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Modelling and control of feed drives Ng, Kenneth 2011

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  MODELLING AND CONTROL OF FEED DRIVES by KENNETH NG  B. A. Sc. (Engineering Physics) University of British Columbia, Canada   A THESIS SUMBITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Mechanical Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2011 ยฉ Kenneth Ng, 2011  ii Abstract Feed drives are used in positioning of machine tools. The drives are actuated either by linear or rotary servo motors. The ball screw drives are driven by rotary motors; hence they have flexibility and added friction due to nut interface. Direct drives are driven by linear motors which have more mechanical stiffness, but less disturbance rejection due to missing load reduction mechanism. This thesis presents the modelling and control of drives with rigid and flexible structures. A single degree of freedom flexible oscillator is mounted on a high speed, rigid feed drive table for experimental illustration of system identification and the active control method proposed in the thesis. The rigid feed drive dynamics include the mechanical component of the rigid body mass and viscous damping, and the electrical component of the power amplifier and motor. The flexible component is modelled by springs, mass and damping elements. Both rigid and flexible dynamics of the system are identified experimentally through unbiased least square, sine sweep and impact model tests. The vibration of the single degree of freedom system is actively damped by an acceleration feedback inserted in the velocity loop. A Kalman filter is used to minimize the drift and noise on the acceleration measurements. The position loop is closed with a proportional controller. It is experimentally demonstrated that the vibrations of the flexible structure can be well damped. However, the acceleration feedback used at the resonance frequency greatly minimizes the bandwidth close to the vibration frequency. Further methods need to be used to expand the bandwidth beyond the natural frequency of the flexible structure by coping with the anti-resonant effect of the acceleration feedback.  iii Table of Contents Abstract ................................................................................................................................ ii Table of Contents ................................................................................................................ iii List of Tables ........................................................................................................................ v List of Figures ..................................................................................................................... vii Nomenclature .................................................................................................................... xii Acknowledgments ........................................................................................................... xviii Chapter 1. Introduction .................................................................................................. 1 Chapter 2. Literature Survey ........................................................................................... 3 2.1 Chapter Overview ................................................................................................... 3 2.2 Identification of Drive Dynamics ............................................................................. 3 2.3 Vibration Control ..................................................................................................... 4 Chapter 3. Experimental Setup ....................................................................................... 9 3.1 Chapter Overview ................................................................................................... 9 3.2 Control System Setup .............................................................................................. 9 3.3 System Hardware Setup ........................................................................................ 10 3.3.1 Linear Motor .................................................................................................. 10 3.3.2 Linear Encoder ............................................................................................... 12 3.3.3 Single Degree Oscillator ................................................................................. 12 3.3.4 Accelerometer Sensor .................................................................................... 13 3.4 Summary ............................................................................................................... 14 Chapter 4. Identification of Feed Drive Dynamics ........................................................ 16 4.1 Chapter Overview ................................................................................................. 16 4.2 System Model ........................................................................................................ 16 4.3 Parameter Identification ....................................................................................... 23 4.3.1 Identification of Motor Dynamics by Ad-Hoc Technique .............................. 25 4.3.2 Identification of Motor Dynamics by Unbiased Least Square Technique ..... 29 4.3.3 Identification of Structural Flexibility by Impact Modal Test ........................ 35  iv 4.3.4 Identification of System Dynamics by Sine Sweep Technique with Nonlinear Least-Square Optimization........................................................................................ 37 4.3.5 Discussion ....................................................................................................... 40 4.4 Summary ............................................................................................................... 47 Chapter 5. Digital Control of Feed Drives ..................................................................... 49 5.1 Chapter Overview ................................................................................................. 49 5.2 State Space Position Control ................................................................................. 49 5.2.1 State Space Pole Placement Controller with Integrator Design .................... 49 5.2.2 Observer Design ............................................................................................. 55 5.2.3 Simulation and Experiment Result ................................................................. 61 5.3 Active Vibration Control ........................................................................................ 64 5.3.1 Velocity Loop Design ...................................................................................... 65 5.3.2 Position Loop Design ...................................................................................... 72 5.3.3 Modification in Experimental Implementation ............................................. 76 5.3.4 Simulation and Experiment Result ................................................................. 88 5.4 Summary ............................................................................................................. 110 Chapter 6. Conclusion ................................................................................................. 112 6.1 Summary ............................................................................................................. 112 6.2 Future Work ........................................................................................................ 113 Bibliography .................................................................................................................... 114 Appendix A. Calculations ........................................................................................... 116 A.1 Unbiased Least Square ........................................................................................ 116 A.2 State Space Pole Placement Controller Design ................................................... 118 A.3 State Space Pole Placement Controller with Integrator Design ......................... 121 A.4 Kalman Filter ....................................................................................................... 121 Appendix B. Oscillator Brackets ................................................................................. 124 Appendix C. System Parameters ............................................................................... 127    v List of Tables Table 3.1: Dimensions of the Aluminium SDOF Oscillator in the Experimental Setup ..... 13 Table 4.1: Time and Laplace Domain Equivalent Stiffness ............................................... 18 Table 4.2: Amplifier and Motor Gain Constant ................................................................. 23 Table 4.3: Viscous and Coulomb Friction Identification Results....................................... 27 Table 4.4: Motor Mass and Friction Identification Results by Ad-Hoc Technique ........... 28 Table 4.5: Motor Mass and Friction Identification Results by ULS Technique ................. 34 Table 4.6: Oscillator Identification Results by Impact Modal Test ................................... 36 Table 4.7: Motor and Oscillator Identification Results by Sine Sweep Technique with NLS Optimization ..................................................................................................................... 38 Table 4.8: Motor Position Frequency Response Maximum Deviation ............................. 44 Table 5.1: Motor Position of Experimental and Simulated Step Response for Motor Position Control System using Pole Placement Controller with Integrator Summary ..... 62 Table 5.2: Proportional Integral Velocity Controller Parameters ..................................... 67 Table 5.3: Kalman Filter Time Update Process Equations ................................................ 85 Table 5.4: Kalman Filter Measurement Update Process Equations ................................. 85 Table 5.5: Noise Covariance Parameters .......................................................................... 86 Table 5.6: Acceleration Measurement and Kalman Filter Output Summary ................... 87 Table 5.7: Simulated Frequency Response from Motor Position Input Command to Motor Position for Active Vibration Control System in Different Position Controller Proportional Gains Summary ............................................................................................ 89 Table 5.8: Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active Vibration Control System with Different Position Controller Proportional Gains Summary .......................................................................... 90 Table 5.9: Motor Position of Experimental Step Response for Active Vibration Control System Summary .............................................................................................................. 92 Table 5.10: Oscillator Acceleration of Experimental Step Response for Active Vibration Control System Summary ................................................................................................. 93 Table 5.11: Motor Position of Experimental Step Response for Active and Non Active Vibration Control System Summary ................................................................................. 96  vi Table 5.12: Oscillator Acceleration of Experimental Step Response for Active and Non Active Vibration Control System Summary....................................................................... 97 Table 5.13: Frequency Response from Motor Position Input Command to Motor Position for Active and Non Active Vibration Control System Summary ..................................... 102     vii List of Figures Figure 1.1: Implementation Work Flowchart ..................................................................... 2 Figure 2.1: Commanding Shaping Vibration Controller with Notch Filter .......................... 5 Figure 2.2: Active Vibration Controller with Resonant Controller Design .......................... 6 Figure 2.3: Active Vibration Controller with Hโˆž Controller Design .................................... 7 Figure 2.4: Active Vibration Controller with Phase Compensator Design .......................... 8 Figure 3.1: SIMODRIVE 611U in Velocity Mode Block Diagram........................................ 10 Figure 3.2: SIMODRIVE 611U in Torque Mode Block Diagram with Position Control ...... 10 Figure 3.3: Siemens Linear Motor and the SIMODRIVE 611U .......................................... 11 Figure 3.4: Heidenhain LIDA18 Linear Encoder ................................................................ 12 Figure 3.5: SDOF Oscillator with Accelerometer Sensor on the Linear Table .................. 14 Figure 3.6: Hardware Configuration Overview ................................................................. 15 Figure 4.1: Single Degree Flexible Drive System Free Body Diagram ............................... 17 Figure 4.2: Time Domain Flexible Drive Model ................................................................. 18 Figure 4.3: Time Domain (Left) and Laplace Domain (Right) Equivalent Stiffness Model 1  ........................................................................................................................................... 19 Figure 4.4: Laplace Domain Equivalent Stiffness Model 2 ................................................ 19 Figure 4.5: Laplace Domain Equivalent Stiffness Model 3 ................................................ 20 Figure 4.6: Flexible Drive Dynamics .................................................................................. 22 Figure 4.7: Friction Model [2] ........................................................................................... 24 Figure 4.8: Motor Force Verses Speed Experimental Result ............................................ 26 Figure 4.9: Ad-Hoc Motor Mass Identification ................................................................. 27 Figure 4.10: Motor Position Frequency Response by Ad-Hoc Technique ........................ 28 Figure 4.11: Typical Voltage Input Signal Command for ULS Technique .......................... 33 Figure 4.12: Estimated Parameters by ULS Technique ..................................................... 34 Figure 4.13: Motor Position Frequency Response by ULS Technique .............................. 35 Figure 4.14: Oscillator Acceleration Frequency Response by Impact Modal Test ........... 37  viii Figure 4.15: Motor Position Frequency Response by Sine Sweep Technique with NLS Optimization ..................................................................................................................... 39 Figure 4.16: Oscillator Acceleration Frequency Response by Sine Sweep Technique with NLS Optimization .............................................................................................................. 40 Figure 4.17: Motor Position Frequency Response Comparison ....................................... 43 Figure 4.18: Oscillator Acceleration Frequency Response Comparison ........................... 45 Figure 4.19: Motor Position Frequency Response by Sine Sweep Technique with NLS Optimization with and without Order Reduction ............................................................. 46 Figure 4.20: Oscillator Acceleration Frequency Response by Sine Sweep Technique with NLS Optimization with and without Order Reduction ...................................................... 46 Figure 4.21: Motor Position Frequency Response by ULS Technique .............................. 48 Figure 4.22: Oscillator Acceleration Frequency Response by Sine Sweep Technique with NLS Optimization .............................................................................................................. 48 Figure 5.1: Open Loop System State Space Model ........................................................... 52 Figure 5.2: Time Update Observer Diagram ..................................................................... 56 Figure 5.3: Measurement Update Observer Diagram ...................................................... 60 Figure 5.4: Pole Placement Controller with Integrator and Measurement Update Observer ............................................................................................................................ 61 Figure 5.5: Motor Position of Experimental and Simulated Step Response for Motor Position Control System using Pole Placement Controller with Integrator ..................... 62 Figure 5.6: Oscillator Acceleration of Experimental Step Response for Motor Position Control System using Pole Placement Controller with Integrator ................................... 64 Figure 5.7: Active Vibration Controller with Phase Compensator Design ........................ 65 Figure 5.8: Open Loop System .......................................................................................... 66 Figure 5.9: Open Loop Motor Velocity (Left) and Oscillator Velocity (Right) Simulated Frequency Response ......................................................................................................... 66 Figure 5.10: Indirect Velocity Loop System ...................................................................... 67 Figure 5.11: Indirect Velocity Loop Motor Velocity (Left) and Oscillator Velocity (Right) Simulated Frequency Response ........................................................................................ 68 Figure 5.12: Direct Velocity Loop System with Phase Compensator ................................ 70  ix Figure 5.13: Direct and Indirect Velocity Loop Motor Velocity (Left) and Oscillator Velocity (Right) Simulated Frequency Response .............................................................. 71 Figure 5.14: Direct and Indirect Velocity Loop Motor Position (Left) and Oscillator Acceleration (Right) Simulated Frequency Response ....................................................... 71 Figure 5.15: Cascaded Position Loop with Velocity Loop ................................................. 73 Figure 5.16: Direct Velocity Loop Motor Position Simulated Frequency Response ......... 73 Figure 5.17: Motor Position (Left) and Oscillator Acceleration (Right) Simulated Frequency Response for Active and Non Active Vibration Control System ..................... 75 Figure 5.18: Active Control System with Disturbance Input ............................................. 76 Figure 5.19: Motor Position (Left) and Oscillator Acceleration (Right) Simulated Disturbance Frequency Response for Active and Non Active Vibration Control System . 76 Figure 5.20: Original Direct Velocity Loop ........................................................................ 78 Figure 5.21: Modified Direct Velocity Loop ...................................................................... 79 Figure 5.22: Kalman Filter Operation Cycle ...................................................................... 85 Figure 5.23: Accelerometer Measurement and Kalman Filter Output ............................. 87 Figure 5.24: Modified Position Loop with Kalman Filter and Feedback Simplification .... 88 Figure 5.25: Simulated Frequency Response from Motor Position Input Command to Motor Position for Active Vibration Control System with Different Position Controller Proportional Gains ............................................................................................................ 