Diversity and non-integer differentiation for system dynamics / Alain Oustaloup.

Format:

Book

Author/Creator:

Oustaloup, Alain, author.

Series:

Control, systems and industrial engineering series

Language:

English

Subjects (All):

Dynamics--Mathematical models.

Dynamics.

System analysis--Mathematical models.

System analysis.

Physical Description:

xxi, 359 pages : illustrations ; 25 cm.

Place of Publication:

London : ISTE ; Hoboken, N.J. : Wiley, 2014.

Summary:

Based on a diversity-structured approach which is notably inspired by various natural forms of diversity (biological among others), this book unquestionably offers a framework, on the one hand, to the introduction of non-integer differentiation (otherwise known as fractional differentiation) as a modeling tool and, on the other hand, to the use of such a modeling form to highlight dynamic performances (and notably of damping) unsuspected in an "integer" approach of mechanics and automatic control. The "non-integer" approach indeed enables us to overcome the mass-damping dilemma in mechanics and, consequently, the stability-precision dilemma in automatic control. This book has been written so that it can be read on two different levels: the first chapter achieves a first level of presentation which goes through the main results while limiting their mathematical development; the five remaining chapters constitute a second level of presentation in which the theoretical passages, deliberately avoided in the first chapter, are then developed at the mathematical level, but with the same goal of simplicity which aspires to make this book an example of pedagogy. Book jacket.

Contents:

Chapter 1 From Diversity to Unexpected Dynamic Performances 1

1.1 Introduction 1

1.2 An issue raising a technological bottle-neck 3

1.3 An aim liable to answer to the issue 4

1.4 A strategy idea liable to reach the aim 5

1.4.1 Why diversity? 5

1.4.2 What does diversity imply? 5

1.5 On the strategy itself 6

1.5.1 The study object 6

1.5.2 A pore: its model and its technological equivalent 7

1.5.3 Case of identical pores 8

1.5.4 Case of different pores 9

1.6 From physics to mathematics 12

1.6.1 An unusual model of the porous face 12

1.6.2 A just as unusual model governing water relaxation 16

1.6.3 What about a non-integer derivative which singles out these unusual models? 17

1.7 From the unusual to the unexpected 22

1.7.1 Unexpected damping properties 22

1.7.2 Just as unexpected memory properties 25

1.8 On the nature of diversity 29

1.8.1 An action level to be defined 29

1.8.2 One or several forms of diversity? 30

1.9 From the porous dyke to the CRONE suspension 33

1.10 Conclusion 35

1.11 Bibliography 36

Chapter 2 Damping Robustness 39

2.1 Introduction 39

2.2 From ladder network to a non-integer derivative as a water-dyke interface model 40

2.2.1 On the admittance factorizing 40

2.2.2 On the asymptotic diagrams at stake 41

2.2.3 On the asymptotic diagram exploiting 43

2.3 From a non-integer derivative to a non-integer differential equation as a model governing water relaxation 47

2.3.1 Flow-pressure differential equation 47

2.3.2 A non-integer differential equation as a model governing relaxation 48

2.3.3 Electrical analogy 49

2.4 Relaxation expression 50

2.5 From a non-integer differential equation to relaxation damping robustness 54

2.5.1 Operational approach 54

2.5.2 Frequency approach 58

2.5.3 On the representation of robustness in a symbolic domain 61

2.6 Validation by an experimental simulation in analog electronics 63

2.6.1 Simulation functional diagram 63

2.6.2 Simulation electronic circuit 64

2.6.3 On the simulation itself: damping robustness tests 69

2.7 Bibliography 70

Chapter 3 Non-Integer Differentiation, its Memory and its Synthesis 71

3.1 Introduction 71

3.2 From integer differentiation to non-integer differentiation 73

3.2.1 Toward non-integer differentiation: the adopted approach 73

3.2.2 Generic form of the order 1 and 2 derivatives 73

3.2.3 Generalization to the integer and non-integer case 74

3.3 From repeated integer integration to non-integer differentiation through non-integer integration 75

3.3.1 Repeated integer integration 76

3.3.2 Non-integer integration 78

3.3.3 Non-integer differentiation 79

3.4 Non-integer differentiation in sinusoidal steady state 82

3.4.1 A definition of the Fresnel vector 82

3.4.2 A direct application to kinematic magnitudes 83

3.4.3 Non-integer derivative of position 84

3.4.4 Verification of the decomposition of x⁽ⁿ⁾(t) 88

3.5 On memory associated with non-integer differentiation 90

3.5.1 From the local to the global by taking into account the past 90

3.5.2 On memory notion 90

3.5.3 On an aspect of human memory (an investigation trail) 91

3.6 On the synthesis of non-integer differentiation 99

3.6.1 Synthesis of a frequency-bounded real non-integer differentiator 99

3.6.2 Synthesis of a frequency-bounded complex non-integer differentiator 105

3.6.3 Stability of the synthesis transmittance 109

3.6.4 Distribution of synthesis zeros and poles: real and imaginary orders of differentiation 112

3.6.5 Determination of the number of synthesis zeros and poles 114

3.6.6 Validation in time domain 117

3.7 Bibliography 120

Chapter 4 On the CRONE Suspension 121

4.1 Introduction 121

4.2 From the porous dyke to the hydropoeumatic version of the CRONE suspension 122

4.2.1 Concept 122

4.2.2 From concept to achievement 122

4.2.3 Vehicle implementation 124

4.3 Metallic version of the CRONE suspension 127

4.3.1 A technological difference in terms of suspensions 127

4.3.2 Performance and robustness objective 127

4.3.3 Strategy 127

4.3.4 Contract collaboration 128

4.3.5 Principle of the CRONE suspension 128

4.3.6 Transfers of the usual and CRONE suspensions 129

4.3.7 Initial behavior: no initial acceleration for the CRONE suspension 130

4.3.8 Stability degree robustness 131

4.3.9 Idea of the synthesis of a non-integer order dashpot 134

4.3.10 Active character of the CRONE suspension 135

4.3.11 Piloted passive CRONE suspension 135

4.4 Bibliography 336

Chapter 5 On the CRONE Control 139

5.1 Introduction 139

5.2 From the porous dyke to the CRONE control of first and second generations 140

5.2.1 First interpretation of the relaxation model: first generation CRONE strategy 141

5.2.2 Second interpretation of the relaxation model: second generation CRONE strategy 144

5.3 Second generation CRONE control and uncertainty domains 147

5.3.1 Uncertainty domains 147

5.3.2 Particular open-loop uncertainty domains 149

5.3.3 Adequacy of the second generation CRONE control template to the particular uncertainty domains 150

5.4 Generalization of the vertical template through the third generation CRONE control 151

5.4.1 First level of generalization 151

5.4.2 Second level of generalization 154

5.4.3 Open-loop transfer integrating the curvilinear template 156

5.4.4 Optimization of the open-loop behavior 158

5.4.5 Structure and parametric estimation of the controller 160

5.4.6 Application 161

5.5 Bibliography 162

Chapter 6 Recursivity and Non-Integer Differentiation 165

6.1 Introduction 165

6.2 Indefinite recursive parallel arrangement of series RC cells 167

6.2.1 Localization of zeros 169

6.2.2 Zero and pole alternating 172

6.2.3 Geometric progression of zeros 174

6.2.4 Position of the zeros relatively to the poles 175

6.2.5 Zero and pole transposition to frequency domain 177

6.2.6 Impulse behavior 178

6.2.7 Step behavior 185

6.3 Recursive arborescent network as a lung respiratory model 199

6.3.1 Modeling by an equivalent electric network 199

6.3.2 Network admittance 202

6.3.3 Non-integer order of the admittance 207

6.3.4 Frequency band corresponding to the non-integer behavior 212

6.4 Unified study of recursive parallel arrangements of RL, RC and RLC cells 214

6.4.1 Recursive parallel arrangement of series RL cells 214

6.4.2 Recursive parallel arrangement of series RC cells 218

6.4.3 Recursive parallel arrangement of series RLC cells 223

6.4.4 Commented synthesis of the main results 231

6.5 A common presentation of results turning on eight RC and RL cell recursive arrangements 238

6.5.1 Systemic recursivity 238

6.5.2 Frequency recursivity 239

6.5.3 Same recursivity on the components 241

6.6 On unit gain frequency in non-integer differentiation or integration 244

6.7 On stored energy in non-integer differentiation or integration 247

6.7.1 Non-integer differentiator 247

6.7.2 Non-integer integrator 250

6.8 Bibliography 252.

Notes:

Includes bibliographical references and index.

ISBN:

9781848214750

1848214758

OCLC:

887912975