Robust Adaptive Control for Fractional-Order Systems with Disturbance and Saturation E-Book

Mou Chen

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley-ASME Press Series
• Sprache: Englisch

Seitenzahl: 260

Veröffentlichungsjahr: 2017

Beschreibung

A treatise on investigating tracking control and synchronization control of fractional-order nonlinear systems with system uncertainties, external disturbance, and input saturation

Robust Adaptive Control for Fractional-Order Systems, with Disturbance and Saturation provides the reader with a good understanding on how to achieve tracking control and synchronization control of fractional-order nonlinear systems with system uncertainties, external disturbance, and input saturation. Although some texts have touched upon control of fractional-order systems, the issues of input saturation and disturbances have rarely been considered together.

This book offers chapter coverage of fractional calculus and fractional-order systems; fractional-order PID controller and fractional-order disturbance observer; design of fractional-order controllers for nonlinear chaotic systems and some applications; sliding mode control for fractional-order nonlinear systems based on disturbance observer; disturbance observer based neural control for an uncertain fractional-order rotational mechanical system; adaptive neural tracking control for uncertain fractional-order chaotic systems subject to input saturation and disturbance; stabilization control of continuous-time fractional positive systems based on disturbance observer; sliding mode synchronization control for fractional-order chaotic systems with disturbance; and more.

• Based on the approximation ability of the neural network (NN), the adaptive neural control schemes are reported for uncertain fractional-order nonlinear systems
• Covers the disturbance estimation techniques that have been developed to alleviate the restriction faced by traditional feedforward control and reject the effect of external disturbances for uncertain fractional-order nonlinear systems
• By combining the NN with the disturbance observer, the disturbance observer based adaptive neural control schemes have been studied for uncertain fractional-order nonlinear systems with unknown disturbances
• Considers, together, the issue of input saturation and the disturbance for the control of fractional-order nonlinear systems in the present of system uncertainty, external disturbance, and input saturation

Robust Adaptive Control for Fractional-Order Systems, with Disturbance and Saturation can be used as a reference for the academic research on fractional-order nonlinear systems or used in Ph.D. study of control theory and engineering.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Series Preface

Symbols and Acronyms

Chapter 1: Introduction

Chapter 2: Fractional Calculus and Fractional-Order Systems

2.1 Fractional Calculus

2.2 Some Typical Fractional-Order Systems

2.3 Conclusion

Chapter 3: Fractional-Order PID Controller and Fractional-Order Disturbance Observer

3.1 Problem Statement

3.2 Fractional-Order PID Controller

3.3 Frequency-Domain Fractional-Order Disturbance Observer

3.4 Conclusion

Chapter 4: Design of Fractional-Order Controllers for Nonlinear Chaotic Systems and Some Applications

4.1 Fractional-Order Control for a Novel Chaotic System Without Equilibrium

4.2 Application of Chaotic System without Equilibrium in Image Encryption

4.3 Synchronization Control for Fractional-Order Nonlinear Chaotic Systems

4.4 Conclusion

Chapter 5: Sliding-Mode Control for Fractional-Order Nonlinear Systems Based on Disturbance Observer

5.1 Problem Statement

5.2 Adaptive Control Design Based on Fractional-Order Sliding-Mode Disturbance Observer

5.3 Simulation Examples

5.4 Conclusion

Chapter 6: Disturbance-Observer-Based Neural Control for Uncertain Fractional-Order Rotational Mechanical System

6.1 Problem Statement

6.2 Adaptive Neural Control Design

6.3 Simulation Example

6.4 Conclusion

Chapter 7: Adaptive Neural Tracking Control for Uncertain Fractional-Order Chaotic Systems Subject to Input Saturation and Disturbance

7.1 Problem Statement

7.2 Adaptive Neural Control Design Based on Fractional-Order Disturbance Observer

7.3 Simulation Examples

7.4 Conclusion

Chapter 8: Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer

8.1 Problem Statement

8.2 Main Results

8.3 Conclusion

Chapter 9: Sliding-Mode Synchronization Control for Fractional-Order Chaotic Systems with Disturbance