89 Figure 5.26: Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active Vibration Control System with Different Position Controller Proportional Gains ........................................................................................... 90 Figure 5.27: Motor Position of Experimental Step Response for Active Vibration Control System ............................................................................................................................... 92 Table 5.9: Motor Position of Experimental Step Response for Active Vibration Control System Summary .............................................................................................................. 92 Figure 5.28: Oscillator Acceleration of Experimental Step Response for Active Vibration Control System .................................................................................................................. 93 Figure 5.29: Position Loop for Active Vibration Control System ...................................... 95 Figure 5.30: Motor Position of Experimental Step Response for Active and Non Active Vibration Control System .................................................................................................. 96  x Figure 5.31: Oscillator Acceleration of Experimental Step Response for Active and Non Active Vibration Control System ....................................................................................... 97 Figure 5.32: Control Input of Experimental Step Response for Active and Non Active Vibration Control .............................................................................................................. 98 Figure 5.33: Motor Position of Experimental Disturbance Rejection Response for Active and Non Active Vibration Control ..................................................................................... 99 Figure 5.34: Oscillator Acceleration of Experimental Disturbance Rejection Response for Active and Non Active Vibration Control ........................................................................ 100 Figure 5.35: Control Input of Experimental Disturbance Rejection Response for Active and Non Active Vibration Control ................................................................................... 101 Figure 5.36: Experimental and Simulated Frequency Response from Motor Position Input Command to Motor Position for Non Active Vibration Control System ........................ 103 Figure 5.37: Experimental and Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Non Active Vibration Control System ............ 103 Figure 5.38: Experimental and Simulated Frequency Response from Motor Position Input Command to Motor Position for Active Vibration Control System ................................ 104 Figure 5.39: Experimental and Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active Vibration Control System .................... 104 Figure 5.40: Experimental Frequency Response from Motor Position Input Command to Motor Position for Active and Non Active Vibration Control System ............................ 105 Figure 5.41: Experimental Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active and Non Active Vibration Control System ................ 105 Figure 5.43: Experimental and Simulated Frequency Response from Disturbance Force at the Oscillator to Oscillator Acceleration for Non Active Vibration Control System ....... 107 Figure 5.44: Experimental and Simulated Frequency Response from Disturbance Force at the Oscillator to Motor Position for Active Vibration Control System ........................... 108 Figure 5.45: Experimental and Simulated Frequency Response from Disturbance Force at the Oscillator to Oscillator Acceleration for Active Vibration Control System ............... 108 Figure 5.46: Experimental Frequency Response from Disturbance Force at the Oscillator to Motor Position for Active and Non Active Vibration Control System ........................ 109 Figure 5.47: Experimental Frequency Response from Disturbance Force at the Oscillator to Oscillator Acceleration for Active and Non Active Vibration Control System ............ 109  xi Figure B.1: Oscillator Brackets Model ............................................................................. 124 Figure B.2: Oscillator Brackets Drawing .......................................................................... 125 Figure B.3: Oscillator Brackets Measurements ............................................................... 126     xii Nomenclature List of Symbols ๐ด State Space Model Dynamics Matrix ๐ต State Space Model Input Matrix ๐‘ Viscous Damping ๐ถ State Space Model Output Matrix ๐ถ๐ธ System Characteristic Equation ๐ท Feed Through Matrix ๐‘‘๐‘“  Disturbance Force Converted to Voltage Equivalent (V) ๐ธ Expected Value Computation ๐น๐‘  Cutting Force (N) ๐น๐‘‘  Disturbance Force (N) ๐น๐‘ก  Motor Force (N) ๐ผ Identity Matrix ๐›ซ Pole Placement Controller Parameters ๐พ1 First Gain Parameter to be Identified through NLS Optimization ๐พ2 Second Gain Parameters to be Identified through NLS  Optimization ๐พ๐‘ก  Motor Force Constant (N/A) or Motor Torque Constant (Nm/A)  ๐พ๐‘Ž  Amplifier Gain Constant (A/V) ๐›ซ๐‘–  Integral Gain ๐พ๐‘ข  Gain Scaling in Unbiased LS Optimization ๐พ๐‘ฃ  Motor Gain per Motor Mass in Continuous Time Domain  xiii ๐พ๐‘ฃ๐‘‘  Motor Gain per Motor Mass in Discrete Time Domain ๐‘˜ Discrete Time Step ๐‘˜2 Oscillator Stiffness (N/m) ๐‘š๐‘Ž  Oscillator Mass (kg) ๐‘š๐‘ก  Motor Mass (kg) ๐‘ Number of Measurements ๐‘๐‘Ÿ  Reference Gain ๐‘๐‘‰ Negative Velocity Function ๐‘‚ Observerability Matrix ๐‘‚๐‘†% Percentage of Overshoot ๐‘ƒ๐‘‰ Positive Velocity Function ๐‘„ Process Noise Covariance ๐‘ Motor Pole ๐‘ƒ๐‘ฃ  Viscous Friction per Motor Mass in Continuous Time Domain ๐‘ƒ๐‘ฃ๐‘‘  Viscous Friction per Motor Mass in Discrete Time Domain ๐‘  Laplace/Continuous Time Domain Transfer Function ๐‘‡๐‘๐‘’๐‘Ž๐‘˜  Peak Time (sec) ๐‘‡๐‘   Sample Time (sec) ๐‘‡๐‘ ๐‘’๐‘ก  Settling Time (sec) ๐‘‰๐‘–๐‘›  Voltage Input (V) ๐‘‹ State Vector ๐‘‹1 Motor State Vector ๐‘ฅ1 Motor Position (m)  xiv ๐‘ฅ2 Oscillator  Position (m) ๐‘ฅ 2 Oscillator Acceleration (m/sec 2) ๐‘… Measurement Noise Covariance ๐‘…๐‘ค  Process Noise Matrix ๐‘…๐‘ฃ  Measurement Noise Matrix ๐‘Ÿ Reference Input ๐‘ข Control Input ๐‘‰ Cost Function in ULS Optimization ๐‘Œ Output Vector in ULS Optimization ๐‘ฆ State Space Model Output ๐‘ง Discrete Time Transfer Function ๐›ผ Equation Coefficient ๐œ Damping Ratio ๐œŽ Sigma Function ๐œƒ Parameters to be Identified in ULS Optimization ๐œ‡ Coulomb Friction ๐œ‡๐‘˜  Coulomb Friction Average ฮฆ Regressor Matrix in ULS Optimization ๐œ”0 Resonant Frequency ๐œ”๐‘›  Natural Frequency     xv List of Scripts 1 Motor Parameters 2 Oscillator Parameters ๐‘‘ Parameters to be Identified through NLS Optimization ๐‘– Integral State ๐‘˜ + 1|๐‘˜ a prori state ๐‘˜|๐‘˜ a posteri state  ๐‘š Measurement Update Process ๐‘› New Parameters (MS) ๐‘  System Parameters (SS) ๐‘  Laplace/Continuous Time Domain (MS) ๐‘ ๐‘‘ Desired System ๐‘ ๐‘  Steady State T Matrix Transpose Operation ๐‘ก Time Update Process ๐‘œ Observer ๐‘œ Desired Observer ๐‘ง Discrete Time Domain State (MS)     xvi List of Notation // Parallel Configuration + Positive Direction - Negative Direction ^ Estimated State ~ Error State     xvii List of Abbreviations 611U Siemens SIMODRIVE 611 Universal Module, Control Motor Driver CEL Control Engineering Laboratory CNC Computer Numerical Control FRF Frequency Response Function DS1103 dSPACE DS1103 Single Board Hardware PPC Controller Board DC Direct Current LHS Left Hand Side LS Least Square MAL Manufacturing Automation Laboratory MS Matrix State MU Measurement Update NLS Nonlinear Least Square NLT Negative Loop Transmission PC Personal Computer RHS Right Hand Side SDOF Single Degree of Freedom SS Scalar State TF Transfer Function TU Time Update     xviii Acknowledgments I wish to express my sincere gratitude to my research supervisors Dr. Yusuf Altintas and Dr. Ryozo Nagamune for their valuable instruction, guidance and support which they have provided throughout my research and coursework. I truly appreciate the great opportunity that they have offered me with. Working on this research project is a valuable experience. I am able to make practical uses of the knowledge that I have learned in class and further strengthen my understanding in the subject.  I also like to thank all members of the Manufacturing Automation Lab, especially to Ehsan Azadi Yazdi, Shixun Fan, Xiaoliang Jin, Anthony Siu and Bernhard Nimmervoll. They made their help available and provided me with many technical advices throughout my research work. Their contributions made the completion of this research work successful. Finally, I want to acknowledge Natural Science and Engineering Research Council of Canadian Network for Research and Innovation in Machining Technology (NSERC CANRIMT) for sponsoring this research work.  Chapter 1. | Introduction 1 Chapter 1. Introduction Precision positioning of tables in machine tools and other servo driven machines is essential for accurate production of parts or measurement of objects using computer control technology. When the mechanical structure of the drive is excited due to high accelerations and disturbance forces, it vibrates, leaving poor surface finish or incorrect measurements. This thesis presents modelling and position control of feed drives with rigid and flexible structures. The application of the proposed methods are rather wide, and can be found in machine tools, material handling, printing and assembly operations. The thesis covers the system modelling, parameter identification, positioning control techniques of a feed drive and active vibration control on a flexible feed drive. The flowchart shown in Figure 1.1 provides an overview of the implementation work. In Chapter 2, a literature survey of past researches on vibration control is presented. Chapter 3 provides the overview of the real time control system implementation and instrumentation, and the experimental setup. A theoretical model and transfer functions of a flexible feed drive system are presented in Chapter 4. Different approaches including an ad-hoc technique, unbiased linear least square technique, impact modal technique, and sine sweep technique with NLS optimization are applied to identify the system parameters in order to select a most suitable model to represent the actual system. Chapter 5 explains the approach to design a position controller through the pole placement technique and a vibration control strategy to suppress the vibrations on the flexible drive. This thesis is concluded in Chapter 6 with a summary of work implemented and recommended future work.  Chapter 1. | Introduction 2 Chapter 3: Experimental Setup Chapter 4.3: Parameters Identification Chapter 5.2: Positioning Controller Design Chapter 5.3: Active Vibration Controller Design Chapter 4.2: System Modelling  Figure 1.1: Implementation Work Flowchart  Chapter 2. | Literature Survey 3 Chapter 2. Literature Survey 2.1 Chapter Overview The focus of this chapter is to review some of the past work by other researchers, which bear the relevance to the areas addressed in this thesis. Section 2.2 reviews the modelling and identification of flexible feed drives, which includes the components of the motor drive and the flexible structure. The detailed technical explanations of the modelling and identification methods published in the literature are presented in Chapter 4. In Section 2.3, the literature on various vibration control strategies are presented.  2.2 Identification of Drive Dynamics Under the increasing demand for more accurate high performance motion systems, numerous system identification techniques appropriate for control purpose of mechanical systems have been proposed in the literatures. Erkorkmaz et al. [2] proposed the use of unbiased estimation of inertia and viscous friction in feed drives. This approach uses the least squares estimation method, and takes account for the presence of Coulomb friction in its formulation. The results from this estimation process are the estimated values for the combined inertia, viscous friction, and dynamic Coulomb friction of the system. In comparison to the simple least square technique, the unbiased least square technique should be more accurate. The other useful dynamics identification approaches suggested by Altintas [1] include the sine sweep and impact modal testing techniques. The sine sweep technique is conducted to excite the system by commanding the open loop feed drive with a sinusoidal input over a wide frequency band containing the system natural modes. The ratio of the structure displacement/acceleration and the exerted force magnitudes, and the phase of the structure displacement/acceleration relative to the force are measured at each frequency. Then the measurements are processed through a parametric fit Chapter 2. | Literature Survey 4 algorithm for modal analysis. While sine sweep technique can deliver excitation force at desired frequency and amplitude, this approach is more time consuming to set up. Alternatively, an impact hammer can be used in exciting the system structure. The impact hammer for modal testing is much easier and quicker to set up, but at the expense of the possible less accurate excitation of the system structure at the desired frequency [1]. For the impact modal testing, the impact force delivered by the hammer can excite a wide frequency range, which contains the system natural modes. The hammer excitation force and resulting vibrations are measured in time domain. Then the measurements are transformed into Fourier domain for modal analysis which is commonly done in modal analysis software. For a comprehensive analysis of the system, all the discussed system identification techniques are applied. The detailed experimental procedures of the different techniques are explained and demonstrated in Chapter 4.  2.3 Vibration Control There are many different vibration control approaches available in the literatures. Vibration control can be broadly divided into two general streams, reference command shaping and active control. Command shaping is a control technique in CNC machines which works by creating a command signal that avoids exciting the system structure natural frequency in order to minimize the vibrations or creates a command signal that cancels its own vibration. Kataoka et al. [15][16] proposed a notch filter to remove the system natural frequency component in the command input to avoid exciting the structure. Most notch filters are known to cause unwanted phase delays. Whereas, the proposed notch filter minimizes the phase delay in the command input with an additional phase compensator gain as shown in Figure 2.1. This approach is a feed forward controller which does not ensure stability of the system.  Chapter 2. | Literature Survey 5  Figure 2.1: Commanding Shaping Vibration Controller with Notch Filter  The common shortcoming of the command shaping method is its vulnerability to reject any disturbance excitation since the control input is designed to only suppress vibrations caused by its reference command. Also, it is very sensitive to the uncertainties in the system model, such as the natural frequency. In comparison, the active vibration control works as a feedback control system, in which the feedback can generate an appropriate compensational control effort to suppress any vibration. Thus, the active vibration control is considered to be more reliable than reference command shaping in vibration control.    Chapter 2. | Literature Survey 6 For active vibration control design, Mahmood et al. [17] has proposed the use of a resonant controller at the flexible structure feedback loop. The control scheme consists of two negative feedback loops as shown in Figure 2.2. The inner loop is designed to increase the damping of the flexible structure and the outer loop provides precise positioning. The proposed approach is designed to suppress vibrations for multimode systems. However, the use of a resonant controller can be insufficient for active vibration control. The system dynamics, specifically the system natural frequency, varies along the feed drive position. A single resonant frequency is selected in the resonant controller to correspond with one vibration mode. Thus, as the system natural frequency varies, the resonant controller cannot ensure high stability robustness. A poor selection of the resonant controller for active control can easily drive the system with more violent vibrations than the system without active control. Moreover, another disadvantage for this proposed control scheme is that leads to a much higher order system due to the resonant controller at the inner loop. Hence, the resulting high order system becomes more difficult to tune at the outer loop.   Figure 2.2: Active Vibration Controller with Resonant Controller Design  Chapter 2. | Literature Survey 7 A different active vibration control approach proposed by Verscheure et al. [10] where Hโˆž controller is used. This control scheme provides the system with good stability robustness and performance robustness. It combines vibration and motion control in a high and low authority control structure as shown in Figure 2.3. The high authority motion controller is built around the system with a low authority vibration controller. The purpose of the vibration controller is to assure that the acceleration of the flexible structure tracks the reference acceleration, which is taken to be zero. The motion controller is a simple controller used to control the position of the motor and to track a reference position. However, the proposed weighting function for the Hโˆž controller appears to be quite specific to their experimental setup, and is not suitable for other systems.   Figure 2.3: Active Vibration Controller with Hโˆž Controller Design Low Authority Controller (Dash Dot) High Authority Controller (Dash)    Chapter 2. | Literature Survey 8 Dietmair et al. [19] proposed the use of a phase compensator to form a damping network at the flexible drive velocity loop as shown in Figure 2.4. This approach is quite similar to other proposed active vibration controller designs, which consists of two control loop. The inner loop is designed for compensating the flexible structure phase delay at the resonant frequency, which effectively adds damping to the flexible structure. The outer loop is designed to track the position reference command. This proposed control strategy is favoured for its simplicity, and can be easily applied to any flexible systems. Therefore, this active phase compensator strategy is further investigated.   Figure 2.4: Active Vibration Controller with Phase Compensator Design   Chapter 3. | Experimental Setup 9 Chapter 3. Experimental Setup 3.1 Chapter Overview This chapter presents the experimental setup used in the thesis. An overview of the real time control system implementation and instrumentation are given in Section 3.2. In Section 3.3, the details on hardware components and its setup, including the linear motor drive and oscillator, for replicating the machine tool vibration, and the use of available sensors measuring the system performance are given.  