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Linear parameter-varying control of CNC machine tool feed-drives with dynamic variations Masih, Hanifzadegan 2014

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LINEAR PARAMETER-VARYING CONTROL OFCNC MACHINE TOOL FEED-DRIVES WITHDYNAMIC VARIATIONSbyMASIH HANIFZADEGANB.Sc., Amirkabir University of Technology (Tehran Polytechnic), 2004M.Sc., Amirkabir University of Technology (Tehran Polytechnic), 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2014c© Masih Hanifzadegan, 2014AbstractThis thesis presents new approaches to feed-drive control of computer numericalcontrol (CNC) machine tools with a significant range of dynamic variations duringmachining operations. Several sources which can cause dynamic variations of feed-drive systems are considered, such as the change of table position, the reductionof workpiece mass, and the variations of tool-path orientation. Feed-drive systemshaving the dynamic variations are modeled as linear parameter varying (LPV) mod-els. For the LPV models, three control methods are proposed to achieve satisfactorycontrol performance of feed-drive systems.In the first method, we propose a parallel structure of an LPV gain-scheduledcontroller which aims at both tracking control and the vibration suppression bytaking into account the resonant modes’ variations which are peculiar to ball-screwdrives. In the second method, instead of designing one LPV controller, a set ofgain-scheduled controllers are designed to compensate for a wide range of dynamicvariations. In this method, switching between two adjacent controllers may resultin a transient jump of control signal at switching instants.In the third method, to ensure a smooth control signal, we present a noveliimethod to design a smooth switching gain-scheduled LPV controller. The movingregion of the gain-scheduling variables is divided into a specified number of localsubregions as well as subregions for the smooth controller switching. Then, onegain-scheduled LPV controller is assigned to each of the local subregions, while foreach switching subregion, a function interpolating local LPV controllers associatedwith its neighbourhood subregions is designed. This interpolating function imposesthe constraint of smooth transition on controller system matrices.The smooth switching controller design problem amounts to solving a feasi-bility problem which involves non-linear matrix inequalities that are solvable by aproposed iterative descent algorithm. The developed smooth switching controller isapplied to control problems in both parallel and serial CNC machine tool mecha-nisms.Finally, for the multi-axis CNC machine tools, a multi-input-multi-output(MIMO) LPV feedback controller is designed to directly minimize contouring errorin the task coordinate frame system.iiiPrefaceThis thesis is an original intellectual property of the author, Masih Hanifzadegan,working under the supervision of Prof. Ryozo Nagamune. This work, which presentsa linear parameter varying framework approach for control of feed-drive systems withdynamic variations in the computer numerical control machine tools, has been com-pleted in the Control Engineering Laboratory of the University of British Columbia.This work was funded by the Natural Sciences and Engineering Research Council ofCanada through the Canadian Network for Research and Innovation in MachiningTechnology.All of the experimental results presented in this thesis were obtained byconducing experiments on the flexible ball-screw drive system in the MechatronicsLaboratory and the Fadal CNC machine tool in the Manufacturing and Automa-tion Laboratory of the University of British Columbia. In all the following publi-cations, the author of this thesis was the lead researcher, responsible for all majorareas of controller design, coding, experimental implementation, data analysis, andmanuscript composition. Prof. Nagamune provided feedback and comments on re-search directions and the manuscript composition. Results in Chapters 7 and 8 willbe submitted for publication.ivJournal Publications• M. Hanifzadegan and R. Nagamune, “Tracking and Structural Vibration Con-trol of Flexible Ball-screw Drives with Dynamic Variations” in IEEE/ASMETransaction on Mechatronics, (in press). (Chapter 4 of this thesis)• M. Hanifzadegan and R. Nagamune, “Switching gain-scheduled control designfor flexible ball-screw drives” in ASME Journal of Dynamic Systems, Measure-ment and Control, vol. 136, no. 1, pp. 014503.1–014503.6, 2014. (Chapter 5of this thesis)• M. Hanifzadegan and R. Nagamune, “Smooth Switching LPV Controller De-sign for LPV Systems” in Automatica, vol. 50, no. 5, pp. 1481–1488, 2014.(Chapter 6 of this thesis)International Conference Publications• M. Hanifzadegan and R. Nagamune, “Robust Switching Control Synthesisfor Flexible Feed Drive Systems” in 1st International Conference on VirtualMachining Process Technology, Montreal, 2012. (Chapters 3 , 5 of this thesis)• M. Hanifzadegan and R. Nagamune, “Simultaneous Tracking and VibrationSuppression Control of Feed Drive Systems with Inertia and Stiffness Varia-tions” in 2nd International Conference on Virtual Machining Process Technol-ogy, Hamilton, 2013. (Chapter 4 of this thesis)• M. Hanifzadegan and R. Nagamune, “A Novel Approach to Smooth SwitchingControl of CNC Machine with Multi-Dimensional Dynamic Variations” in 3ndInternational Conference on Virtual Machining Process Technology, Calgary,2014. (Chapter 7 of this thesis)vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Machine Tool Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 21.2 High Speed Machining . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Feed-Drive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Feed-Drive Systems Control . . . . . . . . . . . . . . . . . . . . . . . 4vi1.4.1 Tracking control . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Contouring control . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Experimental Setups . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.1 A single-axis ball-screw drive system . . . . . . . . . . . . . . 61.5.2 A multi-axis SKM CNC machine tool . . . . . . . . . . . . . 71.5.3 A two-axis bipod PKM mechanism . . . . . . . . . . . . . . . 81.6 Dynamic Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.1 Dynamic variations due to mass removal . . . . . . . . . . . . 91.6.2 Dynamic variations due to table position . . . . . . . . . . . 101.6.3 Dynamic variations due to cutting path direction . . . . . . . 111.7 Controller Design Objective . . . . . . . . . . . . . . . . . . . . . . . 121.8 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 122 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Feed-Drive Systems Modeling and Identification . . . . . . . . . . . . 152.1.1 Modeling methods . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Identification methods . . . . . . . . . . . . . . . . . . . . . . 172.1.3 Dynamic variations and LPV modeling . . . . . . . . . . . . 182.2 Single-Axis Feed-Drive Control . . . . . . . . . . . . . . . . . . . . . 202.2.1 Classical control techniques . . . . . . . . . . . . . . . . . . . 202.2.2 Modern control techniques . . . . . . . . . . . . . . . . . . . . 212.2.3 Vibration control techniques . . . . . . . . . . . . . . . . . . 222.2.4 LPV control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.5 Switching LPV control . . . . . . . . . . . . . . . . . . . . . . 242.2.6 Smooth switching LPV control . . . . . . . . . . . . . . . . . 252.3 Multi-Axis Feed-Drive Control . . . . . . . . . . . . . . . . . . . . . 27vii3 Modeling and Identification of Feed-drive Systems . . . . . . . . 303.1 Ball-Screw Drive Systems . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Rigid-Body Mode Model . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Flexible Mode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Parameter calculations . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.3 Dynamic variations of the analytical model . . . . . . . . . . 423.4 System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.1 Rigid-body mode and friction identification . . . . . . . . . . 463.4.2 Flexible mode identification . . . . . . . . . . . . . . . . . . . 473.5 Dynamic Variations in Ball-Screw Drive Systems . . . . . . . . . . . 503.5.1 One varying parameter (xc) . . . . . . . . . . . . . . . . . . . 513.5.2 Two varying parameters (xc and mwor) . . . . . . . . . . . . 533.6 Linear Parameter Varying System Representation . . . . . . . . . . . 544 Parallel Tracking and Structural Vibration Control of the Ball-Screw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1 Control Objectives for Ball-screw Drive Systems . . . . . . . . . . . 584.2 A Parallel Controller Structure . . . . . . . . . . . . . . . . . . . . . 584.3 Controller Design Procedure . . . . . . . . . . . . . . . . . . . . . . . 604.3.1 Tracking controller design . . . . . . . . . . . . . . . . . . . . 604.3.2 Structural vibration controller design . . . . . . . . . . . . . . 624.3.3 Gain-scheduled controller design . . . . . . . . . . . . . . . . 654.3.4 Guidelines for tuning design parameters . . . . . . . . . . . . 664.3.5 Disturbance observer design . . . . . . . . . . . . . . . . . . . 684.4 Controller Design for Experimental Ball-Screw Drive Setup . . . . . 69viii4.4.1 Tracking controller design . . . . . . . . . . . . . . . . . . . . 694.4.2 Vibration controller design . . . . . . . . . . . . . . . . . . . 704.4.3 PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5.1 Frequency domain analysis . . . . . . . . . . . . . . . . . . . 724.5.2 Time domain analysis . . . . . . . . . . . . . . . . . . . . . . 725 Switching Gain-Scheduled Control of the Ball-Screw . . . . . . . 785.1 The Switching LPV Controller Design Method . . . . . . . . . . . . 795.1.1 Design problem . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.2 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . 835.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2.1 Controller design and analysis . . . . . . . . . . . . . . . . . . 845.2.2 Tracking performance analysis . . . . . . . . . . . . . . . . . 855.2.3 Bandwidth analysis . . . . . . . . . . . . . . . . . . . . . . . 865.2.4 Flexible mode suppression analysis . . . . . . . . . . . . . . . 876 Smooth Switching LPV Controller Design for LPV Systems . . 896.1 A Smooth Switching LPV Controller Design Problem . . . . . . . . 906.1.1 Notation for interval sets . . . . . . . . . . . . . . . . . . . . 906.1.2 Description of an LPV plant and an LPV controller . . . . . 916.1.3 Description of a smooth switching LPV controller . . . . . . 936.1.4 Statement of a controller design problem . . . . . . . . . . . . 936.2 Formulation of a Feasibility Problem . . . . . . . . . . . . . . . . . . 956.2.1 Conditions for system stability (i) and L2 gain (ii) . . . . . . 956.2.2 Conditions for smooth switching (iii) . . . . . . . . . . . . . . 97ix6.2.3 Conditions for rate of change of controller (iv) . . . . . . . . 986.2.4 A non-convex feasibility problem . . . . . . . . . . . . . . . . 1006.2.5 Reduction to a finite-dimensional problem . . . . . . . . . . . 1016.3 An Iterative Descent Algorithm . . . . . . . . . . . . . . . . . . . . . 1026.4 Cases for a Two-Dimensional Parameter Space . . . . . . . . . . . . 1057 Smooth Switching LPV Control of SKM and PKM . . . . . . . . 1117.1 SKM Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.1.1 Control problem and generalized plant . . . . . . . . . . . . . 1127.1.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 1137.1.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 1167.2 PKM Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2.1 A bipod in PKM system . . . . . . . . . . . . . . . . . . . . . 1207.2.2 Control objective and augmented plant . . . . . . . . . . . . 1237.2.3 Control structure and design . . . . . . . . . . . . . . . . . . 1257.2.4 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 1268 LPV Contouring Control of CNC Machine Tools . . . . . . . . . . 1338.1 Serial CNC Machine tool Modeling and Coordinate Transformations 1338.1.1 CNC machine tools mechanism . . . . . . . . . . . . . . . . . 1338.1.2 CNC machine tool model . . . . . . . . . . . . . . . . . . . . 1348.1.3 2D coordinate transformation . . . . . . . . . . . . . . . . . . 1358.1.4 3D coordinate transformation . . . . . . . . . . . . . . . . . . 1378.2 A MIMO Controller Structure and Design . . . . . . . . . . . . . . . 1388.2.1 Control objectives . . . . . . . . . . . . . . . . . . . . . . . . 1388.2.2 Controller structure . . . . . . . . . . . . . . . . . . . . . . . 139x8.2.3 Contouring controller design . . . . . . . . . . . . . . . . . . 1398.3 Identification and System Parameters . . . . . . . . . . . . . . . . . 1428.4 2D Contouring Control Problem . . . . . . . . . . . . . . . . . . . . 1448.4.1 Design of an LPV controller for 2D trajectories . . . . . . . . 1458.4.2 Frequency domain simulation results . . . . . . . . . . . . . . 1468.4.3 Time domain simulation results . . . . . . . . . . . . . . . . . 1478.4.4 Time domain experimental results . . . . . . . . . . . . . . . 1478.5 3D Contouring Control Problem . . . . . . . . . . . . . . . . . . . . 1508.5.1 Design of an LPV controller for 3D trajectories . . . . . . . . 1508.5.2 Frequency domain simulation results . . . . . . . . . . . . . . 1528.5.3 Time domain simulation results . . . . . . . . . . . . . . . . . 1529 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Appendix A Hybrid Method in Ball-Screw Drive Systems Modeling175A.1 Wave Equations with Boundary Conditions . . . . . . . . . . . . . . 175A.2 Motor and Table Equations of Motion . . . . . . . . . . . . . . . . . 176A.3 Solving the Differential Equation Problem . . . . . . . . . . . . . . . 177xiList of Tables3.1 Parameters of the ball-screw drive systems . . . . . . . . . . . . . . . 323.2 Identified rigid-body mode parameters . . . . . . . . . . . . . . . . . 483.3 LPV model of the system parameters as a function of xc . . . . . . . 533.4 LPV model of the system parameters as a function of xc and mwor . 554.1 Design parameters of the gain-scheduled controller . . . . . . . . . . 704.2 Cutting geometry and coefficients . . . . . . . . . . . . . . . . . . . . 764.3 Absolute tracking error in time domain experiments . . . . . . . . . 765.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Tracking error in time domain experiments for f1(t) = 1.8 sin(1340t)and f2(t) = 1.3 sin(1340t) [V] . . . . . . . . . . . . . . . . . . . . . . 875.3 Bandwidth ωc and resonance amplitude reduction GR . . . . . . . . 887.1 Parameters of the bipod system . . . . . . . . . . . . . . . . . . . . . 1277.2 The maximal values in the time domain performance of LPV controllers1318.1 Rigid-body mode parameters of 3-axis Fadal CNC machine tool . . . 1438.2 Flexible mode parameters of 3-axis Fadal CNC machine tool . . . . 1458.3 Weighting function parameters in 2D contouring control . . . . . . . 1468.4 Weighting function parameters in 3D contouring control . . . . . . . 151xiiList of Figures1.1 Two types of CNC mechanisms . . . . . . . . . . . . . . . . . . . . . 31.2 Two types of feed-drive systems . . . . . . . . . . . . . . . . . . . . . 41.3 2D path; tracking and contouring errors . . . . . . . . . . . . . . . . 51.4 The ball-screw system test setup . . . . . . . . . . . . . . . . . . . . 71.5 Three-axis Fadal CNC machine tool system . . . . . . . . . . . . . . 81.6 Two-axis bipod PKM mechanism . . . . . . . . . . . . . . . . . . . . 91.7 FRF of the ball-screw drive system for workpieces with different weights 101.8 FRF of the ball-screw drive system with the table in different positions 101.9 Equivalent mass variations of a bipod mechanism as a function oflinks’ rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1 A ball-screw drive system components . . . . . . . . . . . . . . . . . 313.2 Rigid-body model of the ball-screw drive systems including inertiaand damping, the current amplifier, and the motor winding in torquecontrol mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 FRF of G0 (solid blue line) and G1 (dash-dot green line) . . . . . . . 353.4 Hybrid free-body diagram of the ball-screw drive system . . . . . . . 363.5 Bode plot of the analytical model of the ball-screw system with (solidblue line) and without (dashed red line) structural damping . . . . . 40xiii3.6 Model validation Bode-diagrams of experimentally measured (reddash-dot lines) and calculated (solid blue lines) FRFs where the nutis placed in different xc locations with no workpiece on the table . . 423.7 Model validation Bode-diagrams of experimentally measured (reddash-dot lines) and calculated (solid blue lines) FRFs where the nutis located at xc=0.49 [m] while different workpiece masses mwor areplaced on the table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8 System parameters variations as a function of the nut location xc . . 443.9 System parameters variations as a function of the workpiece mass mwor 453.10 System parameters variations as a function of nut location xc andworkpiece mass mwor . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.11 Rigid-body mode identification . . . . . . . . . . . . . . . . . . . . . 483.12 Comparison of real and imaginary parts of the measured (dash-dotred line) and the estimated (solid green line) flexible mode FRFs . . 493.13 Comparison of the measured (dash-dot red line) and the identified(solid green line) total FRFs . . . . . . . . . . . . . . . . . . . . . . . 503.14 Validation of G22 for identified FRF (blue line), and measured FRF(red line) with hammer test . . . . . . . . . . . . . . . . . . . . . . . 513.15 Estimated parameters at identified positions (red cross marker) andLPV functions in (3.27) (green dashed line) . . . . . . . . . . . . . . 523.16 Comparison of the estimated parameters at identified positions (redcross marker) and LPV model (dashed green line) with the analyticalmodel (solid blue line) . . . . . . . . . . . . . . . . . . . . . . . . . . 543.17 Estimated parameters at identified positions (magenta points) andcontinuous LPV function (surface) . . . . . . . . . . . . . . . . . . . 55xiv4.1 Block diagram of the parallel controller structure . . . . . . . . . . . 594.2 Block diagram of the H∞ tracking controller design . . . . . . . . . . 614.3 Block diagram of the structural vibration controller design . . . . . . 644.4 Weighting function selections, (a) |W−1e | (red dash-dot line) and |Tre|(purple solid line), (b) |W−1f | (red dash-dot line) and |Tfu| (purplesolid line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Gain plots of S transfer function (top) and Tru transfer function (bot-tom) without KVib (dash-dot blue line) and with KVib (solid red line);the dashed lines in top and bottom figures indicate the gain of W−1eand -3 dB, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . 714.6 Gain plots of the transfer function Tfu with just KTrack (blue), withboth KTrack and KVib (red), and the weighting function Wf−1 withdashed line (black) where the table xc is placed at 200 (dash-dot line),350 (dashed line) and 500 [mm] (solid line) . . . . . . . . . . . . . . 714.7 Measured gain plots of the transfer functions from fc to uc in Case I(blue), Case II (red) and Case III (green) where the table is locatedat 250 [mm] (dash-dot line) and 450 (solid line) . . . . . . . . . . . . 734.8 Reference trajectory signal with constant jerk . . . . . . . . . . . . . 744.9 Tracking error comparison of Case I (dash-dot blue line) and Case II(red solid line) (top), and tracking error comparison of Case II (redsolid line) and Case III (dash-dot green line) (bottom) while there isno external disturbance . . . . . . . . . . . . . . . . . . . . . . . . . 75xv4.10 Tracking error comparison of Case I (dash-dot blue line) and Case II(red solid line) (top), and tracking error comparison of Case II (redsolid line) and Case III (dash-dot green line) (bottom) while there isharmonic disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . 754.11 Simulated resultant cutting forces: half immersion up milling force(top) and half immersion down milling force (bottom) . . . . . . . . 774.12 Measured tracking error in emulated half immersion up milling forCase I (dash-dot blue line) and for Case II (solid red line) (top), andCase II (solid red line) and Case III (dash-dot green line) (bottom) 774.13 Measured tracking error in emulated half immersion down milling forCase I (dash-dot blue line) and Case II (solid red line) (top), andCase II (red solid line) and Case III (dash-dot green line) (bottom) 775.1 Block diagram of the switching feedback control system. . . . . . . . 795.2 Augmented block diagram for controller design . . . . . . . . . . . . 815.3 Selected weighting functions Wr and Wf together with Tfe and Tre . 845.4 Reference trajectory and tracking error for no disturbance condition 865.5 Tracking error in Case I (red) and Case II (green) for harmonicdisturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.6 Gain of |Try| in Case I (red) and Case II (green); xc = 250 [mm](solid line) and xc = 450 [mm] (dashed line) . . . . . . . . . . . . . . 875.7 FRF magnitude of the transfer functions from fc to uc with different(xc,mwor); Case I (red), Case II (green) and open loop (black) . . . 886.1 Sub-intervals for switching control (I = 3) . . . . . . . . . . . . . . . 916.2 Controllers assigned to sub-intervals (I = 3) . . . . . . . . . . . . . . 93xvi6.3 An example of transitions of the η-value . . . . . . . . . . . . . . . . 1046.4 Rectangles for switching control (I = 3, J = 2) . . . . . . . . . . . . 1077.1 The output feedback structure in control of the ball-screw drive system1127.2 γ-value and η-value after each iterations . . . . . . . . . . . . . . . . 1147.3 The x-y plane trajectory . . . . . . . . . . . . . . . . . . . . . . . . . 1157.4 Time domain reference position and velocity of x (dashed green line)and y (red solid line) drives . . . . . . . . . . . . . . . . . . . . . . . 1157.5 The simulated tracking error and control signal in case of fc = 0, withthe non-switching controller (dashed blue), the hysteresis switchingcontroller (dash-dot green), the smooth switching controller (solid red),and switching instants (vertical dash-dot lines) . . . . . . . . . . . . 1167.6 The simulated tracking error and control signal in case of fc = f1, withthe non-switching controller (dashed blue), the hysteresis switchingcontroller (dash-dot green), the smooth switching controller (solid red),and switching instants (vertical dash-dot lines) . . . . . . . . . . . . 1177.7 FFT of the tracking error for three cases . . . . . . . . . . . . . . . . 1187.8 Reference trajectory position and velocity . . . . . . . . . . . . . . . 1187.9 The experimental tracking performance when fc = 0 for non-switchingcontroller (dotted black), hysteresis switching controller (dashed red),the smooth switching controller (solid blue), and switching instants(vertical dash-dot lines) . . . . . . . . . . . . . . . . . . . . . . . . . 1187.10 The experimental tracking performance when fc = 50sin(314t) [N] fornon-switching controller (dotted black), hysteresis switching (dashedred), smooth switching controller (solid blue), and switching instants(vertical dash-dot lines) . . . . . . . . . . . . . . . . . . . . . . . . . 119xvii7.11 A bipod PKM system mechanism . . . . . . . . . . . . . . . . . . . . 1207.12 Augmented block diagram for controller design . . . . . . . . . . . . 1247.13 Block diagram of the smooth switch controller structure . . . . . . . 1267.14 The bipod PKM parameters H1 (solid line) and Cm (dashed line)variations as functions of rotation angle φ . . . . . . . . . . . . . . . 1277.15 Gain plot of S (top) and transfer functions from fd to x1 (bottom)for Θ(1) (blue), Θ(1,2) (yellow) and Θ(2) (red), and |We|−1 (dash-dotline) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.16 The boundaries of the switching intervals (dash-dot line) for (I = 3)and the parameter variations trajectory (solid line) . . . . . . . . . . 1297.17 The tracking performance when fd = 0 with the non-switching con-troller (blue), the hysteresis switching controller (green), the smoothswitching controller (red), and switching instants (vertical dash-dotlines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.18 The tracking performance when fd = f0 with the non-switching con-troller (blue), the hysteresis switching controller (green), the smoothswitching controller (red), and switching instants (dash-dot lines) . . 1328.1 A 3-axis CNC machine . . . . . . . . . . . . . . . . . . . . . . . . . . 1348.2 2D trajectory coordinate transformation and contouring error esti-mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.3 3D trajectory coordinate transformation and contouring error esti-mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.4 Block diagram of the MIMO LPV controller structure in contouringerror control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.5 Block diagram of the MIMO controller design and weighting functions 141xviii8.6 Flexible modes measured FRF (solid blue line) botained using ac-celerometer and identified transfer functions (dash-dot green line) . . 1448.7 Block diagram of the MIMO controller structure in 2-axis control . . 1458.8 Range of θ variations (blue line) and gridded points (red cross) in 2Dcontouring controller design . . . . . . . . . . . . . . . . . . . . . . . 1468.9 Gain plots of sensitivity transfer function from rx to et (top) andfrom ry to en (bottom); gain plot of inverse of Wt and Wn weightingfunctions (dashed lines) . . . . . . . . . . . . . . . . . . . . . . . . . 1468.10 Reference trajectories in 2D contouring control . . . . . . . . . . . . 1488.11 Time domain simulation results in 2D countouring control . . . . . . 1498.12 Time domain experimental results in 2D countouring control . . . . 1498.13 Block diagram of the MIMO controller structure in 3-axis control . . 1508.14 Range of θ variations (inside the blue line area), gridded points (redcross), and trajectory of αx and αy (dashed green line) in 3D con-touring controller design . . . . . . . . . . . . . . . . . . . . . . . . . 1528.15 Gain plots of sensitivity transfer functions from rx to et (top), fromry to en (middle), and from rz to en (bottom); gain plots of inverseof Wt and Wn weighting functions (dashed lines) . . . . . . . . . . . 1528.16 Reference trajectories in 3D contouring control . . . . . . . . . . . . 1548.17 Time domain simulation results in 3D contouring control . . . . . . 154xixList of SymbolsAs Average cross section of the screw.Be Viscous damping of the rigid-body mode of the ball-screw.cc Axial viscous damping of the table of ball-screw.chys Structural damping constant.cm Viscous damping of the motor shaft.Cs Damping matrix in equation of motion.ds Diameter of the screw.e Tracking error.en Normal tracking error in the t-n task coordinate frame.