Note: Intro -- Fractional Order Motion Controls -- Contents -- Foreword -- Preface -- Acknowledgments -- Acronyms -- PART I FUNDAMENTALS OF FRACTIONAL ORDER CONTROLS -- 1 Introduction -- 1.1 Fractional Calculus -- 1.1.1 Definitions and Properties -- 1.1.2 Laplace Transform -- 1.1.3 Fractional Order Dynamic Systems -- 1.1.4 Stability of LTI Fractional Order Systems -- 1.2 Fractional Order Controls -- 1.2.1 Why Fractional Order Control? -- 1.2.2 Basic Fractional Order Control Actions -- 1.2.3 A Historical Review of Fractional Order Controls -- 1.3 Fractional Order Motion Controls -- 1.4 Contributions -- 1.5 Organization -- PART II FRACTIONAL ORDER VELOCITY CONTROLS -- 2 Fractional Order PI Controller Designs for Velocity Systems -- 2.1 Introduction -- 2.2 The FOPTD System and Three Controllers Considered -- 2.3 Design Specifications -- 2.4 Fractional Order PI and [PI] Controller Designs -- 2.4.1 Integer Order PID Controller Design -- 2.4.2 Fractional Order PI Controller Design -- 2.4.3 Fractional Order [PI] Controller Design -- 2.5 Simulation -- 2.6 Chapter Summary -- 3 Tuning Fractional Order PI Controllers for Fractional Order Velocity Systems with Experimental Validation -- 3.1 Introduction -- 3.2 Three Controllers to be Designed and Tuning Specifications -- 3.3 Tuning Three Controllers for FOVS -- 3.4 Illustrative Examples and Design Procedure Summaries -- 3.4.1 Fractional Order [PI] Controller Design Procedures -- 3.4.2 Fractional Order PI Controller Design Procedures -- 3.4.3 Integer Order PID Controller Design Procedures -- 3.5 Simulation Illustration -- 3.5.1 Case-1s Simulation Tests for the Designed FOPI and FO[PI] Controllers with ?c =10 rad/s and fm = 50? -- 3.5.2 Case-2s Simulation Tests for the Designed IOPID and FOPI and FO[PI] Controllers with ?c =15 rad/s and fm = 65? -- 3.6 Experimental Validation -- 3.6.1 Experimental Setup. , 3.6.2 HIL Emulation of the FOVS -- 3.6.3 Experimental Results -- 3.7 Chapter Summary -- 4 Relay Feedback Tuning of Robust PID Controllers -- 4.1 Introduction -- 4.2 Slope Adjustment of the Phase Bode Plot -- 4.3 The New PID Controller Design Formulae -- 4.4 Phase and Magnitude Measurement via Relay Feedback Tests -- 4.5 Illustrative Examples -- 4.5.1 High-order Plant P2(s) -- 4.5.2 Plant with an Integrator P5(s) -- 4.5.3 Plant with a Time Delay P6(s) -- 4.5.4 Plant with an Integrator and a Time Delay P7(s) -- 4.6 Chapter Summary -- 5 Auto-Tuning of Fractional Order Controllers with Iso-Damping -- 5.1 Introduction -- 5.2 FOPI and FO[PI] Controller Design Formulae -- 5.2.1 FOPI Controller Auto-Tuning -- 5.2.2 FO[PI] Controller Auto-Tuning -- 5.3 Measurements for Auto-Tuning -- 5.4 Simulation Illustration -- 5.4.1 High-Order Plant P2(s) -- 5.4.2 Plant with an Integrator P5(s) -- 5.4.3 Plant with a Time Delay P6(s) -- 5.5 Chapter Summary -- PART III FRACTIONAL ORDER POSITION CONTROLS -- 6 Fractional Order PD Controller Tuning for Position Systems -- 6.1 Introduction -- 6.2 Fractional Order PD Controller Design for Position Systems -- 6.2.1 Integer Order PD Controller Design -- 6.2.2 Fractional Order PD Controller Design -- 6.3 Design Procedures -- 6.4 Simulation Illustration -- 6.4.1 Step Response Comparison -- 6.4.2 Ramp Response Comparison -- 6.5 Experimental Validation -- 6.5.1 Introduction of the Experimental Platform -- 6.5.2 Experimental Model Simulation -- 6.5.3 Experiments on the Dynamometer -- 6.6 Chapter Summary -- 7 Fractional Order [PD] Controller Synthesis for Position Systems -- 7.1 Introduction -- 7.2 Position Systems and Design Specifications -- 7.3 Fractional Order [PD] Controller Design -- 7.4 Controller Design Examples and Bode Plot Validations -- 7.4.1 FO[PD] Controller Design -- 7.4.2 FOPD Controller Design. , 7.4.3 IOPID Controller Design -- 7.5 Implementation of Two Fractional Order Operators -- 7.5.1 Implementation of s? for FOPD -- 7.5.2 Implementation of (1 + t s)µ for FO[PD] -- 7.6 Simulation Illustration -- 7.6.1 Case-I: Step Response Comparison with T = 0.4s -- 7.6.2 Case-II: Step Response Comparison with T = 0.04s -- 7.6.3 Step Response Comparison with Time Delay -- 7.6.4 Step Response Comparison with Backlash Nonlinearity -- 7.7 Experimental Validation -- 7.7.1 Introduction to the Experimental Platform -- 7.7.2 Experiments on the Dynamometer Platform -- 7.8 Chapter Summary -- 8 Time-Constant Robust Analysis and Design of Fractional Order [PD] Controller -- 8.1 Introduction -- 8.2 Problem Statement -- 8.3 FO[PD] Tuning Specifications and Rules -- 8.3.1 FO[PD] Robustness to Time-Constant Variations -- 8.3.2 Numerical Computation -- 8.4 The Solution Existence Range and An Online Computation Method -- 