3.2 Control System Setup This section provides an overview of the real time controlled system in the experimental setup. An external PC is set up for controller design and provides access to dSPACE DS1103 controller board through ControlDesk software. The Siemens SIMODRIVE 611U board, a control motor driver, can be operated in either the velocity or torque operating mode as it commands its power module to send in the appropriate amount of current to operate the motor drive. When operating in the velocity mode as shown in Figure 3.1, the reference command is sent through the DS1103 controller board to the SIMODRIVE 611U drive module. The system is controlled by the SIMODRIVE 611U onboard velocity controller to ensure a constant motor velocity output which is measured by the embedded motor encoder. For custom design positioning and active vibration controller implementation through the DS1103, the SIMODRIVE 611U board should be operating in torque mode. It is because the SIMODRIVE 611U onboard current controller regulates a constant torque (force) output from the motor. The linear encoder measurement signal is used as a feedback to the custom design controller on the DS1103 controller board to form a closed control loop as shown in Figure 3.2. The experimental setup was prepared by a former M.Eng. candidate in MAL and the details can be found in his M.Eng. project report [21].  Chapter 3. | Experimental Setup 10  Figure 3.1: SIMODRIVE 611U in Velocity Mode Block Diagram   Figure 3.2: SIMODRIVE 611U in Torque Mode Block Diagram with Position Control  3.3 System Hardware Setup 3.3.1 Linear Motor A linear electric motor has its stator and rotor lay out in parallel so the motor can produce a linear force along its length. A linear motor drive has fewer moving components than conventional linear actuator; hence it can achieve high speeds and accelerations with minimal backlash and friction. However, the absence of gear reduction between the linear motor and the table makes linear drives sensitive to the applied load, such as carrying load, cutting force or disturbance force on the motor.  The singled-sided Siemens 1FN1 series linear motor mounted on the Siemens linear table setup is used in this experimental setup. Siemens 1FN1 linear motor is a permanent magnet motor which is capable of maximum thrust of 1720 N, maximum acceleration of 2 g (19.6 m/s2) and correspondingly a maximum free load velocity of 200 m/min. This linear motor is considered as a high force motor at the industrial standard. Chapter 3. | Experimental Setup 11 To ensure safety and protection of the movable part, several rubber dampers are installed at both ends of the linear guide rail. Temperature monitoring circuits are integrated in the motor for protecting the system against impermissible high thermal load. During operation, water of 20ยฐC temperature at 4 L/min flow is directed into the cooling system to remove the heat generated by the motor. Under the rated conditions, the cooling system can remove up to 90% of the generated heat. Another safety feature installed on this linear motor setup is the Balluff proximity sensor limit switch pair. At each end of the guide rail, a limit switch is installed to detect the motor. Once the motor is detected by the limit switch at the end of the guide rail, this normally closed limit switch immediately sends a signal to the SIMODRIVE 611U board deactivating the linear motor drive. This can avoid the motor from launching to the end of the guide rail and damaging the linear motor and its supporting structure.   Figure 3.3: Siemens Linear Motor and the SIMODRIVE 611U Chapter 3. | Experimental Setup 12 3.3.2 Linear Encoder The Heidenhain LIDA 18 exposed linear encoder is used to measure the displacement of the linear motor. The linear encoder consists of a scale tape and a scanning head that operate without mechanical contact. The output of the encoder is a pair of 1 V peak-to- peak sinusoidal analog signals which are 90 deg out of phase from each other. The measured signal is digitized into discrete format which can be used in the DS1103 control module. The linear encoder feedback signal is further enhanced through the use of a Heidenhain IBV606 interpolation box. The resolution of the linear encoder with the quadrature reading after interpolation is 5 micron [21].   Figure 3.4: Heidenhain LIDA18 Linear Encoder  3.3.3 Single Degree Oscillator The machine tool on a milling machine together behaves like a flexible feed drive system. As the motor position moves, it induces a force to the machine tool which the machine tool vibrates. To model and illustrate the active damping of the flexible feed drive system, a thin bar is used as a single degree of freedom (SDOF) oscillator and it is mounted on the linear motor moving platform. The thin bar structure as a SDOF oscillator is selected to ensure the structure flexibility only along the motor drive axis. Using a thin bar structure would also guarantee that its resonance is at low frequency region and the structure has a low mechanical damping property. For demonstration purpose, the oscillation and the damping of the oscillation can be easily observed. The thin bar SDOF oscillator is secured on the linear motor moving platform with a set of Chapter 3. | Experimental Setup 13 custom designed brackets. As the motor moves, the thin bar should experience noticeable vibrations along the motor drive axis.  Height (mm) Width (mm) Thickness (mm) Natural Frequency (Hz) 328.5 14.7 1.6 8.8 Table 3.1: Dimensions of the Aluminium SDOF Oscillator in the Experimental Setup  3.3.4 Accelerometer Sensor Kistler 8702B500 Accelerometer has a measureable range of ยฑ500 g (4900 m/s2). For measuring the acceleration of the SDOF oscillator, the accelerometer is bolted on at the top end of flexible structure as shown in Figure 3.5. The accelerometer output signal is amplified through the signal amplifier before it is fed into the DS1103 control module. The accelerometer has been calibrated with a calibration exciter, Bruel & Kjaer Type 4294, so that the amplifier transducer sensitivity conversion is set to 14.4 mV/g for this experimental setup. In industrial machine tools, either Ferrari sensor [1] is used for direct measurement or double derivative of encoder signals are used for indirect measurement of acceleration of the drives.  Chapter 3. | Experimental Setup 14  Figure 3.5: SDOF Oscillator with Accelerometer Sensor on the Linear Table  3.4 Summary This section summarised the work discussed throughout the chapter. In Section 3.2, the control system overview, including the velocity and torque operating mode, is explained. In Section 3.3, the system hardware including the linear motor, linear encoder, SDOF oscillator and accelerometer sensor are described in detail. The Siemens linear motor is configured through the SIMODRIVE 611U module. The control command is processed through the DS1103 control board. A customised designed SDOF oscillator is mounted on the linear motor to replicate a flexible feed drive system. The linear encoder is used for measuring the motor position and the accelerometer is used for measuring the SDOF oscillator acceleration. The hardware configuration overview is provided in Figure 3.6.  Chapter 3. | Experimental Setup 15  Figure 3.6: Hardware Configuration Overview Controller (Dash Dot) Flexible Feed Drive System (Dash)  Chapter 4. | Identification of Feed Drive Dynamics 16 Chapter 4. Identification of Feed Drive Dynamics 4.1 Chapter Overview The focus of Chapter 4 is on the system characteristic. The theoretical system model transfer function for a flexible motor drive is derived in Section 4.2. In Section 4.3, experimental procedures of the different parameter identification techniques are discussed. The discussion is provided in Section 4.3.5 to compare these identification techniques. The most suitable identified model is applied to represent the actual system for controller design.  4.2 System Model The mechanical system of a single axis flexible drive is commonly modelled as a two degree of freedom mass, spring and damper system. In this section, the theoretical model is derived to obtain the two system transfer functions of interest. The first transfer function is between voltage input, ๐‘‰๐‘–๐‘› , and motor position, ๐‘ฅ1. The second transfer function is between voltage signal, ๐‘‰๐‘–๐‘› , and oscillator acceleration, ๐‘ฅ 2. This is because the motor position and oscillator acceleration are measurable quantities through the linear encoder and accelerometer, respectively. Then the experimentally measured system frequency responses can be compared directly with the derived transfer functions.  Chapter 4. | Identification of Feed Drive Dynamics 17 Fd b1 Ft Fc x1 mt ma x2 Fd Ft Fc x1 mt ma x2 k2 b2 b1  Figure 4.1: Single Degree Flexible Drive System Free Body Diagram ๐‘ฅ1: Motor Position ๐‘ฅ2: Oscillator Position ๐น๐‘ก : Motor Force  Consider the two degree of freedom mass, spring and damper mechanical system as shown in Figure 4.1. The driving motor force, ๐น๐‘ก , is moving the motor mass, ๐‘š๐‘ก , against the viscous friction force which is represented as a damper, ๐‘1, on the motor. The motor mass, ๐‘š๐‘ก , is also subjected to disturbance force, ๐น๐‘‘ . The SDOF oscillator is attached on the motor mass. The mechanical properties of SDOF oscillator is lumped to the top which includes the oscillator mass, ๐‘š๐‘Ž , damper, ๐‘2 , and stiffness, ๐‘˜2 . The SDOF oscillator is also under disturbance force, ๐น๐‘ , which may be considered as the cutting force on the machine tool during machining. To simplify the model analysis, the motor disturbance force, ๐น๐‘‘ , and oscillator disturbance force, ๐น๐‘ , are assumed to be zero. The system transfer functions of interest can be derived by Laplace domain equivalent stiffness model. First obtain the time domain system stiffness model and then transform the model to Laplace domain. Each stiffness element in the Laplace domain model can Chapter 4. | Identification of Feed Drive Dynamics 18 be treated as a spring element in the time domain. The following table provides the time domain and Laplace domain equivalent stiffness presentation for each element.   Time Domain Laplace Domain Motor Mass ๐‘š๐‘ก  ๐‘š๐‘ก โˆ™ ๐‘  2 Motor Damper ๐‘1 ๐‘1 โˆ™ ๐‘  Oscillator Mass ๐‘š๐‘Ž  ๐‘š๐‘Ž โˆ™ ๐‘  2 Oscillator Damper ๐‘2 ๐‘2 โˆ™ ๐‘  Oscillator Stiffness ๐‘˜2 ๐‘˜2 Table 4.1: Time and Laplace Domain Equivalent Stiffness  b1 mt b2 k2 x2x1 Ft ma  Figure 4.2: Time Domain Flexible Drive Model  The mass component in the model is considered to be attached to ground by a spring with a magnitude of its respective mass magnitude. Then the time domain stiffness is replaced with the Laplace domain equivalent stiffness as shown in Figure 4.3.  Chapter 4. | Identification of Feed Drive Dynamics 19 b1 mt b2 k2 ma x2x1 Ft b1 s mt s2 b2 s k2 ma s2 x2x1 Ft  Figure 4.3: Time Domain (Left) and Laplace Domain (Right) Equivalent Stiffness Model 1  The motor mass, ๐‘š๐‘ก , is in parallel with the motor damping, ๐‘1. The oscillator stiffness, ๐‘˜2, is in parallel with the oscillator damping, ๐‘2. The parallel arrangement of the stiffness is the sum of each stiffness component. Then the result of this summation is shown in Figure 4.4.  ma s2  k2 + b2 smt s2 + b1 s x2x1 Ft  Figure 4.4: Laplace Domain Equivalent Stiffness Model 2  The model in Figure 4.4 can be illustrated differently as shown in Figure 4.5, where the motor position, ๐‘ฅ1, is switched with the oscillator position, ๐‘ฅ2, such that obtaining the transfer functions between the motor force, ๐น๐‘ก , and the motor position , ๐‘ฅ1, becomes more obvious. Now, the stiffness model becomes the longer spring at the top and it is at parallel with the two shorter springs arranged in series. Then the motor force is applied at the end. Chapter 4. | Identification of Feed Drive Dynamics 20 mt s2 + b1 s k2 + b2 sma s2 x1x2 Ft  Figure 4.5: Laplace Domain Equivalent Stiffness Model 3  The transfer function between the motor force and the motor position has the following form as shown in Eq.(4.1).  ๐‘ฅ1 ๐น๐‘ก = ๐‘š๐‘Ž โˆ™ ๐‘  2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2 โˆ™  ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  +  ๐‘š๐‘Ž โˆ™ ๐‘ 2 โˆ™  ๐‘˜2 + ๐‘2 โˆ™ ๐‘   (4.1)  Since the Laplace domain equivalent stiffness of the oscillator mass, ๐‘š๐‘Ž , and damping, ๐‘2, magnitudes are expected to be relatively small compared to the stiffness magnitude, ๐‘˜2, within the frequency of interest between 1 Hz to 35 Hz, the stiffness magnitude, ๐‘˜2, should be the dominating term. This transfer function can be further simplified as shown in Eq.(4.2).   ๐‘˜2 โ‰ซ  ๐‘š๐‘Ž โˆ™ ๐‘  2    ๐‘˜2 โ‰ซ  ๐‘2 โˆ™ ๐‘     ๐‘ฅ1 ๐น๐‘ก = ๐‘˜2  ๐‘˜2 โˆ™  ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  = 1 ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘   (4.2)  Chapter 4. | Identification of Feed Drive Dynamics 21 The transfer function between the motor force, ๐น๐‘ก , and the oscillator position, ๐‘ฅ2, can be obtained by rewriting the motor position, ๐‘ฅ1, term with respect to the oscillator position, ๐‘ฅ2, in Eq.(4.3).  ๐‘ฅ1 = ๐‘š๐‘Ž โˆ™ ๐‘  2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2 ๐‘2 โˆ™ ๐‘  + ๐‘˜2 โˆ™ ๐‘ฅ2 (4.3)  The relationship between oscillator position and the motor position is expressed in the following form as shown in Eq.(4.4).  ๐‘ฅ2 ๐‘ฅ1 = ๐‘2 โˆ™ ๐‘  + ๐‘˜2 ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.4)  Since the magnitude of the oscillator damping coefficient, ๐‘2, is considered to be much smaller than its stiffness magnitude, ๐‘˜2, the numerator of Eq.(4.4) can be considered as a constant gain at the low frequency region of interested.   ๐‘˜2 โ‰ซ  ๐‘2 โˆ™ ๐‘     ๐‘ฅ2 ๐‘ฅ1 = ๐‘˜2 ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.5)  Then the transfer function between the motor force and oscillator position is given by Eq.(4.6).  Chapter 4. | Identification of Feed Drive Dynamics 22 ๐‘ฅ2 ๐น๐‘ก = ๐‘ฅ1 ๐น๐‘ก โˆ™ ๐‘ฅ2 ๐‘ฅ1 = 1 ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™ ๐‘˜2 ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.6)  Recall that the transfer function of interest is between the motor force and the oscillator acceleration. Then this is rewritten as in Eq.(4.7).  ๐‘ฅ 2 ๐น๐‘ก = ๐‘ฅ1 ๐น๐‘ก โˆ™ ๐‘ฅ2 โˆ™ ๐‘  2 ๐‘ฅ1 = ๐‘˜2 โˆ™ ๐‘  2  ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™  ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.7)  Now, the dynamics of a flexible feed drive, including the electrical and mechanical system, can be represented as shown in Figure 4.6.   Figure 4.6: Flexible Drive Dynamics Electrical System (Dash Dot) Mechanical System (Dash)  ๐‘‰๐‘–๐‘›  (V) is the voltage input applied to the current amplifier, denoted with the gain, ๐พ๐‘Ž  (A/V). The amplifier produces the current ๐‘– (A) in the motor armature, resulting in the motor force ๐น๐‘ก  (N). The motor force is directly proportional to the current ๐‘– (A) and motor force constant, ๐พ๐‘ก  (N/A). In addition to the force produced by the current running through the armature, the motor is also subject to a disturbance force, ๐น๐‘‘  (N), which may include Coulomb friction from the guide rail or external applied force. Chapter 4. | Identification of Feed Drive Dynamics 23 It should be noted that the electrical system has a much higher bandwidth than the mechanical system. The high amplifier bandwidth implies that at low frequencies, which is the mechanical system operating range, the electrical system can be approximated by the constant amplifier gain, ๐พ๐‘Ž .  ๐พ๐‘Ž  Amplifier Gain Constant (A/V) ๐พ๐‘ก  Motor Force Gain Constant (N/A) 1.1121 142.00 Table 4.2: Amplifier and Motor Gain Constant  The system can be now rewritten as transfer functions between the input voltage, ๐‘‰๐‘–๐‘› , and the motor position, ๐‘ฅ1, and the oscillator acceleration, ๐‘ฅ 2.  ๐น๐‘ก = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‰๐‘–๐‘›    ๐‘ฅ1 ๐‘‰๐‘–๐‘› = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘   (4.8)  ๐‘ฅ 2 ๐‘‰๐‘–๐‘› = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™ ๐‘˜2 โˆ™ ๐‘  2 ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.9)  4.3 Parameter Identification Through the theoretical model, the motor position and oscillator acceleration transfer function structure are obtained. Now experimental identification is required to extract these transfer function parameters. This section presents the procedures of three techniques for identifying the Coulomb friction, viscous friction and motor mass, and two techniques for identifying the SDOF oscillator parameters.  Chapter 4. | Identification of Feed Drive Dynamics 24 There are many types of frictional forces that arise between contacting surfaces, the linear type being Coulomb and viscous friction, and nonlinear type such as the Stricbeck effect and static friction [2]. Here, only the linear frictions are considered in the identification process.   Figure 4.7: Friction Model [2]  Coulomb friction force needs to be overcome to get an object moving initially and is constant regardless the speed of the moving object. Viscous friction, on the other hand, is proportional to the speed of the moving object. The Coulomb and the viscous friction constants have to be identified for both positive and negative direction of the linear motor. The results from the different identification techniques will be compared and the most suitable model is selected for controller design.  Chapter 4. | Identification of Feed Drive Dynamics 25 4.3.1 Identification of Motor Dynamics by Ad-Hoc Technique An ad-hoc technique is applied to identify the motor system parameters. This technique is favourable for its simplicity and it can generally provide good estimates of the motor mass, viscous and Coulomb friction parameters. Consider the motor drive equation of motion, Eq.(4.10), derived from Figure 4.6.  ๐‘š๐‘ก โˆ™ ๐‘ฅ 1 ๐‘ก = โˆ’๐‘1 โˆ™ ๐‘ฅ 1 ๐‘ก + ๐พ๐‘Ž โˆ™ ๐พ๐‘ก โˆ™ ๐‘‰๐‘–๐‘›  ๐‘ก โˆ’ ๐œ‡๐‘˜  (4.10)  ๐‘š๐‘ก : Motor Mass (kg) ๐‘ฅ 1: Motor Acceleration (m/s 2) ๐‘1: Viscous Friction (Nยทm/s) ๐‘ฅ 1: Motor Velocity (m/s) ๐พ๐‘Ž : Amplifier Gain Constant (A/V) ๐พ๐‘ก : Motor Torque Gain Constant (N/A) ๐‘‰๐‘–๐‘› : Voltage Input (V) ๐œ‡๐‘˜ : Coulomb Friction (N)  When the motor drive is moving at a constant velocity, the motor acceleration is zero. Then the left hand side (LHS) of the equation of motion becomes zero. Since only the linear viscous friction is considered, then different voltage inputs correspond to different motor velocity outputs, and the Coulomb friction is only a force offset. Then the viscous friction is identified by jogging the motor back and forth at constant velocities. The corresponding motor current required to generate the force to move the table is recorded. The results are plotted as a motor force versus speed graph and fitted with a linear function. The linear function slope constant is interpreted as the viscous Chapter 4. | Identification of Feed Drive Dynamics 26 friction coefficient. By interpolation, the fitted linear function Y-intercept corresponds to the friction at zero velocity, which it is interpreted as the Coulomb friction, ๐œ‡๐‘˜ .  To ensure the motor travelling at constant velocities, the SIMODRIVE 611U is selected to operate in the velocity mode. In this parameter identification analysis, only the mean values of the steady state motor current and velocity are processed. This avoids the nonlinear friction effect in the transit measurements coupled into the analyzed data. The friction identification analyzed data fitted with a linear function result is presented in Figure 4.8 and Table 4.3.   