~en−b Projection of tracking error on the n-b plane.Es Young’s moduli of the screw.et Tangential tracking error in the t-n task coordinate frame.ex Tracking error of the x-axis in the x-y-z coordinate system.xxey Tracking error of the y-axis in the x-y-z coordinate system.ez Tracking error of the z-axis in the x-y-z coordinate system.F1 Block-diagonal matrix for the condition (6.18).f1 Actuator force applied to the table T1 in the bipod system.F2 Vector of length 2(M + 1)(I − 1) for the condition (6.21).f2 Actuator force applied to the table T2 in the bipod system.F3 Block-diagonal matrix for the condition (6.30).fc External force applied to the table of the ball-screw.fd Disturbance force applied to the table T3 in the bipod.fm Equivalent force applied by the motor.G Open-loop plant transfer function of the feed-drive system.Gflex,k k-th flexible mode model of the feed-drive system.Grigid Rigid-body mode transfer function of the feed-drive system.Nmod k-th flexible mode identified transfer function.Gs Shear moduli of the screw.H∞ Type of controller design approach.I Number of divisions.im Motor current variable.Jcoup Inertia of couplings system between the screw and the DC motor.xxiJe Equivalent moment of inertia of the ball-screw.Jenc Inertia of encoders.Jm Moment of inertia of the motor shaft, coupling and encoder.Jmot Inertia of motor shaft.Js Average second inertia of the screw cross section.Jtach Inertia of tacho-generator.K Set of functions of K(i) for i = 1 to I.ka Motor amplifier gain.kai DC motor amplifier integral gain.kap DC motor amplifier proportional gain.kb Axial stiffness of the thrust bearing and its housing.kc Torsional stiffness of the coupling between the motor and screw.kci Integral gain of the PI current controller.kcp Proportional gain of the PI current controller.KDOB Disturbance observer.ke Total equivalent stiffness of the ball-screw.kemf Back-EMF constant.K(i) Local LPV controller in Θ(i) interval.K(i+1) Local LPV controller in Θ(i+1) interval.xxiiK({i,i+1}) Local LPV controller in Θ({i,i+1}) interval.km Equivalent stiffness of the motor armature.kn Axial stiffness of the nut connection.Ks Stiffness matrix in dynamic equation of motion.kt DC motor torque constant.K(θ) Parameter-varying system matrix of the LPV controller.KTrack Tracking controller.KVib Vibration suppression controller.kW Order of the shaping function W .Lm DC motor inductance.Ls Length of the links in the bipod system.ls Length of the screw.M The control continuity order at the switching surface.L Linear combination of two matrix-valued functions.mc Summation of mass of the workpiece and table.MH High frequency gain of the shaping function W−1.ML Low frequency gain of the shaping function W−1.Ms Mass matrix in dynamic equation of motion.ms Mass of the screw.xxiiimtab Table mass.mwor Mass of the workpiece on the table of the ball-screw.P Parameter-dependent Lyapunov matrix.pt Pitch value of the screw.Q Stable low pass filter in the disturbance observer.R Domain of real number.r Reference signal of the feed-drive axes.rg Gear ratio of the ball-screw.Rm DC motor resistance.rx Reference trajectory of the x-axis.ry Reference trajectory of the y-axis.S Sensitivity transfer function from r to e.S1 Link number one in the bipod system.S2 Link number two in the bipod system.S3 Link number three in the bipod system.T1 Table number one in the bipod system.T2 Table number two in the bipod system.T3 Table number three in the bipod system.Tf Friction torque apply to the motor.xxivTfe Transfer function from fc to e.Tfu Closed-loop transfer functions from fc to uc.Tn−b Transformation matrix to project from the x-y-z coordinate system into n-bplane of the t-n-b coordinate system.TR Rotation matrix transformation.Tre Transfer function from r to e.Tru Complementary sensitivity transfer function from r to uc.Tt Transformation matrix from the x-y-z coordinate system into the tangential di-rection of the t-n-b coordinate system.u Control input in state-space realization.uc Position of the ball-screw table.ucx Table position in x direction.ucy Table position in y direction.ucz Table position in z direction.um Equivalent motor displacement.us Axial deformation of the screw.V Interval set of θ rate of change.vc Disturbance observer output signal.vm Input voltage applied to the DC motor.xxvvt Output of KTrack controller.vv Output of KVib controller.w Exogenous input in state-space realization.We Weighting functions to penalize the tracking error e.Wf Weighting functions to penalize disturbance force fc.Wn Weighting functions to penalize the contouring error.Wr Weighting functions to penalize the reference signal.Wt Weighting functions to penalize the tangential tracking error.Wv Weighting functions to penalize the control effort.x State vector in state-space realization.x1 Position of the table T1 in the bipod system.x2 Position of the table T2 in the bipod system.xc Axial position of the table.xcl Closed-loop system state in the state-space realization.xm State vector.y Measured output in the state-space realization.z Performance output in the state-space realization.xxviα(i,i+1) Scalar-valued function maps θ ∈ Θ({i,i+1}) into a real number.α(i,i+1)P Linearly interpolating function.αx Angle between the tool path and the x-axis.αy Angle between the tool path and the y-axis.αz Angle between the tool path and the z-axis.α Set of functions of α(i,i+1)P for i = 1 to I − 1.β(j,j+1) A scalar-valued function maps θ2 ∈ [θ(j)2 , θ(j+1)2 ] into a real number.γ Bound on the L2-gain of the closed-loop system. Contouring error.ζ Damping ratio.η Bound on the rate of change of the controller matrix.ηd Desired value of η.ηˆ Minimized value of η in (6.45).Θ Interval set of θ.θ Time varying gain-scheduling parameter of the LPV system.Θ(i) Sub-interval of Θ.Θ({i,i+1}) Transitional sub-interval.θm Rotation angle of the motor.θs Torsional deformation of the screw.xxviipi Ratio of circumference of circle to its diameter.ρs Density of the screw.Σcl Closed-loop system matrix.τm Torque applied by DC motor.φ Rotation angle of the bipod links.ωb Unit-gain crossing frequency of the shaping function W−1.ωd Damped resonant frequency.xxviiiList of AbbreviationsCCC cross-coupling control.CNC computer numerical control.DC direct current.DOF degrees-of-freedom.EMF electromotive force.FEM finite element method.FFT fast Fourier transfer.FRF frequency response function.HSM high speed machining.LMI linear matrix inequality.LPV linear parameter varying.LQR linear-quadratic regulator.LTI linear time invariant.xxixMAE mean absolute error.MIMO multi-input-multi-output.PI proportional and integral.PID proportional, integral and derivative.PKM parallel kinematic machine.PWM pulse-width modulation.RL resistor–inductor.SISO single-input single-output.SKM serial kinematic machine.VCNC virtual CNC.VMC vertical machining center.xxxAcknowledgementsThis journey would not have happened without the support of my family, profes-sors, and friends. I would like to thank my father, Ali, my mother, Nahid, mybrother and sisters, Farhad, Zoha and Hoda, and my girlfriend, Mina Arabkhedri,for encouraging me and inspiring me to follow my dreams and take risks.Foremost, I would like to thank my supervisor, Professor Ryozo Nagamune,for his enormous support, advice and patience throughout the entire research pro-cess. He has been a knowledgeable mentor and an incisive guide to me. In addition,I have to thank Professor Yusuf Altintas, the principal investigator of the CanadianNetwork for Research and Innovation in Machining Technology, for his financial andacademic support and help in my research by providing the best environment toconduct my research.Furthermore, I would like to thank the rest of my thesis committee, ProfessorFarrokh Sassani, Professor Steve Feng and Professor Bhushan Gopaluni, for theirgreat academic support of my thesis.My gratitude goes to the previous members of Control Engineering Labora-tory (CEL) at the University of British Columbia, Dr. Ehsan Azadi Yazdi, Dr. Da-nial Sepasi, Mr. Marious Postma, Mr. Moein Javadian and Mr. Omid Bagheriehfor their amazing help and support in the early stage of my research, and currentxxximembers of CEL, Mr. Pan Zhao, Mr. Arman Zandinia, Mr. Jeffrey Homer, Mr. Dil-lon Melamed, and Mr. Alireza Alizadegn for being supportive colleagues and greatfriends. I also want to thank members of Manufacturing and Automation Labo-ratory at the University of British Columbia for great collaborations in providingequipment and conducting experiments including Ms. Sneha Tulsyan, Mr. ByronReynolds, Mr. Fan Chen, Mr. Alexander Yuen, Mr. Mohammad Rezayi Khoshdar-regi.I would like to thank my dear friends, Dr. Ali H. Kashani, Ms. Ajung Moon,Ms. Shadi Shirazi, Mr. Farzad Khademolhossieni, Dr. Amir Rassuli, Mr. BehnamRavazi, Mr. Sina Radmard and Mr. Mahdi Saeidifar for their priceless friendship,extreme support and encouragements.MASIH HANIFZADEGANTHE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2014xxxiiThis thesis is dedicated to my loving parents, Ali and Nahid,my wonderful siblings, Farhad, Zoha, and Hoda,and my supportive and adoring girlfriend, Mina,each of whom inspired me throughtheir courageous strength, persistent faith, and endless love.xxxiiiChapter 1IntroductionA computer numerical control (CNC) machine tool system is an automated ma-chine tool which is controlled by a microprocessor without any human intervention.The programmable CNC microprocessor receives feedback signals from sensors, andbased on the machining processes, sends control signals to the feed-drive and spindlesystems. The machine tool feed-drive systems actuate CNC mechanisms to changethe workpiece position and orientation, while the spindle system holds and spins acutting tool to remove material from the workpiece. In the machining processes,selective parts of the workpiece are removed automatically while the cutting toolfollows a cutting path. The CNC machine tools are equipped with a single or multi-cutting tools to perform required manufacturing processes such as turning, milling,boring, slotting, sawing, drilling and threading.Depending on the applications of the machine tools, the type of the feed-drive mechanisms and the number of degrees-of-freedom (DOF) are selected. Forinstance, a lathe machine has two-DOF translational mechanisms for cylindricalcutting, whereas a milling machine has three-DOF translational mechanisms in order1to place the cutting tool at desired positions in its working volume. Furthermore, afive-axis machine tool system is able to place the cutting tool and the workpiece inany relative positions and orientations.1.1 Machine Tool MechanismsTwo typical machine tool mechanisms, widely used in the industry, are open-loopkinematic chain and closed-loop kinematic chain mechanisms (see Figure 1.1). Inan open-loop mechanism, such as the three-axis serial kinematic machine (SKM)structure shown in Figure 1.1 (a), there is only one kinematic chain to transferthe machining forces to the ground. The SKM system ensures larger work-spacevolume, simpler dynamic behavior and less stiffness compared to PKM. On theother hand, the parallel kinematic machine (PKM) mechanism, such as the tricepsPKM mechanisms of the Loxin company shown in Figure 1.1 (b), has more thanone kinematic chains to the ground. Hence, the closed-loop mechanism has a higherstiffness, therefore, it is more suitable for a heavy duty machining process. However,the PKM system typically operates in a smaller working volume, and the analysisof its dynamic response is more complicated compered to the SKM mechanism.1.2 High Speed MachiningHigh speed machining (HSM) has been introduced in the manufacturing industryto increase the productivity in the machining processes with higher accuracy andquality. There are various industries, such as automotive, electronic componentsmanufacturing, aircraft and die mold, which utilize the HSM to manufacture theirproducts with a minimum possible time and cost. The global demand for higher2(a) Fadal 3-axis CNC with SKM structure, source: www.fadal.com(b) Loxin tricep withPKM structure, source:www.pkmtricept.comFigure 1.1: Two types of CNC mechanismsmanufacturing efficiency has led to the development of the HSM technology in differ-ent fields such as tooling materials, bearing systems, feed-drive systems and digitalprocessing systems. Among these fields, high speed and accurate positioning con-trol of the feed-drive systems provide higher productivity and improves machiningtolerance and surface finish.1.3 Feed-Drive SystemsA feed-drive system locates the workpiece and cutting tool at a desired positionand determines the orientation by actuating the machine tool mechanism. Twocommon feed-drive systems, direct and indirect, are depicted in Figure 1.2. In thedirect drive system (see Figure 1.2 (a)), a linear motor is used to locate a slide dueto a magnetic force between a magnet and a stator-coil. Although the magneticmechanism of this actuator provides high speed and acceleration (up to 10g), themechanism without a gear severely limits the maximum acceleration as the cutting3force and load capacity increases.In the indirect drive system (see Figure 1.2 (b)), a rotary motor drives aworkpiece via a gear, ball-screw and nut system. Rotational motion is translated intolinear motion by rolling steel balls between the screw shaft and the nut. The screwand gear systems support high load capacity and high rigidity of the indirect feed-drive systems against the machining force. Although the contact-based mechanismof the ball-screw system restricts speed and acceleration capacity, the ball-screwsystem maintains its acceleration capacity for a large range of workpiece inertiavariations. High efficiency, high service life, high load capacity, high rigidity, largedisplacement stroke, and low heat dissipation are the reasons for the widespreadapplication of the ball-screw drive systems in the manufacturing industry.(a) Direct feed-drive system: linear motorl q f v Screw Nut Motor (b) Indirect feed-drive: ball-screwFigure 1.2: Two types of feed-drive systems1.4 Feed-Drive Systems ControlIn the machining of a product, the quality of machining strictly depends on theaccurate position control of the tooltip with respect to the workpiece in the pres-ence of friction and machining forces. Achieving higher accuracy and performance4in manufacturing of mass products motivates synthesis of advanced model-basedcontrol approaches to further enhance the performance of machining processes withhigh speed, low tolerance and better surface finishing.The performance of the machine tool feed-drive systems can be evaluatedby its ability to minimize two important errors, i.e., the tracking error and thecontouring error. As it is illustrated in Figure 1.3, the tracking error is the deviationof the workpiece actual position from the desired position, while the contouring erroris the deviation of the cutting tool from the tool path trajectory. In this figure, thet-n is the tangential and normal coordinate system attached to the trajectory andθ is the angle between t and x axes.The servo control of the CNC machine tool system has been split into twocategories, the tracking control and the contouring control. In the tracking control,an interpolator creates the desired workpiece and tooltip trajectory, and calculatesthe reference position of each axis individually. Instead of minimizing the trackingerror of each axis, the overall multi-axis control performance can be fulfilled byminimizing the contouring error.Figure 1.3: 2D path; tracking and contouring errors51.4.1 Tracking controlIn the tracking control approach, table and/or tool motion in each axis is controlledindividually to minimize the tracking error of each axis. In this approach, the mainobjectives of the feed-drive control are high precision tracking, high bandwidth,and disturbance attenuation. Accomplishments of these objectives are hindered byseveral factors such as disturbance forces, frictions, and dynamic variations.1.4.2 Contouring controlAlthough tracking control of each axis reduces the corresponding tracking error, itdoes not guarantee the minimization of the contouring error. To resolve this issue,the contouring control approach directly aims at minimizing the contouring error.Since the direction of the path changes during the cutting process, the directionin which we estimate the contouring error also varies. Hence, the path orientationchange can be considered as a source of dynamic variations in multi-axis CNCmachine tool control.1.5 Experimental SetupsIn this thesis, various control algorithms in following three experimental setupsare tested. These setups include a single-axis ball-screw drive, a multi-axis CNCmachine tool, and a two-axis bipod system.1.5.1 A single-axis ball-screw drive systemThe experimental setup of the single-axis ball-screw drive system in the Mechatron-ics Laboratory at University of British Columbia is illustrated in Figure 1.4. An6Figure 1.4: The ball-screw system test setup820 [mm] long screw is driven directly by a brushless direct current (DC) motorwhich is operated in the current control loop. A table with a 20 [kg] mass slides onthe roller bearings guideway and its location, with a 380 [mm] stroke, is measuredby a linear encoder with 100 [nm] resolution. The rotary encoder at the edge of thescrew measures the angle of the shaft.For real-time control implementation and variable monitoring of the feed-drive systems, we use a microcontroller DS1103 (dSPACE, Inc.) with the ControlDesk software interface. The main advantages of dSPACE system are its high reli-ability, direct compatibility with the MATLAB, SIMULINK, and easy programma-bility. The computed control signal is transferred from the microcontroller to themotors amplifiers via the digital to analogue converter unit on the dSPACE systemand the linear and rotary encoders are connected through the encoder interface unitto the dSPACE system.1.5.2 A multi-axis SKM CNC machine toolIn the multi-axis control, multi-feed-drive systems should be controlled simultane-ously in order to move the cutting tool and the workpiece through cutting trajec-7tories. For the experimentation purposes, we use an industrial 3-axis Fadal 2216vertical machining center (VMC) which is located at the Manufacturing Automa-tion Laboratory in University of British Columbia and shown in Figure 1.5. Thefeed-drive translates the table in the x and y directions, while the column includingthe spindle travels only in the z direction.The Fadal CNC machine has been equipped with open architecture con-trol which allows direct control of the feed-drive systems and spindle by externalcomputer, microprocessor, or microcontroller. The motors are connected to torquecontrol mode amplifiers which receive the control signals from dSPACE.Figure 1.5: Three-axis Fadal CNC machine tool system1.5.3 A two-axis bipod PKM mechanismThere is a bipod set-up to study the coupling of drive shafts in the PKM system atFraunhofer Institute for Machine Tools and Forming Technology (IWU) in Chem-nitz, Germany. The mechanism of the test stand is shown in Figure 1.6. In thissystem, a cutting tool or workpiece is placed on the working table. The workingtable is connected to the linear drive actuators by three rigid links and revolute8joints. Rotations of the parallel links at the revolute joints convert linear motorsmotion into two-dimensional motion of the working table. In the bipod mechanism,the resultant machining and disturbance forces are applied to the working table.Figure 1.6: Two-axis bipod PKM mechanism1.6 Dynamic VariationsOne of important aspects in this work is to design a robust controller for machinetool systems, by considering the following three parameter variations inherent tomost of such systems.1.6.1 Dynamic variations due to mass removalDynamic variations can happen due to cutting of the workpiece during the machin-ing process. These variations can be considerably large. For example, for somemachining applications, the blank workpiece mass can be heavier than the tablemass. If most of the blank mass is removed by the machining process, the totaltable and workpiece mass can decrease a maximum of 90%. This inertia reductionchanges the feed-drive dynamics, and thus affects the closed-loop stability and per-formance. For instance, the effect of the workpiece mass on the system dynamics9has been investigated by measuring the frequency response function (FRF) of a flex-ible ball-screw drive system in Figure 1.7 for five workpieces with different weights.According to this figure, the resonance frequency and amplitude change for eachworkpiece. In addition, since the load is directly applied to the motor in the directdrive systems, the workpiece mass reduction drastically changes the inertia of thedirect drive systems.Figure 1.7: FRF of the ball-screwdrive system for workpieces with dif-ferent weightsFigure 1.8: FRF of the ball-screwdrive system with the table in differ-ent positions1.6.2 Dynamic variations due to table positionIn both SKM and PKM mechanisms, the dynamics of machine tool system changesas a function of its position. In the flexible ball-screw drive system in Section 1.5.1,an important feature is that the dynamics of the flexible modes changes as the tablemoves along the screw. This is because the stiffness of the system varies dependingon the table location, due to the change of the active length of the screw between themotor shaft and the table. In addition, run-out effect is another cause for dynamicvariations. The run-out effect occurs due to the misalignment of the ball bearing axisholding the screw shaft. The variations in FRF of a flexible mode for various table10positions are shown in Figure 1.8. These position variations change the amplitudeand frequency of the flexible modes.In the PKM mechanism, the rigid-body mode parameters change as a func-tion of mechanism movement due to the coupled kinematic chain. For instance, inthe bipod system in Figure 1.6, the equivalent mass of the system varies drasticallyas a function of mechanism rotation as shown in Figure 1.9. According to this figure,we can observe that the equivalent mass increases by about four times as the link’srotation angle changes from 10 to 80 degrees.Figure 1.9: Equivalent mass variations of a bipod mechanism as a function of links’rotation1.6.3 Dynamic variations due to cutting path directionThe third source of variations occurs based on the direction of the cutting trajectory.As mentioned before, in the contouring control, the contouring error is a function ofthe cutting trajectory direction and tracking error. For instance, in the 2D trajectorywhich is shown in Figure 1.3, contouring error can be estimated by transformation ofthe tracking error components from the x-y coordinate system into the t-n coordinatesystem. In 2D trajectory, this coordinate transformation becomes a θ rotation withrespect to the x-axis. For simplicity, the contouring error, ε, is estimated as ε(θ) =− sin(θ)ex + cos(θ)ey, where ex and ey are tracking errors in the x and y directions,11respectively. Therefore, the direction that the contouring error should be minimizedchanges as a function of path direction.1.7 Controller Design ObjectiveThe main goal in this research is to propose a systematic method of designing andimplementing a linear parameter varying (LPV) controller for single-axis and multi-axis control of the CNC machine tool feed-drive systems with SKM and PKM mech-anisms. It is required that, for a large range of dynamic variations, the controllerensures the closed-loop stability and advances the following performances:• minimum tracking error in single-axis and contouring error in multi-axis per-formance,• maximum bandwidth for high speed tracking performance,• resonance vibration suppression of flexible ball-screw drives,• compensation performance for machining and friction forces.1.8 Organization of the ThesisThe rest of this thesis is organized as follows. The literature review has been givenin Chapter 2. The analytical modeling and system identification of the ball-screwdrive system have been presented in Chapter 3. In the analytical modeling, we havemodeled the rigid-body mode dynamic and flexible mode dynamics by consideringthe axial and torsional vibrations. The system dynamics has been identified byleast squares fitting of the model with measured FRF while the table was placed atdifferent locations with different workpiece weights. Finally, the LPV model of the12ball-screw drive system has been derived in order to design controllers for the modelin Chapters 4, 5 and 7.Having the LPV model, we have proposed the parallel structure control de-sign approach in Chapter 4. In this method, we have designed two controllers,one is a linear time invariant (LTI) controller and the second one is an LPV gain-scheduled controller. The LTI controller aims at moving the workpiece table so thatits position tracks reference trajectories with high speed and high accuracy. TheLPV gain-scheduled controller is for suppressing the structural vibration caused bythe cutting force disturbance around system’s resonant frequencies. To take intoaccount the resonant modes’ variations which are peculiar to ball-screw drives, again-scheduled controller has been utilized as the vibration suppression controller.Application of the switching gain-scheduled control technique to the ball-screw drive system with a wide range of operating conditions has been studiedin Chapter 5. To achieve high tracking performance of the table position againstthe dynamic variations and the cutting force disturbance, a set of gain-scheduledcontrollers have been designed so that each controller damps out the resonance ofthe ball-screw system and increases the closed-loop bandwidth for a local operatingrange, and tracking performance is guaranteed under the switching between thesecontrollers.In Chapter 6, to ensure smooth transient signals of the switching controller,a novel smooth switching design approach and an optimization algorithm have beendescribed. In this method, the moving region of the gain-scheduling variables hasbeen divided into a specified number of local subregions as well as subregions forthe smooth controller switching, and one gain-scheduled LPV controller has beenassigned to each of the local subregions. The smooth switching controller design13problem amounts to solving a feasibility problem which involves non-linear matrixinequalities.The usefulness of the proposed controller design method has been demon-strated with a control application of SKM and PKM systems in Chapter 7. Intwo-axis SKM control, we have simulated control of two identical flexible ball-screwsystems to track a line and circle in the x-y plane. We have tested the smooth switch-ing controller on a single-axis experimental test setup. Also, in the PKM system,the machine tool dynamics change significantly depending on the tool tip positionand speed in the workspace. To control the PKM system with a large range ofdynamic variations, we have designed the model-based smooth switching LPV con-troller. The simulated time domain results have been compared with non-switchingand hysteresis switching controllers.In Chapter 8, we have considered the dynamic variations of the augmentedplant due to rotation of the task coordinate frame. In this chapter, we have designeda new method for multi-axis machine tool control which minimizes the contouringerror in the task coordinate frame. We have proposed a method to design a multi-input-multi-output (MIMO) LPV feedback controller for contouring control of twoand three-axis CNC machine tools. The controller design approach has been de-scribed, and frequency and time domain simulations have been developed for multi-axis machine tools. The experimental results have been provided in two-axis CNCmachine tool control.The conclusion and summary of results have been provided in Chapter 9, inaddition to the contributions of this research and possible future works.14Chapter 2Literature ReviewThis chapter reviews the literature relevant to modeling, system identification andcontrol of feed-drive systems.2.1 Feed-Drive Systems Modeling and IdentificationModeling of feed-drive systems is the first step in the controller synthesis. Thismodel can be obtained analytically to predict the system dynamics behavior or itcan be estimated by identification. This section reviews some work in the literaturethat has been done in this area.2.1.1 Modeling methodsIn the modeling of the ball-screw drive system, Kim and Chung [59, 60] proposed alumped modeling approach. In their method, inertia and stiffness of the ball-screwdrive system are modeled as equivalent lumped inertia and springs. They took intoaccount both torsional and axial vibration of the screw, to formulate a two-by-twotransfer matrix between the force on the table and the torque of the motor to the15position of the table and rotation angle of the screw. In order to express the systemresponse in the low frequency range, one can utilize this method. However, a moreaccurate model including the distributed model for the screw is required for theprediction of the system behavior in the high frequency range.In an alternative approach to the lumped modeling in [59, 60], Varanasi andNayfeh in [99] introduced the hybrid modeling of the ball-screw drive system. In thehybrid modeling, the screw is described as a distributed beam while the other partsare treated as lumped components. Then, the axial and torsional vibrations of thescrew are predicted according to the Euler–Bernoulli