9.1 Problem Statement

9.2 Design of Fractional-Order Disturbance Observer

9.3 Disturbance-Observer-Based Synchronization Control of Fractional-Order Chaotic Systems

9.4 Simulation Examples

9.5 Conclusion

Chapter 10: Anti-Synchronization Control for Fractional-Order Nonlinear Systems Using Disturbance Observer and Neural Networks

10.1 Problem Statement

10.2 Design of Disturbance Observer

10.3 Anti-Synchronization Control of Fractional-Order Nonlinear Systems

10.4 Simulation Examples

10.5 Conclusion

Chapter 11: Synchronization Control for Fractional-Order Systems Subjected to Input Saturation

11.1 Problem Statement

11.2 Synchronization Control Design of Fractional-Order Systems with Input Saturation

11.3 Simulation Examples

11.4 Conclusion

Chapter 12: Synchronization Control for Fractional-Order Chaotic Systems with Input Saturation and Disturbance

12.1 Problem Statement

12.2 Design of Fractional-Order Disturbance Observer

12.3 Design of Synchronization Control

12.4 Simulation Examples

12.5 Conclusion

Appendix A: Fractional Derivatives of Some Functions

A.1 Fractional Derivative of Constant

A.2 Fractional Derivative of the Power Function

A.3 Fractional Derivative of the Exponential Function

A.4 Fractional Derivatives of Sine and Cosine Functions

Appendix B: Table of Caputo Derivatives

Appendix C: Laplace Transforms Involving Fractional Operations

C.1 Laplace Transforms

C.2 Special Functions for Laplace Transforms

C.3 Laplace Transform Tables

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Begin Reading

List of Illustrations

Chapter 2: Fractional Calculus and Fractional-Order Systems

Figure 2.1 Chaotic behaviors of fractional-order Lorenz system (2.46): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.2 Bifurcation diagram of fractional-order Lorenz system (2.46) for .

Figure 2.3 Dynamic characteristic of two cycles for fractional-order Lorenz system (2.46): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.4 Simulation result of fractional-order van der Pol oscillator (2.47).

Figure 2.5 Chaotic behaviors of fractional-order Genesio–Tesi system (2.48): (a) – plane; (b) – plane, (c) – plane, (d) –– space.

Figure 2.6 Bifurcation diagram of fractional-order Genesio–Tesi system (2.48) for .

Figure 2.7 Dynamic characteristic of two cycles for fractional-order Genesio–Tesi system (2.48): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.8 Chaotic behaviors of fractional-order Arneodo system (2.49): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.9 Bifurcation diagram of fractional-order Arneodo system (2.49) for .

Figure 2.10 Dynamic characteristic of one cycle for fractional-order Arneodo system (2.49): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.11 Chaotic behaviors of fractional-order Lotka–Volterra system (2.50): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.12 Bifurcation diagram of fractional-order Lotka–Volterra system (2.50) for .

Figure 2.13 Dynamic behaviors of fractional-order Lotka–Volterra system (2.50): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.14 Chaotic behaviors of fractional-order financial system (2.51): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.15 Bifurcation diagram of fractional-order financial system (2.51) for .

Figure 2.16 Dynamic characteristic of one cycle for fractional-order financial system (2.51): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.17 Chaotic behaviors of fractional-order Newton–Leipnik system (2.52): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.18 Bifurcation diagram of fractional-order Newton–Leipnik system (2.52) for .

Figure 2.19 Dynamic behaviors of fractional-order Newton–Leipnik system (2.52): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.20 Simulation result of fractional-order Duffing system (2.53).

Figure 2.21 Bifurcation diagram of fractional-order Duffing system (2.53) for .

Figure 2.22 Dynamic characteristic of one cycle for fractional-order Duffing system (2.53).

Figure 2.23 Chaotic behaviors of fractional-order Lü system (2.54): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.24 Bifurcation diagram of fractional-order Lü system (2.54) for .

