New Trends in Nanotechnology and Fractional Calculus Applications (eBook)

Preface 5
Contents 7
Part I New Trends in Nanotechnology 12
Novel Molecular Diodes Developed by Chemical Conjugation of Carbon Nanotubes with Peptide Nucleic Acid 13
References 23
Hybrid Single Walled Carbon Nanotube FETs for High Fidelity DNA Detection 26
1 Introduction 26
2 Experimental Section 28
2.1 SWNT Purification and Dispersion 28
2.2 Device Fabrication 29
2.3 Synthesis of SWNT-ssDNA Conjugations and Detection of Specific DNA Sequences 30
3 Results and Discussion 30
4 Conclusion 32
References 32
Towards Integrated Nanoelectronic and Photonic Devices 34
1 Introduction 34
2 Integrated Electronic Devices 36
2.1 Scaling of MOSFET Devices 37
2.2 Nanotube Transistors and Interconnects 38
2.3 Ultimate Integrated Devices Based on Graphene 40
3 Photonic Devices and Interconnects 41
3.1 Optical Interconnects 43
3.2 Photonic sources and Bragg Reflectors 44
3.3 Photonic Crystal Waveguides 47
4 Summary 48
References 49
New Noninvasive Methods for `Reading' of Random Sequences and Their Applications in Nanotechnology 51
1 Introduction 51
2 Procedure of the Optimal Linear Smoothing (POLS) 52
3 The Statistics of the Fractional Moments and Detection of the Statistical Proximity 57
4 The Universal Distribution Function of the Relative Fluctuations 60
5 Brief Description of Problems that Can Be Solved by These Noninvasive Methods 61
5.1 Comparison with Pattern Equipment and Self-Verification of the Readiness of Complex Equipment to Measurements 61
5.2 The Diagnosis of Deceases Based on Quantitative ``Reading'' of Noises, Which Are Registered Inside a Human Body 62
5.3 The Increasing of Sensitivity of Existing Spectrometers and Creation of Supersensitive Gas-Sensors and Chromatographers 63
5.4 Applications to Different Nanotechnologies 63
References 63
Quantum Confinement in Nanometric Structures 65
1 Introduction 65
2 2D Structures 66
3 1D Structures 68
4 0D Structures 71
5 Conclusions 74
References 74
Part II Techniques and Applications 76
Air-Fuel Ratio Control of an Internal Combustion Engine Using CRONE Control Extended to LPV Systems 77
1 Introduction 77
2 CRONE Control Principles 78
2.1 Template Definition 78
2.2 Open Loop Transfer Function Including the Template 78
2.3 Parameterization of Optimal Open Loop 79
2.4 Optimization for Optimal Open Loop Determination 80
2.5 Parametric Synthesis of the Controller 81
3 CRONE Control of LPV Systems 81
3.1 Principles 81
3.2 LPV Plant Representation 82
3.3 LPV CRONE Controller Structure 83
3.4 Objectives and Constraints 84
4 Lambda Control of an IC Engine 86
4.1 Presentation and Motivation 86
4.2 Model Description 87
4.3 Delay Approximation 87
4.4 Results 88
4.5 Simulation 90
5 Conclusion 91
References 91
Non Integer Order Operators Implementation via Switched Capacitors Technology 93
1 Circuital Implementation of Fractional Order Integrator 93
2 Switched Capacitors Implementation of Fractional Order Integrator 95
2.1 Switched Capacitors Introduction 95
2.2 Switched Capacitors Implementation 96
3 Results 97
4 Conclusion 101
References 101
Analysis of the Fractional Dynamics of an Ultracapacitorand Its Application to a Buck-Boost Converter 103
1 Introduction 103
2 Ultracapacitor Model 104
3 Buck-Boost Converter 108
4 Simulation Results 109
5 Conclusions 110
References 110
Approximation of a Fractance by a Network of Four Identical RC Cells Arranged in Gamma and a Purely Capacitive Cell 112
1 Introduction 112
2 Fractional Dynamic System 114
3 Forced Response 116
4 Free Response 118
5 Conclusion 125
References 125
Part III Mathematical Tools 126
On Deterministic Fractional Models 127
1 Introduction 128
2 Fractional Integral and Differential Operators 130
3 Time-Fractional Derivative: An Application of Fractional Differential Operators in Modeling Subdiffusive and Superdiffusive Problems 132
4 Fractional Dynamics Systems 137
4.1 Linear Case Using the Caputo Fractional Derivative 137
4.2 Quasi-Linear Case Using the Caputo Fractional Derivative 138
4.3 Conclusion 139
4.4 Examples 139
4.5 Applications 143
5 Spatial–Fractional Derivative: Anomalous Diffusion 149
5.1 Spatial–Fractional Diffusion Equation 149