Figure 4.8: Motor Force Verses Speed Experimental Result  Chapter 4. | Identification of Feed Drive Dynamics 27 Direction Slope: Viscous Friction (Nยทs/m) Y-Intercept: Coulomb Friction (N) Positive 72.6540 38.5 Negative 72.4609 34.4 Table 4.3: Viscous and Coulomb Friction Identification Results  With the Coulomb and the viscous friction identified, the motor mass, ๐‘š๐‘ก , is the only remaining unknown in the motor drive equation of motion. To identify the motor mass, the motor is driven in open loop with a series of square wave commands at different amplitudes while the velocity is measured by the linear encoder. The measured velocity and corresponding voltage input signal are known variables on right hand side (RHS) of the motor drive equation of motion in. The motor acceleration on left hand side (LHS) of the motor drive equation of motion can be obtained by taking the differentiation of the corresponding velocity measurement. The LHS and RHS of the motor drive equation of motion are graphed. The motor mass, ๐‘š๐‘ก , parameter is adjusted until the LHS coincides with the RHS of the motor drive equation of motion.   Figure 4.9: Ad-Hoc Motor Mass Identification Chapter 4. | Identification of Feed Drive Dynamics 28 Motor Mass, ๐’Ž๐’• (kg) 30.5 Motor Viscous Friction Coefficient, ๐’ƒ๐Ÿ (Nยทs/m) 72.6 Positive Coulomb Friction, ๐+ (N) 38.5 Negative Coulomb Friction, ๐โˆ’ (N) 34.4 Average Coulomb Friction, ๐๐’Œ (N) 36.4 Table 4.4: Motor Mass and Friction Identification Results by Ad-Hoc Technique  The identified result of the motor position frequency response by the ad-hoc technique is plotted against the sine sweep experimental result as shown in Figure 4.10.   Figure 4.10: Motor Position Frequency Response by Ad-Hoc Technique  Chapter 4. | Identification of Feed Drive Dynamics 29 4.3.2 Identification of Motor Dynamics by Unbiased Least Square Technique The unbiased least square (ULS) technique presented by Erkorkmaz et al. [2] is a mathematical approach for identifying the system parameters which includes the motor mass, viscous and Coulomb friction. This approach works similar to a simple least square method which the identified parameters have the sum of the square of the errors minimized but also takes account for the presence of Coulomb friction in its formulation. Recall the motor drive position transfer function and convert it to Laplace domain.  ๐‘š๐‘ก โˆ™ ๐‘ฅ1 ๐‘  โˆ™ ๐‘  2 = โˆ’๐‘1 โˆ™ ๐‘ฅ1 ๐‘  โˆ™ ๐‘  + ๐พ๐‘Ž โˆ™ ๐พ๐‘ก โˆ™ ๐‘‰๐‘–๐‘›  ๐‘ก โˆ’ ๐œ‡๐‘˜  (4.11)  ๐‘ฅ1 ๐‘  = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‰๐‘–๐‘›  ๐‘  โˆ’ ๐œ‡๐‘˜ ๐‘  ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘   (4.12)  To facilitate a disturbance estimation, Coulomb friction is assumed as an act of the input signal disturbance, ๐‘‘๐‘“ , as shown in Eq.(4.13).  ๐œ‡๐‘˜ ๐‘  = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‘๐‘“  (4.13)  Now, the motor position transfer function can be expressed in terms of the voltage input with disturbance input as shown in Eq.(4.14),  ๐‘ฅ1 ๐‘  = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™  ๐‘‰๐‘–๐‘›  ๐‘  โˆ’ ๐‘‘๐‘“ ๐‘    (4.14)  Chapter 4. | Identification of Feed Drive Dynamics 30 ๐‘ฅ1 ๐‘  = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™ ๐‘ข ๐‘   (4.15)  and the motor velocity transfer function as shown in Eq.(4.16).  ๐‘ฅ 1 ๐‘  = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘  + ๐‘1 โˆ™ ๐‘ข ๐‘   (4.16)  The voltage input command is processed through an analog to digital converter, and hence the velocity expression in Eq.(4.16) should be transformed into discrete time domain with a zero order hold at the input stage as shown in Eq.(4.17). The details of the derivations are given in Appendix A.  ๐‘ฅ 1 ๐‘ง = ๐พ๐‘ฃ๐‘‘ ๐‘ง โˆ’ ๐‘ƒ๐‘ฃ๐‘‘ โˆ™ ๐‘ข ๐‘ง  (4.17)  ๐พ๐‘ฃ๐‘‘ = ๐พ๐‘ฃ โˆ’๐‘ƒ๐‘ฃ โˆ™  1โˆ’ ๐‘’๐‘ƒ๐‘ฃโˆ™๐‘‡๐‘      ๐‘ƒ๐‘ฃ๐‘‘ = ๐‘’ ๐‘ƒ๐‘ฃโˆ™๐‘‡๐‘     ๐‘ฅ 1 ๐‘ง + 1 = ๐‘ƒ๐‘ฃ๐‘‘ ๐‘ฅ 1 ๐‘ง + ๐พ๐‘ฃ๐‘‘ โˆ™ ๐‘ข ๐‘ง  (4.18)  ๐‘ง is replaced by ๐‘˜ โˆ™ ๐‘‡๐‘   for each sampling instance. For simplicity, ๐‘˜ โˆ™ ๐‘‡๐‘  is written as ๐‘˜.  ๐‘ฅ 1 ๐‘˜ + 1 = ๐‘ƒ๐‘ฃ๐‘‘ โˆ™ ๐‘ฅ 1 ๐‘˜ + ๐พ๐‘ฃ๐‘‘ โˆ™ ๐‘ข ๐‘˜  (4.19)  Chapter 4. | Identification of Feed Drive Dynamics 31 ๐‘ฅ 1 ๐‘˜ + 1 = ๐‘ƒ๐‘ฃ๐‘‘ โˆ™ ๐‘ฅ 1 ๐‘˜ + ๐พ๐‘ฃ๐‘‘ โˆ™ ๐‘‰๐‘–๐‘›  ๐‘˜ โˆ’ ๐พ๐‘ฃ๐‘‘ โˆ™ ๐‘‘๐‘“  (4.19)  Since Coulomb friction is the only disturbance considered in this model, the disturbance, ๐‘‘๐‘“ , is a function of the motor velocity, ๐‘ฅ 1. This can be represented by the equation as shown in Eq.(4.20).  ๐‘‘๐‘“ ๐œ” ๐‘˜  = ๐‘ƒ๐‘‰ ๐‘ฅ 1 ๐‘˜  โˆ™ ๐‘‘๐‘“ + + ๐‘๐‘‰ ๐‘ฅ 1 ๐‘˜  โˆ™ ๐‘‘๐‘“ โˆ’ (4.20)  ๐‘ƒ๐‘‰ = 1 2 ๐œŽ ๐‘ฅ 1 ๐‘˜  โˆ™  1 + ๐œŽ ๐‘ฅ 1 ๐‘˜    (4.21)  ๐‘๐‘‰ = โˆ’ 1 2 ๐œŽ ๐‘ฅ 1 ๐‘˜  โˆ™  1 โˆ’ ๐œŽ ๐‘ฅ 1 ๐‘˜    (4.22)  ๐œŽ =  0 ๐‘–๐‘“ ๐‘ฅ 1 ๐‘˜ = 0 1 ๐‘–๐‘“ ๐‘ฅ 1 ๐‘˜ > 0 โˆ’1 ๐‘–๐‘“ ๐‘ฅ 1 ๐‘˜ < 0   (4.23)  Now, all output is a function of the previous states. Then all ๐‘ measurement samples are presented in a matrix form as shown in Eq.(4.24).  ๐‘Œ = ฮฆ๐œƒ (4.24)  ๐‘Œ =  ๐‘ฅ 1 2 ๐‘ฅ 1 3 โ‹ฎ ๐‘ฅ 1 ๐‘      ๐œƒ =     ๐‘ƒ๐‘ฃ๐‘‘ ๐พ๐‘ฃ๐‘‘ ๐พ๐‘ฃ๐‘‘ ๐‘‘๐‘“ + ๐พ๐‘ฃ๐‘‘ ๐‘‘๐‘“ โˆ’       Chapter 4. | Identification of Feed Drive Dynamics 32 ฮฆ =  ๐‘ฅ 1 1 ๐‘‰๐‘–๐‘› 1 ๐‘ฅ 1 2 ๐‘‰๐‘–๐‘›  2 โ‹ฎ ๐‘ฅ 1 ๐‘ โˆ’ 1 โ‹ฎ ๐‘‰๐‘–๐‘›  ๐‘ โˆ’ 1 โˆ’๐‘ƒ๐‘‰ 1 โˆ’๐‘๐‘‰ 1 โˆ’๐‘ƒ๐‘‰ 2 โˆ’๐‘๐‘‰ 2 โ‹ฎ โˆ’๐‘ƒ๐‘‰ ๐‘ โˆ’ 1 โ‹ฎ โˆ’๐‘๐‘‰ ๐‘ โˆ’ 1     where ๐‘Œ is the output vector. ฮฆ is the regressor matrix. ๐œƒ is the parameter vector to be estimated. The objective in the parameter estimation process is to find the parameter estimate vector, ๐œƒ , that minimizes the cost function, Eq.(4.25).  ๐‘‰ ๐œƒ  = 1 2  ๐‘Œ โˆ’ ฮฆ๐œƒ  ๐‘‡  ๐‘Œ โˆ’ ฮฆ๐œƒ   (4.25)  The optimal parameter estimate vector to minimize ๐‘‰ ๐œƒ   is given in Eq.(4.26).  ๐œƒ =  ฮฆTฮฆ โˆ’1ฮฆY (4.26)  ๐‘ƒ ๐‘ฃ๐‘‘ = ๐œƒ  1     ๐พ ๐‘ฃ๐‘‘ = ๐œƒ  2     ๐‘‘ ๐‘“ + = ๐œƒ  3 ๐พ ๐‘ฃ๐‘‘     ๐‘‘ ๐‘“ โˆ’ = ๐œƒ  4 ๐พ ๐‘ฃ๐‘‘    The parameter estimate vector for extracting the estimate motor mass, viscous friction and both positive and negative Coulomb friction are shown as the following equations accordingly.  ๐‘š ๐‘ก =  ๐‘ƒ ๐‘ฃ๐‘‘ โˆ’ 1 โˆ™ ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‡๐‘  ๐พ ๐‘ฃ๐‘‘ โˆ™ ln ๐‘ƒ ๐‘ฃ๐‘‘  (4.27)  Chapter 4. | Identification of Feed Drive Dynamics 33 ๐‘ 1 =  1โˆ’ ๐‘ƒ ๐‘ฃ๐‘‘ โˆ™ ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐พ ๐‘ฃ๐‘‘  (4.28)  ๐œ‡ + = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‘ ๐‘“ +  (4.29)  ๐œ‡ โˆ’ = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‘ ๐‘“ โˆ’  (4.30)  To identify the parameters, the amplitude of the input signal has been scaled with different gain factors, ๐พ๐‘ข , such that different velocity ranges are measured. Then the parameters, ๐‘š ๐‘ก , ๐‘ 1, ๐‘‘ ๐‘“ +  and ๐‘‘ ๐‘“ โˆ’  are estimated for the cases where sufficient excitation is achieved to overcome the static friction.   Figure 4.11: Typical Voltage Input Signal Command for ULS Technique  Chapter 4. | Identification of Feed Drive Dynamics 34  Figure 4.12: Estimated Parameters by ULS Technique  By increasing the scaling factor, ๐พ๐‘ข , the estimated parameters converge as shown in Figure 4.12. Thus, the converged values are chosen to be the identified system parameters which are listed in Table 4.5.  Motor Mass, ๐’Ž๐’• (kg) 30.7 Motor Viscous Friction Coefficient, ๐’ƒ๐Ÿ (Nยทs/m) 156.1 Positive Coulomb Friction, ๐+ (N) 45.1 Negative Coulomb Friction, ๐โˆ’ (N) -41.7 Average Coulomb Friction, ๐๐’Œ (N) 43.4 Table 4.5: Motor Mass and Friction Identification Results by ULS Technique  The identified result of the motor position frequency response by ULS technique is plotted against the sine sweep experimental result as shown in Figure 4.13.  Chapter 4. | Identification of Feed Drive Dynamics 35  Figure 4.13: Motor Position Frequency Response by ULS Technique  4.3.3 Identification of Structural Flexibility by Impact Modal Test The parameter identification test for the SDOF oscillator is conducted by an impact modal test. An ideal impact to a structure is a perfect impulse, which has an infinitely small duration, creating a constant amplitude input in frequency domain. This would result in all modes of the system to be excited with equal energy. Thus, the impact by the hammer is intended to reproduce such impulse input. During the identification, the test hammer is required to strike the SDOF oscillator as quickly as possible since the excited frequency range is inversely proportional to the contact time between the test hammer and the SDOF oscillator [1]. The PCB Piezotronics 086C80 impulse hammer with the calibrated sensitivity of 22.2 mV/N is used to exert the impact force. The load cell on the impulse hammer obtains a recording of the impact force and the accelerometer sensor on the SDOF oscillator captures the structure acceleration response. CUTPRO 9.0 software from MAL is used to process the acquired data and conduct the modal analysis Chapter 4. | Identification of Feed Drive Dynamics 36 for estimating the SDOF oscillator modal parameters. The identified modal parameters are listed in Table 4.6.  Oscillator Mass, ๐’Ž๐’‚ (kg) 0.0116 Oscillator Damping Coefficient, ๐’ƒ๐Ÿ (Nยทs/m) 0.0041 Oscillator Stiffness, ๐’Œ๐Ÿ (N/m) 35.5 Table 4.6: Oscillator Identification Results by Impact Modal Test  Recall the oscillator acceleration transfer function in Eq.(4.31) which is between the voltage input and oscillator acceleration. The sine sweep experimental measurement is also taken from the voltage input to oscillator acceleration.  ๐‘ฅ 2 ๐‘‰๐‘–๐‘› = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™ ๐‘˜2 โˆ™ ๐‘  2 ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.31)  In order to compare the SDOF oscillator impact modal test result with the sine sweep experimental result, the dynamics of the motor must also be included. As for now, the ULS identified parameters are used with the impact modal test identified parameters to form the oscillator acceleration transfer function. This oscillator acceleration transfer function by impact modal test is plotted against the sine sweep experiment result as shown in Figure 4.14.  Chapter 4. | Identification of Feed Drive Dynamics 37  Figure 4.14: Oscillator Acceleration Frequency Response by Impact Modal Test  4.3.4 Identification of System Dynamics by Sine Sweep Technique with Nonlinear Least-Square Optimization Sine sweep technique with nonlinear least square (NLS) optimization is conducted by commanding the motor drive in open loop with a sinusoidal command input over a frequency range containing the system natural modes. During the experiments, the voltage input, motor position and oscillator acceleration are recorded to construct the motor position and oscillator acceleration frequency response Bode plot. The NLS optimization modal identification algorithm provided by the UBC Control Engineering Laboratory is applied to fit the experimental data and extract the system parameters. The advantage of this optimization algorithm is to ensure the identified parameters to have a real physical interpretation by restricting the parameters to be within a set that is determined by a priori knowledge. In this experimental setup, for instance, the parameters such as natural frequency and damping ratio must be positive and less than unity, respectively. Recall the structure of transfer functions derived in Section 4.2. Chapter 4. | Identification of Feed Drive Dynamics 38 ๐‘ฅ1 ๐‘‰๐‘–๐‘› = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘   (4.32)  ๐‘ฅ 2 ๐‘‰๐‘–๐‘› = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘  2 ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™ ๐‘˜2 ๐‘š๐‘Ž โˆ™ ๐‘ 2 + ๐‘2 โˆ™ ๐‘  + ๐‘˜2  (4.33)  To process the optimization algorithm, the transfer functions are required to be rewritten into a different presentation form with the elements presented in a series of vibration modes as shown in Eq. (4.34) and Eq.(4.35). Therefore, there are five parameters, ๐พ1, ๐พ2, ๐‘, ๐œ๐‘‘ , and ๐œ”๐‘‘ , to be identified at these transfer functions. The identified parameters are listed in Table 4.7.  ๐‘ฅ1 ๐‘‰๐‘–๐‘› = ๐พ1 ๐‘  โˆ™  ๐‘  + ๐‘๐‘‘  (4.34)  ๐‘ฅ 2 ๐‘‰๐‘–๐‘› = ๐พ2 โˆ™ ๐‘  2 ๐‘  โˆ™  ๐‘  + ๐‘๐‘‘ โˆ™  ๐‘ 2 + 2 โˆ™ ๐œ๐‘‘ โˆ™ ๐œ”๐‘‘ โˆ™ ๐‘  + ๐œ”๐‘‘2  (4.35)  ๐‘ฒ๐Ÿ 5.3394 ๐‘ฒ๐Ÿ 12689 ๐’‘๐’… 10.7555 ๐œป๐’… 0.0089 ๐Ž๐’… 55.2283 Table 4.7: Motor and Oscillator Identification Results by Sine Sweep Technique with NLS Optimization Chapter 4. | Identification of Feed Drive Dynamics 39 The identified results of the motor position and oscillator acceleration frequency response by sine sweep technique with NLS optimization are plotted against the sine sweep experiment result as shown in Figure 4.15 and Figure 4.16, respectively.   Figure 4.15: Motor Position Frequency Response by Sine Sweep Technique with NLS Optimization  Chapter 4. | Identification of Feed Drive Dynamics 40  Figure 4.16: Oscillator Acceleration Frequency Response by Sine Sweep Technique with NLS Optimization  4.3.5 Discussion Different techniques have been applied to identify the motor and oscillator parameters. In this section, the advantages and disadvantages, and challenges involved in conducting each identification technique are discussed. The most suitable identified model is selected for controller design. In Section 4.3.1, the presented ad-hoc technique is a simple identification method that is easy to apply for estimating the motor drive parameters. However, it has a crucial limitation in estimating the motor mass. The motor mass estimation is highly dependent on the userโ€™s interpretation of the experimental data because matching the force graph of the LHS plot and RHS plot is done manually by the user. Thus, the accuracy of the motor mass estimate is limited. By inspection, the motor mass matching variation is not noticeable within ยฑ0.5 kg from the experiment result. Chapter 4. | Identification of Feed Drive Dynamics 41 In Section 4.3.2, the presented ULS technique is a mathematical approach for identifying the motor drive parameters. This technique does not rely on the userโ€™s interpretation of the experimental result as opposed to the ad-hoc technique. During the identification process, the greatest challenge is the preparation of an appropriate voltage input signal command waveform as shown in Figure 4.11. Without a proper input command waveform, it is difficult to have the estimate parameters converged by increasing the scaling factor of the voltage input signal. If the time duration of each command waveform step is set too long, the high voltage input command would send the motor to the end of the guide rail. When the motor reaches the end of the guide rail, the motor triggers the limit switch which disables the amplifier and motor drive. Then the identification process is terminated. If the time duration of each command waveform step is set too short, that is same as sending in a high frequency input, then the estimated parameters become unreliable due to high phase delay in the motor drive. In Section 4.3.3, the impact modal test is applied to identify the SDOF oscillator system parameters. The modal parameter estimations were taken from the average of ten separate measurements. However, it is difficult to strike the impact hammer at the same small area on the oscillator during each measurement. Thus, the impact location variation may deteriorate the accuracy of the estimated parameters. Another drawback with this identification approach is that the impact excitation force is preferably applied at the motor moving platform to replicate the motor force. Instead, the excitation force is applied at the oscillator. It is because when the impact excitation force is applied at the motor moving platform, the test result shows a vibration mode occurring at 3 Hz as well. It seems like the impact force excited parts of the motor which it did not with the sine sweep excitation. Then the identified oscillator dynamics must be coupled with a previously identified motor dynamics in order to form the oscillator acceleration transfer function. Hence, the accuracy of the identified oscillator acceleration transfer function is also depended on the accuracy of the identified motor dynamics. In Section 4.3.4, the implementation of the sine sweep technique with NLS optimization is discussed. This is a practical system identification approach in determining all system Chapter 4. | Identification of Feed Drive Dynamics 42 parameters in an efficient manner. Both the motor and oscillator parameters are acquired simultaneously. However, there are two minor drawbacks. The first drawback is that only finite input frequencies are selected in the sine sweep process. It is rare that the exact natural frequency point is selected to capture the maximum resonance response.  This also leads to the second drawback. More frequency points are usually selected at the proximately of the structure resonant frequency. This causes the NLS optimization algorithm to weight in much heavily at that frequency range for experimental data fitting. Thus, the optimization algorithm identified result matches well mostly at the resonance frequency range and shows higher discrepancy outside that frequency range as shown in Figure 4.15. To improve the identified result, data points should be taken evenly across the frequency of interest to avoid the issue with heavy weight in result in the frequency with more data points. All the motor position and oscillator acceleration identified frequency responses are plotted and compared against the sine sweep experimental result in Figure 4.17 and Figure 4.18, respectively. For the motor dynamics identification, the experimentally measured motor position frequency response magnitude matches well against all the identified magnitude responses. The deviations between the experimentally measured motor position frequency response phase and the identified phase responses are more noticeable. Among the three implemented techniques, the NLS optimized and ULS result display a better match to the experimental result. The NLS optimized result shows a better fit near resonance frequency. However, it has a higher phase mismatch at lower frequency in comparison with the ULS identified result. The ULS identified result matches well against the experimental result only with a small constant phase offset which should provide a more conservative model for controller design. Therefore, the ULS identified model is selected to represent the actual motor dynamics for controller design. It should be noted that this phase offset may due to a number of reasons. One reason may be the effect of the nonlinear friction along the guide way. Another reason for this constant phase offset can be the phase delay introduced by the measuring process. Chapter 4. | Identification of Feed Drive Dynamics 43 Nonetheless, for modelling and controller design simplicity, the small constant phase offset in the motor position frequency response should be acceptable.   Figure 4.17: Motor Position Frequency Response Comparison  The maximum deviations between the experimental and identified frequency responses are obtained by inspection. The detail results are listed in Table 4.8.      Chapter 4. | Identification of Feed Drive Dynamics 44 Implemented Technique Magnitude (dB) Phase (deg) Ad-Hoc +1.9 at 1 Hz -31.7 at 1 Hz ULS -0.2 at 1 Hz -7.3 at 6 Hz NLS Optimization -3.6 at 1 Hz +17.2 at 1 Hz Table 4.8: Motor Position Frequency Response Maximum Deviation  For the oscillator acceleration dynamics identification, the sine sweep experimentally measured oscillator acceleration frequency response matches well against the NLS optimized and impact modal test result. Ideally, the impact modal test is capable of capturing a better frequency response of the SDOF oscillator at resonant since the impact force is capable of exciting the full frequency spectrum across the natural frequency range. However, an ideal impact is difficult to achieve. Also, the accuracy of the oscillator acceleration identified result is also limited by the identified motor dynamics that coupled with. As previously discussed, the identified motor dynamics is not exact. Of the two identified oscillator acceleration frequency responses, the frequency response obtained by the NLS optimized result is chosen for controller design because this identified frequency response shows a consistent match to the experimental result throughout the frequency range of interest.  