Figure 2.25 Dynamic characteristic of one cycle for fractional-order Lü system (2.54): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.26 Chaotic behaviors of fractional-order three-dimensional system (2.55): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.27 Bifurcation diagram of fractional-order three-dimensional system (2.55) for .

Figure 2.28 Dynamic characteristic of one cycle for fractional-order three-dimensional system (2.55): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.29 Chaotic behaviors of fractional-order hyperchaotic oscillator (2.56): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.30 Bifurcation diagram of fractional-order hyperchaotic oscillator (2.56) for .

Figure 2.31 Dynamic characteristic of one cycle for fractional-order hyperchaotic oscillator (2.56): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.32 Chaotic behaviors of fractional-order four-dimensional hyperchaotic system (2.57): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.33 Bifurcation diagram of fractional-order four-dimensional hyperchaotic system (2.57) for

Figure 2.34 Dynamic characteristic of one cycle for fractional-order four-dimensional hyperchaotic system (2.57): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.35 Chaotic behaviors of fractional-order hyperchaotic cellular neural network (2.58): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 2.36 Bifurcation diagram of fractional-order hyperchaotic cellular neural network (2.58) for .

Figure 2.37 Dynamic behaviors of fractional-order hyperchaotic cellular neural network (2.58): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Chapter 3: Fractional-Order PID Controller and Fractional-Order Disturbance Observer

Figure 3.1 Integer-order PID control system.

Figure 3.2 Fractional-order control system.

Figure 3.3 Range of values of integer-order PID and fractional-order control system.

Figure 3.4 Unit step response curve of fractional-order system (3.5).

Figure 3.5 Step response of closed-loop fractional-order system (3.5).

Figure 3.6 Bode diagram of fractional-order system (3.5) with fractional-order controller (3.6).

Figure 3.7 Frequency-domain disturbance observer.

Figure 3.8 Bode plots of ideal case and its approximations with .

Figure 3.9 Estimation performance of FODO with filter order .

Figure 3.10 Estimation error for output of FODO with filter order and disturbance .

Figure 3.11 Estimation performance of FODO with filter order .

Figure 3.12 Estimation error for output of FODO with filter order and disturbance .

Figure 3.13 Control performance without FODO.

Figure 3.14 Control performance with FODO.

Chapter 4: Design of Fractional-Order Controllers for Nonlinear Chaotic Systems and Some Applications

Figure 4.1 Chaotic behaviors of the novel chaotic system: (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 4.2 Poincaré map in the – plane.

Figure 4.3 Circuit of the novel chaotic system (4.3).

Figure 4.4 Chaotic behaviors of the chaotic circuit: (a) – plane; (b) – plane; (c) – plane.

Figure 4.5 Numerical simulation results of the system (4.23).

Figure 4.6 Control inputs of the system (4.23).

Figure 4.7 Stabilization of Chen system (4.27).

Figure 4.8 Control inputs of Chen system (4.27).

Figure 4.9 Stabilization of Genesio's system (4.30).

Figure 4.10 Control inputs for Genesio's system (4.30).

Figure 4.11 Stabilization of hyperchaotic Lorenz system (4.33).

Figure 4.12 Control inputs for hyperchaotic Lorenz system (4.33).

Figure 4.13 (a) Original image; (b) corresponding gray distribution histogram.

Figure 4.14 (a) Encrypted image; (b) corresponding gray distribution histogram.

Figure 4.15 Correlations of two horizontal adjacent pixels: (a) in original image; (b) in encrypted image.

Figure 4.16 Attacked images.

Figure 4.17 Decrypted images.

Figure 4.18 Restored images.

Figure 4.19 Decryption attempt results using: (a) correct key; (b) incorrect key.

Figure 4.20 Chaotic behaviors of fractional-order Chen system: (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 4.21 Synchronization results of state variables of two fractional-order Chen systems: (a) and ; (b) and ; (c) and .

Figure 4.22 Synchronization errors , , and of two fractional-order Chen systems.