beam vibration model. UsingGalerkin technique, the distributed model is simplified into a low order dynamicmodel. Similar to [99], Whalley et al. in [103] applied distributed-lumped hybridmodeling. They presented the system modeling in a block diagram format. In thisform of representation, other system characteristics such as backlash and motormodel can be simply added to the model of the ball-screw drive system.Frey et al. in [38] studied the dominant behavior of rotational vibration modeof the ball-screw drive system for different operating and coupling conditions. Theymodeled the ball-screw with the hybrid approach where the screw is modeled as aflexible Timoshenko beam while the other components are lumped. They demon-strated the extreme sensitivity of the axial vibration mode to the workpiece massand the table position even though the rotational vibration mode of the screw wasnot affected significantly by those dynamics variables. Vicente et al. [100] modeledthe ball-screw drive system by considering the screw as a distributed subsystemwith N-DOFs and approximated the solution of the dynamic differential equationsusing the Ritz method. In addition to the axial rotational modes of the screw, Dongand Tang [28] modeled the flexural dynamics of the screw as a Timoshenko beam.16They investigated the changes in the system model with different table positionsand workpiece masses.The finite element method (FEM) in [33, 77, 109] is also a practical methodfor modeling the ball-screw drive system. In the FEM, the complex behavior of thesystem can be modeled by breaking up the system into a large number of elementswith high DOFs. For instance, Zaeh et al. [109] presented the FEM model of theball-screw drive system. In this design, the stiffness matrix between the screw andthe nut represents the balls’ interaction between them. They also considered bothlateral and vertical motions of the balls in their modeling of the system. To capturethe coupling between the torsional and lateral dynamics of the ball-screw drivesystem, Okwudire and Altintas [77] developed a hybrid FEM model. The screwis modeled as Timoshenko beam elements, while other components are modeled aslumped mass and spring. One of the important advantages of [77] is its ability topredict the model as a function of table position.2.1.2 Identification methodsOne of the problems of the analytical modeling methods is that the parameters ofthe model are highly dependent on catalog specifications of the system components.These specifications may not always be available or might change after a while. Inaddition, the FEM model requires very large matrices with possible computationalerrors to accurately model the system. Therefore, system identification can beutilized as an alternative method to realistically estimate the parameters of themodel with high accuracy and lower computational efforts.Erkorkmaz and Altintas [31] proposed an open-loop identification method toestimate the rigid-body mode of the ball-screw drive system. They developed an un-17biased least squares approach to identify the rigid-body mode inertia and dampingby considering the disturbance characteristics in the dynamic model. In addition, byintroducing a Kalman filter, they estimated the non-linear friction model parame-ters. In another attempt to identify the rigid-body mode, Erkorkmaz and Wong [34]presented a rapid closed-loop technique to avoid any considerable down time on theCNC machine tools. In this approach, they identified a closed-loop simple low ordertransfer function between the reference signal and the table position. By imposinginequality constrains on the closed-loop pole locations, they guaranteed stability ofthe identified system, and solved optimization problems with the Lagrange multi-plier technique. In a similar approach, Wong and Erkorkmaz [105] identified theclosed-loop dynamic model by solving the contained optimization problem with thegenetic algorithm method.Supplementary to the rigid-body mode identification of the ball-screw drivesystem, the flexible modes are identified in [76]. Okwudire and Altintas in [76]estimated parameters of a two-by-two transfer matrix from torque and table forceto the screw angle and table position. They measured the FRF of the system andutilized the least squares method to estimate the model parameters. In addition tothe plant modeling, friction modeling as well as identification and compensation inthe ball-screw drive systems have been studied in the literature such as in [8] and[97].2.1.3 Dynamic variations and LPV modelingIn a system with a large range of dynamic variations, modeling of the system witha fixed LTI model can not be satisfactory in predicting the system behavior forthe whole range of variations. To control such system, a time varying model is18required to express the system variations. In the LPV model, the plant is modeledas a linear system while its parameters vary as a function of time or other reference(gain-scheduled) parameters. An LPV plant provides an accurate model of thesystem for any working point in the operating range. However, this high accuracyof the LPV plants means high complexity which leads to dependency of the modelto the gain-scheduled parameters.There have been some attempts to identify dynamic variations in the ma-chine tool systems in [65, 66, 79, 90]. Paijmans et al. [79] proposed a technique foridentifying the LTI models for different operating points and found the LPV modelby interpolation of the system transfer functions’ poles and zeros in the single-inputsingle-output (SISO) system. Law et al. [65, 66] proposed a reduced order methodfor modeling the dynamic variations of a three-axis milling machine for differenttool positions. In this approach, the sub-structures of the machine tool system areconsidered as LTI models while they are connected to each other by adaptations ofconstraint formulations. By moving the tool in the workspace, the constraints areconstantly updated to predict the new model.Particularly in the flexible ball-screw drive system, the dynamics of the flex-ible modes change as the table moves along the screw. This is because the stiffnessof the system varies depending on the table location, due to the change of the ac-tive length of the screw between the motor shaft and the table. In addition, therun-out effect is another cause for dynamic variations. The run-out effect occursdue to the misalignment of the ball bearings on their axis which holds the screwshaft. As demonstrated in [90], the dynamic variations caused by the run-out of theball-screw drive system can be observed by measuring the FRF at table positionswith intervals less than one-fourth of the screw pitch.192.2 Single-Axis Feed-Drive ControlFrom the servo drive control perspective, there are two primary requirements in HSMcontrol: tracking and vibration suppression. For position tracking of the workpiece,it is required to follow the desired cutting trajectory with high speed and accuracy.In addition, vibration suppression means the attenuation of the workpiece vibrationcaused by the cutting force disturbances.There are challenges in the tracking and vibration suppression control of theflexible ball-screw drive systems. Firstly, the bandwidth of the closed-loop system islimited by the flexible modes. This imposes constraints on the achievable trackingspeed. Secondly, the flexible modes can be excited by the applied cutting forces nearresonant frequencies. Such excitation causes vibration of the machine structure anddeteriorates the trajectory tracking performance. Thirdly, the dynamics of the ball-screw drive systems continuously vary during the machining operation as a functionof the varying stiffness and the decreasing workpiece mass due to material removal.The varying dynamics further limit the achievable performance of ball-screw drivesystems. In order to obtain satisfactory performance of the ball-screw drive con-troller design and to meet the tracking and vibration suppression requirements, allof the aforementioned challenges need to be addressed simultaneously. As far asthe ball-screw drive system is concerned, several approaches to tracking and vibra-tion control have been proposed [4]. One can categorize the controller approach forfeed-drive control into classical and modern methods.2.2.1 Classical control techniquesIn the classical control approach, only the rigid-body mode of the system is con-sidered in the controller design. Although the neglect of the flexible modes in the20modeling simplifies the controller design in the classical approach, these flexiblemodes can be excited due to the control signal, reference trajectory and machiningforces. The application of notch filters and low pass filters were proposed in [102],[112] to prevent the flexible modes excitation due to the control signal. Moreover,Jones et al. in [54] proposed a method to generate a smooth trajectory to avoidexcitation of the flexible modes due to discontinuous reference trajectory. Finally,to mitigate excitation of the flexible modes due to machining forces, Chen andTlusty [20] and Chung et al. [23] proposed an active damping approach for controlof the ball-screw drive system. They designed cascade controllers based on the PIfeedback from encoders and an accelerometer to increase stiffness of the closed-loopsystem. The main disadvantage of the classical methods is that they may not berobust against the dynamic variations.2.2.2 Modern control techniquesIn the modern control approaches, flexible vibration modes of the ball-screw drivesystems are explicitly taken into account in the modeling and control synthesis.To solve robustness issues, the modern techniques apply active feedback whichmakes the closed-loop system less sensitive to the parameter variations. Severalapproaches have been proposed as feedback control such as model predictive [29],sliding mode [57, 76, 98], and H∞ control synthesis [52, 98]. Dumur et al. [29] devel-oped a generalized predictive controller to damp out structural modes of a ball-screwdrive system. They also designed an adaptation loop to update the controller pa-rameter vector with a least squares type of strategy.Kamalzadeh and Erkorkmaz [57, 58] utilized an adaptive sliding model con-troller as a framework in high precision and high bandwidth tracking control of the21ball-screw drive systems. The feed-drive system performance in friction compensa-tion, ripple force reduction, and vibration suppression were advanced by means offeedforward compensators, Kalman filters and notch filters, respectively. The de-signed controller significantly enhanced the bandwidth of the closed loop system anddamped out structural vibrations. In another approach, Okwudire and Altintas [76]proposed a robust discrete-time sliding mode controller to compensate for the vi-bration modes by considering a MIMO model and disturbance effect. However, zerotracking error was ensured only with designing of a complex filter.Itoh et al. [52] designed a two-DOF compensator as a combination of a feed-forward and an H∞ feedback controller. While the feedback part deals with therobustness against variations of the resonance modes, the feedforward part uti-lizes coprime factorization to provide the fast and precise positioning. Gordon andErkorkmaz [41] presented the pole-placement method to fulfill both high bandwidthand vibration suppression of the ball-screw drive system by considering the loopshaping and notch filters. The closed-loop poles are located such that they improvethe bandwidth and disturbance rejection of the rigid-body mode, and increase thestiffness of the flexible modes. Tsai et al. [95] proposed a learning controller to filterout the undesired signal from reference commands in order to avoid excitation ofresonant modes in high-precision machining.2.2.3 Vibration control techniquesVibration control of a flexible system has been studied extensively in the contextof various applications with challenges. Examples are servo drives [40], flexiblebeams [55], parallel-kinematic mechanisms [68, 88], single-link flexible manipula-tors [80], hard disk drives [75, 101], micro grippers [43], piezoelectric actuators22[83, 106] and flexible robotic manipulators [86]. Several control approaches asso-ciated with active vibration suppression have been developed, e.g., sliding modecontrol [76], linear quadratic Gaussian (LQG) control [56], robust H∞ control [35],robust µ-synthesis control [37], input shaping control [24], model predictive con-trol [16, 94], integral force feedback (IFF) [10], positive position feedback (PPF) [27]and integral resonant control (IRC) [13].As far as the ball-screw drive system is concerned, several approaches tovibration control have been proposed [4]. Erkorkmaz and Kamalzadeh [33, 57] syn-thesized an adaptive sliding mode control, notch filter and active damping techniquesto suppress axial and torsional vibration of the ball-screw drive systems. Altintasand Khoshdarregi [2] avoided residual vibrations in the CNC machine by applyinginput shaping filters on the reference commands. Erkorkmaz and Hosseinkhani [32]developed a pole placement controller and lead filter to directly minimize the cuttingforce disturbance responses across the frequency spectrum. Also, they designed alead filter to compensate the power electronic system delay. Recently, Hosseinabadiand Altintas [49] introduced active damping in the sliding mode control of the drivesystem. They used Kalman filter to estimate the acceleration and velocity of thesystem and increase system damping to suppress the CNC machine vibration.2.2.4 LPV controlDespite previous studies on the ball-screw drive control mentioned above, its generalshortcoming is the inability of the controller design methods to explicitly deal withdynamic variations of the ball-screw drives. To our best knowledge, Symens et al.[93] were the first who considered stiffness variations of a linear motor driven systemin the modeling and control synthesis. They designed several H∞ controllers by23obtaining dynamic models for the flexible gripper placed in different positions. Thedesigned controllers were linearly interpolated as table moves. Extension on theprevious work, a systematic approach for identification of the dynamic variations anddesigning of a gain-scheduled controller have been presented by Paijmans et al. [79,25] for a SISO plant.Unlike the gain-scheduled control approach for a SISO model, in order todeal with the plant variations in controller design of a MIMO model systematically,Apkarian and Adams [5] developed an advanced LPV gain-scheduled controller de-sign method. Analysis and design for LPV systems have attracted a considerableamount of attention over the last two decades. A number of gain-scheduled LPVcontroller design methods for LPV systems have been developed, and various suc-cessful control applications have been reported. See e.g. [5, 6, 74, 78, 84, 91, 104]and references therein.Based on this technique, Sepasi et al. [89] synthesized a robust gain-schedulingcontroller for tracking control of ball-screw drive systems. However, they did notconsider any solution to avoid excitation of the ball-screw drive system due to cut-ting force disturbances. Fan et al. [36] proposed a two-loop tracking control, whichconsisted of the µ controller and the disturbance observer, to resolve both robusttracking and vibration control problems. However, the fixed disturbance observermostly works in the low frequency range, and it is not effective for the frequencyrange where flexible modes normally exist.2.2.5 Switching LPV controlA common issue of LPV controller design methods based on robust control theoryis the inherent conservatism of the designed controllers. As a means to reduce the24conservatism and to improve the closed-loop performance achieved by a single LPVcontroller, a switching LPV controller design method was proposed in [72, 73] byusing the multiple parameter-dependent Lyapunov functions. In this method, themoving region of the gain-scheduling variables is divided into subregions, and eachsubregion is associated with one local LPV controller. To ensure the performanceeven with controller switching, two strategies for switching, that is, the hysteresisswitching strategy and the average dwell time strategy, were investigated. Theadvantage of utilizing the switching LPV controller in some industrial practices hasbeen demonstrated in [11, 45, 50, 71, 73, 81]. Optimization of the subregion divisionand location of the switching surfaces has been studied in [9, 53].2.2.6 Smooth switching LPV controlA potential drawback of the developed switching LPV controllers is that they maynot provide a smooth transient response after switching of controllers. A non-smooth transient response can lead to mechanical damage, fatigue loading, or signalsaturation, and therefore, it is undesirable in practical applications. To cover thisdrawback, there has been extensive research on a bumpless transfer of controllersswitching in [30, 42, 61, 82, 96, 107, 108] and interpolation of controllers in [12,14, 17, 18, 19, 47, 69]. With a bumpless transfer, continuity of a plant input isensured at the switching time for LTI controllers by different techniques. Someexamples of bumpless transfer methods are to force an off-line controller to track anon-line controller signal with anti-windup bumpless transfer in [30, 42, 107], matchof switching controller signals by an optimal linear quadratic minimization in [96],to follow an ideal control signal target at switching time by minimizing an L2-gainbound in [108], and to set states of an off-line controller before switching to ensure25a continuous control signal for a set of linear single-input-single-output controllersin [61] and for multi-input-multi-output controllers in [82].On the other hand, in the interpolation approaches, local controllers areblended to guarantee the closed-loop and switching stability [69]. For instance, anobserver-based multi-variable gain-scheduled controller has been developed in [12] bylinear interpolation of local controllers. As another attempt, a supervisory controllerhas been synthesized in [47], where control signals coming from local LTI controllersfor different control objectives are interpolated to provide quadratic stability andperformance guarantees. Moreover, in [14], interpolation of LTI controllers has beenpresented to generate a gain-scheduled controller with preservation of the closed-loop quadratic stability and transient performance. Furthermore, in [19], a methodwas developed to design a smooth switching LPV controller by interpolating LPVcontrollers in overlapping regions of any two adjacent subregions. Its applicationshave been presented in [17, 18].Among all of the previous work, the method in [17, 18, 19] directly dealswith smooth switching of the LPV controllers. Instead of switching controllers in-stantaneously, this method interpolates the controllers with a fixed function in anoverlapping region. As a consequence of employing a fixed function for controllerinterpolation, the control signal may not be differentiable at the boundaries of over-lapping regions, which can degrade signals’ smoothness. The other disadvantageof this approach is the lack of a measure to quantitatively evaluate smoothness ofcontroller switching. Moreover, the method in [17, 18, 19] is limited to the statefeedback case, and to the case of one-dimensional space of the scheduling variable.We propose a method in Chapter 6, which is presented in [44], to design asmooth switching LPV controller. The method is novel in the sense that, in con-26trast to the previous work in [12, 14, 47, 69], we design LPV controllers as localcontrollers and interpolate smoothly between them instead of LTI controllers inter-polation. Also, different from the method in [17, 18, 19], we consider the controllerdesign problem for output feedback systems, and for both one- and two-dimensionalspaces of gain-scheduling variables. In addition, unlike [17, 18, 19], we introducea matrix norm of the controller system matrix as a measure of “smoothness” forcontroller switching. Also, we utilize adjustable interpolation functions to improvethe smoothness, and provide a higher order differentiable control signal by takingthem into account in a controller design problem. The controller design problem isformulated to satisfy both the standard L2-gain condition of the closed-loop systemand the smoothness condition of the switching controllers.2.3 Multi-Axis Feed-Drive ControlAlthough the tracking control of individual axes improves the total tracking error,it does not guarantee the decrease of the contouring error. To resolve this issue,the contouring control approach is utilized which directly aims at minimizing thedeviation error. Various approaches for precision control of the CNC machine havebeen reviewed in [51]. The contouring control can be categorized by both the cross-coupling control (CCC) method developed first by [62] and the task coordinate frametransformation with direct contouring control introduced by [21, 22, 48].In the CCC method, the contouring error is calculated on-line while a com-pensator sends additional signals to each axis servo controller to minimize the con-touring error. More advanced techniques have been developed for the design ofthe CCC such as the fuzzy-logic-based regulator in [92], the non-linear adaptivecontroller in [67], and the model predictive controller in [110]. Two of the major27disadvantages of the CCC method are that it has been designed for time-invariantplants and it has low efficiency in the control of non-linear contours with high feed-rate speeds.Alternatively, Davis et al. [22] proposed contouring control in the task coor-dinate frame. In the 2D case, the tangential and normal components of the trackingerror are minimized by designing a robust controller in [48], a model predictivecontroller in [63], and an adaptive controller in [26].For three-axis machine tool control, the axes of the coordinate system aretransformed into the tangential-normal-bidirectional (t-n-b) task frame as in [22]and both the contouring and lag tracking errors are controlled by a varying PD andfeed-forward controller with more emphasis being on the contouring error.In five-axis control, the tool-path coordinate approach has been developedin [70] by calculating the contouring error and tracking lag error in the tool-pathcoordinate system and designing a proportional, integral and derivative (PID) con-troller. In addition, two MIMO sliding mode controllers have been developed in [3]in order to control the tooltip position and the tool-orientation.The idea of designing an LPV controller for contouring error minimizationwas first introduced in [21]. However, this work is limited to the control of circularmotion tracking without a clear control structure or design approach. One of thedisadvantages of the previous methods in the frame-based approach is that the con-troller becomes conservative in the sense of performance by designing several SISOLTI controllers for each independent axis. In addition, the structural flexible modesof the machine tool system have been ignored in the design of most of those con-trollers. In the proposed sliding mode and task coordinate frame based approach,the calculation of the Jacobian is an essential step in designing the contouring con-28troller. However, this calculation is computationally complex and increases therequired time to compute the controller parameters in the real-time implementa-tion.Motivated by the work presented in [21], a MIMO LPV controller for bothtwo-and three-axis CNC machine tool controls is designed in Chapter 9 of this thesisby following the design of the controller in the task frame coordinate system. In thedesign stage, a varying controller as a function of tool-path directions is explicitlydesigned by meeting the required feed-rate. The controller is designed to be stable,to increase the bandwidth, minimize the contouring and tracking lag errors, considerthe flexible modes of machine tools and the disturbance force in the design stage.In addition, we design a three-dimensional controller without using the Jacobianmatrix and its inverse which simplifies the design and implementation of this controlapproach.29Chapter 3Modeling and Identification ofFeed-drive SystemsThis chapter focuses on the modeling and identification of feed-drive systems. Theanalytical model of the ball-screw drive system, i.e., rigid and flexible modes struc-ture, has been obtained by considering the electrical circuits of the motors in therigid-body mode and axial-rotational vibration in the flexible mode. This modelwas validated by FRF measurement. In order to consider the run-out effect andobtain the LPV model, the system parameters based on the FRF measurementsfor different table positions and workpiece masses were identified. At the end, theLPV state-space equation was derived and was used in Chapters 4, 5, and 7 for thedevelopment of the proposed controllers.3.1 Ball-Screw Drive SystemsModeling of the ball-screw drive systems is the first step in the controller synthesis.Different methods such as analytical methods [20, 99], identification methods, [31],30[76], [89], and FEM [33, 77, 109] have been developed to model ball-screw drivesystems. A schematic picture in Figure 3.1 represents a ball-screw drive system.This ball-screw drive consists of a DC motor, a screw, a nut, and a table. Byapplying a voltage vm to the DC motor, it generates a torque τm to the screw whichis directly coupled to the DC motor. The torque rotates the screw by the angle θm,and then this rotation is translated into linear motion uc of the table by the nutsplaced underneath the table. In addition, an external force fc is applied to the tabledue to the resultant component of cutting forces and friction along the screw axis.The screw is supported axially by a thrust bearing close to the motor shaft, and bya radial bearing on the other end. The motor is a permanent magnet synchronousDC motor which can reach a speed of up to 27 [m/min] and an acceleration of1g [m/s2] in a 360 [mm] stroke. The details on the electrical and mechanical systemparameters, obtained from manufacturers catalogs and CAD software [77], can befound in Table 3.1.Figure 3.1: A ball-screw drive system components31Table 3.1: Parameters of the ball-screw drive systemsSymbol Unit Value Symbol Unit ValueElectrical parameters Mechanical parameterska [A/V] 0.887 ds [mm] 20kai [V/A.s] 3.0019×105 Jcoup [kgm2] 9.3212×10−7kap [V/A] 111.55 Jenc [kgm2] 1.7 ×10−4kemf [V/rad/s] 0.4173 Jmot [kgm2] 7.65×10−5kt [Nm/A] 0.72 Jtach [kgm2] 4 ×10−5Lm [h] 0.0375 ls [cm] 820Rm [Ω] 6.5 mtab [kg] 20Stiffness and damping parameterscc [Nm/s] 0 kb [N/m] 1.13×108cm [Nm/s/rad] 0.0156 kc [Nm/rad] 6500rg [m/rad] 0.0032 kn [N/m] 1.37×1083.2 Rigid-Body Mode ModelIn the low frequency range, rigid-body mode is the dominant mode that modelsthe system behavior. In order to model the rigid-body mode, both mechanical andelectrical characteristics of this system were considered as shown in Figure 3.2. Byneglecting the flexible modes, it was assumed that the table position uc and theequivalent displacement of the motor shaft um are equal,um = rgθm, (3.1)where rg is the gear ratio equals topt2pi, and pt is pitch of the ball-screw. Then thedynamics equation of the motor can be written asJed2θmdt2= τm −Bedθmdt, (3.2)where Je and Be are the equivalent inertia and viscous damping of the ball-screwdrive system, respectively. The equivalent inertia can be derived in terms of the32system components’ inertia as follows:Je(mwor(t)) := (mtab +mwor(t))(rgpi)2+mspid2s8+ Jenc + Jcoup + Jtach + Jm, (3.3)where mtab and mwor(t) are the mass of the table and the mass of the work-piece,respectively, and ms, ds, Jenc, Jcoup, Jtach, and Jm are the mass and the diameterof the screw, the moment of inertia of the encoders, the coupling, the tachometersalong rotation axis, and the motor shaft, respectively. According to (3.3), the totalinertia variation is a function of the change in the workpiece mass, i.e.,∆Je(t) =(rgpi)2∆mwor(t), (3.4)while the gear ratio rg2 specifies the significance of this change on the total inertia.In order to obtain the rigid-body mode transfer function between the torqueand the table position, we take the Laplace transformation of (3.2) asG0(s) :=umτm=rgJe(mwor)s2 +Bes. (3.5)In addition to the mechanical model of the ball-screw drive system, the electricmodel of DC motor winding as a resistor–inductor (RL) circuit, back-electromotiveforce (EMF) effect gain, and the pulse-width modulation (PWM) amplifier influenceas a proportional and integral (PI) transfer function were taken into account. Thedifferential equation for the current, im := τm/kt, in the motor winding is interpretedasLmdimdt+Rmim = kcpkavm + kci∫kavmdt− kemfdθmdt, (3.6)where ka, kemf , Rm, Lm, kcp, and kci are the amplifier gain, the back-EMF constant,the motor resistance, the motor inductance, as well as the proportional and the in-tegral gains of the PI current controller, respectively. The transfer function between33Figure 3.2: Rigid-body model of the ball-screw drive systems including inertia anddamping, the current amplifier, and the motor winding in torque control modethe motor current im and the input command vm was constructed by combining(3.2) and (3.6) asG1(s) :=imvm=ka(kcps2 + (Bekcp + Jekci)s+Bekcika)JeLms3 + (BeLm + JeRm + kcp)s2 + (BeRm +Bekci + Jekci + ktkemf )s+Bekci.