Chapter 4. | Identification of Feed Drive Dynamics 45  Figure 4.18: Oscillator Acceleration Frequency Response Comparison  Extended work is conducted to validate the assumption that the oscillator stiffness magnitude is much greater than the magnitude of its mass and damping coefficient. With this assumption, the transfer functions are possible for system order reduction simplification as presented in Section 4.2. The motor position and oscillator acceleration transfer function without simplification are processed through the NLS optimization modal identification algorithm. The NLS optimized transfer function with and without simplification results are compared in Figure 4.19 and Figure 4.20. The results show that the assumption for the system order reduction simplification is valid because the system with simplification matches well against the system without simplification. Between the two results, the magnitude difference is no more than half decibel and the phase difference is no more than 2 deg. Chapter 4. | Identification of Feed Drive Dynamics 46  Figure 4.19: Motor Position Frequency Response by Sine Sweep Technique with NLS Optimization with and without Order Reduction   Figure 4.20: Oscillator Acceleration Frequency Response by Sine Sweep Technique with NLS Optimization with and without Order Reduction Chapter 4. | Identification of Feed Drive Dynamics 47 4.4 Summary This section recaps the work discussed throughout the chapter. In Section 4.2, the flexible drive system is modelled as a theoretical two degree of freedom mass, spring and damper system. Through the theoretical model, the structure of the two transfer functions of interest is obtained. The first transfer function is between the voltage input to the motor position. The second transfer function is between the voltage input to the SDOF oscillator acceleration. In Section 4.3, different system identification techniques are presented. The identified frequency responses are analyzed and compared against the experimental results. The motor position frequency response model by ULS technique and the SDOF oscillator acceleration frequency response model by NLS optimization are selected for controller design. The results of the identified motor position and oscillator acceleration frequency response are shown in Figure 4.21 and Figure 4.22, respectively.   Chapter 4. | Identification of Feed Drive Dynamics 48  Figure 4.21: Motor Position Frequency Response by ULS Technique   Figure 4.22: Oscillator Acceleration Frequency Response by Sine Sweep Technique with NLS Optimization Chapter 5. | Digital Control of Feed Drives 49 Chapter 5. Digital Control of Feed Drives 5.1 Chapter Overview Two types of control strategies are presented in this chapter. The first strategy is the positioning control of a rigid body feed drive. The positioning controller and state observer design are shown in Section 5.2. The second strategy is the active vibration control of a flexible feed drive. The active controller design and implementation modifications are presented in Section 5.3. Both types of controlled system performance are validated through simulations and experiments.  5.2 State Space Position Control The feed drive positioning performance is usually specified by the system overshoot percentage, settling time, and/or peak time. In this section, the focus is to design and implement a state space pole placement position controller to achieve the specified performance. The pole placement control strategy assigns the desired poles to the closed loop system to meet the system performance criteria.  5.2.1 State Space Pole Placement Controller with Integrator Design The system characteristic equation defines the system poles which govern the performance and stability of the system. The idea of pole placement control strategy is to design a controller that manipulates the closed loop poles of the system in order to achieve the desired performance. The continuous time characteristic equation of a closed loop position control system is commonly presented as a second order system as shown in Eq.(5.1).  ๐ถ๐ธ ๐‘  =  ๐‘ 2 + 2 โˆ™ ๐œ โˆ™ ๐œ”๐‘› โˆ™ ๐‘  + ๐œ”๐‘› 2 = 0 (5.1)  Chapter 5. | Digital Control of Feed Drives 50 In Eq.(5.1), the system characteristic equation is written in terms of damping ratio, ๐œ, and natural frequency, ๐œ”๐‘› . Then it is convenient to express the damping ratio and natural frequency in the characteristic equation as functions of the desired performance criteria, such as percentage of overshoot, settling time and/or peak time. The following equations provide the analytical expressions between performance criteria, and damping ratio, ๐œ, and/or natural frequency, ๐œ”๐‘›  [7].   ๐‘‚๐‘†% = ๐‘’ โˆ’ ๐œ‹โˆ™๐œ  1โˆ’๐œ2  ร— 100 (5.2)  ๐‘‡๐‘ ๐‘’๐‘ก = โˆ’ ln 0.02 โˆ™  1 โˆ’ ๐œ2 ๐œ โˆ™ ๐œ”๐‘›  (5.3)  ๐‘‡๐‘๐‘’๐‘Ž๐‘˜ = ๐œ‹ ๐œ”๐‘› โˆ™  1 โˆ’ ๐œ2  (5.4)  where ๐‘‚๐‘†% is the percentage of overshoot, ๐‘‡๐‘ ๐‘’๐‘ก  is the settling time in seconds, and ๐‘‡๐‘๐‘’๐‘Ž๐‘˜  is the peak time in seconds. The system desired characteristic equation in continuous time domain, ๐ถ๐ธ๐‘ ๐‘‘ ๐‘  , is expressed in a general form as shown in Eq.(5.5).  ๐ถ๐ธ๐‘ ๐‘‘ ๐‘  =  ๐‘  2 + 2 โˆ™ ๐œ๐‘ ๐‘‘ โˆ™ ๐œ”๐‘ ๐‘‘ โˆ™ ๐‘  + ๐œ”๐‘ ๐‘‘ 2 = 0 (5.5)  where ๐œ”๐‘ ๐‘‘  is the desired natural frequency and ๐œ๐‘ ๐‘‘  is the desired damping ratio. Recall the derived motor equation of motion in Chapter 4. The system is rewritten in the continuous time state space model form shown in Eq.(5.6) and Eq.(5.7). Chapter 5. | Digital Control of Feed Drives 51 ๐‘‹ 1 ๐‘ก = ๐ด1๐‘  โˆ™ ๐‘‹1 ๐‘ก + ๐ต1๐‘  โˆ™ ๐‘ข ๐‘ก  (5.6)  ๐‘ฆ1 ๐‘ก = ๐ถ1๐‘  โˆ™ ๐‘‹1 ๐‘ก + ๐ท1๐‘  โˆ™ ๐‘ข ๐‘ก  (5.7)  where,  ๐‘‹ 1 ๐‘ก =  ๐‘ฅ 1 ๐‘ก ๐‘ฅ 1 ๐‘ก      ๐‘‹1 ๐‘ก =  ๐‘ฅ 1 ๐‘ก ๐‘ฅ1 ๐‘ก      ๐‘ข ๐‘ก = ๐‘‰๐‘–๐‘›  ๐‘ก     ๐‘ฆ1 ๐‘ก =  ๐‘ฅ1 ๐‘ก    ๐ด1๐‘  =  โˆ’ ๐‘1 ๐‘š๐‘ก 0 1 0      ๐ต1๐‘  =  ๐พ๐‘Ž โˆ™ ๐พ๐‘ก ๐‘š๐‘ก 0      ๐ถ1๐‘  =  0 1     ๐ท1๐‘  = 0  Since the command signal is digitised through the analog to digital converter, the continuous time state space model should be discretised. The discrete equivalent state space model as shown in Figure 5.1 includes the open loop system dynamics equation, Eq.(5.8), and output equation, Eq.(5.9). The detail discretisation calculation is available in Appendix A.  ๐‘‹1 ๐‘˜ + 1 = ๐ด1๐‘ง โˆ™ ๐‘‹1 ๐‘˜ + ๐ต1๐‘ง โˆ™ ๐‘ข ๐‘˜  (5.8)  ๐‘ฆ1 ๐‘˜ = ๐ถ1๐‘ง โˆ™ ๐‘‹1 ๐‘˜  (5.9)  Chapter 5. | Digital Control of Feed Drives 52  Figure 5.1: Open Loop System State Space Model ๐‘‹1(๐‘˜): States (Motor Velocity and Position) ๐‘ฆ1(๐‘˜): Measured Output (Motor Position)  For the controller design, it is desired that the controlled system output, ๐‘ฆ1(๐‘˜), tracks the reference input signal, ๐‘Ÿ(๐‘˜), with zero steady state error. The first step is to define a new state ๐‘ฅ1๐‘–(๐‘˜) that is the integral of the error between the reference and the output of the system as shown in Eq.(5.10).  ๐‘ฅ1๐‘– ๐‘˜ + 1 = ๐‘ฅ1๐‘– ๐‘˜ + ๐‘‡๐‘  โˆ™  ๐‘Ÿ ๐‘˜ โˆ’ ๐‘ฆ1 ๐‘˜   (5.10)  It is important that the integration state, ๐‘ฅ1๐‘–(๐‘˜), reaches steady state when the input signal to the integrator goes to zero. Now, there is an extra dynamics equation describing the error in the system. Augment the integration state from Eq.(5.10) into the original dynamics equation to form the new open loop state space model as shown in Eq.(5.11) and Eq.(5.12).  ๐‘‹1๐‘› ๐‘˜ + 1 = ๐ด1๐‘ง ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜ + ๐ต1๐‘ง ๐‘› โˆ™ ๐‘ข ๐‘˜ + ๐ธ1๐‘ง ๐‘› โˆ™ ๐‘Ÿ ๐‘˜  (5.11)  ๐‘ฆ1 ๐‘˜ = ๐ถ1๐‘ง ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜  (5.12) Chapter 5. | Digital Control of Feed Drives 53 where,  ๐‘‹1๐‘› ๐‘˜ =  ๐‘‹1 ๐‘˜ ๐‘ฅ1๐‘– ๐‘˜      ๐ด1๐‘ง ๐‘› =  ๐ด๐‘ง 0 โˆ’๐‘‡๐‘  โˆ™ ๐ถ๐‘ง 1      ๐ต1๐‘ง ๐‘› =  ๐ต๐‘ง 0        ๐ธ1๐‘ง ๐‘› =  0 ๐‘‡      ๐ถ1๐‘ง ๐‘› =  ๐ถ๐‘ง 0   The new open loop state space model, including an integral state, can be stabilized through performing a full state feedback control since both velocity and position state are accessible from measurement in this experimental setup. The control law is defined as shown in Eq.(5.13).  ๐‘ข ๐‘˜ = โˆ’๐›ซ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜ = โˆ’ ๐›ซ ๐›ซ๐‘–  ๐‘‹1 ๐‘˜ ๐‘ฅ1๐‘– ๐‘˜   (5.13)  where ๐พ is the controller gain and ๐›ซ๐‘–  is the integrator gain. In the presence of the system controller, the closed loop dynamics equation is rewritten in Eq.(5.14).  ๐‘‹1๐‘› ๐‘˜ + 1 =  ๐ด1๐‘ง ๐‘› โˆ’ ๐ต1๐‘ง ๐‘› โˆ™ ๐›ซ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜ + ๐ธ1๐‘ง ๐‘› โˆ™ ๐‘Ÿ ๐‘˜  (5.14)  The new discrete time state matrix, ๐ด1๐‘ง ๐‘› , and the new discrete time input matrix, ๐ต1๐‘ง ๐‘› , are fixed system parameters, and the controller, ๐›ซ๐‘› , is the only adjustable parameter. The controller parameter, ๐›ซ๐‘› , is adjusted to manipulate the new system characteristic equation, ๐ถ๐ธ๐‘–๐‘  ๐‘ง , to match with the new system desired characteristic equation, Chapter 5. | Digital Control of Feed Drives 54 ๐ถ๐ธ๐‘–๐‘ ๐‘‘  ๐‘ง , which contains all the system desired closed loop poles including the integrator pole as shown in Eq.(5.15).  det  ๐‘ง โˆ™ ๐ผ โˆ’  ๐ด1๐‘ง ๐‘› โˆ’ ๐ต1๐‘ง ๐‘› โˆ™ ๐›ซ๐‘›  = ๐ถ๐ธ๐‘–๐‘  = ๐ถ๐ธ๐‘–๐‘ ๐‘‘  ๐‘ง = 0 (5.15)  This new system desired characteristic equation, ๐ถ๐ธ๐‘–๐‘ ๐‘‘  ๐‘ง , is simply the discretised form of the pre-defined continuous time system desired characteristic equation, ๐ถ๐ธ๐‘ ๐‘‘ ๐‘   together with the integrator characteristics. The new system desired characteristic equation, ๐ถ๐ธ๐‘–๐‘ ๐‘‘  ๐‘ง , has a general form as shown in Eq.(5.16).  ๐ถ๐ธ๐‘–๐‘ ๐‘‘  ๐‘ง = ๐‘ง 3 + ๐›ผ๐‘ 1 โˆ™ ๐‘ง 2 + ๐›ผ๐‘ 2 โˆ™ ๐‘ง + ๐›ผ๐‘ 3 = 0 (5.16)  The second step to ensure the system output, ๐‘ฆ1 ๐‘˜ , is equal to the reference input, ๐‘Ÿ ๐‘˜ , at steady state, a reference gain, ๐‘๐‘Ÿ , is introduced to scale the reference input [7]. The controlled input, ๐‘ข ๐‘˜ , with the reference gain, ๐‘๐‘Ÿ , is shown in Eq.(5.17).  ๐‘ข ๐‘˜ = โˆ’๐›ซ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜ + ๐‘๐‘Ÿ โˆ™ ๐‘Ÿ ๐‘˜  (5.17)  Then choose ๐‘๐‘Ÿ  with its zero lines on the closed loop integrator pole to reduce the effect of the integrator dynamics in the closed loop system. This ensures the integrator to have slow dynamics because the integrator is meant to have influence at steady state response for reducing the steady state error. The complete closed loop state space model including the pole placement controller with integrator is shown in Eq.(5.18) and Eq.(5.19).  Chapter 5. | Digital Control of Feed Drives 55 ๐‘‹1๐‘› ๐‘˜ + 1 =  ๐ด1๐‘ง ๐‘› โˆ’ ๐ต1๐‘ง ๐‘› โˆ™ ๐›ซ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜ +  ๐ต1๐‘ง ๐‘› โˆ™ ๐‘๐‘Ÿ + ๐ธ1๐‘ง ๐‘› โˆ™ ๐‘Ÿ ๐‘˜  (5.18)  ๐‘ฆ ๐‘˜ = ๐ถ1๐‘ง ๐‘› โˆ™ ๐‘‹1๐‘› ๐‘˜  (5.19)  5.2.2 Observer Design The presented full state feedback pole placement controller in the previous section is only feasible when all the states are accessible for feedback control. However, if not all states are available or some of the state measurements have high noise content or poor accuracy, the measurements become unavailable or unreliable for feedback control. In this case, an observer is used to estimate the states for feedback. In this section, the two types of observer, time update (TU) and measurement update (MU), are discussed. The idea behind the observer is to place the system model in parallel with the actual system and to drive them both with the same input. When the model and actual system both have the same initial state vector, the state estimate generated by the model should track the actual state vector. Nevertheless, there are always uncertainties in the actual system and in practice, without feedback, the state estimate would diverge from the true state. The solution is to use the measurement output, ๐‘ฆ1 ๐‘˜ , and to compare it with the modelโ€™s estimated measurement output, ๐‘ฆ 1 ๐‘˜ . The difference between the two is then used to correct the state estimate.  5.2.2.1 Time Update Observer Time update observer acts like a predictor which provides the estimate states at the next sampling instance, ๐‘˜ + 1. According to the TU observer diagram in Figure 5.2, the time update observer equation is obtained as in Eq.(5.20).  Chapter 5. | Digital Control of Feed Drives 56  Figure 5.2: Time Update Observer Diagram ๐‘ฆ1 ๐‘˜ : Measured Output (Motor Position) ๐‘ฆ1๐‘’๐‘ ๐‘ก  ๐‘˜ : TU Estimated Output (Motor Position) ๐‘ฆ 1๐‘ก  ๐‘‹1๐‘’๐‘ ๐‘ก  ๐‘˜ : TU Estimated States ๐‘‹ 1๐‘ก   ๐‘‹ 1๐‘ก ๐‘˜ + 1 = ๐ด1๐‘ง โˆ™ ๐‘‹ 1๐‘ก ๐‘˜ + ๐ต1๐‘ง โˆ™ ๐‘ข ๐‘˜ + ๐ฟ๐‘ โˆ™  ๐‘ฆ1 ๐‘˜ โˆ’ ๐ถ1๐‘ง โˆ™ ๐‘‹ 1๐‘ก ๐‘˜   (5.20)  where ๐‘‹ 1๐‘ก ๐‘˜  is the TU estimated state vector and ๐ฟ๐‘  is the TU observer feedback gain. The TU state estimate error, ๐‘‹ 1๐‘ก ๐‘˜ , is defined as in Eq.(5.21). Now, the TU state estimate error dynamics is defined at Eq.(5.22).  ๐‘‹ 1๐‘ก ๐‘˜ = ๐‘‹1 ๐‘˜ โˆ’ ๐‘‹ 1๐‘ก ๐‘˜  (5.21)  Chapter 5. | Digital Control of Feed Drives 57 ๐‘‹ 1๐‘ก ๐‘˜ + 1 = ๐ด1๐‘ง โˆ™  ๐‘‹1 ๐‘˜ โˆ’ ๐‘‹ 1๐‘ก ๐‘˜  โˆ’ ๐ฟ๐‘ โˆ™ ๐ถ1๐‘ง โˆ™  ๐‘‹1 ๐‘˜ โˆ’ ๐‘‹ 1๐‘ก ๐‘˜   (5.22)  ๐‘‹ 1๐‘ก ๐‘˜ + 1 =  ๐ด1๐‘ง โˆ’ ๐ฟ๐‘ โˆ™ ๐ถ1๐‘ง โˆ™ ๐‘‹ 1๐‘ก ๐‘˜  (5.23)  Eq.(5.23) describes the state estimate error dynamics as the difference between the actual and estimated state vectors changes over time. Ideally, this error should decay to zero as fast as possible. Thus the poles of the error dynamics system matrix should be set to fast and stable. Define the desired error dynamics poles, ๐ถ๐ธ๐‘œ๐‘‘  ๐‘ง  as Eq.(5.24).  ๐ถ๐ธ๐‘œ๐‘‘  ๐‘ง = ๐‘ง 2 + ๐›ผ๐‘œ1 โˆ™ ๐‘ง + ๐›ผ๐‘œ2 = 0 (5.24)  The actual observer error dynamics poles are stated in the observer characteristic equation, ๐ถ๐ธ๐‘œ , which is obtained by Eq.(5.25).  det  ๐‘ง โˆ™ ๐ผ โˆ’  ๐ด1๐‘ง โˆ’ ๐ฟ๐‘ โˆ™ ๐ถ1๐‘ง  = ๐ถ๐ธ๐‘œ = 0 (5.25)  Then the observer characteristic equation, ๐ถ๐ธ๐‘œ , should be adjusted to equate with the desired observer characteristic equation, ๐ถ๐ธ๐‘œ๐‘‘ , by selecting an appropriate observer feedback gain, ๐ฟ๐‘ .  det  ๐‘ง โˆ™ ๐ผ โˆ’  ๐ด1๐‘ง โˆ’ ๐ฟ๐‘ โˆ™ ๐ถ1๐‘ง  = ๐ถ๐ธ๐‘œ = ๐ถ๐ธ๐‘œ๐‘‘  ๐‘ง  (5.26)  Chapter 5. | Digital Control of Feed Drives 58 Note that ๐ฟ๐‘  has many elements as there are error dynamics poles which means that the poles can be placed anywhere as long as the system is observable through the output,๐‘ฆ1 ๐‘˜ . This can be verified by checking the determinant of the observability matrix, Eq.(5.27), does not equal zero. For better observer performance, the observer poles are set to have a faster dynamics than the closed loop system.  ๐‘‚ =  ๐ถ1๐‘ง ๐ถ1๐‘ง โˆ™ ๐ด1๐‘ง โ‹ฎ ๐ถ1๐‘ง โˆ™ ๐ด1๐‘ง nโˆ’1   (5.27)  5.2.2.2 Measurement Update Observer The measurement update (MU) observer works like a corrector. It provides a state vector estimate at the current sampling instance, ๐‘˜, based on the measurement at ๐‘˜. This type of observer is expected to achieve a better estimate for feedback purpose since it minimizes the delay in the system. The MU observer design strategy is similar to TU observer design approach. At the current sampling instance, ๐‘˜ , the MU observer updates the estimate state using measurement ๐‘ฆ1 ๐‘˜  as shown in Eq.(5.28).  ๐‘‹ 1๐‘š ๐‘˜ = ๐‘‹ 1๐‘ก ๐‘˜ + ๐ฟ๐‘ โˆ™  ๐‘ฆ1 ๐‘˜ โˆ’ ๐ถ1๐‘ง โˆ™ ๐‘‹ 1๐‘ก ๐‘˜   (5.28)  where ๐‘‹ 1๐‘š ๐‘˜  is the estimated state vector at ๐‘˜ using measurements up to ๐‘˜ and ๐ฟ๐‘  is the MU observer feedback gain. To generate the state vector at the next sampling instance, ๐‘˜ + 1, the whole process goes through the system dynamics equation as shown in Eq.(5.29).  Chapter 5. | Digital Control of Feed Drives 59 ๐‘‹ 1๐‘ก ๐‘˜ + 1 = ๐ด1๐‘ง โˆ™ ๐‘‹ 1๐‘š ๐‘˜ + ๐ต1๐‘ง โˆ™ ๐‘ข ๐‘˜  (5.29)  Note that Eq.(5.29) is similar to the TU observer equation, Eq. (5.20), since the next sampling instance, ๐‘˜ + 1 , estimate is based on measurements up to ๐‘˜  only. By substituting the MU estimated state vector, ๐‘‹ 1๐‘š ๐‘˜  from Eq.(5.28) into Eq.(5.29), it becomes obvious in how to relate the current estimator gain, ๐ฟ๐‘ , with the predictive estimator gain, ๐ฟ๐‘ .  ๐‘‹ 1๐‘ก ๐‘˜ + 1 = ๐ด1๐‘ง โˆ™ ๐‘‹ 1๐‘ก ๐‘˜ + ๐ต1๐‘ง โˆ™ ๐‘ข ๐‘˜ + ๐ด1๐‘ง โˆ™ ๐ฟ๐‘ โˆ™  ๐‘ฆ1 ๐‘˜ โˆ’ ๐ถ1๐‘ง โˆ™ ๐‘‹ 1๐‘ก ๐‘˜   (5.30)  ๐ฟ๐‘ = ๐ด1๐‘ง โˆ™ ๐ฟ๐‘  (5.31)  ๐ฟ๐‘ = ๐ด1๐‘ง โˆ’1 โˆ™ ๐ฟ๐‘  (5.32)    Chapter 5. | Digital Control of Feed Drives 60  Figure 5.3: Measurement Update Observer Diagram ๐‘ฆ1(๐‘˜): Measured Output (Motor Position) ๐‘ฆ1๐‘’๐‘ ๐‘ก (๐‘˜): MU Estimated Output (Motor Position) ๐‘ฆ 1๐‘š  ๐‘‹1๐‘’๐‘ ๐‘ก (๐‘˜): MU Estimated States ๐‘‹ 1๐‘š     Chapter 5. | Digital Control of Feed Drives 61 Then the pole placement controller with integrator and measurement update observer are combined as shown in Figure 5.4.   Figure 5.4: Pole Placement Controller with Integrator and Measurement Update Observer ๐‘ฆ1 ๐‘˜ : Measured Output (Motor Position) ๐‘ฆ1๐‘’๐‘ ๐‘ก (๐‘˜): MU Estimated Output (Motor Position) ๐‘ฆ 1๐‘š  ๐‘‹1๐‘’๐‘ ๐‘ก (๐‘˜): MU Estimated States ๐‘‹ 1๐‘š   5.2.3 Simulation and Experiment Result The pole placement controller with integrator is implemented to the linear motor drive. The performance of the positioning control is validated through experiments and simulations. In the experiment, the feedback states to the controller are estimated through the measurement update observer. In the simulation, the average Coulomb Chapter 5. | Digital Control of Feed Drives 62 friction identified through the ULS technique and uniformly distributed random noises of ยฑ5 micron from the linear encoder are considered as the system disturbance.   Figure 5.5: Motor Position of Experimental and Simulated Step Response for Motor Position Control System using Pole Placement Controller with Integrator   Desired Experiment Simulation Overshoot (%) 5.0 5.4 6.7 Setting Time (sec) 0.75 0.77 0.91 Table 5.1: Motor Position of Experimental and Simulated Step Response for Motor Position Control System using Pole Placement Controller with Integrator Summary  Chapter 5. | Digital Control of Feed Drives 63 The position command step input response experimental and simulated result of the pole placement controller with integrator have been compared. As shown in Figure 5.5, neither of the result meets the desired controller performance criteria because of a number of reasons which are explained as follows. The pole placement controller is designed based on the desired system overshoot percentage and settling time as corresponds with a specified system natural frequency and damping ratio. Then the designed controller manipulates the system poles to obtain the specified system natural frequency and damping ratio in order to meet the design criteria. However, this design approach does not put any disturbance in consideration. Therefore, the addition of disturbance, such as friction force and noise, in the experiment and simulation alters the system performance from the designed model. For better positioning performance, a feedforward controller is recommended for compensating some of the disturbances. The system model uncertainty is also a contributing factor that causes the experimental and simulated result to be different from the design model. For that same reason, it can be seen that the experimental result does not match with simulated result as well. The experimental result moves much gradually compared to the simulated result. This may be an indication that the Coulomb friction model is not accurate. From Chapter 4, the identified system only considers the linear friction and discards the effect of the nonlinear friction behaviour, which that may underestimate the actual friction in the system. The experimental result shows that the actual system reaches steady state before the simulated system. The reason may be either the actual motor mass is less than the identified motor mass such that the motor is easier to slow down or the actual viscous friction is higher than the model such that the friction force slows down motion of the motor quicker. To improve the simulation accuracy, a more extensive system model should be included in the simulation. During the experiment, the oscillator acceleration measurement is also captured. The oscillator vibrations continue even after the motor position reaches steady state as shown in Figure 5.6. This vibration is a representation of the machine tool vibration when vibration control is not considered in the controller design. Thus, this causes poor Chapter 5. | Digital Control of Feed Drives 64 surface on a machined part. Therefore, this issue leads to the study for vibration control in feed drive system.   