Figure 4.23 Synchronization results of state variables of two fractional-order Lorenz systems: (a) and ; (b) and ; (c) and .

Figure 4.24 Synchronization errors , , and of two fractional-order Lorenz systems.

Figure 4.25 Chaotic masking technology.

Figure 4.26 Simulation results: (a) information signal ; (b) mixture signal ; (c) recovered signal ; (d) error signal .

Chapter 5: Sliding-Mode Control for Fractional-Order Nonlinear Systems Based on Disturbance Observer

Figure 5.1 Output of fractional-order system (5.24) follows desired trajectory : (a) and ; (b) and .

Figure 5.2 Tracking errors of fractional-order system (5.24): and .

Figure 5.3 Disturbance and approximate output of for the simulation of fractional-order system (5.24): (a) and ; (b) and .

Figure 5.4 Estimation errors and for disturbances and in fractional-order system (5.24).

Figure 5.5 Control inputs and of fractional-order system (5.24).

Figure 5.6 Output of fractional-order system (5.27) follows desired trajectory : (a) and ; (b) and .

Figure 5.7 Disturbance estimation results for the simulation of fractional-order system (5.27): (a) and ; (b) and .

Figure 5.8 Estimation errors and for disturbances and in fractional-order system (5.27).

Figure 5.9 Control inputs and of fractional-order system (5.27).

Chapter 6: Disturbance-Observer-Based Neural Control for Uncertain Fractional-Order Rotational Mechanical System

Figure 6.1 Comparison result of and .

Figure 6.2 Output of the system (6.32) follows the desired trajectory : (a) and ; (b) and ; (c) and .

Figure 6.3 Tracking errors , , and for desired trajectories , and .

Figure 6.4 Disturbance and approximation output of : (a) and ; (b) and ; (c) and .

Figure 6.5 Disturbance estimation errors , , and .

Figure 6.6 Control inputs , , and of the system (6.32).

Chapter 7: Adaptive Neural Tracking Control for Uncertain Fractional-Order Chaotic Systems Subject to Input Saturation and Disturbance

Figure 7.1 Chaotic behaviors of fractional-order chaotic electronic oscillator model (7.61): (a) –– space; (b) –– space.

Figure 7.2 Tracking control results of the fractional-order chaotic electronic oscillator (7.61) (a) Output follows desired trajectory ; (b) tracking error .

Figure 7.3 Disturbance estimation results of the fractional-order chaotic electronic oscillator (7.61) (a) Disturbance and approximation output of ; (b) observation error .

Figure 7.4 Control input of the fractional-order chaotic electronic oscillator (7.61).

Figure 7.5 Chaotic behavior of (7.64): (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 7.6 Tracking control results of the fractional-order modified jerk system (7.64) (a) Output follows desired trajectory ; (b) tracking error .

Figure 7.7 Disturbance estimation results of the fractional-order modified jerk system (7.64) (a) Disturbance and approximation output of ; (b) observation error .

Figure 7.8 Control input of the fractional-order modified jerk system (7.64).

Chapter 8: Stabilization Control of Continuous-Time Fractional Positive Systems Based on Disturbance Observer

Figure 8.1 Whole closed-loop control system.

Figure 8.2 Disturbance estimation errors and for the simulation of fractional-order system (8.1) and (8.39).

Figure 8.3 Responses of state variables and for the fractional-order system (8.1) and (8.39) with FODO.

Figure 8.4 Responses of state variables and for the fractional-order system (8.1) and (8.39) without FODO.

Figure 8.5 Fractional electrical circuit.

Figure 8.6 Disturbance estimation errors and for the simulation of the fractional electrical circuit (8.46).

Figure 8.7 Responses of state variables and for the fractional electrical circuit (8.46) with FODO.

Figure 8.8 Responses of state variables and for the fractional electrical circuit (8.46) without FODO.

Chapter 9: Sliding-Mode Synchronization Control for Fractional-Order Chaotic Systems with Disturbance

Figure 9.1 Chaotic behaviors of modified fractional-order jerk system: (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 9.2 Comparison result of and .