(3.7)Finally, the total transfer function between control signal vm and the table locationum is described as:umvm(s) = ktG0(s)G1(s). (3.8)By referring to the system parameters in Table 3.1, the equivalent systeminertia (3.3) is determined as Je = 5.9 × 10−4 [kg.m2] by assuming mwor = 0 [kg].Also, for the experimental ball-screw drive setup, we can replace the transfer func-tion G1 in (3.7) with a single amplifier gain ka. This simplified model is justifiedby the Bode diagrams of G0 and G1 in (3.5) and (3.7), respectively, illustrated inFigure 3.3. According to this figure, bandwidth of G1, is 4700 [rad/s] which is muchgreater than bandwidth of G0 equal 10 [rad/s]. Hence, the only effect of the electricfield in the low frequency range is its DC gain which is equal to ka and thus, thehigh frequency dynamics of the G1 model can be neglected.34Figure 3.3: FRF of G0 (solid blue line) and G1 (dash-dot green line)3.3 Flexible Mode ModelIn addition to the rigid-body mode model of the ball-screw drive system, modelingof its flexible mode is an important task in predicting the system vibration in thehigh frequency range. Therefore, by considering the torsional and axial vibrations,the hybrid method developed in [99] was utilized. The advantage of this method isthat the dynamic equation of the system is known as a function of the nut locationxc and the workpiece mass mwor. Therefore, variation of the system dynamics canbe analytically expressed based on those parameters.The free-body diagram of the ball-screw drive system in both axial and tor-sional modes is depicted in Figure 3.4. In this illustration, the screw is modeled asa distributed Euler-Bernoulli beam where its axial and torsional deformations aredenoted by us(x, t) and θs(x, t), respectively, in location x from the origin of thethrust bearing and time t. Also, the table position and the motor shaft rotation35(a) Axial mode vibration model(b) Torsional mode vibration modelFigure 3.4: Hybrid free-body diagram of the ball-screw drive systemangle are denoted by uc and θm, respectively, while the disturbance force to thetable and input torque from the motor are represented by fc and τm, respectively.In this approach, the interactions of the connections are modeled as lumpedtorsional and axial springs, and dash-pot dampers. In the axial vibration, the screwis supported by a thrust bearing at the side close to the motor with its stiffnessbeing equal to kb, while there is no axial constraint present on the other side of thescrew. The compliance of nut and screw connection is considered as a spring kn atthe nut position xc. A dash-pot damper cc represents the viscous damping of thetable. In the torsional vibration, the inertia, torsional damping, and the equivalentstiffness of the motor shaft are considered as Jm, cm, and km, respectively. Thecoupling between the screw and the motor is modeled by kc. Finally, the torsionalcompliance of the nut and screw connection equals to r2gkn.The axial and torsional vibration of the screw is interpreted by the longitu-36dinal and the rotational wave equations. The connections between the motor andthe table at the nut location have been taken into account as the boundary condi-tions of these wave equations in Appendix A. The wave equations which are coupledwith the dynamic equation of the motor and the table can be simplified with thequasi-static method, and can be solved analytically with the Lagrange method. Thefinal differential equation of motion of the ball-screw system was obtained in [99] asfollows:Msu¨m(t)u¨c(t)+ Csu˙m(t)u˙c(t)+Ksum(t)uc(t) =fm(t)fc(t) , (3.9)whereMs :=m11 m12m21 m22 , Ks := ke1 −1−1 1 , Cs :=cm/r2g 00 cc , (3.10)Ms, Cs, Ks being the overall inertia, damping and stiffness matrices, respectively.The other parameters include um which was defined in (3.1), fm := τm(t)/rg whichis the equivalent force applied by the motor, and ke which is the total equivalentstiffness of the ball-screw drive system. The components of the total inertia matrix,Ms, as well as the equation for ke follow:m11(xc) :=J1(xc)r2g+ Jm(xc)/r2g +me(xc),m12(xc) = m21(xc) :=J12(xc)r2g−me(xc),m22(xc,mwor) :=J2(xc)r2g+me(xc) +mwor +mtab,ke(xc) :=[1kb+xcEsAs+2kn+r2gkc+r2gxcGsJs]−1,(3.11)where Es, Gs, As, and Js are the Young’s and shear modulus of the screw, theaverage cross section, and the second moment of inertia of the screw cross section,37respectively. The rest of the parameters including J1, J2, J12, Je and me are givenin Appendix A.3.In order to consider the structural damping characteristic of the complianceelements associated with hysteresis energy loss in the material, we can replace allsprings with complex springs, each having a ki(1 + jηi) stiffness where ηi is the lossfactor associated with the i-th spring. As a result, ke in (3.11) becomes a complexequivalent stiffness. With this in mind, the total inertia matrix Ms would notchange, while the new stiffness and damping matrices by considering the hysteresisloss in the martial can be determined in [99] as:Khys := Re(ke)1 −1−1 1 ,Chys(xc,mc) :=cmr2g+ chys −chys−chys cc + chys ,(3.12)wherechys(xc,mwor) := ηe√kˆe(m11m22 −m212)m11m22 −m212. (3.13)Here, kˆe and the total loss factor ηe can be determined by solving the followingequation:kˆe(xc)(1 + jηe) :=[1kb(1 + jηi)+xcEsAs+2kn(1 + jηi)+r2gkc(1 + jηi)+r2gxcGsJs]−1.(3.14)3.3.1 Parameter calculationsWe use the system parameters in Table 3.1 to calculate the inertia, damping andstiffness matrices in (3.10) by assuming xc = 0.59 [m] and mwor = 0 [kg]. Theonly parameter which is not given in that table to calculate the system matrices38is the viscous damping. We will estimate this parameter in Section 3.4.1 based onrigid-body mode identification. By neglecting the structural damping effect chys andequivalent damping of the table cc, the system matrices are calculated as:Ms :=45.52 0.730.73 23.45 [kg], Ks := 2.579× 1071 −1−1 1 [N/m],Cs :=1540 00 0 [kg/s].(3.15)This system has a resonant frequency at 1305 [rad/s] and a damping ratio of 0.46%.The estimated ζ value is smaller than our expectation for the ball-screw drive system(which is between 1% to 3%). Therefore, to have a more realistic damping effect,we include the structural damping effect in (3.12). For this purpose, the loss factorof the spring is estimated as ηi = 0.053 based on system identification. In this case,the inertia matrix Ms does not change and the other matrices are calculated as:Khys := 2.581× 1071 −1−1 1 [N/m], Chys :=2146 −606−606 606 [kg/s].(3.16)This new system has a resonant frequency of 1305 [rad/s] and a damping ratio of1.99%. We plot the Bode diagram of the transfer functions between the motor forceand the motor position G11, the motor force and the table position G12 = G21, andthe table force and the table position G22 in two cases with and without structuraldamping in Figure 3.5. The low damping ratio of the first case leads to the highamplitude of the system gain at the resonant frequency. This amplitude is reducedin the second case by adding the structural damping into our calculations.39Figure 3.5: Bode plot of the analytical model of the ball-screw system with (solidblue line) and without (dashed red line) structural damping3.3.2 Model validationModel validation is conducted by comparing the model and the FRF measurementsby Dynamic Signal Analyzer device. In order to plot the Bode diagram of thesystem, we take the Laplace transform of the model (3.9) to obtain the transferfunction between the inputs and the outputs as:um(s)uc(s) = G(s)fm(s)fc(s) , (3.17)whereG(s) :=(Ms (xc,mwor) s2 + Chys (xc,mwor) s+Khys(xc))−1, (3.18)while Ks and Cs in (3.9) are replaced by Khys and Chys as introduced in (3.12).Among the four elements of matrix G(s) we measure FRF of G11 and G12 which40are the transfer functions between the motor force as the input and the motor shaftposition and the table position as the outputs, respectively, whereG11(s) G12(s)G21(s) G22(s) := G(s), (3.19)withG12 = G21. We compare the Bode diagram of the calculated and measured FRFof G11 and G12 in Figure 3.6 by placing the nut in different positions, i.e., xc = 0.205,0.290, 0.390, and 0.590 [m] while there is no workpiece on the table, i.e., mwor=0[kg]. In addition to the table location variations, we investigate the workpiece massvariations in the theoretical model and the experimental measurements in Figure 3.7by placing four different workpiece masses on the table, i.e., mwor = 2.27, 4.54, 6.80,and 9.07 [kg].By comparing the results of the measured and the calculated FRFs in Fig-ures 3.6 and 3.7, it can be seen that the measured gain plot of each transfer function(i.e., G11 and G12 functions) is similar to the calculated gain plot of that function.It could also be observed that although the resonance frequency changes for dif-ferent table positions and workpiece masses; for both of these cases, the resonancefrequency value is almost identical for the experimental and analytical FRF. On thecontrary, there are some discrepancies between the estimated and measured phasevalues. The measured FRF shows a 10 to 30 [deg] lower phase value at the resonancefrequency and higher frequency ranges. This mismatch can be explained as a resultof high frequency vibration modes that are ignored in the analytic modeling of thesystem. The next structural mode occurs at 3900 [rad/s] which is three times higherthan the axial-torsional mode.41(a) xc=0.205 [m] (b) xc=0.290 [m](c) xc=0.390 [m] (d) xc=0.590 [m]Figure 3.6: Model validation Bode-diagrams of experimentally measured (red dash-dot lines) and calculated (solid blue lines) FRFs where the nut is placed in differentxc locations with no workpiece on the table3.3.3 Dynamic variations of the analytical modelAs stated earlier, one of the advantages of the analytical method is that it hasdynamic equation as a function of the nut (table) location xc and the workpiecemass mwor. Having the system validated by the FRF measurements, we simulatethe influence of xc and mwor on the parameters of the modeled system in (3.10-3.11). The changes of the system parameters as a function of the table location isshown in Figure 3.8 in the interval of 100 to 600 [mm]. Based on Figure 3.8 (a),the system stiffness ke, damping chys, resonant frequency ωd and damping ratio ζdecrease as the table moves away from the motor. The decrease in those parameters42(a) mwor=2.27 [kg] (b) mwor=4.54 [kg](c) mwor=6.80 [kg] (d) mwor=9.07 [kg]Figure 3.7: Model validation Bode-diagrams of experimentally measured (red dash-dot lines) and calculated (solid blue lines) FRFs where the nut is located atxc=0.49 [m] while different workpiece masses mwor are placed on the table43is intuitively expected, since the system becomes more flexible due to the increaseof the active length of the screw between the motor and the table. However, thesimulation results in Figure 3.8 (b) do not show a significant change in the elementsof the inertia matrix Ms as a function of the table position.(a) Variations in stiffness and damp-ing constants, resonance frequency anddamping ratio.(b) Variations in elements of the in-ertia matrix MsFigure 3.8: System parameters variations as a function of the nut location xcAdditionally, we describe the system variations according to the changes ofthe table mass mwor from 0 to 100 [kg] in Figure 3.9. According to (3.11), m22 is theonly element of the inertia matrix Ms that is a function of mwor. As the mass mworincreases, the resonant frequency ωd decreases since the stiffness remains constant.In addition, the structural damping chys, which is a function of m22 (3.13), shows adecrease in value as mwor increases.Finally, we plot the 3D diagram of the ωd and chys as a function of xc andmwor in Figure 3.10. This figure shows that the ωd increases from 800 [rad/s]to 1600 [rad/s] in the operating range of the varying parameters xc and mwor.Furthermore, the chys value changes from its minimum value at 600 [kg/s] to its44maximum value at 1500 [kg/s] in the operating range of the varying parameters.Figure 3.9: System parameters variations as a function of the workpiece mass mwor3.4 System IdentificationIn the previous sections, the ball-screw drive system was analytically modeled. As itcan be seen, during the modeling, several simplifying assumptions were consideredin order to be able to derive the system motion equations. In addition, the modelassumes the catalog values presented in Table 3.1. Also, some system characteristics,such as the damping constants, as well as the compliance of the connections and thesupport system, may not be provided in the catalog.To resolve this issue, we identify the system transfer function G in (3.17)by considering the dynamic variations as a function of the table position and theworkpiece mass. The system transfer function matrix is a two-input-two-outputsystem whose inputs are fm and fc, while its outputs are uc and um, measured bya linear encoder and a rotary encoder, respectively.The identified transfer matrix G should precisely and consistently representthe system dynamics in order to achieve a good performance in the feed-drive control45(a) Damped resonant frequency (b) Damping constantFigure 3.10: System parameters variations as a function of nut location xc andworkpiece mass mworsystem. The general structure of the identified transfer matrix G can be written asG(xc,mwor)(s) = Grigid(mwor)(s)1 11 1+Nmod∑k=1Gflex,k(xc,mwor)(s), (3.20)where the transfer matrices Grigid and Gflex,k respectively designate the rigid-bodymode and the k-th flexible mode, and the number of the flexible modes to be takeninto consideration is denoted by Nmod. The rigid-body mode and the flexible modeparameters are identified by assuming a fixed table mass and Nmod= 1 in (3.20)1.3.4.1 Rigid-body mode and friction identificationSince FRF measurements in the low frequency band are dominated by friction, analternative approach for estimating the rigid-body mode parameters Je and Be wasproposed by Erkorkmaz and Altintas in [31] by jogging the screw axis back andforth at various speeds. As it is shown in Figure 3.11 (a), a square command signal,1Here we assumed that the working frequency range is up to 550 Hz. Since it turned out thatthe only first resonant mode is within this range, we decided to select Nmod= 1. The next resonantfrequency occurs at 620 [Hz].46vm, with a different amplitude and direction is applied to the motor. The transferfunctions of these modes are written explicitly byGrigid(s) :=r2gs(Jes+Be). (3.21)By writing the motor force fm as a function of the motor input voltage vm asfm =kaktvmrg, and by measuring the velocity and acceleration of the table, weutilize the unbiased least squares approach [31] to estimate Je and Be from therigid-body mode differential equation below:Jex¨m = rgkaktvm −Beθ˙m −Tfrg, (3.22)where Tf is the Coulomb friction torque defined asTf :=T−coul x˙m < 0,0 x˙m = 0,T+coul x˙m > 0,(3.23)which is applied to the motor. Adding Tf in the rigid-body mode identificationremoves the bias of the least squares estimation. As stated in [31], by solving thediscretized differential equation, the identified rigid-body mode parameters are ob-tained and listed in Table 3.2. The comparison between the measured and simulatedtable acceleration is depicted in Figure 3.11 (b). In this figure, the simulated resultsare obtained by the identified rigid-body mode parameters.3.4.2 Flexible mode identificationThe flexible mode identification in the two-input-two-output system is achieved bymeasuring the FRF of the transfer function between the motor input force and the47Table 3.2: Identified rigid-body mode parametersSymbol and unit Identified parametersJe [kg.m2] 7.1432 × 10−4Be [Nm/rad/s] 0.0156T+coul [Nm] 0.3686T−coul [Nm] -0.2551(a) Input voltage to the motor(b) Measured (dashed red line) and estimated(solid blue line) acceleration of the tableFigure 3.11: Rigid-body mode identificationmotor shaft and table locations. The dynamic model below explains the matrixtransfer function from the input channels to the output channels:Gflex =G11(s) G12(s)G21(s) G22(s)−1 11 1Grigid, (3.24)where Gij , defined in (3.19), is a proper rational transfer function from the i-th inputchannel to the j-th output channel. By neglecting higher order vibration modes, thetransfer function could be considered as the summation of the rigid and the flexiblemodes asGflex,ij(s) :=αij + βijss2 + 2ζωds+ ω2d, (3.25)48where ωd and ζ are the natural frequency and the damping ratio of the flexiblemode. Having the identified Grigid from the previous section, we measure the FRFfrom the motor force to the screw and table positions, i.e., G11(ω) and G21(ω).By subtracting Grigid from Gij , the Gflex,ij is obtained. Then by the least squaresapproach, similar to [77], we identify α11, α21, β11, β21, ωd, and ζ in order tominimize the error between the measured and estimated real and imaginary parts ofGflex,11 and Gflex,21 as shown in Figure 3.12. Adding the identified Gflex,ij to Grigid,we compare the total identified G11 and G21 with the measured FRFs in Figure 3.13.(a) Gflex,11 real and imaginary parts (b) Gflex,21 real and imaginary partsFigure 3.12: Comparison of real and imaginary parts of the measured (dash-dot redline) and the estimated (solid green line) flexible mode FRFsHaving G11(ω) and G21(ω), and by knowing that G12 = G21, the calculationof G22(ω) using the estimation of mode shape vectors becomes possible as explainedin [1]:α22 + jωdβ22 =(α21 + jωdβ21)2α11 + jωdβ11. (3.26)We validated the estimated G22 by measuring FRF from force to the table position49(a) G11 gain and phase (b) G21, G12 gain and phaseFigure 3.13: Comparison of the measured (dash-dot red line) and the identified(solid green line) total FRFsto the table position using hammer test in Figure 3.14. According to this figure,there is acceptable agreement between measured and identified FRF of G22.3.5 Dynamic Variations in Ball-Screw Drive SystemsAs mentioned before, the system dynamics varies depending on the table position,due to the change of the active length of the screw between the motor shaft and thetable. In addition, the run-out effect is another non-linearity in the system. Therun-out effect occurs due to the misalignment of the ball bearing axes holding thescrew shaft. As demonstrated in [90], the run-out effect can also be represented interms of parameter variations in the flexible modes. To observe these variations, werepeat the identification procedure in the previous sections by measuring the FRFat different table locations and workpiece masses.50Figure 3.14: Validation of G22 for identified FRF (blue line), and measured FRF(red line) with hammer test3.5.1 One varying parameter (xc)Since the model parameters vary with respect to the table location, FRF measure-ments were taken by placing the table in different positions and changing the positionat every 5 [mm] in the interval xc ∈ [0.2, 0.6] [m]. The identified parameters andthe estimated LPV model are shown in Figure 3.15. By inspecting the parametervariations in Figure 3.15, the following function form for parameter variations isadopted:o(xc) = c0 + c1xc + c2x2c + c3 sin(xcrg+ c4), (3.27)where o(xc) is a representation of ωd, ζ, αij , and βij for ij equals 11 and 21. Thisfunction form consists the following two parts: the sinusoidal part and the quadraticpart. The sinusoidal part is used in order to represent the periodic parameter varia-tions and it corresponds to the periodic run-out effect due to bearings misalignment;whilst the quadratic part is employed by an inspection of the FRF data in Figure 1.4.51(a) (b)Figure 3.15: Estimated parameters at identified positions (red cross marker) andLPV functions in (3.27) (green dashed line)Since ωd shows negligible harmonic behaviors among the estimated param-eters, c3 in (3.27) is ignored. The identified functions of (3.27) are shown in Fig-ure 3.15, and the estimated parameters are listed in Table 3.3.Having the LPV system transfer function, we calculate the physical param-eters of the system in (3.12), and compare the estimated LPV model with theanalytical model results in Figure 3.16. In this comparison, we can see that theanalytical model predicts the stiffness, damping, and resonant frequency very closeto the experimental LPV model. On the contrary, the m11 and m12 elements of theinertia matrix in (3.11) model are quite different from the estimated LPV modelwhile the m22 element in the that model is almost identical to the estimated LPV.52Table 3.3: LPV model of the system parameters as a function of xcSymbol and unit Flexible mode identified parametersζ [%] +2.70 −2.36(xc) +3.05(x2c) −0.15 sin(xc/rg − 1.90)ω1 [103rad/s] +1.57 −0.49(xc) +0.17(x2c) +0.00 sin(xcrg− 0.90)α11 [10−3 m/s2/N] +5.05 +7.05(xc) −3.83(x2c) −0.22 sin(xcrg− 2.90)α12 [10−3m/s2/N] −11.3 −5.2(xc) +2.90(x2c) −0.4 sin(xcrg− 6.19)β11 [10−6 m/s/N] −0.99 −2.52(xc) +2.60(x2c) +0.396 sin(xc/rg + 1.90)β12 [10−6m/s/N] +2.77 +1.98(xc) −2.28(x2c) −0.71 sin(xcrg− 1.90)3.5.2 Two varying parameters (xc and mwor)Similar to the previous section, we measure the FRF by changing the table positionxc at every 5 mm intervals while the workpiece mass mwor is set to 0, 5, and 10 [kg]for all the table positions. Among the estimated parameters, we depict the ωdestimation in Figure 3.17 using a linear and a quadratic function.For the linear part, it is assumed that ωd is a linear function of xc and mwor,i.e. ωd := c1 + c2(xc) + c3(mwor). As a quadratic harmonic function, we determineωd :=(c1 + c2(xc) + c3(x2c) + c4 sin(c5xc + c6)) (c7 + c8(mwor) + c9(m2wor)). By com-paring the mean and the root mean square (rms) error of each function, we noticethat the mean and the rms errors are reduced from 4.70 and 5.86 in the linear modelto 3.50 and 4.32 in the quadratic harmonic model, respectively. This means thatthere is only a 25% reduction in the mean error and a 26% reduction in the rms error;which is obtained at the expense of increasing the LPV parameters from three in thelinear model to nine parameters in the non-linear model. Theretofore, to maintainthe simplicity of the control design and implementation, we choose the linear modelto represent the dynamics of the system. The other identified parameters are listed53(a) Stiffness, damping, and resonant frequency (b) Inertia matrix elementsFigure 3.16: Comparison of the estimated parameters at identified positions (redcross marker) and LPV model (dashed green line) with the analytical model (solidblue line)in Table 3.4.3.6 Linear Parameter Varying System RepresentationFor controller design purposes, we will transform the model into a linear parametervarying system. The transfer matrix G in (3.20) can be utilized in an LPV system.We assume Nmod = 1 with an introduction of a state vector xm, asG(θ) :x˙m(t) = Am(θ)xm(t) +Bm(θ)fm(t)fc(t) ,um(t)uc(t) = Cmxm(t).(3.28)54(a) Fitted as a linear function(b) Fitted as a quadratic harmonic func-tionFigure 3.17: Estimated parameters at identified positions (magenta points) andcontinuous LPV function (surface)Table 3.4: LPV model of the system parameters as a function of xc and mworSymbol Identified parameters Unitζ 2.3 [%]ωd 1299− 15(mwor) + 0.32(0.64− xc) [rad/s]α12 −15.35 + 0.41(mwor) + 2.8(0.64− xc) 10−3 [m/s2/N]α11 9.17− 0.11(mwor) + 3.9(0.64− xc) 10−3 [m/s2/N]β12 5.15− 0.08(mwor)− 0.48(0.64− xc) 10−6 [m/s/N]β12 −2.825(mwor)− 0.820(0.64− xc) 10−6 [m/s/N]Here, θ is a function of the following LPV parametersθ(t) := [xc(t),mwor(t)]. (3.29)55The system matrices Am, Bm and Cm in (3.28) consist of state space realizations ofthe transfer matrices for modes in (3.21) and (3.25) as:Am :=Arigid(mwor(t)) 00 Aflex(xc(t),mwor(t)) , (3.30)Bm :=Brigid(mwor(t))Bflex(xc(t),mwor(t)) , (3.31)Cm :=[Crigid Cflex], (3.32)Grigid(s) = Crigid(sI −Arigid)−1Brigid, (3.33)Gflex(s) = Cflex(sI −Aflex))−1Bflex. (3.34)Here, the matrices of the state-space realizations in (3.33) and (3.34) areArigid = −BeJe(mwor(t))02 I202 I2 , Brigid =r2gJe(mwor(t))01 11 1,Aflex =−2ζωd(xc(t),mwor(t)))I2 I2−ω2d(xc(t),mwor(t))I2 02 , Bflex =β11 β12β21 β22α11 α12α21 α22(xc(t),mwor(t)),Crigid = [I2 02] , Cflex,k = [I2 02] .