Figure 5.6: Oscillator Acceleration of Experimental Step Response for Motor Position Control System using Pole Placement Controller with Integrator  5.3 Active Vibration Control In the system without active vibration control, translating the motor position induces vibrations onto the SDOF oscillator which the vibrations persist for a long period of time as shown in the positioning controller experimental result in the previous section. The continuous vibrations deteriorate the accuracy and precision of the machine tool. Therefore, the presented active vibration control is intended to suppress the vibrations at the SDOF oscillator while the motor drive tracks the reference position command accurately. Chapter 5. | Digital Control of Feed Drives 65 The presented active vibration control is in a classic cascade control structure where it is divided into the velocity and position loop as shown in Figure 5.7. The velocity loop is designed to minimize the vibrations at the oscillator through the use of a phase compensator. Then a cascaded position controller is applied to track the motor position to its reference command.   Figure 5.7: Active Vibration Controller with Phase Compensator Design Velocity Loop (Dash Dot) Position Loop (Dash)  5.3.1 Velocity Loop Design To begin the velocity loop design, consider the open loop system in Figure 5.8 and the motor and oscillator velocity frequency response in Figure 5.9 which are derived from the identified transfer functions in Chapter 4.  Chapter 5. | Digital Control of Feed Drives 66  Figure 5.8: Open Loop System   Figure 5.9: Open Loop Motor Velocity (Left) and Oscillator Velocity (Right) Simulated Frequency Response  In the velocity loop, both motor and oscillator velocity output are used as feedback signals. When only the motor velocity output is used as the feedback, this is known as the indirect velocity loop as shown in Figure 5.10. When both motor velocity and oscillator velocity output are used as the feedback, this is known as the direct velocity Chapter 5. | Digital Control of Feed Drives 67 loop. Here, a proportional integral (PI) velocity controller is used to ensure a fast system response with no steady state error in tracking the velocity command.  Proportional Integral (PI) Velocity Controller Parameters Proportional Gain: 41.8 Integral Gain: 130.0 Table 5.2: Proportional Integral Velocity Controller Parameters   Figure 5.10: Indirect Velocity Loop System  Chapter 5. | Digital Control of Feed Drives 68  Figure 5.11: Indirect Velocity Loop Motor Velocity (Left) and Oscillator Velocity (Right) Simulated Frequency Response  The indirect velocity loop oscillator velocity frequency response simulation shown in Figure 5.11 indicates that there is a phase delay of 90 deg at the oscillator resonance. To minimize the oscillator resonant response, a +90 deg phase shift is needed in the direct velocity feedback loop. Therefore, the phase compensator for the direct velocity feedback loop is required to be at unity gain and +90 deg when it crosses the resonance frequency. Then this compensator can be designed in two methods. The first method is to use a combination of an all pass filter and band pass filter. The band pass filter limits the actuation of the phase compensator to be within the resonant frequency range. Then the all pass filter is a unity filter with a varying phase which is tuned such that the phase of the compensator reaches +90 deg at the oscillator resonant frequency, ๐œ”0. The second method is the use of the ๐‘  transfer function with a phase compensator gain of 1/๐œ”0. This method provides the +90 deg phase shift at all frequencies and the constant gain of 1/๐œ”0 ensures the unity gain at the oscillator resonant frequency.  The main issue with the first method is the tuning of the band pass filter and the all pass filter. Since the expected motor positioning operating range is up to 35 Hz, the band pass filter should have a narrow passband which is difficult to achieve with a low order Chapter 5. | Digital Control of Feed Drives 69 band pass filter. Then using a high order band pass filter is possible to create a narrow passband but the phase delay of a high order band pass changes quickly across the passband frequency range. Thus, when a slight variation in the resonant frequency occurs, the feedback phase is not guaranteed to be accurate. The feedback without an accurate phase at the resonant reduces the active vibration control performance and defeats the purpose of this phase compensation active control approach. The second method turns out to be easier to implement. However, there is a minor drawback in this method since the use of ๐‘  transfer function and the constant phase compensator gain of 1/๐œ”0 can only ensure the unity gain at the resonant frequency, ๐œ”0. This means that when the system operates at frequency range below the resonant frequency, the phase compensator gain is less than unity, and at frequency range above the resonant frequency, the phase compensator gain is greater than unity. Thus, when the system operates at frequency range away from the resonant frequency, the active control may perform worse than the system without active control since the phase compensator gain is either less or greater than unity. Nonetheless, as shown in simulations, the second method provides a decent active vibration control performance. It is because given the expected operating frequency range up to 35 Hz, the operating range remains close to the natural frequency range. At low frequency operating range as below the resonant frequency, the excitation of the oscillator is relatively low. Thus, the vibrations on flexible structure should be negligible. Then the phase compensator for active vibration control is not crucial. At high frequency operating range as above the resonant frequency, the magnitude of the constant phase compensator gain, 1/๐œ”0, is greater than unity. The feedback signal becomes overly amplified and can cause overshoot in the system. The overly amplified signal can be amended by restraining the maximum feedback signal using a saturation block at the direct velocity feedback loop. Through simulations as shown in Figure 5.13, even without limiting the maximum feedback signal in the direct velocity loop, the overly amplified feedback signal only increases the oscillation velocity frequency response magnitude no more than a couple of decibels compared to the indirect velocity Chapter 5. | Digital Control of Feed Drives 70 feedback system at high frequency range. To summarize the earlier discussion, as long as the system operating frequency range is close to the resonance frequency range of the oscillator, the constant compensator gain is reasonable to apply in the active vibration control system. The result of indirect and directive velocity loop with phase compensator velocity frequency response simulation is compared in Figure 5.13.   Figure 5.12: Direct Velocity Loop System with Phase Compensator   Chapter 5. | Digital Control of Feed Drives 71  Figure 5.13: Direct and Indirect Velocity Loop Motor Velocity (Left) and Oscillator Velocity (Right) Simulated Frequency Response  Since the motor and oscillator velocity are not direct measurable quantities in this experimental setup, it is much useful to look at the motor position and oscillator acceleration frequency response as shown in Figure 5.14.   Figure 5.14: Direct and Indirect Velocity Loop Motor Position (Left) and Oscillator Acceleration (Right) Simulated Frequency Response  It should also be noted that when the system has multiple modes, the use of band pass filter at the resonant frequency, ๐œ”0, ensures that the phase compensation is being Chapter 5. | Digital Control of Feed Drives 72 active only around the vibration mode at ๐œ”0 and decouples the control effort to other vibration modes [19]. The band pass filter implementation is also useful to minimize the drawback of using a constant phase compensator gain as previously described. It is because when the system is operating away from the resonant frequency, the effect of the improper amplified feedback signal is limited. For this experimental setup, since only the first resonance mode is considered for active vibration control, the use of band pass filter is not necessary.  5.3.2 Position Loop Design With the completion of the velocity loop design, a cascaded position controller is implemented to close the position loop as shown in Figure 5.15. The motor position negative loop transmission (NLT) frequency response simulation shown in Figure 5.16 indicates that the anti-resonant introduced by oscillator dynamic response may impose challenges to the position controller design.    Chapter 5. | Digital Control of Feed Drives 73  Figure 5.15: Cascaded Position Loop with Velocity Loop Indirect Velocity Loop (Dash Dot) Direct Velocity Loop (Dash)   Figure 5.16: Direct Velocity Loop Motor Position Simulated Frequency Response Chapter 5. | Digital Control of Feed Drives 74 At first, a loop shaping controller is designed to set the motor positioning bandwidth above the anti-resonant frequency and increase the phase margin to be at least 60 deg to ensure stability. This is not possible due to the rapid phase change between the two phase extrema across the anti-resonant frequency. The minimum phase must be increased to -120 deg in order to satisfy the +60 deg of phase margin. However, this leads to the loop shaping controller shifting the maximum phase above 0 deg, where the system becomes unstable. Instead, a proportional position controller is applied. In theory, a high proportional gain position controller can increase the motor positioning bandwidth above the anti- resonant frequency. However, the phase margin is less than 60 deg near the anti- resonant frequency, which the system can easily become unstable. Therefore, it is best to avoid extending the motor positioning bandwidth above the anti-resonant frequency. Now, the system bandwidth is limited by the anti-resonant behaviour since the motor position frequency response drops rapidly at that frequency range. In practice, another limitation must put into consideration which is the motor drive saturation. Thus, an appropriate proportional gain position controller is implemented to obey the motor drive saturation requirement.  The motor position and oscillator acceleration frequency response simulation in closed position loop are shown in Figure 5.17. The system with direct velocity loop is denoted as an active control system and the system with only indirect velocity loop, equivalently as without direct velocity loop, is denoted as a non active control system.  Chapter 5. | Digital Control of Feed Drives 75  Figure 5.17: Motor Position (Left) and Oscillator Acceleration (Right) Simulated Frequency Response for Active and Non Active Vibration Control System  The disturbance force, which considered as Coulomb friction and external applied force, are included in the active control system as shown in Figure 5.18. The Coulomb friction is working against the motor force, and the external applied force is applied at the flexible oscillator. An addition feed forward controller can be used to compensate for the Coulomb friction, since it has been identified in the parameter identification process. The disturbance frequency responses are derived as shown in Figure 5.19. For the non active control system, it is assumed that motor position is not affected by the excitation applied at the oscillator. Thus, the disturbance frequency response has zero magnitude and does not appear on the motor position disturbance frequency response.  Chapter 5. | Digital Control of Feed Drives 76  Figure 5.18: Active Control System with Disturbance Input     Figure 5.19: Motor Position (Left) and Oscillator Acceleration (Right) Simulated Disturbance Frequency Response for Active and Non Active Vibration Control System  5.3.3 Modification in Experimental Implementation  Two types of sensor used to measure the system feedback: linear encoder for motor position, and accelerometer for oscillator acceleration. Since the linear encoder is a high Chapter 5. | Digital Control of Feed Drives 77 accuracy sensor, the velocity measurement obtained by taking the difference in positions per sampling time is reliable as a feedback signal in the indirect velocity loop. On the other hand, the accelerometer's high noise and DC offset cause the accelerometer measurement to be undesirable as a feedback signal in the direct velocity loop. The accelerometer noisy feedback is amplified through velocity controller which constantly saturates the motor drive power. Because the oscillator velocity cannot be measured directly, the accelerometer measurement must be numerically integrated to obtain the velocity measurement. The problem comes in when taking the numerical integration of the accelerometer signal with a DC offset, this causes the derived oscillator velocity signal to drift. Thus, the derived oscillator velocity signal becomes unsuitable as a feedback signal in the direct velocity feedback loop. For these reasons, modifications are made to overcome the two implementation challenges.  5.3.3.1 Feedback Simplification The accelerometer must be fastened securely at the tip of the SDOF oscillator. However, the holding force of the senor may have transmitted to the sensor itself. Thus, the accelerometer picks up a small offset signal. By taking the numerical integration of the accelerometer DC offset signal introduces a drift into the derived oscillator velocity signal. The solution to overcome this challenge is to avoid taking numerical integration of the accelerometer signal. The reason for numerically integrating the acceleration signal is to convert the acceleration signal to velocity signal used in the direct velocity loop feedback. It is because the direct velocity feedback signal needs to go through a +90 deg phase shift by an ๐‘  transfer function at the phase compensator. This phase shift processed by an ๐‘   transfer function is equivalent to processing the signal by a differentiation operation. Now, this can be seen that these two processes are actually redundant operations which cancel each other. Therefore, modification can be made by cancelling the numerical integration with the phase shift operation, and simply scaling the acceleration signal with the phase compensator gain of 1/๐œ”0  as the phase Chapter 5. | Digital Control of Feed Drives 78 compensated direct velocity feedback signal. Then the accelerometer sensorโ€™s DC offset no longer causes the feedback signal to drift. For comparison, the original and modified direct velocity loop are shown in Figure 5.20 and Figure 5.21, respectively.   Figure 5.20: Original Direct Velocity Loop   Chapter 5. | Digital Control of Feed Drives 79  Figure 5.21: Modified Direct Velocity Loop  5.3.3.2 Kalman Filter Another implementation issue is due to the accelerometer noisy measurement signal used in the direct velocity feedback loop. The amplified noisy measurement signal can easily saturate the motor drive power which causes the motor to overheat. The initial attempt is by applying a low pass filter to remove the high frequency measurement noise. However, using a high cut off frequency low pass filter cannot remove most of the noise, but using a low cut off frequency low pass filter introduces an undesirable phase delay at the feedback signal. The feedback signal with the addition phase delay deteriorates the active vibration control performance. To remove the noise at the accelerometer signal without scarifying the active vibration control performance, the Kalman filter is implemented. The Kalman Filter algorithm developed by R. Kalman is a recursive filter using measurements containing sensor noise and other inaccuracies to estimate the states of a linear system in time domain. The filter is applied to the states of a discretised system Chapter 5. | Digital Control of Feed Drives 80 to estimate the new states at each discrete time increment. The two independent types of noise considered are measurement noise and process noise. Measurement noise is introduced at the system output and caused by poor measurement sensors. The process noise is introduced at the system input and caused by model uncertainty. Through the Kalman filter algorithm, the filter produces a priori state estimate at sampling instance ๐‘˜ given knowledge of the process prior to sampling instance ๐‘˜, and a posteriori state estimate at sampling instance ๐‘˜  given measurement at sampling instance ๐‘˜  with weighted average, and computes the uncertainty of the estimates. The weight of the weighted average is distributed between the estimate value through the time update process and measurement update process. More weight is given to the value with lesser uncertainty. The estimate produced by the filter tends to be more accurate than the original measurements because the weighted average value has a better estimated uncertainty than either of the values that went into the weighted average. The Kalman filter estimates a process by using a form of feedback control. The filter estimates the process state at each sampling instance and then obtains feedback in the form of noisy measurements. As such, the Kalman filter algorithm is categorised into two processes, time update and measurement update. The time update process is responsible for projecting forward the current state and error covariance estimates to obtain the a priori estimates for the next sampling instance, ๐‘˜ + 1. The measurement update process is responsible for the feedback for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. The time update equations can also be thought of as predictor equations, while the measurement update equations can be thought of as corrector equations.  The motor position and oscillator acceleration transfer function are converted into discrete time state space system with the consideration of the process noise, ๐‘…๐‘ค , and measurement noise, ๐‘…๐‘ฃ. The detail calculation of converting the transfer functions into discrete time state space system is given in Appendix A. The discrete time state space system model is represented by the dynamics and output equation as shown in Eq.