Figure 9.3 Synchronization control results of modified fractional-order jerk system: (a) and ; (b) and ; (c) and ; (d) synchronization errors , , and .

Figure 9.4 Disturbance observer results of the modified fractional-order jerk system: (a) and ; (b) and ; (c) and ; (d) observation errors , , and .

Figure 9.5 Dynamic behaviors of fractional-order Liu system: (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 9.6 Synchronization control results of fractional-order Liu system: (a) and ; (b) and ; (c) and ; (d) synchronization errors , , and .

Figure 9.7 Disturbance observer results of the fractional-order Liu system: (a) and ; (b) and ; (c) and ; (d) observation errors , , and .

Chapter 10: Anti-Synchronization Control for Fractional-Order Nonlinear Systems Using Disturbance Observer and Neural Networks

Figure 10.1 Anti-synchronization states of and of the master system (10.32) and the slave system (10.33).

Figure 10.2 Anti-synchronization states of and of the master system (10.32) and the slave system (10.33).

Figure 10.3 Anti-synchronization states of and of the master system (10.32) and the slave system (10.33).

Figure 10.4 Anti-synchronization errors , , and of the master system (10.32) and the slave system (10.33).

Figure 10.5 Disturbance estimation result of and for fractional-order Lorenz system.

Figure 10.6 Disturbance estimation result of and for fractional-order Lorenz system.

Figure 10.7 Disturbance estimation result of and for fractional-order Lorenz system.

Figure 10.8 Disturbance observation errors , , and of fractional-order Lorenz system.

Figure 10.9 Anti-synchronization states of and of the master system (10.35) and the slave system (10.36).

Figure 10.10 Anti-synchronization states of and of the master system (10.35) and the slave system (10.36).

Figure 10.11 Anti-synchronization states of and of the master system (10.35) and the slave system (10.36).

Figure 10.12 Anti-synchronization errors , , and of the master system (10.35) and the slave system (10.36).

Figure 10.13 Disturbance estimation result of and for the fractional-order Lü system.

Figure 10.14 Disturbance estimation result of and for the fractional-order Lü system.

Figure 10.15 Disturbance estimation result of and for the fractional-order Lü system.

Figure 10.16 Observation errors , , and of the fractional-order Lü system.

Chapter 11: Synchronization Control for Fractional-Order Systems Subjected to Input Saturation

Figure 11.1 Chaotic behaviors of fractional-order modified Chua's circuit with sine function: (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 11.2 Synchronization control results of the master system (11.17) and the slave system (11.21): (a) and ; (b) and ; (c) and .

Figure 11.3 Time responses of synchronization errors , , and of the master system (11.17) and the slave system (11.21).

Figure 11.4 Time responses of synchronization control inputs , , and of the master system (11.17) and the slave system (11.21).

Figure 11.5 Chaotic behaviors of fractional-order four-dimensional modified Chua's system: (a) – plane; (b) – plane; (c) –– space; (d) –– space.

Figure 11.6 Synchronization control results of the master system (11.28) and the slave system (11.30): (a) and ; (b) and ; (c) and ; (d) and .

Figure 11.7 Time responses of synchronization errors , , , and of the master system (11.28) and the slave system (11.30).

Figure 11.8 Time responses of synchronization control inputs , , , and of the master system (11.28) and the slave system (11.30).

Chapter 12: Synchronization Control for Fractional-Order Chaotic Systems with Input Saturation and Disturbance

Figure 12.1 Chaotic behaviors of fractional-order Chua's circuit (12.31) with disturbance: (a) – plane; (b) – plane; (c) – plane, (d) –– space.

Figure 12.2 Synchronization states of and for the systems (12.31 and 12.33).

Figure 12.3 Synchronization states of and for the systems (12.31 and 12.33).

Figure 12.4 Synchronization states of and for the systems (12.31 and 12.33).

Figure 12.5 Synchronization errors , , and for the systems (12.31 and 12.33).

Figure 12.6 Disturbance estimation in system (12.33) versus actual in system (12.31).