6 Fractional Differential Equations and Chaos 151
References 154
A New Approach for Stability Analysis of Linear Discrete-Time Fractional-Order Systems 155
1 Introduction 155
2 Linear Discrete-Time Fractional-Order Systems 156
3 Asymptotic Stability 159
4 Numerical Example 163
5 Conclusion 164
References 164
Stability of Fractional-Delay Systems: A Practical Approach 167
1 Introduction 167
2 The Proposed Numerical Algorithm 169
3 Numerical Example 172
References 173
Comparing Numerical Methods for Solving Nonlinear Fractional Order Differential Equations 175
1 Introduction 175
2 Numerical Methods 176
2.1 Quadrature Method (QM) 176
2.2 Predictor Corrector Method (PCM) 177
2.3 Fixed Length Integral Principle (FLIP) 177
2.4 Nested Memory Principle (NMP) 178
2.5 Higher Order Methods (Lubich) 178
2.6 Kumar Agrawal Method (KAM) 178
3 Numerical Simulations 178
4 Conclusions 181
References 182
Fractional-Order Backward-Difference Definition Formula Analysis 184
1 Introduction 184
2 Mathematical Preliminaries 185
2.1 Notation 185
2.2 Fractional-Order Backward-Difference/Sum 185
3 Relation Between Coefficients ai() 186
3.1 Numerical Example 187
4 Coefficients ai() Values Analysis 189
5 Coefficients ai() Increasing Rate 190
6 Concluding Remarks 194
References 194
Fractional Differential Equations on Algebroids and Fractional Algebroids 195
1 Introduction 195
2 Rabinovich Equations on a Leibniz Algebroid 196
3 Fractional Rabinovich Equations 198
4 The Fractional Equations on a Fractional Leibniz Algebroid 199
5 Numerical Simulations and Conclusions 201
References 202
Generalized Hankel Transform and Fractional Integralson the Spaces of Generalized Functions 204
1 Introduction 204
2 Notations and Definitions 205
2.1 Notations 205
2.2 The Spaces Fp, and Their Duals 206
3 Fractional Integrals 207
4 Generalized Hankel Transform on Fp, 207
5 GHT of Fractional Integrals on Fp, 210
6 Fractional Integral of GHT on Fp, 212
7 Concluding Remarks 213
References 213
Some Bounds on Maximum Number of Frequencies Existing in Oscillations Produced by Linear Fractional Order Systems 214
1 Introduction 214
2 Problem Statement 215
3 An Upper Bound 217
4 A Lower Bound 218
5 Conclusion 220
References 221
Fractional Derivatives with Fuzzy Exponent 222
1 Introduction 222
2 Fuzzy Sets and Numbers 223
3 Ordered Fuzzy Numbers 225
4 Fuzzy Fractional Derivative 227
5 Conclusions 230
References 230
Game Problems for Fractional-Order Systems 233
1 Fractional-Order Systems 233
2 Game Problems 237
3 Example 239
References 240
Synchronization Analysis of Two Networks 242
1 Introduction 242
2 Synchronization Analysis 243
3 Numerical Examples 247
3.1 Directed Connection 247
3.2 Undirected Connection 248
4 Conclusion 251
References 251
Part IV Fractional Modelling 253
Modeling Ultracapacitors as Fractional-Order Systems 254
1 Introduction 254
2 Modeling and Identification 256
3 Experiment and Results 256
4 Circuit Realization 258
5 Conclusions 259
References 259
IPMC Actuators Non Integer Order Models 260
1 Introduction 260
2 Experimental Setup 261
3 Experimental Results 264
4 Conclusion 268
References 268
On the Implementation of a Limited Frequency Band Integrator and Application to Energetic Material Ignition Prediction 270
1 Introduction 270
2 A Model to Predict Energetic Materials Ignition 271
3 A Testbench for Ignition Prediction Model Validation 272
4 Fractional Integration Algorithm 274
4.1 Fractional Integration Numerical Method Specifications 274
4.2 Algorithm Description 275
5 Application to Ignition Prediction of Energetic Materials 276
5.1 Implementation Description 276
5.2 Some Rules of Thumb 277
5.3 Reference Signal Computation 278
5.4 Application and Result Analysis 279
6 Conclusion 280
References 281
Fractional Order Model of Beam Heating Processand Its Experimental Verification 283
1 Introduction 283
1.1 Transfer Function e(Ts) and Its Bode Diagram 284
2 Mathematical Description of Heating Process 285
3 Experimental Verification of Heating Process Model 287
3.1 Experimental Setup 287
3.2 Thermoelectric Module – Peltier Module 288
3.3 Modelling Results 289
4 Conclusion 290
References 290