(5.33) and Eq.(5.34), respectively. Chapter 5. | Digital Control of Feed Drives 81 ๐‘‹ ๐‘˜ + 1 = ๐ด๐‘ง โˆ™ ๐‘‹ ๐‘˜ + ๐ต๐‘ง โˆ™ ๐‘ข ๐‘˜ + ๐‘…๐‘ค ๐‘˜  (5.33)  ๐‘ฆ ๐‘˜ = ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜ + ๐‘…๐‘ฃ ๐‘˜  (5.34)  The process and measurement noise are assumed to be zero mean Gaussian white noise. The noise matrices are defined as the expected values of the corresponding noise vectors described mathematically by Eq.(5.35) and Eq.(5.36).  ๐‘„ = ๐ธ ๐‘…๐‘ค โˆ™ ๐‘…๐‘ค ๐‘‡  (5.35)  ๐‘… = ๐ธ ๐‘…๐‘ฃ โˆ™ ๐‘…๐‘ฃ ๐‘‡  (5.36)  where ๐‘„ and ๐‘… are the covariance matrices of the process and measurement noise, respectively. Note that ๐‘…๐‘ค  and ๐‘…๐‘ฃ  are mutually independent, then the expected value of ๐‘…๐‘ค  and ๐‘…๐‘ฃ  is zero.  0 = ๐ธ ๐‘…๐‘ค โˆ™ ๐‘…๐‘ฃ ๐‘‡  (5.37)  The Kalman filter performs the estimation in a predictor-corrector approach with time and measurement update. With the given system model, the a priori state estimate vector, ๐‘‹ ๐‘˜+1|๐‘˜ ๐‘˜ + 1 , at the next sampling instance is computed based on the a posteriori state estimate, ๐‘‹ ๐‘˜|๐‘˜ ๐‘˜ , at the current sampling instance through the time update equation as show in Eq.(5.38).  Chapter 5. | Digital Control of Feed Drives 82 ๐‘‹ ๐‘˜+1|๐‘˜ ๐‘˜ + 1 = ๐ด๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜ ๐‘˜ + ๐ต๐‘ง โˆ™ ๐‘ข ๐‘˜  (5.38)  where ๐‘ข ๐‘˜  is the known system input. The residual, ๐‘…๐‘Ÿ๐‘’๐‘  , is defined as the difference between the sensor measurement, ๐‘ฆ ๐‘˜ , and a priori output estimate, ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜ , as shown in Eq.(5.39). The residual reflects the discrepancy between the a priori output estimate and the actual output measurement. When the residual, ๐‘…๐‘Ÿ๐‘’๐‘  , reaches zero, the output estimate equals to the actual output.  ๐‘…๐‘Ÿ๐‘’๐‘   ๐‘˜ = ๐‘ฆ ๐‘˜ โˆ’ ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜  (5.39)  The measurement update equation uses the a priori state estimate, ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜ , at the current sampling instance ๐‘˜ obtained at the previous sampling instance, ๐‘˜ โˆ’ 1, and with the weighted average residual as a correction term to compute the a posteriori state estimate, ๐‘‹ ๐‘˜|๐‘˜ ๐‘˜  as shown in Eq.(5.40).  ๐‘‹ ๐‘˜|๐‘˜ ๐‘˜ = ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜ + ๐พ๐‘˜ โˆ™ ๐‘…๐‘Ÿ๐‘’๐‘   ๐‘˜  (5.40)  where ๐พ๐‘˜  is the weighted average Kalman gain. The Kalman gain is defined for minimizing the a posteriori estimate error covariance and thus an accurate estimation of the system states can be achieved. The a priori and a posteriori estimate error are defined as in Eq.(5.41) and Eq.(5.42), respectively.  ๐‘’๐‘˜|๐‘˜โˆ’1 โ‰ก ๐‘‹ โˆ’ ๐‘‹ ๐‘˜|๐‘˜โˆ’1 (5.41)  Chapter 5. | Digital Control of Feed Drives 83 ๐‘’๐‘˜|๐‘˜ โ‰ก ๐‘‹ โˆ’ ๐‘‹ ๐‘˜|๐‘˜  (5.42)  Then the a priori and a posteriori estimate error covariance are as shown in Eq.(5.43) and Eq.(5.44).  ๐‘ƒ๐‘˜+1|๐‘˜ = ๐ธ ๐‘’๐‘˜+1|๐‘˜ โˆ™ ๐‘’๐‘˜+1|๐‘˜ ๐‘‡  (5.43)  ๐‘ƒ๐‘˜|๐‘˜ = ๐ธ ๐‘’๐‘˜|๐‘˜ โˆ™ ๐‘’๐‘˜|๐‘˜ ๐‘‡  (5.44)  Referring to [12], the resulting function for the Kalman gain is as show in Eq.(5.45).  ๐พ๐‘˜ = ๐‘ƒ๐‘˜|๐‘˜โˆ’1 โˆ™ ๐ถ๐‘ง ๐‘‡ โˆ™  ๐ถ๐‘ง โˆ™ ๐‘ƒ๐‘˜ |๐‘˜โˆ’1 โˆ™ ๐ถ๐‘ง ๐‘‡ + ๐‘… โˆ’1  (5.45)  At Eq.(5.45), as the measurement error covariance, ๐‘…, approaches zero, the Kalman gain, ๐พ๐‘˜ , weights the residual more heavily as shown in Eq.(5.46).  lim ๐‘…โ†’0 ๐พ๐‘˜ = ๐ถ๐‘ง โˆ’1 (5.46)  On the other hand, as the a priori estimate error covariance, ๐‘ƒ๐‘˜+1|๐‘˜ , approaches zero, the Kalman gain, ๐พ๐‘˜ , weights the residual less heavily as shown in Eq.(5.47).  lim ๐‘ƒ๐‘˜+1|๐‘˜โ†’0 ๐พ๐‘˜ = 0 (5.47)  Chapter 5. | Digital Control of Feed Drives 84 Another way of thinking about the weighting by the Kalman gain, ๐พ๐‘˜ , is that as the measurement error covariance, ๐‘…, approaches zero, the actual output measurement, ๐‘ฆ ๐‘˜ , is trusted more, while the a priori output estimate, ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜ , is trusted less. In contrary, as the a priori estimate error covariance, ๐‘ƒ๐‘˜ |๐‘˜โˆ’1, approaches zero the actual output measurement, ๐‘ฆ ๐‘˜ , is trusted less, while the a priori output estimate, ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜ , is trusted more. The minimized a priori estimate error covariance is as shown in Eq.(5.48),  ๐‘ƒ๐‘˜+1|๐‘˜ = ๐ด๐‘ง โˆ™ ๐‘ƒ๐‘˜|๐‘˜ โˆ™ ๐ด๐‘ง ๐‘‡ + ๐‘„ (5.48)  and the minimized a posteriori estimate error covariance is as shown in Eq.(5.49).  ๐‘ƒ๐‘˜|๐‘˜ =  ๐ผ โˆ’ ๐พ๐‘˜ โˆ™ ๐ถ๐‘ง โˆ™ ๐‘ƒ๐‘˜โˆ’1|๐‘˜  (5.49)  Figure 5.22 provides an overview of the Kalman filter operation cycle. At first, the initial state and error covariance are predefined in the filter. Then the time update projects the current state estimate ahead in time. The measurement update adjusts the projected estimate by an actual measurement at that time. The cycle repeats with time update process and follows again with measurement update process.  Chapter 5. | Digital Control of Feed Drives 85 Measurement Update (Correct) Time Update (Predict) Initial State and Covariance  Figure 5.22: Kalman Filter Operation Cycle  Time Update (Predict) Process Project the state ahead ๐‘‹ ๐‘˜+1|๐‘˜ ๐‘˜ + 1 = ๐ด๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜ ๐‘˜ + ๐ต๐‘ง โˆ™ ๐‘ข ๐‘˜  Project the error covariance ahead ๐‘ƒ๐‘˜+1|๐‘˜ = ๐ด๐‘ง โˆ™ ๐‘ƒ๐‘˜|๐‘˜ โˆ™ ๐ด๐‘ง ๐‘‡ + ๐‘„ Table 5.3: Kalman Filter Time Update Process Equations  Measurement Update (Correct) Process Compute the Kalman gain ๐พ๐‘˜ = ๐‘ƒ๐‘˜|๐‘˜โˆ’1 โˆ™ ๐ถ๐‘ง ๐‘‡ โˆ™  ๐ถ๐‘ง โˆ™ ๐‘ƒ๐‘˜|๐‘˜โˆ’1 โˆ™ ๐ถ๐‘ง ๐‘‡ + ๐‘… โˆ’1  Update estimate with measurement ๐‘‹ ๐‘˜|๐‘˜ ๐‘˜ = ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜ + ๐พ๐‘˜ โˆ™  ๐‘ฆ ๐‘˜ โˆ’ ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜|๐‘˜โˆ’1 ๐‘˜   Update the error covariance ๐‘ƒ๐‘˜|๐‘˜ =  ๐ผ โˆ’ ๐พ๐‘˜ โˆ™ ๐ถ๐‘ง โˆ™ ๐‘ƒ๐‘˜โˆ’1|๐‘˜  Table 5.4: Kalman Filter Measurement Update Process Equations  The noise level during measurement and the accuracy of sensors together with the modelling uncertainties are essential to the derivation of the noise covariance matrices. In the actual implementation of the filter, the measurement noise covariance, ๐‘…, is measured prior to operation of the filter. To determine the measurement error Chapter 5. | Digital Control of Feed Drives 86 covariance, ๐‘…, is generally straightforward since it can be done by taking some offline senor measurement. The determination of the process noise covariance, ๐‘„, is more difficult because typically it is not possible to directly observe the estimate process. Nevertheless, a relatively poor process model can still produce an acceptable result if higher uncertainty is injected into the process via the selection of ๐‘„. In general, whether or not a rational approach is possible for determining the noise covariance parameters, strong filter performance statically speaking can be achieved by tuning the filter parameters ๐‘„ and ๐‘…. Thus, for this experimental setup, the measurement noise covariance, ๐‘…, is set by computing the covariance of the offline accelerometer signal recordings. The process noise covariance, ๐‘„, is tuned until the Kalman filter output coincides well with system excitation by input command and exogenous disturbance. When ๐‘„ is set to too low, the filter output is unable to track exogenous disturbance, and when ๐‘„ is set to too high, the filter is unable to reduce the sensor noise. The tuned noise covariance parameters are listed in Table 5.5. The accelerometer and Kalman filter measurement update output measurements and comparison summary are shown in Figure 5.23 and Table 5.6, respectively.  ๐‘… Measurement Noise Covariance (m/s2)2 ๐‘„ Process Noise Covariance (m/s2)2 0.0453 0.0500 Table 5.5: Noise Covariance Parameters  Chapter 5. | Digital Control of Feed Drives 87  Figure 5.23: Accelerometer Measurement and Kalman Filter Output  Oscillator Acceleration Accelerometer Measurement Kalman Filter Mean (m/s2) -0.214 0.000205 Covariance (m/s2)2 0.0453 0.00102 Table 5.6: Acceleration Measurement and Kalman Filter Output Summary  The use of Kalman filter has successfully reduced the offset and the covariance of the acceleration signal. With the implementation modifications of feedback simplification and Kalman filter, the active vibration control system has been implemented and operated properly. The accelerometer signal offset and noise no more causes feedback signal drift and saturation to the amplifier. The complete control system structure with the modifications is shown in Figure 5.24.  Chapter 5. | Digital Control of Feed Drives 88  Figure 5.24: Modified Position Loop with Kalman Filter and Feedback Simplification  5.3.4 Simulation and Experiment Result 5.3.4.1 Frequency Response Simulation Simulations are conducted to compare the motor position and oscillator acceleration frequency response of the active vibration control system with different proportional position controller. The proportional gain magnitudes include 10, 15, 20 and 30 which are denoted by P10, P15, P20 and P30, respectively.  Chapter 5. | Digital Control of Feed Drives 89  Figure 5.25: Simulated Frequency Response from Motor Position Input Command to Motor Position for Active Vibration Control System with Different Position Controller Proportional Gains  Position Controller Gain P10 P15 P20 P30 Bandwidth (Hz) 2.0 3.6 5.0 6.4 Table 5.7: Simulated Frequency Response from Motor Position Input Command to Motor Position for Active Vibration Control System in Different Position Controller Proportional Gains Summary  Chapter 5. | Digital Control of Feed Drives 90  Figure 5.26: Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active Vibration Control System with Different Position Controller Proportional Gains  Position Controller Gain P10 P15 P20 P30 Maximum Magnitude (dB) 54.6 58.2 61.2 65.7 Table 5.8: Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active Vibration Control System with Different Position Controller Proportional Gains Summary  With the flexible oscillator dynamics in the direct velocity feedback system, an anti- resonant effect is introduced into the motor position frequency response at the oscillator resonant frequency. The anti-resonant frequency region imposes a great Chapter 5. | Digital Control of Feed Drives 91 limitation in extending the motor positioning bandwidth since the motor position frequency response magnitude drops to as much as -50 dB. The anti-resonant effect at the motor position frequency response occurs because the negative oscillator acceleration feedback signal cancels with the position command signal. Thus, there is no control input signal directing into the motor. For position controller tuning, the motor position frequency response simulation shown in Figure 5.25 indicates that further increase of the proportional gain of the position controller can only provide marginal improvement to the motor positioning bandwidth but at the cost of reducing the active vibration control performance. It is because with a higher proportional gain position controller, the magnitude of the oscillator acceleration frequency response increases as shown in Figure 5.26.  5.3.4.2 Step Response Experiment The intent of the step response experiment is to replicate the induced vibrations on the machine flexible structure when the machine position is translated. During the experiment, both the motor position and SDOF oscillator acceleration response are measured. Two sets of step response experiment are conducted. The first experiment is comparing the 0.1 m position command step input response of the active vibration control system with different proportional position controllers. The second experiment is comparing the 0.1 m position command step input response of the system with and without active vibration control.  Chapter 5. | Digital Control of Feed Drives 92  Figure 5.27: Motor Position of Experimental Step Response for Active Vibration Control System Input: Motor Position Step Input Command Output: Motor Position  Position Controller Gain P10 P15 P20 P30 Overshoot (%) 0.3 2.3 11.5 23.5 Table 5.9: Motor Position of Experimental Step Response for Active Vibration Control System Summary  Chapter 5. | Digital Control of Feed Drives 93   Figure 5.28: Oscillator Acceleration of Experimental Step Response for Active Vibration Control System Input: Motor Position Step Input Command Output: Oscillator Acceleration  Position Controller Gain P10 P15 P20 P30 Max Acceleration Magnitude (m/s2) 24.2 32.9 40.1 47.8 Table 5.10: Oscillator Acceleration of Experimental Step Response for Active Vibration Control System Summary  Chapter 5. | Digital Control of Feed Drives 94 As listed in Table 5.9 and Table 5.10 experiment summary, P15 position controller should be the most suitable controller for this active control system. The system response to the step input is 2.3% of motor position overshoot and 32.9 m/sec2 of maximum oscillator acceleration. The experimental result agrees with the simulated result that setting motor positioning bandwidth below the oscillator resonant frequency ensures system stability. It should be noted that for demonstration purpose, this particular flexible oscillator is used to ensure that the oscillator vibrations can be easily identified. The oscillator has a low resonant frequency at 8 Hz which greatly limits the motor positioning bandwidth. In practice, the machine tool is much stiffer, corresponded to a higher resonant frequency, than the flexible oscillator in this experimental setup. Thus, a much higher motor positioning bandwidth is achievable in the actual machine. In Figure 5.29, the highlighted component is the phase compensator gain. When the gain is set to zero, the active vibration control is disabled. The system performance with and without active vibration control are compared. The motor position and oscillator acceleration step input response are shown in Figure 5.30 and Figure 5.31, respectively. Experimental result has verified with the simulation that the active control system increases the motor position overshoot compared to the system without active control. The oscillator acceleration response in the active control system reaches steady state after half free oscillation cycle and the maximum oscillator acceleration is also reduced. Note that the first oscillation cycle captured at the oscillator acceleration measurement is due to the acceleration and deceleration of positioning the motor. Thus, the oscillator also experiences a similar magnitude of acceleration and deceleration. In comparison, for the system without active control, denoted as the non active control system, the oscillator continues to vibrate even after the motor position reaches steady state. The control input comparison shown in Figure 5.32 indicates that the active control system requires a higher control input to control the motor position since the motor needs to suppress the vibrations on the oscillator at the same time.  Chapter 5. | Digital Control of Feed Drives 95  Figure 5.29: Position Loop for Active Vibration Control System Phase Compensator Gain (Dash)    Chapter 5. | Digital Control of Feed Drives 96  Figure 5.30: Motor Position of Experimental Step Response for Active and Non Active Vibration Control System Input: Motor Position Step Input Command Output: Motor Position  Position Controller P15 Active Control Non Active Control Overshoot (%) 2.3 1.5 Table 5.11: Motor Position of Experimental Step Response for Active and Non Active Vibration Control System Summary  Chapter 5. | Digital Control of Feed Drives 97  Figure 5.31: Oscillator Acceleration of Experimental Step Response for Active and Non Active Vibration Control System Input: Motor Position Step Input Command Output: Oscillator Acceleration  Position Controller P15 Active Control Non Active Control Max Acceleration Magnitude (m/s2) 32.9 63.5 Table 5.12: Oscillator Acceleration of Experimental Step Response for Active and Non Active Vibration Control System Summary  Chapter 5. | Digital Control of Feed Drives 98  Figure 5.32: Control Input of Experimental Step Response for Active and Non Active Vibration Control Input: Motor Position Step Input Command Output: Control Input  5.3.4.3 Disturbance Rejection Experiment The intent of the disturbance rejection experiment is to replicate the vibrations at the machine flexible structure when it comes in contact with a workpiece during operation. The disturbance rejection performance is conducted to compare the capacity of the active and non active control system to withstand external disturbance. Exogenous disturbance force is applied at the oscillator by tapping the tip of the oscillator. The active and non active control system response of the motor position response, oscillator Chapter 5. | Digital Control of Feed Drives 99 acceleration response and control input are captured and compared as shown in Figure 5.33, Figure 5.34 and Figure 5.35, respectively.   Figure 5.33: Motor Position of Experimental Disturbance Rejection Response for Active and Non Active Vibration Control Input: Exogenous Disturbance Force Output: Motor Position  The active control system has small back and forth translation as a compensational effort to reduce the oscillator vibrations. In comparison, the non active control system remains stationary, such that no compensational control effort is available.  Chapter 5. | Digital Control of Feed Drives 100  Figure 5.34: Oscillator Acceleration of Experimental Disturbance Rejection Response for Active and Non Active Vibration Control Input: Exogenous Disturbance Force Output: Oscillator Acceleration  At Figure 5.34, the initial negative oscillator acceleration spike is due to the exogenous disturbance force applied at the oscillator tip. Even though the disturbance applied to the active vibration control system is higher than the non active vibration control system, the oscillator vibrations in the active control system have been suppressed immediately. The oscillator vibrations at the non active control system continue for a much longer time. The comparison between the two system control inputs is shown in Figure 5.35. There is no control input going into the non active control system. The non active control system Chapter 5. | Digital Control of Feed Drives 101 is not aware of the vibration at the oscillator because no accelerometer signal is fed back into the control loop. Since there is some noise in the accelerometer feedback signal which gets amplified through the controller, the active control system voltage input does not reach zero even at steady state.   Figure 5.35: Control Input of Experimental Disturbance Rejection Response for Active and Non Active Vibration Control Input: Exogenous Disturbance Force Output: Control Input  Chapter 5. | Digital Control of Feed Drives 102 5.3.4.4 Position Input Command Frequency Response Experiment The comparison is made between the simulated and experimental result with position input command frequency responses for the system with and without active control. The experiment is conducted by taking a sine sweep experiment from 1 Hz to 35 Hz, and the motor position and oscillator acceleration frequency responses are captured. As shown in Figure 5.36, Figure 5.37, Figure 5.38 and Figure 5.39, the simulated result matches well against the experimental result up until the higher frequency range above 20 Hz. The reason for this frequency response mismatch is due to the unmodelled system dynamics at the higher frequency. The experimental result shows that the implemented active control system limits motor positioning bandwidth up to the oscillator resonant frequency region as shown in Figure 5.40, and reduces the oscillator acceleration magnitude from 80 dB to 60 dB at resonant frequency as shown in Figure 5.41.  