Figure 12.7 Disturbance estimation error for in system (12.33) and in system (12.31).

Figure 12.8 Synchronization control input for the systems (12.31 and 12.33).

Figure 12.9 Chaotic behaviors of fractional-order hyperchaos Chua's circuit: (a) – plane; (b) – plane; (c) – plane; (d) –– space.

Figure 12.10 Synchronization states of and for the systems (12.42 and 12.43).

Figure 12.11 Synchronization states of and for the systems (12.42 and 12.43).

Figure 12.12 Synchronization states of and for the systems (12.42 and 12.43).

Figure 12.13 Synchronization states of and for the systems (12.42 and 12.43).

Figure 12.14 Synchronization errors , , , and for the systems (12.42 and 12.43).

Figure 12.15 Disturbance estimate in system (12.43) versus actual in system (12.42).

Figure 12.16 Disturbance estimation error for in system (12.43) and in system (12.42).

Figure 12.17 Synchronization control input for the systems (12.42 and 12.43).

List of Tables

Chapter 4: Design of Fractional-Order Controllers for Nonlinear Chaotic Systems and Some Applications

Table 4.1 Correlation coefficients in original and encrypted images

Chapter 9: Sliding-Mode Synchronization Control for Fractional-Order Chaotic Systems with Disturbance

Table 9.1 Equilibrium points of the modified fractional-order jerk system

Appendix B: Table of Caputo Derivatives

Table B.1 Caputo derivatives of particular functions for the fractional orders , , and

Table B.2 List of , , and

Appendix C: Laplace Transforms Involving Fractional Operations

Table C.1 Some special functions for Laplace transforms

Table C.2 Inversions of Laplace transforms with fractional and irrational operators

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Robust Adaptive Control for Fractional-Order Systems with Disturbance and Saturation

Mou Chen

Nanjing University of Aeronautics and Astronautics, Jiangsu, Nanjing, China

Shuyi Shao

Nanjing University of Aeronautics and Astronautics, Jiangsu, Nanjing, China

Peng Shi

University of Adelaide, Adelaide, South Australia

This edition first published 2018

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Library of Congress Cataloging-in-Publication Data

Names: Chen, Mou, author. | Shao, Shuyi, author | Shi, Peng, 1958- author.

Title: Robust adaptive control for fractional-order systems with disturbance and saturation / by Professor Mou Chen, Doctor Shuyi Shao, Professor Peng Shi.

Description: Hoboken, NJ : John Wiley & Sons, 2017. | Includes bibliographical references and index. |

Identifiers: LCCN 2017027175 (print) | LCCN 2017041115 (ebook) | ISBN 9781119393337 (pdf) | ISBN 9781119393313 (epub) | ISBN 9781119393276 (cloth)

Subjects: LCSH: Adaptive control systems.

Classification: LCC TJ217 (ebook) | LCC TJ217 .S53 2017 (print) | DDC 629.8/36-dc23

LC record available at https://lccn.loc.gov/2017027175

Cover Design: Wiley

Cover Image: © CaryllN/Gettyimages

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Preface

This book is devoted to an investigation of some issues of tracking control and synchronization control for fractional-order nonlinear systems in the presence of system uncertainty, external disturbance, and input saturation. On the basis of definitions of the fractional integral and fractional derivatives, lemmas of stability analysis for fractional-order systems, design techniques of disturbance observers, approximation methods of system uncertainty, and handling methods of input saturation, the main research motives of this book are given as follows:

1.

In the modeling process, there exist a vast amount of uncertainties caused by modeling error, which may not only degrade the performance of the control system but also even lead to instability of the dynamics system. Therefore, the uncertainty should be considered in the control design to improve the closed-loop system performance of fractional-order systems. Furthermore, neural networks can approximate any continuous uncertain dynamics with an arbitrary accuracy. Many adaptive neural control schemes have been reported for uncertain integer-order nonlinear systems. However, the neural network approximation technique has rarely been considered in uncertain fractional-order nonlinear systems in past decades.

2.