Analytical Design Method for Fractional Order Controller Using Fractional Reference Model 291
1 Introduction 291
2 Fractional Order Systems 292
3 The Proposed Design Method 293
4 Illustrative Example 295
5 Conclusions 297
References 299
On Observability of Nonlinear Discrete-Time Fractional-Order Control Systems 300
1 Introduction 300
2 Fractional Discrete-Type Calculus 301
3 Observability Concepts 303
4 Fractional Order and Observability for Two-Dimensional Systems 306
References 307
Chaotic Fractional Order Delayed Cellular Neural Network 308
1 Introduction 308
2 Fractional Operators and its Approximation 309
3 Fractional Order Delayed Cellular Neural Network 311
4 Chaos in the Fractional Order Delayed Cellular Neural Network 312
5 Conclusions 314
References 315
Fractional Wavelet Transform for the Quantitative Spectral Analysis of Two-Component System 316
1 Introduction 317
2 Signal Processing Method 317
2.1 B-spline 318
2.2 Fractional B-spline 318
2.3 Fractional B-spline Wavelets 319
3 Experimental 320
3.1 Apparatus and Software 320
3.2 Preparation of Standard Solutions 320
3.3 Commercial Veterinary Products 320
4 Method Applications 320
4.1 Fractional Wavelet Transform of the Absorption Spectra 320
4.2 Calibration Graphs 321
4.3 Method Validation 324
4.4 Analysis of the Commercial Veterinary Formulation 324
5 Conclusions 325
References 326
Fractional Wavelet Transform and Chemometric Calibrations for the Simultaneous Determination of Amlodipineand Valsartan in Their Complex Mixture 327
1 Introduction 327
2 Experimental 328
2.1 Instruments and Software 328
2.2 Commercial Tablet Preparation 328
2.3 Standard Solutions 329
2.4 Preparation of the Commercial Tablet Samples 329
3 Results and Discussions 329
3.1 PLS Calibration Models and Prediction 331
3.2 Method Applications to Tablet Samples 332
4 Conclusions 333
References 333
Part V Fractional Control Systems 335
Analytical Impulse Response of Third Generation CRONE Control 336
1 Introduction 336
2 Third Generation CRONE Control 337
3 Analytical Response Using Residue Method 337
3.1 Integration upon 2 and 6 Curves 339
3.2 Integration upon 3 and 5 Curves 341
3.3 Integration upon 4 Curve 343
4 Poles Characterization 344
5 Analytical Impulse Response 345
6 Simulations 346
7 Conclusion 348
Appendix 348
References 349
Stability Analysis of Fractional Order Universal Adaptive Stabilization 350
1 Introduction 350
2 Preliminaries 351
2.1 Fractional Calculus and Mittag–Leffler Function 351
2.2 Nussbaum Function 352
3 Stability Analysis of Fractional UAS 352
3.1 Fractional Dynamics with Integer-Order Control Strategy 352
3.2 Fractional Dynamics with Fractional Control 355
3.3 Application of Viscoelastic and Electromagnetic Theories in UAS 356
4 Nussbaum Function with Mittag–Leffler Form 357
5 Simulations 358
6 Conclusion 360
References 360
Position and Velocity Control of a Servo by Using GPCof Arbitrary Real Order 362
1 Introduction 362
2 Generalized Predictive Control 363
2.1 Introduction 363
2.2 Fractional-Order Generalized Predictive Control 364
3 System Description 365
4 Controllers Design and Results 366
4.1 Velocity Control 366
4.2 Position Control 367
5 Conclusions 368
References 369
Decentralized CRONE Control of mxn Multivariable System with Time-Delay 370
1 Introduction 370
2 Mathematical Backgrounds 371
3 CRONE Control Methodology and MIMO Systems 372
3.1 Definition of the Diagonal Open-Loop Transfer Function Elements 373
3.2 Decoupling and Optimised Controller 375
3.3 Research of Time-Delay of Each Open-Loop 375
3.4 Open-Loop Transfer Matrix Optimization 376
3.5 Decentralized CRONE Controller 378
4 Application 379
4.1 Research of Time-Delay 380
4.2 Specifications 381
4.3 Optimisation 381
4.4 Results 382
5 Conclusion 383
References 384
Fractional Order Adaptive Control for Cogging Effect Compensation 385
1 Introduction 385
2 Fractional Order Adaptive Compensation of Cogging Effect 386
2.1 Motivations and Problem Formulation 386
2.2 Stability Analysis of Integer Order Case 389
2.3 Stability Analysis of the Fractional Order Case 391
3 Simulation Illustrations Using PMSM Position Servo System Model 395
3.1 Case-1: Integer Order Adaptive Compensation for Multi-Harmonics Cogging Effect 396