(3.35)56Chapter 4Parallel Tracking and StructuralVibration Control of theBall-ScrewThe goal of this chapter is to propose a controller structure which is suitablefor ball-screw drive systems with dynamic variations to attain high performanceof tracking and vibration suppression. The controller consists of two parallel con-trollers, which take charge of different objectives. The first controller is the linear-time-invariant one synthesized to improve the tracking accuracy and speed of theworkpiece table for reference trajectories. This controller is designed so that it re-duces the low frequency gain of the sensitivity function and increases the bandwidthof the closed-loop system. On the other hand, a gain-scheduled controller suppressesthe flexible mode. The controller is designed to damp out the flexible mode withvarying resonant frequencies. As the types of these two controllers, we will adoptthe H∞ controller and the LPV controller in this chapter. The advantages of the57proposed parallel controller structure in controller design and implementation willbe stated. The experimental result will be compared with the conventional PIDcontroller.4.1 Control Objectives for Ball-screw Drive SystemsIn feed-drive control of the ball-screw drives, the ultimate goal is to accuratelycontrol the position of the workpiece which is specified by reference trajectories.However, cutting force disturbances in the machining processes can increase vibra-tion and tracking error. Hence, it is essential to design a feedback controller whichaccomplishes the fast and accurate tracking to the reference trajectories, and atthe same time, suppresses structural vibration of the ball-screw drive uniformly forvarying resonant modes.4.2 A Parallel Controller StructureBall-screw drive systems are typically designed and manufactured to have naturalfrequencies higher than the servo bandwidth [4]. In this case, it is possible to sep-arate the controller design problem into two sub-problems associated with differentfrequency ranges, that is 1) to design a controller to track a reference signal in thelow frequency range, and 2) to design another controller to suppress vibration nearthe resonant frequencies.The separation of one multi-objective control problem into two single-objectivesub-problems leads to the proposal of a parallel controller structure depicted in Fig-ure 4.1. The parallel controller consists of two controllers, a tracking controllerKTrack, a vibration suppression controller KVib and a disturbance observer KDOB.58The controller KTrack is used to minimize the tracking error e between a referencetrajectory r and the workpiece table location uc, as well as to maximize the closed-loop bandwidth for fast tracking. On the other hand, the controller KVib mitigatesthe structural vibration due to the cutting force disturbances fc by feeding backtwo signals, the table position uc and the motor shaft equivalent position um. Twocontroller outputs, denoted by vt and vv, are summed up and applied as the controlcommand vm to the DC motor. KDOB is a disturbance observer to estimate andcancel out the effect of friction by receiving both table position uc and motor signalvm as inputs and to send a correction signal vc to the motor.Figure 4.1: Block diagram of the parallel controller structureThe parallel structure of controllers in Figure 4.1 offers some advantages overMIMO controllers without any specific structure, by facilitating both controller de-sign and implementation. In most controller design techniques, we design param-eters to be tuned iteratively. For the proposed parallel structure, to improve oneperformance (i.e., tracking or vibration suppression), we can focus on tuning designparameters which are relevant to that performance, without changing the other de-sign parameters which influences the other performance. This decoupling betweendesign parameters and control objectives simplifies the controller design. In ad-59dition, the design of a general MIMO controller often generates a controller withunduly large degrees, while the controllers in the proposed structure tend to haverelatively low degrees. Such low degree controllers reduce computational burdens inthe design and implementation with a microcontroller. Moreover, in the manufac-turing industry, most of the servo drives have already been equipped with a trackingcontroller.4.3 Controller Design ProcedureHere, we will propose a controller design procedure for the parallel controller struc-ture. For the controller design, an LPV model of the ball-screw drive system is givenin Section 3.6. The ball-screw drive system presented in Chapter 3 can be modeledas a two-input-two-output system whose inputs are fm and fc, and whose outputsare um and uc. For the subsequent controller design, we transform the transfermatrix model G in (3.20) into an LPV system, by introducing a state vector xm, in(3.28).With the LPV model, Section 4.3.1 and Section 4.3.2 respectively reduce thedesign problems of KTrack and KVib to an H∞ controller design problem and anLPV gain-scheduled controller design problem. We provide a guideline for tuningdesign parameters in Section 4.3.4.4.3.1 Tracking controller designFor the model (3.28), we will first design the tracking controller KTrack in Figure 4.1.Although the model depends on the nut position xc, for the tracking controllerdesign, we consider only the nominal rigid-body model by ignoring dynamic vari-ations. This simplification can be considered to be legitimate in ball-screw drive60applications, because the reference signal for the table movement is normally givenover frequencies lower than resonant modes. For the linear-time-invariant nominalplant, there are a number of tracking controller design methods, such as PID, linear-quadratic regulator (LQR), sliding mode, and H∞ control. Next, we explain how toreduce the tracking controller design problem to an H∞ controller design problem.The objectives in designing the tracking controller KTrack are to guaranteeclosed-loop stability, to minimize the tracking error, and to increase the closed-loopbandwidth for fast machining. In order to meet these objectives, in the H∞ con-troller design framework, the feedback controller KTrack is synthesized by selectingappropriate weighting functions We and Wv in Figure 4.2. In the figure, two weight-Figure 4.2: Block diagram of the H∞ tracking controller designing functions We and Wv are introduced to penalize the tracking error e and thecontrol signal vt, respectively. The general approach to the tuning of the weighingfunctions is provided in Section 4.3.4. The combination of the plant G in (3.20)61with weighting functions We and Wv creates an augmented plant asx˙(t) = Ax(t) +B1r(t) +B2vt(t),z(t) = C1x(t) +D11r(t) +D12vt(t),e(t) = C2x(t) +D21r(t),(4.1)where x is the state vector consisting of the plant states and the states of weightingfunctions, and z := [ze zv]T . For the LPV model of the system in (3.28) andassuming mwor=0, a dynamical transfer function We(s) := Ce(sI − Ae)−1Be + Deand a constant gain Wv, system matrices for the generalized plant in (4.1) areA :=Am 0−BeCml Ae , B1 :=0Be , B2 :=ktkargBmv0 , D21 := 1,C1 :=−DeCml Ce0 0 , C2 :=[Ce 0], D11 :=De0 , D12 :=0Wv ,(4.2)where Bmv is the first column of the matrix Bm, and Cml is the second row of thematrix Cm. Having the augmented plant, the controller KTrack is synthesized sothat the closed-loop system is internally stable and the H∞ norm of the transferfunction Trz from r to z is minimized. This H∞ control problem can be solved bythe well-known theory in [39] and the available MATLAB command hinfsyn.m.4.3.2 Structural vibration controller designAs mentioned in Section 4.1, the ball-screw drive system has flexible modes withvarying dynamics depending on the table location. Machining forces may excitethese varying flexible modes. Damping of the varying flexible modes can be achievedby the second controller KVib(xc) which is gain-scheduled by the table position xc.The objectives in designing a gain-scheduled controller KVib(xc) are to reduce the62peak amplitude of the flexible modes at varying resonant frequencies, to satisfyclosed-loop stability, and to maintain the tracking performance which has alreadybeen achieved by KTrack.To fulfill the control objectives, we consider the feedback structure in Fig-ure 4.3. To obtain system matrices for the augmented plant in (4.5), we first com-bine the ball-screw plant G(xc) and the tracking controller KTrack(s) := CKt(sI −AKt)−1CKt +DKt asx˙n(t) = Anxn(t) +Bnktkargvv(t)fc(t) ,um(t)uc(t) = Cnxn(t),(4.3)whereAn :=Am − ktkarg BmvDKtCmktkargBmvCKt−BKtCm AKt , Bn :=Bm0 ,Cn := [Cm 0].(4.4)Then, weighting functions Wf := Cf (sI−Af )−1Bf+Df , and Wv shown in Figure 4.3are respectively to penalize the vibration amplitude and the control signal vv at thespecified frequency ranges due to disturbance forces fc. The combination of theLPV plant G(xc) in (3.20) with KTrack, Wf and Wv yields an augmented plant as˙˜x(t) = A˜(xc)x˜(t) + B˜1(xc)f(t) + B˜2(xc)vv(t),z˜(t) = C˜1(xc)x˜(t) + D˜11(xc)f(t) + D˜12(xc)vv(t),y(t) = C˜2(xc)x˜(t) + D˜21(xc)fc(t),(4.5)where x˜ is the total state vector, z˜ := [zf zv]T , y := [um uc]T , and with a dynamicalweighting function Wf (s) := Cf (sI − Af )−1Bf + Df and a constant gain Wv, the63system matrices in (4.5) can be derived asA˜ :=An 0BfCm Af , B˜1 :=Bf0 , B˜2 :=Bmv0 , D˜21 := 0,C˜1 :=DfCn Cf0 0 , C˜2 :=[Cm 0], D˜11 :=00 , D˜12 :=0Wv .(4.6)Figure 4.3: Block diagram of the structural vibration controller designFor the augmented plant (4.5), the controller design problem to meet theobjectives mentioned above can be stated as designing a gain-scheduled controllerin (4.8) which guarantees exponential stability of the closed-loop system, and a givenL2-gain bound as∫ T0z˜T (τ)z˜(τ)dτ < γ˜2∫ T0fT (τ)f(τ)dτ, ∀T > 0, (4.7)for any trajectory of the time-varying parameter xc within pre-specified ranges ofthe position xc and the rate of change x˙c for the table. To solve this controllerdesign problem, the advanced gain-scheduled controller design method, developedin [5] and reviewed in the following section, has been used.644.3.3 Gain-scheduled controller designThe gain-scheduled controller KVib(xc) in Section 4.3.2x˙k(t) = AKv(xc(t))xk(t) +BKv(xc(t))um(t)uc(t) ,vv(t) = CKv(xc(t))xk(t) +DKv(xc(t))um(t)uc(t) ,(4.8)is synthesized by solving an optimization problem using the linear matrix inequality(LMI) approach [5]. The optimization problem for satisfying closed-loop stabilityand L2-gain bound is:minAˆK ,BˆK ,CˆK ,DK ,X,Yγsubject toX II Y (xc) > 0, xc ∈ L,N11(xc) ∗ ∗ ∗N21(xc) N22(xc) ∗ ∗N31(xc) N32(xc) −γI ∗N41(xc) N42(xc) N43(l) −γI< 0, xc ∈ L(4.9)whereN11(xc) := XA˜+ BˆK + (∗), N21(xc) := AˆTK + A˜+ B˜2DKC˜2,N22(xc) := −Y˙ + A˜Y + B˜2CˆK + (∗), N31(xc) := (XB˜1 + BˆKD˜21)T ,N32(xc) := (B˜1 + B˜2DKD˜21)T , N41(xc) := C˜1 + D˜12DKC˜2,N42(xc) := C˜1Y + D˜12CˆK , N43(xc) := D˜11 + D˜12DKD˜21,(4.10)and A˜, B˜1, B˜2, C˜1, C˜2, D˜11, D˜12 and D˜21 correspond to the matrices given in (4.6),and L denotes the moving region of the parameter xc. To reduce the optimization65problem into a finite-dimensional one, all parameter-varying matrices AˆK , BˆK , CˆK , DK ,and Y are parameterized in terms of xc, with the same dependency on xc as in (3.27),while X is assumed to be a constant matrix variable. After solving the optimiza-tion problem above, the controller matrices in (4.8) can be reconstructed from [5,Equation (20)] asAKv(xc) BKv(xc)CKv(xc) DKv(xc) =X 00 I−1Q(xc)X−1 − Y (xc) 00 I−1, (4.11)whereQ(AˆK , BˆK , CˆK , DK , Y ) :=AˆK BˆKCˆK DK−X(A˜− B˜2DKC˜2)Y + BˆKC˜2Y +XB˜2CˆK XB˜2DKDKC˜2Y 0 .(4.12)4.3.4 Guidelines for tuning design parametersNext, we will provide the guidelines to select the weighting functions We, Wf andWv in Sections 4.3.1 and 4.3.2 by a conventional heuristic method in the contextof ball-screw drive control applications. Although methods have been developed toautomatically determine the weighting functions based on optimization in, e.g., [64,7], they also have some tuning parameters, such as lower and upper bounds offrequency responses in [64]. The tuning parameters need to be iteratively modifiedfor performance improvement. The presented guidelines will be useful even for suchtuning.Figure 4.4 (a) and Figure 4.4 (b) illustrate how the gain plots of the functionsWe−1 and Wf−1 typically affect those of the closed-loop transfer functions from r toe in Figure 4.2, denoted by Tre, and from fc to uc in Figure 4.3, denoted by Tfu. As66(a) (b)Figure 4.4: Weighting function selections, (a) |W−1e | (red dash-dot line) and |Tre|(purple solid line), (b) |W−1f | (red dash-dot line) and |Tfu| (purple solid line)indicated in Figure 4.4, the performances of the ball-screw drive systems are relatedto the closed-loop gain plots as follows:• Tracking error is decreased by reducing the low frequency gain of Tre.• Tracking speed, related to the closed-loop bandwidth, is increased by increas-ing the unit-gain crossing frequency of Tre.• Stability margin of the closed-loop system is enlarged by reducing the peakgain of Tre.• Structural vibration is reduced by damping out the resonant modes ofTfu.One way of parameterizing the weighting functions was given in [111] asW (s) =(s/M1/kWH + ωbs+ ωbM1/kWL)kW, (4.13)where tuning parameters in controller design are ML, low frequency gain of W−1,MH , high frequency gain of W−1, ωb, unit-gain crossing frequency of W−1, and kWdegree of the function W .67For the function We, to reduce tracking error and to improve the trackingspeed of the drive system, one can try to reduce ML and increase ωb, respectively.However, ωb needs to be smaller than the first principal resonant mode because alarge ωb may lead to structural vibration. In many applications including ball-screwdrives, the parameters MH and kW are typically set to 2 and 1, respectively. If thefirst order kW was not suitable, an appropriate degree of kW is searched from lowto high degree.On the other hand, for the function Wf , to improve the vibration suppressionby damping out resonant modes, one can try to decrease MH and increase ML, bysetting ωb below the first principal resonant frequency. The degree kW is againsearched in an ascending order.Finally, for the function Wv, to reduce the motor voltage input and/or toavoid voltage saturation, one should try to increase the gain of Wv, which is eithera static gain or a dynamic transfer function.4.3.5 Disturbance observer designTime-varying and position-dependent characteristics of friction forces justify designof an observer to estimate and compensate friction forces. The disturbance observer,KDOB, is synthesized to estimate and cancel out friction and disturbance forces inthe low frequency range. There are two inputs vm and uc, and one output vc fromKDOB (see Figure 4.1). The structure of KDOB is suggested by Umeno and Hori [97]asKDOB :=[−QQGˆrigid], Q(s) :=1a2τ2s2 + a1τs+ 1, (4.14)where Gˆrigid, Q, τ , a1 and a2 are an estimated model of the rigid-body mode, astable low pass filter, the cut-off frequency and two positive constants, respectively.68The second-order low pass filter Q is added to make Q/Gˆrigid as a proper transferfunction. In the frequency range lower than the cut-off frequency of the filter Q, theinner-loop system from vc and fc to fˆc can be written asfˆc =ktkarg(1−GˆrigidG)vm + fc. (4.15)The estimated disturbance fˆc approaches to the actual disturbance fc when theGˆrigid is a close estimation of the rigid-body mode of the G 1.4.4 Controller Design for Experimental Ball-Screw DriveSetupWe will apply the modeling and controller design methods presented in Sections 4.2and 4.3 to the experimental ball-screw drive setup shown in Figure 1.4. Recall thatthe screw with 820 [mm]-length is driven directly by a brushless DC motor whichis operated in current control loop. The table with mc =20 [kg] mass slides on theroller bearing guideway and its location xc = uc is measured by a linear encoderwith 100 [nm] resolution. The rotary encoder at the edge of the screw measures theangle of the shaft θm and we calculate its equivalent translation with um = rgθm.4.4.1 Tracking controller designIn designing the tracking controller KTrack, the parameters of weighting functionsWe and Wv are selected according to the guidelines in Section 4.3.4. The selectedparameters for We in (4.13) are given in Table 4.1 and Wv = 0.0006. Additionally,the transfer function Q in (4.14), which is the low pass filter in KDOB, is selected1Due to small gain of Q in the high frequency range, we ignore effect of KDOB in the controllerdesign stage.69to be a stable filter with a cut-off frequency less than the system natural frequency.Table 4.1: Design parameters of the gain-scheduled controllerWeights kW ML [dB] MH [dB] ωb [rad/s]We 3 -140 4 750Wf 1 -110 -175 6To assess the closed-loop performance of the designed controller, the mag-nitude of the sensitivity function S, which is the transfer function from r to e andthe complementary sensitivity function Tru, which is the transfer function from r touc, are shown with blue lines in Figure 4.5. As shown in this figure, the weightingfunction We penalizes the gain of S in the low frequency range. Reduction of thetracking error about −130 [dB] in the low frequency range is satisfactory for achiev-ing the micro-meter tracking error. In addition, the bandwidth, which correspondsto crossing frequency of Tre over −3 [dB], is about 450 [rad/s].4.4.2 Vibration controller designHaving obtained the tracking controller KTrack, the next step is to synthesize again-scheduled controller KVib(xc) to compensate for vibration of the system in theresonant frequencies. The parameters of the weighting function Wf in (4.13) arechosen as in Table 4.1, and Wv = 100.To evaluate the effect of the designed controller, the gain plots of the transferfunction from fc to uc for open-loop and closed-loop systems are shown in Figure 4.6while the table is located at three positions 200, 350 and 500 [mm]. As can be seen inFigure 4.6, the amplitude of the gain in the resonant frequency is reduced by about16 [dB] for the three table positions. This result confirms satisfactory performance of70100 101 102 103 104−100−500Gain (dB)100 101 102 103 104−40−200Gain (dB)Frequency (rad/s)Figure 4.5: Gain plots of S transferfunction (top) and Tru transfer func-tion (bottom) without KVib (dash-dotblue line) and with KVib (solid redline); the dashed lines in top and bot-tom figures indicate the gain of W−1eand -3 dB, respectively100 101 102 103 104−180−160−140−120Gain (dB)Frequency (rad/s)    KTrack KTrack & KVibFigure 4.6: Gain plots of the transferfunction Tfu with just KTrack (blue),with both KTrack and KVib (red),and the weighting functionWf−1 withdashed line (black) where the tablexc is placed at 200 (dash-dot line),350 (dashed line) and 500 [mm] (solidline)the controller in compensation of structural vibration near the resonant frequenciesand avoidance of violent vibration.The effect of the vibration suppression controller KVib on the tracking con-troller KTrack is evaluated by plotting the transfer functions S and Tru in Figure 4.5with and without KVib. The similarity of two sensitivity transfer functions in thelow frequency range indicates that addition of KVib does not degrade the trackingperformance in the low frequency range. On the other hand, the designed KVibcontroller improves the bandwidth drastically by damping out the flexible mode. Infact, the closed-loop bandwidth with KVib increased to 1500 [rad/s] which is threetimes more than the closed-loop bandwidth without KVib.4.4.3 PID controllerA PID controller is a conventional controller which is designed to compare its per-formance with the advanced controllers. In the PID controller, input and output of71the controller are the tracking error and the motor voltage, respectively. By usingthe MATLAB Control Systems Toolbox, the proportional, integral and derivativegains are tuned as 5× 105, 3× 106 and 120, respectively.4.5 Experimental ResultsThe controllers designed in the previous section are implemented by a microcon-troller, DS1103 (dSPACE Inc.), with a sampling period of 0.36 [ms]. The controllerevaluation is conducted in both the frequency domain and the time domain. Inall the experimental results, the performance of the following three cases can becompared with each other, Case I, with only KTrack controller, Case II, with bothKTrack and KVib controllers, Case III, with only PID controller.4.5.1 Frequency domain analysisIn the frequency domain, the performance of controllers in the vibration suppressionis evaluated by measuring the FRF from the input disturbance to the table location.The magnified measured FRFs (Gain×107) in three cases are illustrated in Figure 4.7for the table positions at xc=250 and 450 [mm]. One can see in the figure that theresonance modes which appear in Case I has been successfully damped out by KVibin Case II, by about 14 [dB] at various table positions. However, the PID controllerhas a higher gain and flexibility in the low frequency range.4.5.2 Time domain analysisIn the time domain experiment, the table travels 300 mm to track a reference signal.The reference signal , r(t) = 650−xc(t) , is a smooth trajectory with a constant jerkas shown in Fig. 4.8. This trajectory specifies position of the table while traveling72630 940 1250 1560 1870−20−10010Gain (dB)Frequency (rad/s)Case IICase III Case IFigure 4.7: Measured gain plots of the transfer functions from fc to uc in Case I(blue), Case II (red) and Case III (green) where the table is located at 250 [mm](dash-dot line) and 450 (solid line)from 150 to 450 [mm] with maximum speed of 480 [mm/s] which is velocity limitof the test bed. Additionally, the transfer function Q transfer function in (4.14) isselected to be a stable filter with a cut-off frequency less than the system naturalfrequency as Q = (10−5s2 + 0.05s+ 1)−1.No disturbanceThe tracking error without disturbance is shown in Figure 4.9 (top and bottomfigures) for the three cases. The maximum tracking error occurs due to frictioneffects when the table direction of motion changes at the beginning and middle ofthe trajectory. The maximum tracking error was reduced by about 60% and 40%in Case II in comparison with Case I and Case III, respectively. These resultsdemonstrate the effect of the vibration suppression controller KVib in Case II toreduce disturbance due to friction, by extending the close loop bandwidth in trackingperformance.730 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2150250350450Position    (mm)     0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−5000500Velocity (mm/s)   0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−200002000Acceleration (mm/s2 )   0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−202 x 104Jerk    (mm/s3 )Time (sec)Figure 4.8: Reference trajectory signal with constant jerkHarmonic disturbanceAs another example to consider the effect of disturbance, a harmonic disturbanceinput voltage signal from the motor fc = 3ktkarg sin(1350t) [N] is applied to excite theflexible mode of the ball-screw which is between 210 to 240 [Hz] (see ωd in Figure 4.6)while following the trajectory in Figure 4.9 (top figure). The disturbance signalcaused the resonance in the system. The tracking error is plotted in Figure 4.10for the three cases. Compare to Case I, the adverse effect of resonance in thetracking performance reduced by about 50%. This result highlights the importanceof vibration suppression to reduce the error in the feed-drive control and avoidresonance.Emulated cutting disturbanceNext, the performance of the designed controller was tested with two cutting pro-cesses, i.e., half immersion up milling and half immersion down milling. In order to74Figure 4.9: Tracking error compar-ison of Case I (dash-dot blue line)and Case II (red solid line) (top), andtracking error comparison of Case II(red solid line) and Case III (dash-dot green line) (bottom) while thereis no external disturbance0 0.5 1 1.5 2−75075  Time (sec)Error (µm)0 0.5 1 1.5 2−75075Error (µm)  Figure 4.10: Tracking error compar-ison of Case I (dash-dot blue line)and Case II (red solid line) (top), andtracking error comparison of Case II(red solid line) and Case III (dash-dot green line) (bottom) while thereis harmonic disturbanceemulate the real cutting processes, it is required to estimate a cutting force effecton the feed-drive control system. Cutting coefficients are estimated by conductinga real cutting test on AISI 4140 steel. The cutting conditions, tool geometries andestimated cutting coefficients are listed in Table 4.2. Having the estimated cuttingcoefficients Ktc, Krc and tool geometries, the cutting force is obtained by followingthe procedure in [1, p. 40]. Using parameters in Table 4.2, cutting forces for twocutting processes, i.e., half immersion up milling and half immersion down millingare simulated. The simulated resultant cutting forces are plotted in Figure 4.11.The cutting processes are simulated by applying the equivalent cutting forcesas a disturbance input signal into the DC motor of the ball-screw test setup. Forthis experiment, in order to generate the maximum vibration of the ball-screw drivein the cutting process, we assume the frequency of the spindle to be very close tothe resonant frequency of the system. The tracking errors of these experiments forhalf immersion up milling and down milling processes are shown in Figure 4.12 and75Table 4.2: Cutting geometry and coefficientsTitle Symbol and Unit ValueDepth of cut acut [mm] 2.7Spindle speed nspin [rpm] 3400Feed rate ccut [mm/rev/tooth] 0.1Cutting diameter Dcut [mm] 18Number of teeth Nteeth 4Helix angle βhelix [deg] 30Tangential cutting force const. Ktc [N/mm2] 1980Normal cutting force const. Krc [N/mm2] 540Tangential cutting force edge const. Kte [N/mm2] 35Normal cutting force edge const. Kre [N/mm2] 30Figure 4.13 for Case I to Case III.Table 4.3: Absolute tracking error in time domain experimentsDis. Case Ave. Max. Dis. Case Ave. Max.NoI 6.7 34.6Up millingI 26.9 70.8II 3.3 14.2 II 9.9 35.9III 7.5 23.5 III 31.7 111HarmonicI 25.3 70Down millingI 28.1 70.3II 13.3 39.1 II 9.5 28III 29 74 III 32.0 105760 100 200 300 360200400600Down milling (N)Spindle rotaion (deg)0 100 200 300 360200400600Up milling  (N)Figure 4.11: Simulated resultant cutting forces: half immersion up milling force(top) and half immersion down milling force (bottom)0 0.5 1 1.5 2−1000100Time (sec)Error (µm)  0 0.5 1 1.5 2−75075Error (µm)  Figure 4.12: Measured tracking errorin emulated half immersion up millingfor Case I (dash-dot blue line) andfor Case II (solid red line) (top), andCase II (solid red line) and Case III(dash-dot green line) (bottom)0 0.5 1 1.5 2−75075Error (µm)  0 0.5 1 1.5 2−1000100Time (sec)Error (µm)  Figure 4.13: Measured tracking er-ror in emulated half immersion downmilling for Case I (dash-dot blue line)and Case II (solid red line) (top), andCase II (red solid line) and Case III(dash-dot green line) (bottom)77Chapter 5Switching Gain-ScheduledControl of the Ball-ScrewIn this chapter, we demonstrate experimentally that the switching gain-scheduledcontrol technique can achieve required objectives in the ball-screw drive system byexplicitly considering the dynamic variations in controller design. To this end, aLPV model is obtained to represent plant parameter variations as a function of thetable position and the workpiece mass. Then, the stroke of the ball-screw is dividedinto a fixed number of subregions. Given the LPV model, a switching controllerwhich consists of the fixed number of gain-scheduled controllers is designed. Eachgain-scheduled controller accomplishes the objectives in each subregion. Finally, themodeling and controller design methods are experimentally tested on a laboratory-scale ball-screw drive.78Figure 5.1: Block diagram of the switching feedback control system.5.1 The Switching LPV Controller Design MethodGiven the LPV model of the ball-screw drive system G(xc,mwor) in (3.20) which rep-resents the dynamic variations, our objective is to design a switching gain-scheduledcontroller depicted in Figure 5.1. The switching gain-scheduled controller will havethe advantage of fast and precise tracking of the reference signal r over conventionalcontrollers, especially in large-scale industrial machining systems with a long strokeand with a large amount of mass reduction during the cutting operation. The inputsof the controller are tracking error e(t) :=r(t)−uc(t) and the equivalent displacementof the motor shaft is um := rgθm where θm is rotational angle of the shaft, while itsoutput is the motor voltage vm. Furthermore, the transient jump after switchingbetween controllers is avoided by a adding a filter after the controller. However,this filter may impose limitations on the controller performance by smoothing thecontrol signal. The friction effect is compensated by design of a disturbance observerdescribed in Section 4.3.5.79The switching gain-scheduled controller consists of n controllers, K(i)1 fori = 1, . . . , n, associated with i-th subregion of the working range of the table L,and a switch. Each controller is itself a gain-scheduled controller which adjusts itsparameters as a function of (xc,mwor). The switch specifies an operating controlleramong those n controllers at each time t depending on the xc(t) value.5.1.1 Design problemIn order to satisfy required objectives, a controller is designed by considering anaugmented plant with weighting functions Wr, Wf and Wv, as shown in Figure 5.2.The weighting functions are tuning parameters in controller design. They penalizethe tracking error, flexible modes vibration amplitude and restrict control signalover specific frequency. The combination of the LPV plant G(xc,mwor) in (3.24)with Wr, Wf and Wv creates an augmented plant as:x˙(t) = A(xc,mwor)x(t) +B1(xc,mwor)w(t) +B2(xc,mwor)vm(t),z(t) = C1(xc,mwor)x(t) +D11(xc,mwor)w(t) +D12(xc,mwor)vm(t),y(t) = C2(xc,mwor)x(t) +D21(xc,mwor)w(t),(5.1)where x is the state vector, and w := [wr f ]T , z := [e zv]T and y := [e um]T .By considering the varying parameters (xc,mwor) and their derivative (x˙c, m˙wor)as bounded parameters, the goal is to design a gain-scheduled controller asK(i)(xc,mwor) :=x˙(i)K (t) = A(i)K (xc,mwor)x(i)K (t) +B(i)K (xc,mwor)y(t),y(i)K (t) = C(i)K (xc,mwor)x(i)K (t) +D(i)K (xc,mwor)y(t),(5.2)for i = {1, 2, ..., n}, where x(i)K is the state vector of the controller K(i)(xc,mwor)such that it guarantees exponential stability of the closed-loop system, and a given1In this chapter, we use the superscript (i) to indicate the numbering of a subregion.80Figure 5.2: Augmented block diagram for controller designL2 gain bound γ from the disturbance input w to the performance signal z as∫ ∞0zT (τ)z(τ)dτ ≤ γ2∫ ∞0wT (τ)w(τ)dτ. (5.3)In the switching gain-scheduled controller design, the region L is dividedinto n overlapped subregions as L =⋃ni=1 S(i). Assuming the hysteresis switchingbetween two adjacent regions S(i) and S(i) presented in [72], there are two switchingsurfaces S(ij) and S(ji). As explained in [72], [39] and [5], the design of a switchinggain-schedule controller is a convex optimization problem which can be convertedinto an optimization problem with LMIs by assuming X as a constant matrix:minimizeAˆ(i)K ,Bˆ(i)K ,Cˆ(i)K ,D(i)K ,X,Y(i) γsubject toΓ(i)1 > 0, xc ∈ S(i), mwor ∈M,Γ(i)2 < 0, xc ∈ S(i), mwor ∈M,Y (i) − Y (j) < 0, xc ∈ S(ij), mwor ∈M,Y (j) − Y (i) < 0, xc ∈ S(ji), mwor ∈M,(5.4)81for i, j = 1, . . . , n and i 6= j, where 2Γ(i)1 (xc,mwor) :=X II Y (i)(xc,mwor) ,Γ(i)2 (xc,mwor) :=N11(xc,mwor) ∗ ∗ ∗N21(xc,mwor) N22(xc,mwor) ∗ ∗N31(xc,mwor) N32(xc,mwor) −γI ∗N41(xc,mwor) N42(xc,mwor) N43(xc,mwor) −γI,(5.5)andN11 := XA+ Bˆ(i)K + (∗), N21 := Aˆ(i)TK +A+B2DKC2,N22 := −Y˙ (i) +AY (i) +B2Cˆ(i)K + (∗), N31 := (XB1 + Bˆ(i)K D21)T ,N32 := (B1 +B2D(i)K D21)T , N41 := C1 +D12D(i)K C2,N42 := C1Y (i) +D12Cˆ(i)K , N43 := D11 +D12D(i)K D21.