8.4.1 The Solution Existence Range -- 8.4.2 Numerical Computation Example and Simulation Tests -- 8.4.3 Simulation Comparison -- 8.4.4 Online Computation -- 8.5 Experimental Validation -- 8.6 Chapter Summary -- 9 Experimental Study of Fractional Order PD Controller Design for Fractional Order Position Systems -- 9.1 Introduction -- 9.2 Fractional Order Systems and Fractional Order Controller Considered -- 9.3 FOPD Controller Design Procedure for the Fractional Order Position Systems -- 9.3.1 Preliminary and Design Specifications -- 9.3.2 Numerical Computation Process -- 9.3.3 Summary of Design Procedure -- 9.4 Simulation Illustration -- 9.4.1 Case-1: IOS-Based Design for IOS -- 9.4.2 Case-2: IOS-Based Design for FOS -- 9.4.3 Case-3: FOS-Based Design for FOS -- 9.5 Experimental Validation -- 9.5.1 HIL Experimental Setup -- 9.5.2 HIL Emulation of the FOS -- 9.5.3 Experimental Results -- 9.6 Chapter Summary. , 10 Fractional Order [PD] Controller Design and Comparison for Fractional Order Position Systems -- 10.1 Introduction -- 10.2 Fractional Order Position Systems and Fractional Order Controllers -- 10.3 Fractional Order [PD] Controller Design -- 10.3.1 Numerical Computation Process -- 10.3.2 Design Procedure Summary -- 10.3.3 Design Example and Bode Plot Validation of FO[PD] Design -- 10.4 Integer Order PID Controller and Fractional Order PD Controller Designs -- 10.5 Simulation Comparisons -- 10.6 Chapter Summary -- PART IV STABILITY AND FEASIBILITY -- 11 Stability and Design Feasibility of Robust PID Controllers for FOPTD Systems -- 11.1 Introduction -- 11.1.1 Research Questions -- 11.1.2 Previous Work -- 11.1.3 Contributions in This Chapter -- 11.2 Stability Region and Flat Phase Tuning Rule for the Robust PID Controller Design -- 11.2.1 Preliminary -- 11.2.2 Stability Region of PID Controller for FOPTD Plants -- 11.3 PID Controller Design with Pre-Specifications on fM and ?C -- 11.3.1 Design Scheme -- 11.3.2 Flat Phase Tuning Rule for the Robust PID Controller Design -- 11.3.3 Design Procedures Summary with An Example -- 11.3.4 How to Find the Achievable Region of the Two Specifications? -- 11.4 Simulation Illustration -- 11.5 Chapter Summary -- 12 Stability and Design Feasibility of Robust FOPI Controllers for FOPTD Systems -- 12.1 Introduction -- 12.2 Stabilizing and Robust FOPI Controller Design for FOPTD Systems -- 12.2.1 The Plant and Controller Considered -- 12.2.2 Stability Region Analysis of the FOPI Controller -- 12.2.3 FOPI Parameters Design with Two Specifications -- 12.2.4 FOPI Parameters Design with an Additional Flat Phase Constraint -- 12.2.5 Achievable Region of Two Design Specifications for the FOPI Controller Design -- 12.3 Design Procedures Summary with an Illustrative Example. , 12.4 Complete Information Collection for Achievable Region of ?c and fm -- 12.5 Simulation Illustration -- 12.6 Chapter Summary -- PART V FRACTIONAL ORDER DISTURBANCE COMPENSATIONS -- 13 Fractional Order Disturbance Observer -- 13.1 Introduction -- 13.2 Disturbance Observer -- 13.3 Actual Design Parameters in DOB and their Effects -- 13.4 Loss of the Phase Margin with DOB -- 13.5 Solution One: Rule-Based Switched Low Pass Filtering with Varying Relative Degree -- 13.6 The Proposed Solution: Guaranteed Phase Margin Method using Fractional Order Low Pass Filtering -- 13.7 Implementation Issues: Stable Minimum-Phase Frequency Domain Fitting -- 13.8 Chapter Summary -- 14 Fractional Order Adaptive Feed-Forward Cancellation -- 14.1 Introduction -- 14.2 Fractional Order Adaptive Feed-Forward Cancellation -- 14.3 Equivalence Between Fractional Order Internal Model Principle and Fractional Order Adaptive Feed-Forward Cancellation -- 14.3.1 Single-Frequency Disturbance Cancellation -- 14.3.2 Generalization to Multi-Frequency Disturbance Cancellation -- 14.4 Frequency-Domain Analysis of the FOAFC Performance for the Periodic Disturbance -- 14.5 Simulation Illustration -- 14.6 Experiment Validation -- 14.6.1 Introduction to the Experiment Platform -- 14.6.2 Experiments on the Dynamometer -- 14.7 Chapter Summary -- 15 Fractional Order Adaptive Compensation for Cogging Effect -- 15.1 Introduction -- 15.2 Fractional Order Adaptive Compensation of Cogging Effect -- 15.2.1 Cogging Effect Analysis -- 15.2.2 Motivations and Problem Formulation -- 15.2.3 IOAC Stability Analysis -- 15.2.4 FOAC Stability Analysis -- 15.3 Simulation Illustration -- 15.3.1 Case-1: IOAC with Constant Reference Speed -- 15.3.2 Case-2: FOAC with Constant Reference Speed -- 15.3.3 Case-3: IO/FOAC with Varying Reference Speed -- 15.4 Experimental Validation. , 15.4.1 Introduction to the Experimental Platform.