Position Controller P15 Active Control Non Active Control Bandwidth (Hz) 3.6 2.5 Table 5.13: Frequency Response from Motor Position Input Command to Motor Position for Active and Non Active Vibration Control System Summary   Chapter 5. | Digital Control of Feed Drives 103  Figure 5.36: Experimental and Simulated Frequency Response from Motor Position Input Command to Motor Position for Non Active Vibration Control System   Figure 5.37: Experimental and Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Non Active Vibration Control System Chapter 5. | Digital Control of Feed Drives 104  Figure 5.38: Experimental and Simulated Frequency Response from Motor Position Input Command to Motor Position for Active Vibration Control System   Figure 5.39: Experimental and Simulated Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active Vibration Control System  Chapter 5. | Digital Control of Feed Drives 105  Figure 5.40: Experimental Frequency Response from Motor Position Input Command to Motor Position for Active and Non Active Vibration Control System   Figure 5.41: Experimental Frequency Response from Motor Position Input Command to Oscillator Acceleration for Active and Non Active Vibration Control System Chapter 5. | Digital Control of Feed Drives 106 5.3.4.5 Disturbance Frequency Response Experiment The simulated and experimental disturbance to output frequency responses at the motor position and oscillator acceleration are captured and compared. The external impact force disturbance is applied at flexible oscillator with an impact force hammer. The impact force and time responses of motor position and oscillator acceleration are measured and processed through the fast Fourier transform algorithm to obtain the system disturbance frequency responses. Simulated and Experimental results for the system without active control system are compared and shown in Figure 5.42 and Figure 5.43. Since the simulated result is based on the assumption that the motor position is not affected by the excitation at the oscillator, the simulated motor position disturbance frequency response is at zero magnitude. For this reason, the simulated motor position disturbance frequency response does not appear on the logarithmic scale magnitude graph in Figure 5.42. For the system with active control disturbance, the simulated and experimental frequency responses are compared and shown in Figure 5.44 and Figure 5.45. The system with and without active control disturbance frequency responses experimental comparisons are shown in Figure 5.46 and Figure 5.47. In Figure 5.46, the active control system motor position shows a higher motor position magnitude response than the non active control system because the active control system provides compensational effort driving the motor to suppress the vibrations induced by the disturbance force. In Figure 5.47, the oscillator acceleration disturbance frequency response has been suppressed with the active control system from 70 dB to 40 dB at resonance frequency. It should be noted that the oscillator acceleration is measured with an accelerometer, which the measured signal is noisy and has a DC offset. This drawback becomes an issue because the poor measured signal remains as the predominated signal in the lower frequency range for below 5 Hz. Therefore, the experimental result captured at the low frequency range is unreliable and deviates from the simulated result. Nonetheless the experimental results show a strong agreement with the simulated results in the higher frequency range.  Chapter 5. | Digital Control of Feed Drives 107   Figure 5.42: Experimental Frequency Response from Disturbance Force at the Oscillator to Motor Position for Non Active Control System    Figure 5.43: Experimental and Simulated Frequency Response from Disturbance Force at the Oscillator to Oscillator Acceleration for Non Active Vibration Control System Chapter 5. | Digital Control of Feed Drives 108  Figure 5.44: Experimental and Simulated Frequency Response from Disturbance Force at the Oscillator to Motor Position for Active Vibration Control System   Figure 5.45: Experimental and Simulated Frequency Response from Disturbance Force at the Oscillator to Oscillator Acceleration for Active Vibration Control System  Chapter 5. | Digital Control of Feed Drives 109  Figure 5.46: Experimental Frequency Response from Disturbance Force at the Oscillator to Motor Position for Active and Non Active Vibration Control System   Figure 5.47: Experimental Frequency Response from Disturbance Force at the Oscillator to Oscillator Acceleration for Active and Non Active Vibration Control System  Chapter 5. | Digital Control of Feed Drives 110 5.4 Summary In this chapter, the positioning and active vibration controller implementation strategies are presented. In Section 5.2, the state space pole placement controller with integrator and the observer design are discussed. Simulations and experiments have been conducted to validate the system performance. The presented pole placement controller design approach does not consider the effect of disturbance. When the disturbance effects, such as Coulomb friction and noise, are included, the control system can no longer perform as expected. Therefore, neither the simulated nor experimental result achieves the desired system performance. The modelling uncertainty is the main reason that the simulated result differs from the experimental result. Through simulations and experiments, the rigid body feed drive control approach possesses a major drawback in ensuring an accurate positioning of the SDOF oscillator because vibration control on the SDOF oscillator is not possible. In Section 5.3, the active vibration controller is implemented. It is designed for suppressing the SDOF oscillator vibrations at the feed drive. This active controller works in a cascaded structure which includes the velocity loop and position loop. In the velocity loop, a phase compensator is used to compensate the phase delay of the oscillator at resonant which effectively increases the damping of the oscillator. Two implementation modifications are proposed for this active vibration controller for this experimental setup. The first modification is the simplification at the phase compensator to minimize the drift effect at the feedback signal. The second modification is the use of a Kalman filter to improve the sensor measurement accuracy and reduce the measurement noise. In Section 5.3.4, experimental and simulated result are presented and compared to validate the active vibration control performance. A trade-off between oscillator vibration suppression and motor positioning control performance is required. For a higher motor positioning bandwidth, the oscillator vibration suppression performance is reduced. The active vibration control system is capable of rejecting exogenous Chapter 5. | Digital Control of Feed Drives 111 disturbance at the oscillator while the non active vibration control system is sensitive to such disturbance as shown experimentally. The implemented active vibration control system reduces the oscillator acceleration by 20 dB and 30 dB at resonant compared to the system without active vibration control for input command frequency response and disturbance frequency response, respectively. For this experimental setup, the motor positioning bandwidth is relatively low because it is limited by the natural frequency of the flexible structure. In practice, the system is capable of achieving higher motor positioning bandwidths since the natural frequency of the flexible structure should be much higher.  Chapter 6. | Conclusion 112 Chapter 6. Conclusion 6.1 Summary Vibrations reduce the positioning accuracy of machine tools; hence, their suppression by active control methods is important. In this thesis, a classical control method for active damping of machine tool drive is demonstrated on a high speed linear drive system. This thesis has presented the detail work of system modelling, system identification and active vibration controller design implementation. In system modelling, a theoretical model of a two degree of freedom system is used to model the flexible feed drive system. Different experimental identification methods are applied to find a suitable identified model for controller design. A pole placement position controller with integrator and observer are designed for a rigid feed drive. The presented pole placement controller has not considered the system disturbance in the design process. Thus, the experimental and simulated result validate that this positioning controller is unable to achieve the desired performance when there is disturbance presented in the system. Moreover, this type of positioning controller for rigid feed drive design approach does not have any control measure to minimize the flexible structure vibrations. It is shown experimentally that the flexible structure vibration continues even after the motor position reaches steady state. An active vibration damping controller is implemented using a phase compensation approach. The position tracking and disturbance rejection performance of the controllers are investigated through simulations and experiments. The results indicate that there is a trade-off between active vibration control and high positioning bandwidth. The achievable positioning bandwidth of the active vibration control system is limited by the flexible structure natural frequency. This is because when the flexible structure dynamics becomes a part of the control system, the flexible structure dynamics introduces an anti-resonant effect at the motor position frequency response. To have the active controller implemented on the experimental setup for this research, design modifications are made to improve the sensor signal. This requires the use of Chapter 6. | Conclusion 113 Kalman filter in the control system, which it has significantly enhanced the quality of accelerometer measurement for the flexible structure vibration.  6.2 Future Work Vibration control is an active research field in terms of both theoretical problems and practical implementation challenges. In the future, this work can be extended in the following directions for further improvements in active damping: ๏‚ท Introduce a more detailed system model which includes an accurate model of the friction force. ๏‚ท Extend the active vibration control to a multimode flexible drive system. ๏‚ท Use gain scheduled active vibration control to compensate for the changes in the natural frequency of the flexible drive systems. ๏‚ท Implement the same active vibration control strategy to an indirect drive system and investigate the performance. ๏‚ท Develop new algorithms to extend the system bandwidth beyond the resonance mode.  Bibliography 114 Bibliography [1] Altintas, Y., 2000, โ€œManufacturing Automation,โ€ Cambridge University Press. [2] Erkorkmaz, K., and Altintas, Y., 2000, โ€œHigh Speed CNC System Design. Part II: Modelling and Identification of Feed Drives,โ€ International Journal of Machine Tools & Manufacture 41 (2001), page 1487-1509. 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Calculations A.1 Unbiased Least Square The detail derivation of the unbiased least square (ULS) is presented in this section.  ๐‘ฅ1 = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‰๐‘–๐‘› โˆ’ ๐œ‡๐‘˜ ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘   (A.1)  ๐œ‡๐‘˜ = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘‘๐‘“  (A.2)  ๐‘ฅ1 = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™  ๐‘‰๐‘–๐‘› โˆ’ ๐‘‘๐‘“ = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘  โˆ™ ๐‘ข (A.3)  ๐‘ฅ1 ๐‘ข = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘ 2 + ๐‘1 โˆ™ ๐‘   (A.4)  Then the motor position transfer function can be rewritten as voltage input with disturbance to motor velocity transfer function.  ๐บ๐‘ก ๐‘  = ๐‘ฅ 1 ๐‘ข = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก โˆ™ ๐‘  + ๐‘1 = ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก ๐‘  + ๐‘1 ๐‘š๐‘ก = ๐พ๐‘ฃ ๐‘  +  โˆ’๐‘ƒ๐‘ฃ  (A.5)  Discretise the transfer function by a zero order hold transform.  Appendix A. | Calculations 117 ๐บ๐‘ก ๐‘ง =  1โˆ’ ๐‘ง โˆ’1 โˆ™ ๐‘  ๐บ๐‘ก ๐‘  ๐‘    (A.6)  ๐บ๐‘ก ๐‘  ๐‘  = 1 ๐‘  โˆ™ ๐พ๐‘ฃ ๐‘  +  โˆ’๐‘ƒ๐‘ฃ = ๐ถ1 ๐‘  + ๐ถ2 ๐‘  +  โˆ’๐‘ƒ๐‘ฃ  (A.7)  ๐ถ1 and ๐ถ2 can be obtained by partial fraction technique.  ๐ถ1 = lim ๐‘ โ†’0 1 ๐‘  โˆ™ ๐พ๐‘ฃ ๐‘  +  โˆ’๐‘ƒ๐‘ฃ โˆ™ ๐‘  = โˆ’๐พ๐‘ฃ ๐‘ƒ๐‘ฃ  (A.8)  ๐ถ2 = lim ๐‘ โ†’๐‘ƒ๐‘ฃ 1 ๐‘  โˆ™ ๐พ๐‘ฃ ๐‘  +  โˆ’๐‘ƒ๐‘ฃ โˆ™  ๐‘  +  โˆ’๐‘ƒ๐‘ฃ  = ๐พ๐‘ฃ ๐‘ƒ๐‘ฃ  (A.9)  Take Z transform of the argument.  ๐บ๐‘ก ๐‘ง =  1โˆ’ ๐‘ง โˆ’1 โˆ™ ๐‘  โˆ’๐พ๐‘ฃ ๐‘ƒ๐‘ฃ โˆ™ ๐‘  + ๐พ๐‘ฃ ๐‘ƒ๐‘ฃ โˆ™  ๐‘  +  โˆ’๐‘ƒ๐‘ฃ    (A.10)  ๐บ๐‘ก ๐‘ง = ๐พ๐‘ฃ ๐‘ƒ๐‘ฃ โˆ™  1โˆ’ ๐‘งโˆ’1 โˆ™ ๐‘  โˆ’1 ๐‘  + 1 ๐‘  +  โˆ’๐‘ƒ๐‘ฃ   (A.11)  ๐บ๐‘ก ๐‘ง = ๐พ๐‘ฃ โˆ’๐‘ƒ๐‘ฃ โˆ™  1โˆ’ ๐‘งโˆ’1 โˆ™  1 1 โˆ’ ๐‘งโˆ’1 โˆ’ 1 1โˆ’ ๐‘’๐‘ƒ๐‘ฃ โˆ™๐‘‡๐‘  โˆ™ ๐‘งโˆ’1   (A.12)  Appendix A. | Calculations 118 ๐บ๐‘ก ๐‘ง = ๐พ๐‘ฃ โˆ’๐‘ƒ๐‘ฃ โˆ™  1 โˆ’ ๐‘’๐‘ƒ๐‘ฃโˆ™๐‘‡๐‘  ๐‘ง โˆ’ ๐‘’๐‘ƒ๐‘ฃโˆ™๐‘‡๐‘   = ๐พ๐‘ฃ๐‘‘ ๐‘ง โˆ’ ๐‘ƒ๐‘ฃ๐‘‘  (A.13)  ๐พ๐‘ฃ๐‘‘ = ๐พ๐‘ฃ โˆ’๐‘ƒ๐‘ฃ โˆ™  1โˆ’ ๐‘’๐‘ƒ๐‘ฃโˆ™๐‘‡๐‘      ๐‘ƒ๐‘ฃ๐‘‘ = ๐‘’ ๐‘ƒ๐‘ฃโˆ™๐‘‡๐‘     ๐‘ฅ 1 ๐‘ง ๐‘ข ๐‘ง = ๐บ๐‘ก ๐‘ง = ๐พ๐‘ฃ๐‘‘ ๐‘ง โˆ’ ๐‘ƒ๐‘ฃ๐‘‘  (A.14)  ๐‘ฅ 1 ๐‘ง + 1 = ๐‘ƒ๐‘ฃ๐‘‘ ๐‘ฅ 1 ๐‘ง + ๐พ๐‘ฃ๐‘‘ โˆ™ ๐‘ข ๐‘ง  (A.15)  A.2 State Space Pole Placement Controller Design The detail calculation of the state space model discretisation is presented in this section. The discretised state space model is used in the pole placement controller. Eq.(A.16) allows computing the state vector at time ๐‘ก given the state vector at the starting ๐‘ก0 and the control input signal between ๐‘ก0  and ๐‘ก. The first term of the equation is the homogenous solution and the second term is the particular solution.  For the discrete equivalent of the continuous state space model, it finds the state vector at the next sampling instance given the state vector and input at the current sampling instance. Then ๐‘ก0 can be represented by ๐‘˜ โˆ™ ๐‘‡๐‘   at current sampling instance and the next sampling instance at ๐‘‡๐‘   seconds later.  ๐‘‹ ๐‘ก = ๐‘’ ๐ด๐‘ โˆ™ ๐‘กโˆ’๐‘ก0 โˆ™ ๐‘‹ ๐‘ก0 + ๐‘’ ๐ด๐‘ โˆ™ ๐‘กโˆ’๐œ ๐‘ก ๐‘ก0 โˆ™ ๐ต๐‘  โˆ™ ๐‘ข ๐œ ๐‘‘๐œ (A.16)  Appendix A. | Calculations 119 Since a zero order hold circuit holds the control constant over the entire sampling period, the control input u(tau) and the input matrix ๐ต๐‘   can be move out of the integration. Changing integration variables and simplifying the equation, the equation is shown in Eq.(A.17).  ๐‘‹ ๐‘˜ โˆ™ ๐‘‡๐‘  + ๐‘‡๐‘  = ๐‘’ ๐ด๐‘ โˆ™๐‘‡๐‘  โˆ™ ๐‘‹ ๐‘˜ โˆ™ ๐‘‡๐‘  + ๐‘’ ๐ด๐‘ โˆ™ ๐‘˜โˆ™๐‘‡๐‘ +๐‘‡๐‘ โˆ’๐œ ๐‘˜โˆ™๐‘‡๐‘ +๐‘‡๐‘  ๐‘˜ โˆ™๐‘‡๐‘  โˆ™ ๐ต๐‘  โˆ™ ๐‘ข ๐œ ๐‘‘๐œ (A.17)  For simplicity, ๐‘˜ โˆ™ ๐‘‡๐‘   is replaced with ๐‘˜.  ๐‘‹ ๐‘˜ + 1 = ๐‘’ ๐ด๐‘ โˆ™๐‘‡๐‘  โˆ™ ๐‘‹ ๐‘˜ + ๐‘’ ๐ด๐‘ โˆ™๐œ‚ ๐‘‡๐‘  0 ๐‘‘๐œ‚ โˆ™ ๐ต๐‘  โˆ™ ๐‘ข ๐‘˜  (A.18)  ๐‘‹ ๐‘˜ + 1 = ๐ด๐‘ง โˆ™ ๐‘‹ ๐‘˜ + ๐ต๐‘ง โˆ™ ๐‘ข ๐‘˜  (A.19)  ๐ด๐‘ง = ๐‘’ ๐ด๐‘ โˆ™๐‘‡๐‘  = ๐ผ + ๐ด๐‘  โˆ™ ๐‘‡๐‘  +  ๐ด๐‘  โˆ™ ๐‘‡๐‘  2 2! +  ๐ด๐‘  โˆ™ ๐‘‡๐‘  3 3! +โ‹ฏ (A.20)  ๐ด๐‘ง = ๐ผ + ๐ด๐‘  โˆ™ ๐‘‡๐‘  โˆ™ ฮจ (A.21)  ฮจ = ๐ผ +  ๐ด๐‘  โˆ™ ๐‘‡๐‘  2! +  ๐ด๐‘  โˆ™ ๐‘‡๐‘  2 3! +  ๐ด๐‘  โˆ™ ๐‘‡๐‘  3 4! +โ‹ฏ (A.22)  ๐ต๐‘ง =  ๐‘’ ๐ด๐‘ โˆ™๐œ‚ ๐‘‡๐‘  0 ๐‘‘๐œ‚ โˆ™ ๐ต๐‘   (A.23)  Appendix A. | Calculations 120 ๐ต๐‘ง =  ๐‘’ ๐ด๐‘  โˆ™๐œ‚ ๐‘‡๐‘  0 ๐‘‘๐œ‚ โˆ™ ๐ต๐‘  = ๐ด๐‘  โˆ’1 โˆ™  ๐ด๐‘ง โˆ’ ๐ผ โˆ™ ๐ต๐‘  = ๐‘‡๐‘  โˆ™ ฮจ โˆ™ ๐ต๐‘   (A.24)  Then apply Eq.(A.20) and Eq.(A.24) to obtain the motor drive discrete equivalent state space model, which includes the motor dynamics and output equation as shown in Eq.(A.25) and Eq.(A.26), respectively.  ๐‘‹1 ๐‘˜ + 1 = ๐ด1๐‘ง โˆ™ ๐‘‹1 ๐‘˜ + ๐ต1๐‘ง โˆ™ ๐‘ข ๐‘˜  (A.25)  ๐‘ฆ ๐‘˜ = ๐ถ1๐‘ง โˆ™ ๐‘‹1 ๐‘˜  (A.26)  ฮจ = ๐ผ +  ๐ด1๐‘  โˆ™ ๐‘‡๐‘  2! +  ๐ด1๐‘  โˆ™ ๐‘‡๐‘  2 3! +  ๐ด1๐‘  โˆ™ ๐‘‡๐‘  3 4! +โ‹ฏ (A.27)  ๐ด1๐‘ง = ๐ผ + ๐ด1๐‘  โˆ™ ๐‘‡๐‘  โˆ™ ฮจ1 (A.28)  ๐ต1๐‘ง = ๐‘‡๐‘  โˆ™ ฮจ1 โˆ™ ๐ต1๐‘  (A.29)  ๐ถ1๐‘ง = ๐ถ1๐‘  (A.30)  ๐ท1๐‘ง = ๐ท1๐‘  = 0 (A.31)   Appendix A. | Calculations 121 A.3 State Space Pole Placement Controller with Integrator Design From Figure 5.4, the close loop system has a zero according to Eq.(A.5).  ๐‘ง๐‘– = 1 + ๐‘‡๐‘  โˆ™ ๐›ซ๐‘– ๐‘๐‘Ÿ  (A.32)  Then the reference gain, ๐‘๐‘Ÿ , can be calculated as shown in Eq.(A.33).  ๐‘๐‘Ÿ = ๐‘‡๐‘  โˆ™ ๐›ซ๐‘– ๐‘ง๐‘– โˆ’ 1  (A.33)  A.4 Kalman Filter The detail calculation of the discretised state space model used for Kalman filter design is presented in this section. Eq.(A.34) is the continuous state space dynamics equation that includes the motor and oscillator dynamics. The first two rows are the same dynamics parameters used in pole placement controller design. The last three rows are addition parameters that describe the oscillator dynamics. Eq.(A.35) is the continuous state space output equation that includes the motor position and oscillator acceleration as output.  ๐‘‹  ๐‘ก = ๐ด๐‘  โˆ™ ๐‘‹ ๐‘ก + ๐ต๐‘  โˆ™ ๐‘ข ๐‘ก  (A.34)  ๐‘ฆ ๐‘ก = ๐ถ๐‘  โˆ™ ๐‘‹ ๐‘ก + ๐ท๐‘  โˆ™ ๐‘ข ๐‘ก  (A.35)  Appendix A. | Calculations 122 ๐‘‹ ๐‘ก =      ๐‘ฅ 1 ๐‘ก ๐‘ฅ1 ๐‘ก ๐‘ฅโ€ฒโ€ฒโ€ฒ2 ๐‘ก ๐‘ฅ 2 ๐‘ก ๐‘ฅ 2 ๐‘ก          ๐‘ข ๐‘ก = ๐‘‰๐‘–๐‘›  ๐‘ก     ๐‘ฆ ๐‘ก =  ๐‘ฅ1 ๐‘ก ๐‘ฅ 2 ๐‘ก    ๐ด๐‘  =        โˆ’ ๐‘1 ๐‘š๐‘ก 0 0 0 0 1 0 0 0 0 0 0 โˆ’ ๐‘1 โˆ™ ๐‘š๐‘Ž + ๐‘š๐‘ก โˆ™ ๐‘2 ๐‘š๐‘ก โˆ™ ๐‘š๐‘Ž โˆ’ ๐‘1 โˆ™ ๐‘2 + ๐‘š๐‘ก โˆ™ ๐‘˜2 ๐‘š๐‘ก โˆ™ ๐‘š๐‘Ž โˆ’ ๐‘1 โˆ™ ๐‘˜2 ๐‘š๐‘ก โˆ™ ๐‘š๐‘Ž 0 0 1 0 0 0 0 0 1 0           ๐ต๐‘  =        ๐พ๐‘ก โˆ™ ๐พ๐‘Ž ๐‘š๐‘ก 0 ๐พ๐‘ก โˆ™ ๐พ๐‘Ž โˆ™ ๐‘˜2 ๐‘š๐‘ก โˆ™ ๐‘š๐‘Ž 0 0            ๐‘๐‘  =  0 0 0 0 0 0 0 0 1 0      ๐ท๐‘  =  0 0 0 0     Same rules from Appendix A.3 are applied to discretise the continuous state space model. Then the discretised state space model with the motor and oscillator dynamics can be obtained.  ๐‘‹ ๐‘˜ + 1 = ๐ด๐‘ง โˆ™ ๐‘‹ ๐‘˜ + ๐ต๐‘ง โˆ™ ๐‘ข ๐‘˜  (A.36)  ๐‘ฆ ๐‘˜ = ๐ถ๐‘ง โˆ™ ๐‘‹ ๐‘˜  (A.37)  ๐ด๐‘ง = ๐ผ + ๐ด๐‘  โˆ™ ๐‘‡๐‘  โˆ™ ฮจ (A.38) Appendix A. | Calculations 123 ๐ต๐‘ง = ๐‘‡๐‘  โˆ™ ฮจ โˆ™ ๐ต๐‘   (A.39)  ๐ถ๐‘ง = ๐ถ๐‘   (A.40)  ๐ท๐‘ง = ๐ท๐‘  = 0 (A.41)  Appendix B. | Oscillator Brackets 124 Appendix B. Oscillator Brackets   Figure B.1: Oscillator Brackets Model  Appendix B. | Oscillator Brackets 125  Figure B.2: Oscillator Brackets Drawing Note: The flexible bar used in the experiment is longer than as shown in the figure.  Appendix B. | Oscillator Brackets 126  Figure B.3: Oscillator Brackets Measurements Units: Millimetres (mm) Note: The flexible bar used in the experiment is longer than as shown in the figure. Appendix C. | System Parameters 127 Appendix C. System Parameters The following table presents the parameters applied in the control system.  Proportional Integral (PI) Velocity Controller P: 41.8 I: 130.0  Proportional (P) Position Controller P: 15  Compensator Gain ๐œ”0 = 55.2 ๐‘Ÿ๐‘Ž๐‘‘/๐‘ ๐‘’๐‘  Motor Position Transfer Function Oscillator Acceleration Transfer Function ๐‘ฅ1 ๐‘‰๐‘–๐‘› = 157.9 30.7 โˆ™ ๐‘ 2 + 156.1 โˆ™ ๐‘   ๐‘ฅ 2 ๐‘‰๐‘–๐‘› = 1281.2 โˆ™ ๐‘  ๐‘ 3 + 13.1 โˆ™ ๐‘ 2 + 3060.6 โˆ™ ๐‘  + 36962.1   Note: The above oscillator acceleration transfer function does not consider the dynamics coupling between the motor and oscillator. It is obtained by applying the NLS optimization only on the oscillator acceleration frequency response result. 

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