(5.6)The first and second inequality conditions in (5.4) guarantee stability and L2 gaininside the i-th region, while the third and forth conditions ensure switching stabilitybetween adjacent regions i and j. Finally, the controller matrices in (5.2) can bereconstructed asA(i)K B(i)KC(i)K D(i)K =X 00 I−1Q(i)X−1 − Y (i) 00 I−1, (5.7)2In this chapter, we use the notation * to indicate a block matrix that makes the total matrixsymmetric.82for i = {1, 2, ..., n}, whereQ(i)(Aˆ(i)K , Bˆ(i)K , Cˆ(i)K , D(i)K , Y(i)) :=Aˆ(i)K Bˆ(i)KCˆ(i)K D(i)K−X(A−B2D(i)K C2)Y(i) + Bˆ(i)K C2Y(i) +XB2Cˆ(i)K XB2D(i)KD(i)K C2Y(i) 0 .(5.8)Here, all parameter-varying matrices Aˆ(i)K , Bˆ(i)K , Cˆ(i)K , D(i)K and Y(i) can be parame-terized in an affine form of xc an mwor as o(i)(xc,mwor) = o(i)0 + o(i)1 xc + o(i)2 mwor.5.1.2 Design parametersThe weighting functions Wr, Wf and Wv are design parameters to advance perfor-mances with presence of the dynamic variations. The functions Wr and Wf havetypically gain shapes as shown in Figure 4.4, and can be formulated as [111] in (4.13).As can be seen in Figure 4.4, W−1r and W−1f are respectively used to shape Tre andTfe, where Tre (Tfe) denotes the transfer function from r to e (from fc to e). Wedescribed the role of each parameter in the closed-loop performance as guidelinesfor tuning these parameters in Section 4.3.4.5.2 Experimental ResultsA ball-screw drive test setup is shown in Figure 1.4. A screw with length xc ∈[200, 600] [mm] is driven directly by a brushless DC motor which is operated inthe current control loop for workpiece mass variations of mwor∈ [0, 10] [kg] with asampling period of 0.36 [ms].835.2.1 Controller design and analysisGiven the LPV model of the ball-screw, a switching gain-scheduled controller wasdesigned with two subregions S(1) ∈ [0, 270] [mm] and S(2) ∈ [250, 600] [mm]. Theparameters of weighting functions Wr and Wf are selected by trial and error, andlisted in Table 5.1. Also, Wv is set to 0.0006 after tuning. The gains of W−1r andW−1f together with those of Tre and Tfe are plotted in Figure 5.3.10−1 100 101 102 103−150−100−500Gain (dB)Frequency (Hz)  (a) |W−1r | (dash-dot line) and |Tre| (solid lines) for various xcand mwor.10−1 100 101 102 103−150−100−50Gain (dB)Frequency (Hz)    (b) |W−1f | (dash-dot line) and |Tfe| (solid lines) for various xcand mworFigure 5.3: Selected weighting functions Wr and Wf together with Tfe and TreThe weighing functions are selected according to the guidelines in Section 4.3.4,in order to penalize Tre and Tfe. For instance, the desired tracking error was 10−6[m] (-120 [dB]) in the low frequency range. Therefore, as can been in Figure 5.3 (a),ML = −140 [dB] is chosen to penalize Tre. Moreover, in order to achieve a band-width greater than 100 [Hz], ωb = 140 [Hz] is selected to advance bandwidth ofthe closed-loop system. On the other hand, to suppress the peaks of the resonance84Table 5.1: Design parametersWeighing functions k ML [dB] MH [dB] ωb [Hz]Wr 3 -140 4 160Wf 1 -113 -178 6mode around 214 [Hz] for various table positions and masses, parameters in Wf areselected so that the larger weight |Wf | (i.e., smaller |Wf |−1) is imposed around thatfrequency, as in Figure 5.3 (b).The performance of the synthesized controller is evaluated by analyzing theexperimental results. In all the experimental results, the performance of two cases,i.e., switching controller (Case I ) and non-switching controller (Case II ) are com-pared with each other.5.2.2 Tracking performance analysisThe tacking performance is evaluated in the time domain experiment. A smoothreference signal r is specified to command the table in r∈ [100, 440] [mm] as shownin Figure 5.4 (a) where xc(t) = 650 − r(t). This experiment is implemented intwo conditions, that is, tracking without a disturbance signal and tracking witha harmonic disturbance signal to excite the flexible modes. The tracking error inCase I and II without disturbance (with disturbance) are shown in Figure 5.4 (inFigure 5.5).3Mean absolute error (MAE) and maximum error for Case I and II are listedin Table 5.2. According to the results, Case I outperforms Case II by up to 61%in MAE and 52% in maximum tracking error.3Since in the test setup, real cutting implementation was not possible, the workpiece massvariations is emulated by placing mwor equals 0 and 10 [kg] (V2) on the table.850.5 1 1.5 20100200300400500r (mm)Time (sec)(a) Reference trajectory (solid line) andswitching instant (dash-dot line).0.5 1 1.5 2−20−1001020Time (sec)Error (µm)(b) Tracking error in Case I (red) andCase II (green) for mwor = 0 [kg] andfc = 0 [N]Figure 5.4: Reference trajectory andtracking error for no disturbance con-dition0.5 1 1.5 2−50050Time (sec)Error (µm)(a) mwor = 0 [kg] andrgktkafc(t) =1.8 sin(1340t) [V]0.5 1 1.5 2−50050Time (sec)Error (µm)(b) mwor = 10 [kg] andrgktkafc =1.3 sin(1340t) [V]Figure 5.5: Tracking error in Case I(red) and Case II (green) for har-monic disturbance5.2.3 Bandwidth analysisThe bandwidth of the closed-loop system is evaluated in the frequency domainexperiment by placing the table in various positions, and measuring the FRF ofTry from the reference to the table position in Case I and II. The input signal isgenerated and swept in the range of 10 to 300 [Hz]. The FRFs in Case I and IIare plotted in Figure 5.6. According to these figures, the bandwidth of Case I is20 [Hz] higher than Case II. The measured bandwidth for various xc and mwor aregiven in Table 5.3.86Table 5.2: Tracking error in time domain experiments for f1(t) = 1.8 sin(1340t) andf2(t) = 1.3 sin(1340t) [V]mwor [kg]rgktkafc [V]MAE [µm] Max error [µm]Case I Case II Case I Case II00 2.57 3.19 14.65 16.37f1(t) 8.06 20.57 25.16 53.1210 f2(t) 5.56 12.00 18.10 44.58101 102−20−1001020  Gain (dB)Frequency (Hz)105 125−3  Figure 5.6: Gain of |Try| in Case I (red) and Case II (green); xc = 250 [mm] (solidline) and xc = 450 [mm] (dashed line)5.2.4 Flexible mode suppression analysisThe vibration suppression of the varying flexible mode is evaluated by measuringthe FRF from the input disturbance to the table position. The measured FRFs forthe open-loop case, Case I and II with various table positions and various workpiecemasses are illustrated in Figure 5.7. According to these results, the flexible modeamplitude at resonance frequency has decreased up to 45% more in Case I thanCase II. The reduction of flexible mode amplitude at resonant frequency is listed inTable 5.3.87100 150 200 250 300−160−150−140−130Gain (dB)Frequency (Hz)  (a) mwor=0 [kg]; xc=250 [mm] (solid line) and xc=450 [mm](dashed line)100 150 200 250 300−170−160−150−140−130Gain (dB)Frequency (Hz)  (b) mwor=10 [kg]; xc=250 [mm] (solid line) and xc=450[mm] (dashed line)Figure 5.7: FRF magnitude of the transfer functions from fc to uc with different(xc,mwor); Case I (red), Case II (green) and open loop (black)Table 5.3: Bandwidth ωc and resonance amplitude reduction GRmwor [kg] xc [mm]ωc [Hz] GR [dB]Case I Case II Case I Case II0440 122.8 101.6 19.4 10.5240 124.4 103.8 18.0 9.610440 123.5 121.1 13.6 9.1240 124.4 123.8 13.0 10.588Chapter 6Smooth Switching LPVController Design for LPVSystemsIn this chapter, we propose a method to design a smooth switching LPV con-troller. The method is novel in the sense that, in contrast to the previous workin [12, 14, 47, 69], we design LPV controllers as local controllers and interpolatesmoothly between them instead of LTI controllers interpolation. Also, differentfrom the method in [17, 18, 19], we consider the controller design problem foroutput feedback systems, and for both one- and two-dimensional spaces of gain-scheduling variables. In addition, unlike [17, 18, 19], we introduce a matrix norm ofthe controller system matrix as a measure of “smoothness” for controller switching.Moreover, we utilize adjustable interpolation functions to improve the smoothness,and provide a higher order differentiable control signal by taking them into accountin a controller design problem. The controller design problem is formulated to satisfy89both the standard L2-gain condition of the closed-loop system and the smoothnesscondition of the switching controllers. Since the formulated problem boils down toa feasibility problem with non-linear matrix inequalities in general, we develop analgorithm in which Lyapunov variables, the local controllers and the interpolatingfunctions are optimized alternately to find a feasible solution.6.1 A Smooth Switching LPV Controller Design Prob-lemIn this section, we will formulate a smooth switching LPV controller design problemto be dealt with in this chapter.6.1.1 Notation for interval setsBefore formulating the smooth switching controller design problem, we will introduceset notation which is necessary for the formulation.Let us consider an interval set in R:Θ :={θ ∈ R : θ ≤ θ ≤ θ}. (6.1)This interval set will become a region where a parameter θ of an LPV system isassumed to vary in time in the controller design problem to be formulated later.For the switching LPV controller design purpose, as shown in Figure 6.1, we divide90the interval θ into (2I − 1)-number of sub-intervals12:Θ(i) :={θ ∈ R : θ(i) ≤ θ ≤ θ(i)}, i = 1, . . . , I,Θ({i,i+1}) :={θ ∈ R : θ(i)≤ θ ≤ θ(i+1)}, i = 1, . . . , I − 1.(6.2)Figure 6.1: Sub-intervals for switching control (I = 3)6.1.2 Description of an LPV plant and an LPV controllerWe will consider a standard LPV generalized plant described byx˙(t) = A(θ(t))x(t) +B1(θ(t))w(t) +B2(θ(t))u(t),z(t) = C1(θ(t))x(t) +D11(θ(t))w(t) +D12(θ(t))u(t),y(t) = C2(θ(t))x(t) +D21(θ(t))w(t),(6.3)where the vectors x, w, u, z and y are respectively the state, the exogenous input, thecontrol input, the performance output and the measured output. The dimensionsof these vectors are not displayed explicitly in the chapter. All the system matricesin (6.3) are assumed to be compatible in size. In addition, we assume that theparameter θ is one-dimensional, that θ is measurable in real time, and that it variesin time within an interval Θ in (6.1) with a constraint of the rate of change:θ(t) ∈ Θ, θ˙(t) ∈ V, ∀t ≥ 0, (6.4)1Throughout this chapter, superscripts with parentheses, such as Θ(i) and Θ({i,i+1}), indicatethe numbering for sub-intervals.2The notation I is used to indicate both the number of divisions and the identity matrix in thischapter, but its indication at each appearance is clear from the context.91where the interval V is defined byV := {v ∈ R : v ≤ v ≤ v} , (6.5)for given real values v and v satisfying v ≤ v.To the LPV generalized plant (6.3), we connect the feedback gain-schedulingLPV controller represented asx˙K(t)u(t) = K(θ(t))xK(t)y(t) , (6.6)where xK is the controller state and K(θ) is the parameter-varying system matrix.Then, it is well-known (see e.g. [39]) that the closed-loop system matrices are affinefunctions of the controller system matrix K(θ). Namely, the closed-loop system isexpressed by x˙cl(t)z(t) = Σcl(K, θ(t))xcl(t)w(t) , (6.7)where xcl :=[xT , xTK]Tand the closed-loop system matrix isΣcl(K, θ) := X0(θ) +XL(θ)K(θ)XR(θ), (6.8)X0(θ) :=A(θ) 0 B1(θ)0 0 0C1(θ) 0 D11(θ), XL(θ) :=0 B2(θ)I 00 D12(θ),XR(θ) :=0 I 0C2(θ) 0 D21(θ) .926.1.3 Description of a smooth switching LPV controllerIn this chapter, we are interested in designing a special type of LPV controllers,which is a smooth switching LPV controller described byK(θ) =K(i)(θ), if θ ∈ Θ(i), i = 1, . . . , I,K({i,i+1})(θ), if θ ∈ Θ({i,i+1}), i = 1, . . . , I − 1.(6.9)Assignment of each controller to each sub-interval is illustrated in Figure 6.2. Here,Figure 6.2: Controllers assigned to sub-intervals (I = 3)the controller K(i) is interpreted as a local LPV controller, to which the LPV con-trollerK(θ) in (6.6) is switched when the parameter θ lies in the sub-interval Θ(i). Onthe other hand, the controller K({i,i+1}) plays a role in smoothly switching betweentwo local LPV controllersK(i) and K(i+1) while the parameter θ is in a transitionalsub-interval Θ({i,i+1}). The precise definition of “smoothness” of controller switchingwill be provided in the next subsection.6.1.4 Statement of a controller design problemFor a given LPV plant in (6.3) with the constraints (6.4) and the sub-intervals in(6.2), our objective is to design an LPV controller (6.6) with the switching rule (6.9)that satisfies the following requirements for any trajectory θ(·) with θ(t) ∈Θ andθ˙(t) ∈ V for all t ≥ 0.(i) The closed-loop system (6.7) is exponentially stable.93(ii) For zero initial condition, the L2-gain of the closed-loop system from w to zis bounded by a specified value γ> 0, i.e.,∫ ∞0zT (t)z(t)dt < γ2∫ ∞0wT (t)w(t)dt. (6.10)(iii) The controllers switch continuously and smoothly between K(i) and K({i,i+1}),as well as between K({i,i+1}) and K(i+1), i.e., for i = 1, . . . ,I−1, and form = 0, 1, . . . ,M for a specified integer M≥ 0,dmdtmK(i)(θ(t)) =dmdtmK({i,i+1})(θ(t)), at θ(t) = θ(i),dmdtmK(i+1)(θ(t)) =dmdtmK({i,i+1})(θ(t)), at θ(t) = θ(i+1).(6.11)(iv) The rate of change of the controller matrix is bounded by a specified valueη> 0 in the transitional sub-intervals, i.e.,∥∥∥∥ddtK(θ(t))∥∥∥∥ < η, (6.12)for any θ ∈ Θ({i,i+1}), i = 1, . . . ,I−1, and any θ˙ ∈ V . Here, ‖·‖ denote the`2-induced matrix norm, i.e., the largest singular value of a matrix.The requirements (i) and (ii) are standard and common for most of the ad-vanced gain-scheduling control techniques; see e.g. [5, 72, 73, 74]. To guaranteecontinuous and smooth switching between controllers, the requirements (iii) and(iv) impose constraints on the controller system matrix K(θ). Especially, the condi-tions (6.11) and (6.12) indicate exactly what we mean by “smoothness of controllerswitching” in this chapter.Note that the conditions (6.11) and (6.12) is indirect means to avoid poortransient behaviors of relevant signals, such as plant states and outputs and controlinputs, while a controller is switching from one to another. We also remark that,94since the value η has no physical meaning, its selection is by trial and error. Afterdesigning a smooth switching LPV controller K(θ) for a selected η, if transientbehaviors of relevant signals are undesirable, then we need to redesign a controllerwith a reduced value of η.Instead of imposing the smoothness constraint on the controller system ma-trix as in (6.12), one may want to bound the time derivative of the closed-loopsystem matrix as∥∥∥∥WL(ddtΣcl(K, θ(t)))WR∥∥∥∥ < η, (6.13)where WL and WR are weighting matrices to be used for appropriately weightingand/or selecting relevant matrix components. It turns out that the replacement ofthe constraint (6.12) with (6.13) will not increase the complexity of the optimizationproblem presented in Section 6.2, since the closed-loop system matrix Σcl dependsaffinely on the controller system matrix K(θ).6.2 Formulation of a Feasibility ProblemIn this section, we will write the requirements (i)-(iv) in the previous section in termsof matrix inequality and equality conditions, and formulate a feasibility problem todesign a controller for satisfying the requirements.6.2.1 Conditions for system stability (i) and L2 gain (ii)As presented in [5], the requirements (i) and (ii) are guaranteed to be fulfilled if thereexists a parameter-dependent matrix P (θ) > 0 that satisfies the matrix inequality33The notation (∗) indicates entries that make the whole matrix symmetric.95M11(K, θ) P (θ)Bcl(K, θ) CTcl(K, θ)(∗) −γI DTcl(K, θ)(∗) (∗) −γI< 0, (6.14)for all θ ∈ Θ and θ˙ ∈ V , whereM11(K, θ) := P˙ (θ) + P (θ)Acl(K, θ) + (∗), (6.15)and Acl, Bcl, Ccl and Dcl are closed-loop system matrices which form Σcl in (6.8).The matrix inequality (6.14) can be rewritten asY0(P, γ, θ, θ˙) + YL(P, θ)Σcl(K, θ)YR + (∗) < 0, (6.16)whereY0(P, γ, θ, θ˙) :=P˙ (θ) 0 00 −γI 00 0 −γI, YL(P, θ) :=P (θ) 00 00 I,YR :=I 0 00 I 0 .(6.17)By substituting (6.8) into (6.16), we haveY0(P, γ, θ, θ˙) + YL(P, θ)X0(θ)YR + YL(P, θ)XL(θ)K(θ)XR(θ)YR + (∗) < 0.(6.18)It is known (see e.g. [5]) that this matrix inequality holds for any θ˙ ∈ V if and onlyif it holds for extreme points of V , i.e., θ˙ = v and θ˙ = v.Due to the coupling between matrix variables P and K(θ) in (6.18), a fea-sibility problem involving this matrix inequality is non-convex. One can convexify96such an optimization problem by using non-linear matrix variable changes or pro-jection/elimination lemma [5, 39, 85]. Here, we will leave the inequality condition(6.18) as it is, and a non-convex feasibility problem is tackled numerically, becausethe convexification is not trivial when we include the requirement (iv) in the con-troller design specification.6.2.2 Conditions for smooth switching (iii)The controller system matrix K({i,i+1}) for the sub-interval Θ({i,i+1}) is assumed tobe given byK({i,i+1})(θ) = L(K(i),K(i+1), α(i,i+1))(θ), (6.19)for i = 1, . . . ,I−1. Here, a function L for the linear combination of two matrix-valued functions K(i) and K(i+1) with respect to α(i,i+1) is defined byL(K(i),K(i+1), α(i,i+1)) := (1− α(i,i+1))K(i) + α(i,i+1)K(i+1), (6.20)where a scalar-valued function α(i,i+1) maps θ ∈ Θ({i,i+1}) into a real number.The continuity and smoothness conditions (6.11) hold if the following equal-ities hold for i = 1, . . . ,I−1 and m = 0, 1, . . . ,M :dmdθmα(i,i+1)(θ) = 0, at θ = θ(i),dmdθm(1− α(i,i+1)(θ)) = 0, at θ = θ(i+1).(6.21)Substitution of (6.19) and (6.20) into (6.11) yields conditions equivalent to (6.11)as dmdtm[α(i,i+1)(θ(t))K∆(θ(t))]∣∣∣∣θ(t)=θ(i)= 0,dmdtm[(1− α(i,i+1))(θ(t))K∆(θ(t))]∣∣∣∣θ(t)=θ(i+1)= 0,(6.22)whereK∆(θ) := K(i+1)(θ)−K(i)(θ). (6.23)97Due to the chain rule, it is straightforward to show that the left-hand sides of thefirst equation, and of the second equation, in (6.22) are linear functions ofdmdθmα(i,i+1)(θ)∣∣∣∣θ=θ(i), m = 0, 1, . . . ,M, (6.24)anddmdθm(1− α(i,i+1)(θ))∣∣∣∣θ=θ(i+1), m = 0, 1, . . . ,M, (6.25)respectively. Therefore, the conditions (6.21) lead to the continuity and smoothnessconditions (6.11).A simple function meeting the conditions (6.21) for M = 0 is a linearlyinterpolating function:α(i,i+1)(θ) =θ − θ(i)θ(i+1) − θ(i). (6.26)In this chapter, we will parameterize the function α(i,i+1) with a general structure,and optimize the closed-loop performance γ with respect to not only the controllerparameters but also the function α(i,i+1), without violating the smoothness con-straint.6.2.3 Conditions for rate of change of controller (iv)Using the chain rule, the time derivative of K({i,i+1}) in (6.19) can be obtained byddtK({i,i+1})(θ(t)) = θ˙(t)M(K(i),K(i+1), α(i,i+1))(θ(t)), (6.27)where the function M(K(i),K(i+1), α(i,i+1)) is defined byM(K(i),K(i+1), α(i,i+1)) :=∂K(i,i+1)∂θ, (6.28)and computed asM(K(i),K(i+1), α(i,i+1)) =∂α(i,i+1)∂θ(K(i+1) −K(i)) + L(∂K(i)∂θ,∂K(i+1)∂θ, α(i,i+1)).(6.29)98The smoothness condition (6.12) holds for any θ ∈ Θ({i,i+1}) and any θ˙ ∈V if and only if the following matrix inequality conditions are satisfied for anyθ ∈ Θ({i,i+1}):ηI vM(K(i),K(i+1), α(i,i+1))(θ)(∗) ηI > 0,ηI vM(K(i),K(i+1), α(i,i+1))(θ)(∗) ηI > 0.(6.30)Due to (6.27), the smoothness condition (6.12) is rewritten as∥∥∥θ˙M(K(i),K(i+1), α(i,i+1))(θ)∥∥∥ < η, (6.31)for any θ ∈ Θ({i,i+1}) and any θ˙ ∈ V . By the Schur complement and the linearityof θ˙M with respect to θ˙, the condition (6.31) is equivalent to the matrix inequalityconditions in (6.30) which correspond to two extremes of the set V .Suppose that the functions K(i), K(i+1) and α(i,i+1) are affine with respectto θ. Then, the smoothness condition (6.12) holds for any θ ∈ Θ({i,i+1}) and anyθ˙ ∈ V if and only if the matrix inequality conditions (6.30) are satisfied for θ = θ(i)and θ = θ(i+1). If the functions K(i), K(i+1) and α(i,i+1) are affine with respect to θ,so is the matrix-valued function M(α(i,i+1),K(i),K(i+1)). Consequently, the matrixinequalities in (6.30) are linear ones with respect to θ, leading to the assertion ofthe corollary.In this chapter, we consider a controller design problem where the smoothnesscondition is imposed only in the sub-intervals Θ({i,i+1}). However, it is easy tomodify the problem, and the design algorithm presented later, into those for thecase when the derivative constraint on the controller matrix is applied to the wholeinterval Θ.996.2.4 A non-convex feasibility problemSo far, we have derived the conditions for the control requirements (i)-(iv) asCondition (6.18), ∀θ ∈ Θ, ∀θ˙ ∈ {v, v} ,Condition (6.21), m = 0, 1, . . . ,M, i = 1, . . . , I − 1,Condition (6.30), ∀θ ∈ Θ({i,i+1}), i = 1, . . . , I − 1.(6.32)To deal with the infinitely many conditions (6.18) and (6.30) over Θ and Θ({i,i+1})respectively, a standard technique is to grid the parameter spaces Θ and Θ({i,i+1}),and approximate the infinitely many conditions with a finite number of conditions[5]. The finitely many conditions can be gathered into matrix inequality and equalityconditions as F1(P,K, α, γ) < 0,F2(α) = 0,F3(K, α, η) > 0.(6.33)where notation for two sets of functions is introduced asK :={K(i)}Ii=1, α :={α(i,i+1)}I−1i=1. (6.34)Here, F1 is a block-diagonal matrix consisting of matrices corresponding to thecondition (6.18), as well as −P , for gridded θ ∈ Θ and θ˙ ∈ {v, v}, F2 is a vector oflength 2(M + 1)(I − 1) for the condition (6.21), and F3 is another block-diagonalmatrix associated with the condition (6.30) for gridded θ ∈ Θ({i,i+1}), i = 1, . . . ,I−1.To achieve high closed-loop performance, we should minimize the value γ.Therefore, the smooth switching LPV controller design problem amounts to findinga feasible solution to the following feasibility problem:Find γ < γd such thatF1(P,K, α, γ) < 0,F2(α) = 0,F3(K, α, η) > 0.(6.35)100To achieve a better L2-gain performance, it is possible to reduce γd by intertwiningthe feasibility problem (6.35) and the bisection search of γd. After solving thisfeasibility problem with the algorithm to be proposed in Section 6.3, we need toverify that the designed P , K and α satisfy the conditions (6.18) and (6.30) withdense grids of the corresponding parameter sets. In case some of the conditions (6.18)and (6.30) fail to be satisfied, we need to resolve the feasibility problem (6.35) withdenser grids than the original grids of the parameter spaces Θ and Θ({i,i+1}).6.2.5 Reduction to a finite-dimensional problemIn order to reduce the infinite-dimensional feasibility problem (6.35) into a finiteone, we need to parameterize the matrix-valued functions P and K(i), i = 1, . . . ,I,as well as scalar-valued functions α(i,i+1), i = 1, . . . ,I−1. One way to parameterizethe functions K(i) and α(i,i+1) is as follows:K(i)(θ) = K(i)0 +L∑`=1ρ`(θ)K(i)` ,α(i,i+1)(θ) = α(i,i+1)0 +N∑n=1gn(θ)α(i,i+1)n .(6.36)Here, matrix variables for optimization are K(i)` , ` = 0, 1, . . . , L, whereas scalarvariables are α(i,i+1)n , n = 0, 1, . . . , N . The functions ρ`, ` = 1, . . . , L, are typicallyselected as copies of features that appear in the LPV plant in (6.3), as explained in[5]. On the other hand, a simple selection of the functions gn, n = 1, . . . , N , canbe the basis functions of polynomials, i.e., gn(θ) = θn. The positive integers L andN are determined by users. Especially, the integer N is chosen to be greater thanor equal to 2M + 1 so that the linear system of equations (6.21) has at least onesolution.101For the parametrization of P over the parameter space Θ, we adoptP (θ) =P (i)(θ), if θ ∈ Θ(i), i = 1, . . . , I,P ({i,i+1})(θ), if θ ∈ Θ({i,i+1}), i = 1, . . . , I − 1,(6.37)whereP (i)(θ) = P (i)0 +L∑`=1ρ`(θ)P(i)` ,P ({i,i+1})(θ) = L(P (i), P (i+1), α(i,i+1)P )(θ),(6.38)and α(i,i+1)P is the linearly interpolating function given in (6.26). Note that thefunction α(i,i+1)P is fixed once the sub-interval Θ({i,i+1}) is determined, but the func-tion α(i,i+1) in (6.36) for controller interpolation is not. With the parametrizationspresented above, numerical optimization to solve (6.35) will be conducted with re-spect to matrix variables{P (i)`}L`=0and{K(i)`}L`=0, i = 1, . . . ,I, and scalar vari-ables{α(i,i+1)n}Nn=0, i = 1, . . . ,I−1. We presented in (6.36)-(6.38) just one way ofparametrization of functions K(i), α(i,i+1) and P , but other parametrizations mayalso be possible.6.3 An Iterative Descent AlgorithmSince the feasibility problem (6.35) is a non-convex one, to find a feasible solutionnumerically, we will propose an alternating algorithm which solves a series of convexoptimization problems by fixing a part of optimization variables. We note that thefeasibility problem (6.35) reduces to a convex one if two out of three variables P ,K and α are fixed. In the algorithm presented below, a given desired η-value isdenoted by ηd. Each step will be elaborated after presenting the algorithm.Step 0: Initialization.1021. Design a single LPV controller K for the whole interval Θ.2. Set the initial local LPV controllers asK(i)(θ) := K(θ), i = 1, . . . , I. (6.39)3. Set the initial interpolating functions asα(i,i+1)(θ) :=θ − θ(i)θ(i+1) − θ(i), i = 1, . . . , I − 1. (6.40)4. Set the initial η-value asη := max {ηd, ηˆ} , (6.41)where ηˆ is defined byηˆ := min η subject to F3(K, α, η) > 0. (6.42)Step 1: Update of Lyapunov variables P .For the fixed K and α (see (6.34)), solve an optimization problem:minPγ subject to F1(P,K, α, γ) < 0. (6.43)Step 2: Update of LPV controllers K.For the fixed P , α and η, solve an optimization problem:minKγ subject toF1(P,K, α, γ) < 0,F3(K, α, η) > 0.(6.44)Step 3: Update of interpolating functions αFor the fixed P , K and γ, solve an optimization problem:ηˆ := minαη subject toF1(P,K, α, γ) < 0,F2(α) = 0,F3(K, α, η) > 0.(6.45)Using the minimized value ηˆ, redefine η by (6.41).103Step 4: Termination criteria.If the γ becomes less than γd, or if the γ value has not decreased during oneiteration of Steps 1-3, then terminate. Otherwise, iterate Steps 1-3.In Step 0-(1), an LPV controller for the whole region Θ is designed. This canbe done by the method using the convex optimization technique in [5]. However,since the dependency of the controller matrix on θ in [5] and the dependency takenin this method are different, we will approximate the designed controller by an LPVcontroller with the structure (6.36) by using least squares optimization. In Steps0-(2) and 0-(3), the initial LPV controller designed in Step 0-(1) is assigned to thesub-intervals Θ(i) and Θ({i,i+1}) as local LPV controllers.In Step 0-(4), as well as in Step 3, the ηˆ-value indicates the largest possiblenorm of the time derivative of the latest LPV controller matrix. If ηˆ is less thanor equal to ηd, then the smoothness constraint is satisfied. On the other hand, ifηˆ is greater than ηd, then we aim at reducing η in Step 3 to meet the smoothnessrequirement. Figure 6.3 shows an example of transitions of the η-value due to (6.41)and (6.45) in the algorithm, and pictorially clarifies the meanings of η, ηˆ and ηd.Figure 6.3: An example of transitions of the η-valueAll the optimization problems (6.42)-(6.45) are convex, and therefore, nu-merically tractable by using efficient convex optimization algorithms [15]. Step 1104and Step 2 update the Lyapunov variables and the local LPV controllers to improvethe L2 gain performance index γ, without worsening the controller smoothness overthe subregions Θ({i,i+1}), i = 1, . . . ,I−1. Step 3 is used in two ways, that is, eitherto improve the controller smoothness index η if η-value in the previous iterationis greater than ηd, or otherwise, to increase the smoothness margin by a betterselection of α, securing a larger search region of K in the next iteration.Theoretically, by the proposed algorithm, the L2 gain performance index γdecreases after each iteration of Steps 1-3. So does the smoothness index η as longas η is greater than the desired value ηd. Once η reaches ηd, then the η remains atηd after each iteration.In Step 3 of the algorithm above, if the optimized η-value in (6.45) does notdecrease so much from the η-value in the previous iteration, then one could try toupdate the controllers K by solving another optimization problem for the fixed Pand α, and a slightly relaxed γ-value, kγ, k > 1:ηˆ := minKη subject toF1(P,K, α, kγ) < 0,F3(K, α, η) > 0.