3.2 Case-2: Fractional Order Adaptive Compensation for Multi-Harmonics Cogging Effect 397
4 Chapter Conclusion 400
References 400
Generalized Predictive Control of Arbitrary Real Order 402
1 Introduction 402
2 Generalized Predictive Control 403
3 Some Mathematical Results 404
4 Fractional-Order Generalized Predictive Control 405
5 FGPC: An Application Example 406
6 Conclusions 408
References 408
Frequency Response Based CACSD for Fractional Order Systems 410
1 Introduction 410
2 Graphical Interface 411
3 Generalized Order Transfer Function 412
4 Illustrative Examples 413
4.1 Ventilatory Support in COPD Patients 414
4.2 Ventilation in Anesthetised Animals 416
5 Conclusions 417
References 417
Resonance and Stability Conditions for Fractional Transfer Functions of the Second Kind 419
1 Introduction 419
2 Stability of Fractional Transfer Functions of the Second Kind 421
2.1 Case || 1 421
2.2 Case || < 1
2.2.1 Subcase -1< 0
2.2.2 Subcase 0 < 1
2.3 Summary 423
3 Resonance of Fractional Transfer Functions of the Second Kind 423
4 Open-Loop – Closed-Loop Analysis 426
5 Numerical Examples 431
5.1 Example 1 431
5.2 Example 2 432
6 Conclusion 434
References 434
Synchronization of Fractional-Order Chaotic Systemvia Adaptive PID Controller 435
1 Introduction 435
2 Adaptive PID Controller Design 436
3 Synchronization of Uncertain Hard Spring 6-Van der Pol Oscillator 438
3.1 System Description 438
3.2 Implementation 440
4 Conclusion 442
References 442
On Fractional Control Strategy for Four-Wheel-Steering Vehicle 443
1 Introduction 443
2 Improvement of Feedforward Control for 4WS Vehicle by Introducing of Fractional Derivative 444
2.1 Two Degrees of Freedom Vehicle Model 444
2.2 Fractional Order Feedforward Control Strategy 445
3 Yaw Feedback Control for 4WS Vehicle 448
3.1 Fractional Order Control Strategy with Yaw Feedback 448
3.2 Yaw Rate Tracking Control Strategy for 4WS Vehicle 449
4 Computational Simulation 450
5 Conclusion 452
References 452
Fractional Order Sliding Mode Controller Designfor Fractional Order Dynamic Systems 453
1 Introduction 453
2 Integer Order VSC for Integer Order Systems 454
3 Fractional Order VSC for Fractional Order State Space Systems 455
4 Two Application Examples 456
4.1 A Second Order System (Two State Variables) 456
4.2 A Third Order System (Three State Variables) 458
5 Concluding Remarks 459
References 460
A Fractional Order Adaptation Law for Integer Order Sliding Mode Control of a 2DOF Robot 461
1 Introduction 461
2 Robot Dynamics and the Control Problem 462
3 Sliding Mode Control Through a Fractional Order Adaptation Scheme 462
4 Simulation Results 466
5 Conclusions 468
References 468
Synchronization of Chaotic Nonlinear Gyros Using Fractional Order Controller 469
1 Introduction 469
2 System Description 470
3 Synchronization of Fractional Modelled System by Fractional PD SMC 471
4 Fuzzy Controller 473
5 Conclusion 475
References 475
Nyquist Envelope of Fractional Order Transfer Functionswith Parametric Uncertainty 476
1 Introduction 476
2 Construction of the Value Set of FOIP 477
3 Nyquist Envelope of FOITF 479
4 A Numerical Example 480
5 Conclusions 482
References 483
Synchronization of Gyro Systems via Fractional-Order Adaptive Controller 484
1 Introduction 484
2 Fractional Adaptive Controller Design 485
3 Synchronization of Uncertain Fractional-Order Gyro System 488
3.1 System Description 488
3.2 Implementation 489
4 Conclusion 491
References 491
Controllability and Minimum Energy Control Problemof Fractional Discrete-Time Systems 492
1 Introduction 492
2 Fractional Systems 493
3 Controllability 494
4 Minimum Energy Control 496
5 Concluding Remarks 498
References 498
Control of Chaos via Fractional-Order State Feedback Controller 499
1 Introduction 499
2 Stability Analysis For Fractional-Order Systems 500
3 Proposed Controller Design 501
3.1 Simple Fractional Controller 501
3.2 State Feedback Controller 502
3.3 Combination of State Feedback and Fractional-Order Controllers 502
3.4 The Stability Analysis 503
4 Implementation of Proposed Controller in Genesio-Tesi Chaotic System 503
5 Conclusion 506
References 506
Index 508