of ordinary differential equations
175(7)
6.3.1 Introduction
175(1)
6.3.2 Open-loop transfer function
176(1)
6.3.3 Drawing of HOL(jω) graph in the complex plane
177(1)
6.3.4 Stability of the third-order ODE
178(4)
6.3.5 Conclusion
182(1)
6.4 Stability analysis of FDEs
182(13)
6.4.1 Introduction
182(1)
6.4.2 Drawing of HOL(jω) graph in the complex plane
182(2)
6.4.3 Stability of the one-derivative FDE
184(3)
6.4.4 Stability of the two-derivative FDE
187(7)
6.4.5 Stability of the N-derivative FDE
194(1)
6.4.6 Conclusion
195(1)
6.5 Stability analysis of ODEs with time delays
195(5)
6.5.1 Introduction
195(1)
6.5.2 Definitions
196(1)
6.5.3 Stability analysis
196(2)
6.5.4 Application to an example
198(2)
6.6 Stability analysis of FDEs with time delays
200(5)
6.6.1 Definitions
200(1)
6.6.2 Stability
201(1)
6.6.3 Application to an example
202(3)
Chapter 7 Fractional Energy
205(42)
7.1 Introduction
205(1)
7.2 Pseudo-energy stored in a fractional integrator
206(5)
7.3 Energy stored and dissipated in a fractional integrator
211(23)
7.3.1 Introduction
211(1)
7.3.2 Electrical distributed network
211(3)
7.3.3 Stored energy
214(1)
7.3.4 Power dissipated in the fractional integrator
215(1)
7.3.5 Energy storage
216(3)
7.3.6 Integer order and fractional order integrators
219(7)
7.3.7 Characterization of fractional energy and its dissipation
226(5)
7.3.8 Fractional energy invariance
231(3)
7.4 Closed-loop and open-loop fractional energies
234(13)
7.4.1 Introduction
234(1)
7.4.2 Energy of the closed-loop model
234(3)
7.4.3 Energy of the open-loop model
237(2)
7.4.4 Stored energies with a step input excitation
239(8)
Chapter 8 Lyapunov Stability of Commensurate Order Fractional Systems
247(46)
8.1 Introduction
247(2)
8.2 Lyapunov stability of a one-derivative FDE
249(9)
8.2.1 Problem statement
249(2)
8.2.2 Numerical simulation
251(2)
8.2.3 Physical interpretation
253(1)
8.2.4 Theoretical interpretation
254(4)
8.3 Lyapunov stability of an N-derivative FDE
258(11)
8.3.1 Introduction
258(1)
8.3.2 The integer order case
258(3)
8.3.3 Lyapunov function of N-derivative systems
261(4)
8.3.4 Stability condition
265(4)
8.4 Lyapunov stability of a two-derivative commensurate order FDE
269(12)
8.4.1 Introduction
269(1)
8.4.2 State space model of the open-loop representation
270(1)
8.4.3 State space models of the closed-loop representation
271(1)
8.4.4 Energy and stability of the open-loop representation
272(2)
8.4.5 Energy and stability of the closed-loop representation
274(2)
8.4.6 Definition of a stability test for a > 0
276(5)
8.5 Lyapunov stability of an N-derivative FDE (TV > 2)
281(12)
8.5.1 Introduction
281(1)
8.5.2 Problem statement
282(1)
8.5.3 LMI generalization for N = 3
283(6)
8.5.4 Application example
289(1)
A.8 Appendix
290(1)
A.8.1 Lemma
290(1)
A.8.2 Matignon's criterion
291(2)
Chapter 9 Lyapunov Stability of Non-commensurate Order Fractional Systems
293(50)
9.1 Introduction
293(2)
9.2 Stored energy, dissipation and energy balance in fractional electrical devices
295(7)
9.2.1 Usual capacitor and inductor devices
295(1)
9.2.2 Fractional capacitor and inductor
296(3)
9.2.3 Energy storage and dissipation in fractional devices
299(2)
9.2.4 Reversibility of energy and energy balance
301(1)
9.3 The usual series RLC circuit
302(4)
9.3.1 Introduction
302(1)
9.3.2 Analysis of the series RLC circuit
302(2)
9.3.3 Stability analysis
304(2)
9.4 The series RLC* fractional circuit
306(9)
9.4.1 Introduction
306(1)
9.4.2 Analysis of the series RLC* circuit
306(1)
9.4.3 Experimental stability analysis
307(3)
9.4.4 Theoretical stability analysis
310(4)
9.4.5 Conclusion
314(1)
9.5 The series RLL*C* circuit
315(5)
9.5.1 Circuit modeling
315(2)
9.5.2 Stability analysis
317(3)
9.6 The series RL*C* fractional circuit
320(5)
9.6.1 Introduction
320(1)
9.6.2 Analysis of the series RL*C* circuit
320(2)
9.6.3 Theoretical stability analysis
322(3)
9.7 Stability of a commensurate order FDE: energy balance approach
325(3)
9.7.1 Introduction
325(1)
9.7.2 Analysis of the commensurate order FDE
325(2)
9.7.3 Application to stability
327(1)
9.8 Stability of a commensurate order FDE: physical interpretation of the usual approach
328(15)
9.8.1 Introduction
328(1)
9.8.2 Commensurate order system
329(1)
9.8.3 Lyapunov function of a fractional differential system
329(2)
9.8.4 Stability analysis
331(3)
9.8.5 Conclusion
334(1)
A.9 Appendix
335(1)
A.9.1 The infinite length LG line
335(4)
A.9.2 Energy storage and dissipation in the fractional capacitor
339(2)
A.9.3 Some integrals
341(2)
Chapter 10 An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems
343(34)
10.1 Introduction
343(1)
10.2 Indirect Lyapunov method
344(9)
10.2.1 Introduction
344(1)
10.2.2 Linearization
344(1)
10.2.3 Nonlinear system analysis
345(4)
10.2.4 Local stability of a one-derivative nonlinear fractional system
349(4)
10.3 Lyapunov direct method
353(10)
10.3.1 Introduction
353(1)
10.3.2 The variable gradient method
353(1)
10.3.3 Nonlinear system with one derivative
354(3)
10.3.4 Nonlinear system with two fractional derivatives
357(6)
10.4 The Van der Pol oscillator
363(3)
10.4.1 Electrical nonlinear system
363(1)
10.4.2 Van der Pol oscillator
364(1)
10.4.3 Simulation of the nonlinear system
364(1)
10.4.4 Limit cycle
365(1)
10.5 Analysis of local stability
366(5)
10.5.1 Linearization
366(1)
10.5.2 Local stability
367(2)
10.5.3 Validation of stability results
369(2)
10.6 Large signal analysis
371(6)
10.6.1 Introduction
371(1)
10.6.2 Approximation of the first harmonic [ MUL 09]
371(1)
10.6.3 Lyapunov function and oscillation frequency
372(1)
10.6.4 Amplitude of the limit cycle
372(2)
10.6.5 Prediction of the limit cycle
374(3)
References 377(18)
Index 395
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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