(6.46)This may possibly improve the controller smoothness index η drastically by sacri-ficing the L2 gain bound γ slightly.6.4 Cases for a Two-Dimensional Parameter SpaceHere, we present the extension of the smooth switching controller design for a one-dimensional gain-scheduling parameter space into that for a two-dimensional space.The analogous discussions to the ones for one-dimensional cases are omitted, andonly the differences from one-dimensional cases are highlighted.105Let us consider a rectangular set in R2 (cf. (6.1)):Θ :=θ :=θ1θ2 ,θ1 ∈[θ1, θ1]θ2 ∈[θ2, θ2]. (6.47)The rectangular set Θ is divided into (2I − 1) × (2J − 1) number of rectangles(cf. (6.2)) asΘ(i,j) :=θ ∈ R2 :θ1 ∈ [θ(i)1 , θ(i)1 ]θ2 ∈ [θ(j)2 , θ(j)2 ],i = 1, . . . , I, j = 1, . . . , J,Θ({i,i+1},j) :=θ ∈ R2 :θ1 ∈ [θ(i)1 , θ(i+1)1 ]θ2 ∈ [θ(j)2 , θ(j)2 ],i = 1, . . . , I − 1, j = 1, . . . , J,Θ(i,{j,j+1}) :=θ ∈ R2 :θ1 ∈ [θ(i)1 , θ(i)1 ]θ2 ∈ [θ(j)2 , θ(j+1)2 ],i = 1, . . . , I, j = 1, . . . , J − 1,Θ({i,i+1},{j,j+1}) :=θ ∈ R2 :θ1 ∈ [θ(i)1 , θ(i+1)1 ]θ2 ∈ [θ(j)2 , θ(j+1)2 ],i = 1, . . . , I − 1, j = 1, . . . , J − 1.(6.48)Notation for rectangular sets is illustrated in Figure 6.4 (cf. Figure 6.1). The rateof change of the parameter vector θ is bounded as (cf. (6.5))V :=v :=v1v2 ,v1 ∈ [v1, v1]v2 ∈ [v2, v2]. (6.49)The smooth switching LPV controller in two-dimensional cases is described106Figure 6.4: Rectangles for switching control (I = 3, J = 2)by (cf. (6.9))K(θ) =K(i,j)(θ), if θ ∈ Θ(i,j),K({i,i+1},j)(θ), if θ ∈ Θ({i,i+1},j),K(i,{j,j+1})(θ), if θ ∈ Θ(i,{j,j+1}),K({i,i+1},{j,j+1})(θ), if θ ∈ Θ({i,i+1},{j,j+1}),(6.50)where K(i,j) is interpreted as a local controller for the subregion Θ(i,j), and othersare considered to be controllers in transitional subregions. The controller systemmatrices K({i,i+1},j), K(i,{j,j+1}) and K({i,i+1},{j,j+1}), for the rectangular regionsΘ({i,i+1},j), Θ(i,{j,j+1}) and Θ({i,i+1},{j,j+1}) respectively, are assumed to be the linearcombination of controller system matrices K(i,j) as (cf. (6.19))K({i,i+1},j) = L(K(i,j),K(i+1,j), α(i,i+1)),K(i,{j,j+1}) = L(K(i,j),K(i,j+1), β(j,j+1)),K({i,i+1},{j,j+1}) = L(K({i,i+1},j),K({i,i+1},j+1), β(j,j+1)).(6.51)107Alternatively, the controller K({i,i+1},{j,j+1}) can be expressed asK({i,i+1},{j,j+1}) = L(K(i,{j,j+1}),K(i+1,{j,j+1}), α(j,j+1)). (6.52)Here, the functions α(i,i+1) and β(j,j+1) respectively map θ1 ∈ [θ(i)1 , θ(i+1)1 ] and θ2 ∈[θ(j)2 , θ(j+1)2 ] into a real number. The conditions on the functions α(i,i+1) and β(j,j+1)to satisfy the requirement of continuous and smooth controller switching at switchinginstances (cf. (6.11) and (6.21)) aredmdθm1α(i,i+1)(θ1) = 0, at θ1 = θ(i)1 ,dmdθm1(1− α(i,i+1)(θ1)) = 0, at θ1 = θ(i+1)1 ,(6.53)dmdθm2β(i,i+1)(θ2) = 0, at θ2 = θ(i)2 ,dmdθm2(1− β(i,i+1)(θ2)) = 0, at θ2 = θ(i+1)2 .(6.54)The smoothness conditions of the switching LPV controller in rectanglesΘ({i,i+1},j), Θ(i,{j,j+1}) and Θ({i,i+1},{j,j+1}) are respectively obtained, by straight-forward calculations using the chain rule of derivatives and the triangular inequalityfor the matrix norm, as follows (cf. (6.31)):∥∥∥θ˙1M({i,i+1},j)1∥∥∥+∥∥∥θ˙2M({i,i+1},j)2∥∥∥ < η, ∀θ ∈ Θ({i,i+1},j),∀θ˙ ∈ V,∥∥∥θ˙1M(i,{j,j+1})1∥∥∥+∥∥∥θ˙2M(i,{j,j+1})2∥∥∥ < η, ∀θ ∈ Θ(i,{j,j+1}),∀θ˙ ∈ V,∥∥∥θ˙1M({i,i+1},{j,j+1})1∥∥∥+∥∥∥θ˙2M({i,i+1},{j,j+1})2∥∥∥ < η, ∀θ ∈ Θ(i,{j,j+1}),∀θ˙ ∈ V.(6.55)108Here, matrix-valued functions are defined as (cf. (6.28))M ({i,i+1},j)1 :=∂L∂θ1(K(i,j),K(i+1,j), α(i,i+1)),M ({i,i+1},j)2 := L(∂K(i,j)∂θ2,∂K(i+1,j)∂θ2, α(i,i+1)),M (i,{j,j+1})1 := L(∂K(i,j)∂θ1,∂K(i,j+1)∂θ1, β(j,j+1)),M (i,{j,j+1})2 :=∂L∂θ2(K(i,j),K(i,j+1), β(j,j+1)),M ({i,i+1},{j,j+1})1 := L(∂K({i,i+1},j)∂θ1,∂K({i,i+1},j+1)∂θ1, β(j,j+1)),M ({i,i+1},{j,j+1})2 :=∂L∂θ2(K({i,i+1},j),K({i,i+1},j+1), β(j,j+1)).All the conditions in (6.55) are in the form of∥∥∥θ˙1M1∥∥∥+∥∥∥θ˙2M2∥∥∥ < η. (6.56)Due to the analogous argument for deriving (6.30), this is equivalent to the followingmatrix inequality conditions:η1I v1M1v1MT1 η1I > 0,η1I v1M1v1MT1 η1I > 0,η2I v2M2v2MT2 η2I > 0,η2I v2M2v2MT2 η2I > 0,η1 + η2 < η.(6.57)Note that there are coupling terms between α and β in both M ({i,i+1},{j,j+1})1and M ({i,i+1},{j,j+1})2 , because K{i,i+1},j and K{i,i+1},j+1 are functions of α(i,i+1).Therefore, the optimization problem corresponding to (6.45) in Step 3 of the pro-posed algorithm in Section 6.3 will become non-convex with respect to α and β forthe case of the two-dimensional parameter space. To solve a series of convex opti-mization problems, in Step 3, we can solve the following two optimization problemsalternately until the value η stops decreasing.109Step 3-a: Update of the function α.For the fixed P , K, β and γ, solve an optimization problem:minαη subject toF1(P,K, α, β, γ) < 0,F2(α, β) = 0,F3(K, α, β, η) > 0.(6.58)Step 3-b: Update of the function β.For the fixed P , K, α and γ, solve an optimization problem:minβη subject toF1(P,K, α, β, γ) < 0,F2(α, β) = 0,F3(K, α, β, η) > 0.(6.59)Steps other than Step 3 of the algorithm in Section 6.3 are unchanged for the two-dimensional parameter case.110Chapter 7Smooth Switching LPV Controlof SKM and PKMIn this chapter, we will design a smooth switching LPV servo controller developedin Chapter 6 for both SKM and PKM machine tools to demonstrate its efficiency.7.1 SKM ControlThe SKM is driven by a ball-screw drive system in Chapter 3. For the SKM sys-tem, the closed-loop system performance with a smooth switching LPV controlleris compared with the performance of a hysteresis switching LPV controller in [72],and a non-switching LPV controller in [5]. For the hysteresis switching controller,controller states are reset at switching instants so that the control signal becomescontinuous at those instants [61].As was already explained in Chapter 3, a plant to be dealt with in this sectionis the ball-screw drive system in (3.28), [45]. Its inputs are fm and fc, and outputsare um and uc. Recall that the stiffness of the ball-screw depends on nut location,111xcwhich is the active length between the DC motor and the table position.7.1.1 Control problem and generalized plantFor a given ball-screw drive plant G(θ) in (3.28), θ := xc and mwor=0, our goal is tominimize the tracking error e between the reference trajectory r and the actual tableposition uc, even in the face of a cutting-force disturbance fc, dynamic variationsof the plant G(θ) with respect to θ, and a saturation constraint on the motor inputvoltage. The output feedback structure to resolve this control problem is depictedin Figure 7.1, where K(θ) is the gain-scheduled LPV controller to be designed.Figure 7.1: The output feedback structure in control of the ball-screw drive systemIn the controller design, the weighting functions Wf , We and Wv, which areshown in Figure 7.1, will be exploited as design parameters. In this example, weselected the weighting functions as Wv = 7× 10−4 andWf (s) = 108 ×1.66s+ 0.47s+ 1.59× 104, We(s) =s2 + 1153s+ 3.32× 1051.58s2 + 1.948s+ 0.59. (7.1)By a combination of G(θ) in (3.28) and the weighting functions, the generalizedLPV plant was obtained in the form of (6.3), where the signals are w := [f, r]T ,z := [ze, zv]T , u := vm, y := e.112We designed the proposed smooth switching LPV controller for the inter-val (6.1) and the constraint (6.5), where the range of the table position and itsvelocity are assumed to beΘ = [0.2, 0.6] [m], V = [−0.45, 0.45] [m/s]. (7.2)The interval Θ was divided into sub-intervals asΘ(1) = [0.20, 0.38] , Θ(2) = [0.42, 0.6] , Θ(1,2) = [0.38, 0.42] . (7.3)In addition, we utilized the parametrizations (6.36) and (6.38) of the optimizationvariables by setting L = 1 andN = 6. Moreover, the integerM in (6.11) was set to 2.Finally, the desired L2-gain and smoothness indexes γd and ηd were selected, by trialand error, as γd = 13 and ηd = 5× 104, respectively. With the selected parameters,we ran the algorithm presented in Section 6.3. γ-values throughout the algorithm areplotted in Figure 7.2, where we can verify the monotonic non-increasing property ofγ-values and η-values. The termination of the algorithm occurred after 20 iterationswhen γ = 13. For interested readers, the controller parameters are given in aMATLAB data file at: www.sites.mech.ubc.ca/%7enagamune/Kpara.zip.7.1.2 Simulation resultsTo verify the performance of the closed-loop system, we applied the reference signalr to the output feedback system in Figure 7.1 without weighting functions. Weassume there is an X-Y table with two uncoupled identical ball-screw drives withparameters given in Table 3.1. The reference trajectory is the combination of twostraight lines and a circle with 0.28 [m] diameter as shown in Figure 7.3. It can beverified that this trajectory satisfies the conditions in (7.2). The signal r commandswhere xc(t) = 0.65 − r(t) in x and y directions were generated by virtual CNC113(a) γ-values after Step 1 (dashed line)and after Step 2 (solid line).5 10 15 200.511.522.533.5 x 105Iteration number15 2048 x 104ηηd=(b) η-values after Step 3Figure 7.2: γ-value and η-value after each iterations(VCNC) software developed in Manufacturing Automation Laboratory at the Uni-versity of British Columbia to provide a quadratic smooth trajectories as shown inFigure 7.4. The table travels 0.38 and 0.29 [m] in x and y directions with maximumand minimum speeds of 0.40 [m/s], fulfilling the assumptions on Θ and V in (7.2).In this figure, the narrow regions between the dash-dot vertical lines indicate thesmooth switching interval regions, whereas the hysteresis switching happens at theend of these intervals.We run simulation by VCNC software for two cases; without disturbanceforce, i.e, fc = 0, and with 50 [N] harmonic disturbance forces that sweeps from 20to 400 [Hz] on the both axes, i.e, fc = f1 wheref1(t) = 50 sin(2pi(20t+ k + 190t2)). (7.4)For the first part when fc = 0, as it is illustrated in the zoom shots of Figure 7.5, thecontroller switch of the hysteresis switching controller signal in the x-axis shortlyafter t = 3.3 (t = 4.64) causes a drastic change of e from 4 to −5 [µm] (−7 to 5[µm]). Also in the y-axis shortly after t = 4.1 (t = 5.2), e jumps from 2 to −1 [µm]114Figure 7.3: The x-y plane trajectoryFigure 7.4: Time domain referenceposition and velocity of x (dashedgreen line) and y (red solid line) drives(−4 to 4 [µm]). Unlike the hysteresis switching, in the smooth switching, smoothvariations of vm in the x and y axes lead to smooth rise of e. In the non-switchingcontroller, the maximum of |e| cannot be minimized less than 6 [µ m] and 4.5 [µm]in the x and y axes.For the second part when fc = f1, the signals e and vm became high frequencysignals, as depicted in Figure 7.6. Here, in the hysteresis switching simulation, thetransient jump of vm shortly after t = 4.65 increases the upper bound of e from−10 to 2 [µm] in x-axis and from −7 to 12 [µm] in y axis. However, the hysteresisswitching controller has a better performance in reducing the maximum trackingerror from 28 [µm] (46 [µm]) than non-switching controller whose maximum trackingerror was 20 [µm] (18 [µm]) in the x-axis (y-axis). The smooth switching controllerachieved the best results among three cases with maximum tracking error of 10 [µm]in both x and y axes.Finally, we take fast Fourier transfer (FFT) of tracking error for three casesplus different controllers obtained in each iteration and illustrate the results in Fig-115(a) The x-axis(b) The y-axisFigure 7.5: The simulated tracking error and control signal in case of fc = 0, withthe non-switching controller (dashed blue), the hysteresis switching controller (dash-dot green), the smooth switching controller (solid red), and switching instants (ver-tical dash-dot lines)ure 7.7. This figure describes gradual improvement of the smooth switching con-troller in the developed descent algorithm, and its comparison with non-smooth andgain-scheduled controllers.7.1.3 Experimental resultsIn the experiments, since the setup is only one axis, we chose the reference signal rcommands, as shown in Figure 7.8, the table to travel 0.40 [m], forward and thenbackward, with maximum and minimum speeds of±0.45 [m/s], fulfilling the assump-116(a) The x-axis(b) The y-axisFigure 7.6: The simulated tracking error and control signal in case of fc = f1,with the non-switching controller (dashed blue), the hysteresis switching controller(dash-dot green), the smooth switching controller (solid red), and switching instants(vertical dash-dot lines)tions on Θ and V in (7.2). In this figure, the narrow regions between the dash-dotvertical lines indicate the smooth switching interval regions at T1 := [0.69, 0.79] andT2 := [1.90, 2.00] [s], whereas the hysteresis switching happens at the end of theseintervals when t1 := 0.79 and t2 := 2.00 [s]. We ran experiments to evaluate thetracking performance for two cases; without disturbance force, i.e, fc = 0, and withharmonic disturbance force, i.e, fc = f1 wheref1(t) = 50 sin(2pi(314t)). (7.5)117Figure 7.7: FFT of the tracking errorfor three casesFigure 7.8: Reference trajectory posi-tion and velocityFor the first part when fc = 0, as it is illustrated in the zoom shots ofFigure 7.9, a sudden change of the hysteresis switching controller signal vm shortlyafter t = t1 change e drastically from 5 to −15 [µm]. Unlike the hysteresis switching,in the smooth switching, no transient jump occurred at t ∈ T1 and t ∈ T2 regionsin vm and e. The maximum of |e| cannot be minimized less than −13, −9 and −6[µm] in non-switching, hysteresis switching and smooth switching respectively.Figure 7.9: The experimental tracking performance when fc = 0 for non-switchingcontroller (dotted black), hysteresis switching controller (dashed red), the smoothswitching controller (solid blue), and switching instants (vertical dash-dot lines)For the second case when fc = 50sin(314t)[N ], the signals e and vm becamehigh frequency signals which are shown in Figure 7.10. Here, in the hysteresis118Figure 7.10: The experimental tracking performance when fc = 50sin(314t) [N] fornon-switching controller (dotted black), hysteresis switching (dashed red), smoothswitching controller (solid blue), and switching instants (vertical dash-dot lines)switching simulation, the transient jump of vm shortly after t = t1 increases theupper bound of e from 18 to −129 [µm]. In the smooth switching result, on theother hand, e smoothly varies from 0 to 1 [µm] at t ∈ T1. In the non-switchingcontroller simulation, the maximum of |e| cannot be minimized less than 95 [µm].7.2 PKM ControlIn this section, we design a smooth switching LPV controller for a PKM system toachieve fast and accurate tracking performance and to reject the machining processforce for any position of the tool tip in the whole workspace. The smooth switchingLPV controller consists of two local LPV controllers; one controller covers the smalland medium range of the PKM structure rotation angle and the other one works inthe large range. The switching between these local controllers occurs smoothly in aswitching interval by a high order polynomial interpolation function. The interpo-lation function and controllers’ parameters are synthesized by solving a non-convexoptimization. The designed controller ensures a smooth switching of controllers in119contrast to a sudden jump of the conventional hysteresis switching [72] or heuristicinterpolation of LTI controllers [19]. By utilizing the proposed smooth switchingcontroller in [44], the synthesized controller performance is simulated in both timeand frequency domain. The simulated time domain results are compared with thatof non-switching [5] and hysteresis switching [72] controllers.7.2.1 A bipod in PKM systemIn the PKM system, a cutting tool is connected through a mechanism to the bodyto cover the whole working volume. In a bipod PKM mechanism, illustrated inFigure 7.11, three legs with a fixed length connect the tool on the table T3 to theframe. In this mechanism, the joints are actuated by two linear forces, f1 and f2.The legs S1 and S2 are connected to tables T1 and T3 by revolute joints. Rotationsof the parallel links S1, S2 and S3 with angle φ convert motion of T1 and T2 in x1and x2 directions into two-dimensional motion of T3 in the x and y directions. It isassumed that resultant machining and disturbance force fd is applied to the tableT3. In the bipod mechanism, the rotation angle φ and its derivative can be writtenFigure 7.11: A bipod PKM system mechanism120as a function of x1 and x2:φ = arcsinx2 − x1 − as2Ls, φ˙ :=x˙2 − x˙12Ls cosφ, (7.6)where Ls is the length of links S1, S2 and S3.The control objective of the bipod PKM is to follow a desired trajectory ofthe table T3 in presence of fd by applying forces f1 and f2. In the tracking controlof the PKM, the main challenge is form significant change of the system dynamicsas a function of φ. To model the dynamic variations, the non-linear model of thePKM is derived in the following part of this section.Non-linear model of the bipod systemThe dynamic equations of motion can be obtained by Lagrange and virtual workmethod as:H1(φ)x¨1 +H12(φ)x¨2 + bx1x˙1 − Cφ(φ)φ˙− C12(φ)φ˙21 +G(φ) = f1 + fd(tanφ)/2,H12(φ)x¨1 +H2(φ)x¨2 + bx2x˙2 + Cφ(φ)φ˙+ C12(φ)φ˙2 −G(φ) = f2 − fd(tanφ)/2,(7.7)where H1, H2 and H12 are equivalent inertias, Cφ, C12 and G are viscous damp-ing, structural damping and gravitational force, respectively. These parameters arecalculated as follows:121mamcmd:=1 014916−120 114116120 0143160m1m2m33∑i=1msims3mb(φ) :=12L2s3∑i=1Jsi +[12m3 +18ms]sin2 φ,Je :=3∑i=1Jsi +m3L2s +14msL2s,(7.8)where constants mi, msi, Jsi, bx1 and bφ designate (cf. Figure 7.11) mass of table Ti,mass and moment of inertia of link Si for i = 1, 2, 3, and viscous friction on table T1and rotational joints, respectively. Note that H1, Cm, Cd and G are the functionsof tanφ, tan2 φ, etc. Note that H1, Cm, Cd and G are the functions of tanφ, tan2 φ,etc. We will use this function dependency later in design of the controller.In this section, for the sake of simplicity, it is assumed that the table T2 isfixed. By this assumption, x¨2 = x˙2 = 0. By this assumption (7.9) simplifies asH1(φ)x¨1 + Cd(φ)x˙1 + Cm(φ)x˙21 +G(φ) = f1 + fd(tanφ)/2, (7.9)whereCd(φ) := bx1 + bφ(1 + tan2 φ)2/(4L2s),Cm(φ) := −me(tanφ)(1 + tan2 φ)/(4Ls cosφ).(7.10)LPV model of the bipod systemDesign of an LPV controller requires derivation of the system equations in the state-space representation. Using (7.9), we can write the differential equation of motion122as:P (φ, x˙1) :x˙pod = Apod(φ, x˙1)xpod +Bpod(φ)fdfˆ1 ,y = Cpodxpod,(7.11)where xpod := [x1 x˙1]T , fˆ1 := f1 −G(φ), andApod(φ, x˙1) :=0 10 −Cd(φ) + Cm(φ)x˙1H1(φ) , Bpod(φ) :=1H1(φ)0 0−0.5 tanφ 1 ,Cpod :=[1 0].(7.12)Due to the physical limitation of the PKM mechanics and drives, the variables φ,x˙1 and their derivatives are bounded parameters as:Φ := {φ ∈ R : 10 ≤ φ ≤ 80 [deg]} , V := {x˙1 ∈ R : −1 ≤ x˙1 ≤ 1 [m/s]} ,Φ˙ :={φ˙ ∈ R : −4.11 ≤ θ ≤ 4.11 [rad/s]}, V˙ :={x¨1 ∈ R : −8 ≤ x¨1 ≤ 8 [m/s2]}.(7.13)7.2.2 Control objective and augmented plantGiven the model P (φ, x˙1) in (7.11), the parameter variations of the PKM systemchanges the system dynamics considerably as a function of tan(φ). In presence ofsuch dynamic variations, the main objectives are to design an LPV controller toprovide a fast and accurate tracking performance of a reference signal r and rejectthe disturbance force fd. The control signal is the error e(t) := r(t)− x1(t) and itsoutput is the actuating force, i.e, f1 to the table T1. In order to meet the requiredobjectives, an augmented plant is formed by adding some weighting functions We,and Wv to the plant, as shown in Figure 7.12. The weighting functions are tuning123r- K(f , x1)ef1WeWv z2z1fdx1P(f , x1)x1Figure 7.12: Augmented block diagram for controller designparameters in controller design to penalize the tracking error, and restrict controlsignal over specific frequencies. The combination of the LPV plant P (φ, x˙1) in (7.11)with We, and Wv provides the augmented plant as presented in (6.3), where x is thestate vector, and w := [r fd]T , z := [z1 z2]T , y := e and the system matrices areA(φ, x˙1) B1(φ) B2(φ)C1 D11 D12C2 D21 0:=ApodA(φ, x˙1) 0 0 B1,pod(φ) B2,pod(φ)−BeCpod Ae Be 0 0−DeCpod Ce De 0 00 0 0 0 Wv−Cpod 0 1 0 0.(7.14)Here, we consider dynamic weighting functions We, and a static gain Wv, given byWe :x˙e = Aexe +Bee,z1 = Cexe +Dee,Wv : z2 = Wvf1, (7.15)where the vector xe is state vector for the systems We, and the signals z1 and z2 areauxiliary signals introduced to describe the control problem mathematically.1247.2.3 Control structure and designWe connect the feedback gain-scheduling LPV controller represented asx˙K(t)f1(t) = K(θ(t))xK(t)e(t) , (7.16)to the augmented LPV plant (6.3) where a parameter θ of an LPV system is assumedto vary in time as a vector including φ(t) and x˙1(t) in a bounded region Θ asθ(t) :=φ(t)x˙1(t) , for θ ∈ Θ :=ΦV , (7.17)and xK is the controller state, K(θ) is the parameter-varying system matrix. Toencounter such a large range of dynamic variations θ ∈ Θ especially as a function ofφ, we divide the interval Θ in (7.17) in φ direction into 3-number of sub-intervals:Θ(i) :={θ ∈ R2 : θ(i) ≤ θ ≤ θ(i)}, i = 1, 2,Θ(1,2) :={θ ∈ R2 : θ(1)≤ θ ≤ θ(2)}.(7.18)In this section, to compensate for the large range of parameter variations, we designa special type of LPV controller, which is the smooth switching LPV controllerproposed in Chapter 6 and described by:K(θ) =K(1)(θ), if θ ∈ Θ(1),K(1,2)(θ), if θ ∈ Θ(1,2),K(2)(θ), if θ ∈ Θ(2).(7.19)Assignment of each controller to each sub-interval is illustrated in Figure 7.13. Here,the controller K(i) is interpreted as a local LPV controller, to which the LPV con-troller K(θ) in (7.16) is switched when the parameter θ lies in the sub-interval Θ(i).On the other hand, the controller K(1,2) plays a role in smoothly switching between125two local LPV controllers K(1) and K(2) while the parameter θ is in a transitionalsub-interval Θ(1,2). The precise definition of “smoothness” of controller switchingwas provided in Chapter 6..f, x1rfd- x1e .,f, x1.f, x14T4,4,f1P(f , x1)x1Figure 7.13: Block diagram of the smooth switch controller structure7.2.4 Simulation resultsWe will design a smooth switching LPV servo controller for a bipod PKM, andcompare its performance with that of a hysteresis switching LPV in [72], and withthat of a non-switching LPV controller in [5]. For the simulation purposes, weused the parameters of a test setup, at Fraunhofer Institute for Machine Tools andForming Technology (IWU) in Chemnitz Germany, shown in Figure 1.6, and itsparameters are listed in Table 7.1. For this system, the parameter variations of H1and Cm in (7.9) are simulated as a function of φ in Figure 7.14. As shown in thisfigure, these parameters vary exponentially as a function of φ.Solving the control problem and the generalized plantFor a given system P (φ, x˙1), our goal is to minimize the tracking error e betweenthe reference trajectory r and the actual table position x1, even in the face of acutting-force disturbance fd, dynamic variations of the plant with respect to θ, and a12620 40 60 80306090120H 1 [N/m/s2 ]φ [deg] −1600−1200−800−4000Cm [N/m2 /s2 ]Figure 7.14: The bipod PKM parameters H1 (solid line) and Cm (dashed line)variations as functions of rotation angle φTable 7.1: Parameters of the bipod systemSymbol and unit Value Symbol and unit Valuem1 [kg] 25.36 Ls [m] 0.7m2 [kg] 22.67 as [m] 0.125m3 [Hz] 10.56 bx [N/m/s] 50ms1,ms2,ms3 [kg] 4.16 bθ [N/m/s] 10Js1, Js2, Js3 [kg.m2] 0.37 – –saturation constraint on the maximum input force as |f1| ≤ 4 [kN ]. In the controllerdesign, the weighting functions We and Wv, which are shown in Figure 7.12, will beexploited as design parameters. In this example, we selected the weighting functionsby following the guideline in [46] as Wv = 1× 10−5 andWe(s) =s3 + 3359s2 + 3.76× 106S + 1.40× 1091.13s3 + 134s2 + 5325s+ 7.01× 104.We designed the proposed smooth switching LPV controller for the inter-val (7.17) and the constraint (7.13). Since the bipod parameters in (7.11) are func-tions of tan(φ), we divide the interval Θ on the φ direction so that for the small valueof tan(φ) there is one interval and for the larger value of tan(φ), another interval.127Therefore, three parameter subregions are selected as:Θ(1)(φ, x˙1) = {10 ≤ φ ≤ 70 [deg], −1 ≤ x˙1 ≤ 1 [m/s]},Θ(2)(φ, x˙1) = {75.4 ≤ φ ≤ 80 [deg], −1 ≤ x˙1 ≤ 1 [m/s]},Θ(1,2)(φ, x˙1) = {70 ≤ φ ≤ 75.4 [deg], −1 ≤ x˙1 ≤ 1 [m/s]}.(7.20)Furthermore, the parametrizations of the controller in (6.36) was utilized by settingL = 5 and N = 5. By trial and error, ρ(θ) was selected in (6.36) asρ(θ) := [1, tan2(φ), tan4(φ), tan6(φ), tan2(φ)x˙1], (7.21)to have a stable linear smooth switching controller for the whole range of θ ∈ Θ.Frequency-domain simulation resultsBy setting these parameters and following the developed descent algorithm in Chap-ter 6, the smooth switching controller was designed. To evaluate the closed-loopperformance of the designed controller, the magnitude of the sensitivity function S,the transfer function from r to e and the gain plot of T , the transfer function fromfd to x1 for the closed-loop systems are shown in Figure 7.15 while the φ angle waslocated at three subregions. According to this figure, the low gain of S varies from−110 to −150 [dB], and the maximum flexibility of the system −120 [dB] occurs atabout 270 [rad/s]. As it expected, both S and T transfer functions show the worstcases for the parameter variations of θ ∈ Θ(2).Time domain simulation resultsTo verify the performance of the closed-loop system, we applied the reference signalr to the output feedback system in Figure 7.13. The signal r commands the tableto travel 1.13 [m], forward and then backward while φ rotates from 10 to 80 [deg],128100 102 104−150−100−500|e(t)/r(t)| [dB]  100 102 104−200−175−150−125Frequency [rad/s]|x 1(t)|/fd(t) [dB]  Figure 7.15: Gain plot of S (top) and transfer functions from fd to x1 (bottom) forΘ(1) (blue), Θ(1,2) (yellow) and Θ(2) (red), and |We|−1 (dash-dot line)with maximum and minimum speeds of ±1 [m/s], fulfilling the assumptions on Θand V in (7.13). The parameters x˙1 and φ trajectory and switching intervals aredepicted in Figure 7.16.10 45 70 75.4 80−101x 1 [m/s]φ [deg]Θ(2)Θ(1,2)Θ(1)Figure 7.16: The boundaries of the switching intervals (dash-dot line) for (I = 3)and the parameter variations trajectory (solid line)We simulated the tracking performances for two cases; without disturbanceforce, i.e, fd = 0, and with harmonic disturbance force, i.e, fd = f0 wheref0(t) = 200 sin(2pi(10t+ 17t2)), (7.22)is a chirp signal from 10 to 90 [Hz] as a disturbance force. The closed-loop perfor-mances of the controllers without disturbance (and with disturbance) are shown in129Figure 7.17 (Figure 7.18). In these figures, the narrow regions between the dash-dotvertical lines indicate the smooth switching interval regions at Ta := [1.19, 1.25] andTb := [1.43, 1.49] [s], whereas the hysteresis switching happens at the end of theseintervals when t1 := 1.25 and t2 := 1.49 [s].For the first case when fd = 0, as it is illustrated in the zoom shots ofFigure 7.17, a sudden change in the hysteresis switching controller signal f1 shortlyafter t = t1 (t = t2) increases e drastically from 15 to 28 [µm] (38 to −39 [µm]).Unlike the hysteresis switching, in the smooth switching, smooth variations of f1 att ∈ Ta (t ∈ Tb) region leads to a smoother rise of e from 5 to 10 [µm] (0 to −11 [µm]).In the non-switching controller, although there is no sudden transient behavior, themaximum of |e| cannot be minimized to less than 18 [µm].For the second case when fd = f0, the signals e and f1 became high frequencysignals in Figure 7.18. Here, in the hysteresis switching simulation, the maximumerror reaches 133 [µm] with f1 = 2.39 [kN] while it reduces to 83 [µm] with lesseffort as 2.29 [kN]. In the non-switching controller simulation, the conservativecharacteristic of the controller provides a maximum |e| which cannot be minimizedless than 171 [µm] with a large maximum effort of 3.80 [kN].Finally, the maximal values of the tracking performance for both cases aresummarized in Table 7.2. According to Table 7.2, when fd = 0 (fd = f0), thesmooth switching controller reduced the maximum |e| by 13% (51%) and by 62%(38%) in comparison with the non-switching controller, and the hysteresis switchingcontroller, respectively. As a matter of energy efficiency, the smooth switchingcontroller decreased the maximum of the controller effort, i.e., max(|f1| by about40% (4%) for the non-switching (hysteresis switching) controller.130Table 7.2: The maximal values in the time domain performance of LPV controllersForce Switching max |e| max |f1|[N] type [µm] [kN]no switch 17.45 1.17fd = 0 hysteresis 39.88 1.23smooth 15.03 1.22no switch 171.10 3.80fd = f0 hysteresis 133.80 2.39smooth 83.45 2.291310 0.5 1 1.5 2 2.5 2.71080φ [deg]0 0.5 1 1.5 2 2.5 2.7−40−2002040Error [µm]0 0.5 1 1.5 2 2.5 2.705001000Force [N]Time [sec]1.4 1.5201,0001.18 1.28010001.18 1.28−50301.4 1.52−40040X: 1.44Y: −11.46Figure 7.17: The tracking performance when fd = 0 with the non-switching con-troller (blue), the hysteresis switching controller (green), the smooth switching con-troller (red), and switching instants (vertical dash-dot lines)0 0.5 1 1.5 2 2.5 2.71080φ [deg]0 0.5 1 1.5 2 2.5 2.7−200−1000100200Error [µm]0 0.5 1 1.5 2 2.5 2.7−2000020004000Force [N]Time [sec]Figure 7.18: The tracking performance when fd = f0 with the non-switching con-troller (blue), the hysteresis switching controller (green), the smooth switching con-troller (red), and switching instants (dash-dot lines)132Chapter 8LPV Contouring Control ofCNC Machine ToolsIn this chapter, we design a MIMO LPV contouring controller for both two-andthree-axis CNC machine tools. The design problem is formulated by estimationof the contouring error as a function of the reference trajectory directions. Theobjective and design process of the MIMO LPV controllers are stated and presented.Finally, the performance of the designed controller has been tested in the frequencyand time domain simulations and experimental results.8.1 Serial CNC Machine tool Modeling and CoordinateTransformations8.1.1 CNC machine tools mechanismThe mechanism of a three-axis Fadal CNC machine is shown in Figure 8.1. Threex, y and z axes are equipped with three ball-screw feed-drive systems connected to133DC motors. By applying voltages vmx, vmy and vmz to the current amplifiers, rep-resenting current commander for the three DC motors, three torques are generatedand transmitted to the screws. As a result, the tables move ucx, ucy and ucz [m] inthe x, y and z directions, respectively. In addition, external forces fcx, fcy and fczare applied to the tables due to the resultant component of the cutting forces andfriction along each axis.Figure 8.1: A 3-axis CNC machine8.1.2 CNC machine tool modelOur focus in this chapter is to model the path direction variations as a function ofthe reference trajectory. Therefore, we ignore dynamic variations of each individualaxis due to table position and workpiece mass. In the first step, we formulatethe three-axis CNC machine tool model as an LTI transfer function which can be134described with the following state-space representationG :x˙m(t) = Amxm(t) +Bmvm(t)fc(t) ,uc(t) = Cmxm(t),(8.1)where xm is a state vector, and Am, Bm and Cm are the system matrices identifiedin Section 8.3, andfc :=fcxfcyfcz, vm :=vmxvmyvmz, uc :=ucxucyucz. (8.2)8.1.3 2D coordinate transformationIn order to control the system in the task coordinate frame, we choose the tangential-normal (t-n) coordinate system which is attached to the desired position of theworkpiece on the trajectory path. In 2D trajectory in [21] as shown in Figure 8.2,the tracking error signals ex and ey are calculated asex := rx − ucx,ey := ry − ucy,(8.3)where rx and ry are the reference positions in the x and y directions. We transfer thetracking error from the x-y coordinate frame into t-n coordinate frame by rotationtransformation matrix TR aseten = TR(αx)exey , (8.4)where en and et are normal and tangential tracking error, respectively, andTR(αx) :=sin(αx) − sin(αx)sin(αx) cos(αx) . (8.5)135The angle αx is defined as the angle between the tangent vector to the trajectorypath and the x axis. αx is calculated as a function of the reference velocities r˙x andr˙y in αx = tan−1(r˙y/r˙x).In a trajectory with a slow and smooth curvature change, or when et isconsiderably smaller than the curvature parameters, we can assume the contouringerror  ≈ en. More precise approximation of contouring error can be obtained byintroducing the delayed desired position and delayed task coordinate frame (td-nd)in Figure 8.2. The delayed time can be estimated [87] astdelay :=etr˙. (8.6)Having the delayed time, we can find the delayed desired position and transformtracking error into the td-nd coordinate frame using TR(αd) with delayed rotationangle αd. αd is the angle between the tangent vector to the path at the delayedposition and x-axis.Figure 8.2: 2D trajectory coordinatetransformation and contouring errorestimationFigure 8.3: 3D trajectory coordinatetransformation and contouring errorestimation1368.1.4 3D coordinate transformationSimilar to 2D trajectory, in tracking a 3D trajectory as shown in Figure 8.3, wetransform the tracking error from the x-y-z coordinate system into the task coordi-nate frame, i.e., t-n-bet~en−b :=TtTn−bexeyez, (8.7)where et is a scalar which is calculated by projecting the tracking error e on the taxis, and ~en−b is a vector which is obtained by the projection of e on the n-b plane.The transformation matrices Tt and Tn−b are formulated asTt :=[cosαx cosαy cosαz]T,Tn−b :=sin2 αx − cosαx cosαy − cosαx cosαz− cosαx cosαy sin2 αy − cosαy cosαz− cosαx cosαz − cosαy cosαz sin2 αz,(8.8)where αx, αy and αz are the angles between the trajectory path and the x, y and zaxes. These angles are describedαx = cos−1(r˙x|r˙|), αy = cos−1(r˙y|r˙|), αz = cos−1(r˙z|r˙|), (8.9)where r˙ is the instant reference velocity, and r˙x, r˙y and r˙z are components of thereference velocity in the x, y and z axes, respectively. In linear algebra, the relationbetween these three angles is given ascos2 αx + cos2 αy + cos2 αz = 1. (8.10)Hence, αz can be calculated as a function of αx and αy.137In 3D trajectory, the contouring error  is estimated to be equal to the mag-nitude of the ~en−b vector. Similar to the 2D case, this estimation can be improved bycalculating the delayed time in (8.6), and the delayed desired position consequently.In the above approximation, minimizing the contouring error  is equivalentto minimizing the ~en−b vector. Having the following relationship for the x, y and zcomponents of ~en−b vector as:en−b,z = tanαz√e2n−b,x + e2n−b,y, (8.11)we can conclude thatmin  ≈ min en−b =min en−b,x and en−b,y, for αz < pi/4,min en−b,z, for αz > pi/4.(8.12)This relationship is useful in the design of the contouring controller for 3D trajec-tories.8.2 A MIMO Controller Structure and DesignIn the SKM CNC machine tools, each axis is actuated independently by a feed-drivesystem. In the proposed LPV control design approach, instead of designing severalSISO controllers, we design a single MIMO controller which varies as a function ofpath direction θ to compensate for the contouring error.8.2.1 Control objectivesIn contouring control of the CNC machine tools, the goal is to minimize the con-touring error  for any desired trajectory. Hence, it is essential to design a MIMOfeedback controller which by minimizing contouring error, which accomplishes fastand accurate trajectory following, and at the same time, suppresses vibration of the138CNC machine due to external forces. As formulated in the previous section, theLPV controller parameters are functions of the trajectory path direction, i.e., αx inthe 2D case, and αx and αy in 3D case.8.2.2 Controller structureThe general structure of the proposed MIMO controller is illustrated in Figure 8.4.The MIMO controller K(θ) is a varying controller as a function of the gain-scheduledparameter θ whereθ :=αx, 2D trajectory,[αx, αy]T , 3D trajectory.(8.13)In this structure, the controller KTrack is used to minimize the contouring error and tangential error et for any path direction with more emphasis on minimizing in the task coordinate frame. These errors are obtained by the transformationmatrix T (θ)T (θ) :=TR(θ), 2D trajectory,[Tt(θ), Tn−b(θ)]T , 3D trajectory,(8.14)which transfers the axial tracking error e into et and .The effect of external forces fc is considered as disturbance forces which areapplied to the plant G. The MIMO controllers’ outputs, denoted by vm, are appliedas the control commands to the feed-drive systems.8.2.3 Contouring controller designFor the model G in (8.1), we design the general controller for the contouring con-troller K(θ) in Figure 8.4. Although the model G is an LTI plant, in order to139Figure 8.4: Block diagram of the MIMO LPV controller structure in contouringerror controlminimize , the controller K should be a MIMO LPV controller as a function of thepath direction θ.To fulfill the control objectives, we consider the feedback structure in Fig-ure 8.5. In Figure 8.5, weighting functions Wf , Wv, Wt and Wn are respectivelyassigned to penalize the system vibration, the control signal vm, the tangential er-ror et and the contouring error  at specified frequency ranges due to disturbanceforce vector fc and the reference input vector r. The combination of the plantG in (8.1) with the transformation matrix T (θ) in (8.14) and dynamical trans-fer functions Wf (s) := Cf (sI − Af )−1Bf + Df , Wt(s) := Ct(sI − At)−1Bt + Dt,Wn(s) := Cn(sI−An)−1Bn+Dn, and a constant diagonal matrix gain Wv yield thefollowing augmented plantGAug(θ) :x˙(t) = A(θ)x(t) +B1(θ)w(t) +B2vm(t),z(t) = C1(θ)x(t) +D11(θ)w(t) +D12vm(t),y(t) = C2(θ)x(t) +D21w(t),(8.15)where x is the total state vector, z := [zt zn zv]T , y := [et ]T , w := [r fc]T , and the140system matrices can be derived asA :=Am BmfCf 00 Af 0−BrT (θ)Cm 0 Ar, B1 :=0 BmfDf0 BfBrT (θ) 0, B2 :=Bmv00,C1 :=−DrT (θ)Cm 0 Cr0 0 0 , D11 :=DrT (θ) 00 0 , D12 :=0Wv ,C2 :=[−T (θ)Cm 0 0], D21 := 0,(8.16)andAr :=AtAn , Br :=BtBn , Cr :=CtCn ,Dr :=DtDn , Bm := [Bmv Bmf ].(8.17)Figure 8.5: Block diagram of the MIMO controller design and weighting functionsFor the augmented plant (8.15), the controller design problem can be stated141as designing a gain-scheduled controller in (4.8), to meet the objectives mentionedabove, and guarantees the exponential stability of the closed-loop system, and agiven L2-gain bound as (5.3) for any trajectory of the time-varying parameter θwithin pre-specified ranges of variations and the rate of change. To solve this con-troller design problem, the advanced gain-scheduled controller design method, re-viewed in Section 4.3.3. can be applied.8.3 Identification and System ParametersIn order to identify the SKM dynamic models in (8.1), it is assumed that the axesare independent and uncoupled. Hence, we can write the G matrices in (8.1) asAm :=AmxAmyAmz, Bmv :=BmvxBmvyBmvz,Cm :=CmxCmyCmz, Bmf :=BmfxBmfyBmfz.(8.18)Here, we write the transfer function of i-axis asGi(s) := Cmi(sI −Ami)−1Bmi +Dmi, for i = x, y, z, (8.19)where Bmi := [Bmvi Bmfi]. The transfer function Gi is considered as a combinationof the both rigid-body and the flexible modes:Gi(s) := Grigid,i(s)×Gflex,i(s), for i = x, y, z, (8.20)where the transfer matrices Grigid,i and Gflex,i respectively designate the rigid-bodyand the flexible modes of i-axis, respectively. These transfer functions are written142asGrigid,i(s) :=rg,ikt,ika,iJe,is2 +Be,is,Gflex,i(s) := Cflex,i(sI −Aflex,i)−1Bflex,i +Dflex,i,for i = x, y, z, (8.21)where rg,i, kt,i, Ka,i, Je,i and Be,i are the gear ratio, motor constant, amplifier gain,inertia, and viscous damping of the i-axis for i = x, y, z, respectively. The rigid-body mode and flexible modes parameters of the three axes are listed in Table 8.1and 8.2 according to the catalog values and the identification technique describedin Section 3.4.1.On the other hand, the flexible modes of the system are identified by fittingthe measured FRF of the flexible modes from motor signals vm to the table positionsuc using the constrained maximum likelihood approach1. The constraints are placedas an inequality on the pole locations of the transfer functions to ensure stability ofthe plants. The estimated and measured Gflex,i for x, y and z axes are depicted inFigure 8.6.Table 8.1: Rigid-body mode parameters of 3-axis Fadal CNC machine toolSymbol Unit x-axis y-axis z-axiska [A/V] 6.4898 7.5768 6.4841kt [Nm/A] 0.4769 0.4769 0.4769pt [mm] 10 10 8Je [kg.m2] 0.0071 0.00979 0.021887Be [Nm/rad/s] 0.0678 0.0287 0.0081341In the MATLAB system identification toolbox, we use the prediction error/maximum likelihoodmethod (PEM) to fit a linear model to the given measured data points by iteratively minimizingthe absolute error. In this toolbox, we set constraints on the locations of the eigenvalues to ensurestability of the estimated model.143(a) The x-axis (b) The y-axis(c) The z-axisFigure 8.6: Flexible modes measured FRF (solid blue line) botained using accelerom-eter and identified transfer functions (dash-dot green line)8.4 2D Contouring Control ProblemIn 2D trajectory, the closed-loop structure is simplified to actuate only the x andy axes in Figure 8.7. The tracking error e is transformed into the tangential andnormal error by matrix T (θ) := TR(αx) in (8.5). Since T (θ) is the only varyingmatrix in (8.16), and since it is a function of cosαx and sinαx, all varying parametersof the controller in the optimization problem (5.4) and (5.5) are defined as affinefunctions of cosαx and sinαx.144Table 8.2: Flexible mode parameters of 3-axis Fadal CNC machine toolSymbol Transfer functionsGx−17.06s5 − 3.041e04s4 + 2.396e08s3 + 6.038e11s2 + 1.689e15s+ 2.155e18s6 + 463.5s5 + 9.845e06s4 + 3.014e09s3 + 2.409e13s2 + 3.699e15s+ 2.6e18Gy53.38s4 + 472.6s3 + 3.023e08s2 + 1.346e11s+ 5.354e13s5 + 169.5s4 + 4.263e06s3 + 4.736e08s2 + 4.927e11s+ 3.874e13Gz−114.5s10 − 6.2e4s9 − 9.4e8s8 − 6.4e11s7 − 2.1e15s6 − 1.9e18s5 + . . .s11 + 584.9s10 + 1e7 + s9 + 4.8e9s8 + 3.5e13s7 + 1.3e16s6 + 4.9e19s5 + . . .−4.6e19s4 − 1.7e24s3 + 2.3e27s2 − 1.1e29s+ 4.3e32+1.5e22s4 + 2.5e25s3 + 5.4e27s2 + 2.4e30s+ 3.6e32Figure 8.7: Block diagram of the MIMO controller structure in 2-axis control8.4.1 Design of an LPV controller for 2D trajectoriesIn designing the tracking controller K(θ) for the x-y table, the weighting functionsWf , Wv, Wt and Wn in the augmented plant (8.15) are selected by trial and er-ror, along with the guidelines in Section 4.3.4 and the transfer function structurein (4.13). The selected parameters for Wt and Wn in (4.13) are given in Table 4.1and Wv = 0.0135I2, Wf = 4.2 × 105I2. In order to penalize en more than et, thetransfer function Wn is selected with one degree higher kW , 30 [dB] smaller MLgain, and 5700 [rad/s] faster ωb.The variation range of θ (αx) is defined as θ ∈ [−pi, pi] [rad] so that thecontroller can track any trajectory in the x-y plane. The rate of change of θ isassumed to be limited as θ˙ ∈ [−11, 11] [rad/s]. For a limited number of gridded145points inside the θ region as depicted in Figure 8.8 by cross marker, we designedthe LPV controller by solving the optimization problem (5.4) with the LMI toolboxof MATLAB software. The simulation and experimental results of the designedcontroller will be discussed later in this chapter.Table 8.3: Weighting function parameters in 2D contouring controlWeights kW ML [dB] MH [dB] ωb [rad/s]Wt 2 -94 1 1250Wn 3 -128 1 6970Figure 8.8: Range of θ variations(blue line) and gridded points (redcross) in 2D contouring controller de-signFigure 8.9: Gain plots of sensitivitytransfer function from rx to et (top)and from ry to en (bottom); gain plotof inverse of Wt and Wn weightingfunctions (dashed lines)8.4.2 Frequency domain simulation resultsTo evaluate the closed-loop performance of the designed controller K(θ), the mag-nitude of the sensitivity function S, which is the transfer function from rx to et, isillustrated with blue lines in the top graph of Figure 8.9. As shown in this figure,the weighting function Wt penalizes the gain of S in the low frequency range. Asimilar transfer function from ry to en is shown in the bottom graph of Figure 8.9.As observed in this figure, the gain of en in the low frequency range is considerably146less than et in the same frequency range. Hence, it is expected that the contour-ing error performance would outperform the tangential tracking performance in thetime domain simulation as well.8.4.3 Time domain simulation resultsIn the time domain simulation, we assess the controller performance in tracking aRose shape trajectory in Figure 8.10 (a) in five seconds. For this trajectory, thesignal θ and its rate of change are plotted in Figure 8.10 (b). Here, θ varies between−180 to 180 [deg] while its maximum rate of change remains less than 630 [deg/s](11 [rad/s]) to be consistent with the design constraints in the previous section. Thereference trajectory positions and velocities are illustrated in Figures 8.10 (c) and(d), respectively. According to these figures, the table travels between −0.05 to 0.05[m] in the x and y directions.By simulating the controller performance using the MATLAB SIMULINKsoftware, we plot the tracking error signals et and en in Figure 8.11 (a). In thisfigure, the maximum tangential tracking error et is less than 450 [µm] while themaximum normal tracking error en is less than 15 [µm]. Furthermore, we plot thex and y axes tracking error ex and ey in Figure 8.11 (b). These results demonstratethe ability of the controller to minimize the contouring error  while ex and ey aremuch bigger values. The control effort signals are plotted in Figure 8.11 (c).8.4.4 Time domain experimental resultsWe apply the modeling and controller design methods presented in the previoussections to the three-axis industrial Fadal CNC machine tool system shown in Fig-ure 8.1.147(a) Trajectory in the x-yplane (b) Variations of θ and its rate of change(c) Position trajectories of ucx (solid blue) anducy (dashed green)(d) Velocity trajectories of u˙cx (solid blue) andu˙cy (dashed green) in timeFigure 8.10: Reference trajectories in 2D contouring controlTo experimentally evaluate the controller performance and compare withsimulation results, we implement the designed controller K(θ) in tracking of thesame Rose shape trajectory in Figure 8.10 (a). The tracking error and controlsignals of the experimental test have been illustrated in Figure 8.12. Similar to thesimulation results, we illustrate the tracking error signals et and en in Figure 8.12 (a).According to this figure, the maximum value of en is successfully minimized to lessthan 12 [µm] while maximum of et reaches over 300 [µm]. The maximum trackingerrors of the x and y axes are shown in Figure 8.12 (b). These axial tracking errorsare 10 to 20 times greater than contouring error en.According to Figure 8.12 (b), the maximum tracking error reaches up to 300[µm] in the x-axis and over 100 [µm] in the y-axis. The control effort signal is148(a) Tracking errors of et (solid blue) and en(dashed green) in the t-n frame(b) Tracking errors of ex (solid blue) and ey(dashed green) in the x-y frame(c) Control signals of vmx and vmy in x and yaxesFigure 8.11: Time domain simulationresults in 2D countouring control(a) Tracking errors of et (solid blue) and en(dashed green) in the t-n frame(b) Tracking errors of ex (solid blue) and ey(dashed green) in the x-y frame(c) Control signals of vmx and vmy in x andy axesFigure 8.12: Time domain experimen-tal results in 2D countouring control149plotted in Figure 8.12 (c). The high frequency oscillation of the control signal inthis figure is observed since it compensates vibration of the structural modes of thesystem and friction forces. The similarity of the simulation and the experimentalresults in Figures 8.11 and 8.12 are indications of the successful contouring controlimplementation in 2D trajectory.8.5 3D Contouring Control ProblemThe LPV controller structure in 3D contouring control is shown in Figure 8.13. Inthis case, the tracking error vector e is transformed into a vector ~en−b and a scalaret by a transformation matrix, T (θ) in (8.14), where θ := [αx, αy]T . In the 3Dcase, the disturbance force fc, the reference signal r, the control signal vm, and thetable position uc are three-dimensional vectors with three elements in x, y and zdirections.Figure 8.13: Block diagram of the MIMO controller structure in 3-axis control8.5.1 Design of an LPV controller for 3D trajectoriesIn designing the tracking controller K(θ) in the x-y-z axes, the parameters of theweighting functions Wv, Wt and Wn in (4.13) are selected by trial and error, along150with the guidelines in Section 4.3.4. The selected parameters for Wt and Wn in (4.13)are given in Table 8.4 and Wv := 0.0150I3 and Wf := 1.4×107I3. In the 3D case, wepicked a smaller ML gain, and a higher corner frequency ωb for Wn in comparisonwith Wt to penalize the contouring error more than the tracking error.Table 8.4: Weighting function parameters in 3D contouring controlWeights kW ML [dB] MH [dB] ωb [Hz]Wt 2 -126 1.01 1570Wn 2 -160 1.06 2200In the 3D case, αz can be calculated as a function of αx and αy in (8.10). Inthe design stage, we found that designing only one LPV controller for the whole 3Dspace is computationally very complicated or even infeasible. As a result, multipleLPV controllers should be designed for smaller sub-spaces and switch between themwith the method developed in Chapter 6.For the sake of simplicity, we design one LPV controller by explicitly consid-ering a small sub-space where 0 < αz < pi/4. The considered sub-space is projectedinto the θ plane (αx-αy plane) by (8.10) as an area inside the circle shape depictedin Figure 8.14. In this figure, the red cross markers indicate the points in whichLPV controllers are designed. The other constraint is placed on rate of change ofαx and αy as α˙i ∈ [−1.5, 1.5] [rad/s] for i = x, y.In the selected subregion of θ, all varying parameters of the controller in theoptimization problem (5.4) and (5.5) are considered to be affine functions of cosαx,cosαy, cosαz and sinαz.151Figure 8.14: Range of θ variations (in-side the blue line area), gridded points(red cross), and trajectory of αx andαy (dashed green line) in 3D contour-ing controller designFigure 8.15: Gain plots of sensitiv-ity transfer functions from rx to et(top), from ry to en (middle), andfrom rz to en (bottom); gain plots ofinverse of Wt and Wn weighting func-tions (dashed lines)8.5.2 Frequency domain simulation resultsTo evaluate the closed-loop performance of the designed controller, the magnitude ofthe sensitivity function Sii is shown with blue lines in Figure 8.15 for various valuesof θ. The Sii is the transfer function from r to et, en−b,x and en−b,y for i = 1, 2 and3, respectively. In this figure, the gain of S22 and S33 is 40 [dB] smaller than thatof S11 in the low frequency range. Hence, the contouring error element en−b,x anden−b,y are much smaller than the tangential error et.8.5.3 Time domain simulation resultsIn the time domain simulation, the CNC machine tool is simulated to track a 3Dpath in Figure 8.16 (a). This path is selected such that αz ∈ [0, pi/4] which resultsin the dashed green line trajectory of gain-scheduled parameters variations on theαx-αy plane in Figure 8.14. The time domain reference signals of the x, y and z-152axes are depicted in Figure 8.16 (b). The change of αx, αy and αz are plotted inFigure 8.16 (c) and their derivatives are shown in Figure 8.16 (d).The tracking error in the t-n-b coordinate system is shown in Figure 8.17 (a).According to this figure, the maximum of the tangential tracking error et is about180 [µm] while the maximum of en does not exceed 8 [µm]. This small value ofthe contouring error is an indication of the success of the controller performancein minimizing the contouring error. On the other hand, the axial tracking errorsex, ey and ez, which are illustrated in Figure 8.17 (b), are about 250, 100 and 80[µm], respectively. These axial tracking errors are 10 to 20 times higher than thecontouring error. As it is shown in Figure 8.17 (c), the three components of en−berror on x, y and z axes remain less than 10 [µm]. Finally, the control effort signalin Figure 8.17 (d) remains less than 1 [V] to avoid the saturation of DC motors.153(a) Trajectory in the x-y-z plane (b) Velocity trajectories of ucx (solidblue), ucy (dashed green) and ucz (dash-dot red)(c) Variations of θ; αx (solid blue), αy(dashed green) and αz (dash-dot red)(d) Variations of θ˙; α˙x (solid blue), α˙y(dashed green) and α˙z (dash-dot red)Figure 8.16: Reference trajectories in 3D contouring control(a) Tracking errors of et (solid blue) and~en−b (dashed green) in the t-n-b frame(b) Tracking errors of ex (solid blue),ey (dashed green) and ez (dash-dotred) in the x-y-z frame(c) Tracking errors of en−b,x (solid blue),en−b,y (dashed green) and en−b,z (dash-dotred)(d) Control signals of vmx, vmy and vmzin x, y and z axesFigure 8.17: Time domain simulation results in 3D contouring control154Chapter 9Conclusion9.1 SummaryIn this thesis, the design of gain-scheduled LPV controllers in feed-drive control ofCNC machine tool systems has been studied. This design considered the dynamicvariations due to table position and workpiece mass in the single-axis machine tool,and the trajectory path directional variations in the multi-axis feed-drive systems.The analytical modeling and identification of the ball-screw drive system inthe SKM mechanism have been conducted to obtain an LPV model. The LPVmodel represents the mathematical model of the system’s dynamic variations as afunction of both table position and workpiece mass in various operating conditionsand any reference trajectory orientation.A new parallel controller structure has been proposed to handle both track-ing and vibration suspension problems in the flexible feed-drive systems. The meritsof this method over classical controllers were compared in both simulation and ex-perimental results in the motion control of a single-axis ball-screw drive system.155For a large range of dynamic variations, we proposed the application of theswitching gain-scheduled LPV controller design technique to the ball-screw drivesystems. The experimental results demonstrated that the switching controllers aremore efficient in achieving accurate tracking, high bandwidth performance and dis-turbance rejection. However, poor transient behaviors of the switching controllerscan occur at the switching instants between local controllers.To ensure smooth transient signals, we proposed a novel method to designa smooth switching LPV controller for one and two gain-scheduled parameters. Inthe design stage, a smooth transition region was defined between local controllersand the smoothness was explicitly taken into account as constraints on the controllerrate of change. We formulated the control design problem as a non-convex feasibilityproblem and solved it by a proposed iterative descent algorithm. The effectivenessof the developed method was demonstrated through fast motion control of bothSKM and PKM feed-drive mechanisms.In the multi-axis CNC machine tool control, instead of minimizing the track-ing error, we formulated the synthesis of a MIMO LPV controller in the task coor-dinate frame to minimize the contouring error directly. The proposed method wasexperimentally tested on an industrial machine-tool system, in which the contouringerror was minimized successfully in high speed tracking of a trajectory.9.2 ContributionsThe main contributions of this thesis are outlined as follows:• Unified LPV modeling and control frameworks have been established for gen-eral CNC machine tool drive systems which have significant dynamic varia-156tions. Both tracking and contouring control problems were considered in theframeworks. The frameworks have been validated in simulations and experi-ments on some serial mechanism CNC machine tools, and in simulations on aparallel kinematic machine tools.• To explicitly take into consideration the variations of the system parametersdue to the run-out effect, workpiece position and mass during the machiningprocesses, LPV modeling and system identification of a flexible ball-screw drivesystem have been developed.• Parallel controller structures have been introduced for existing and future CNCmachine tools. This structure makes controller design easier, by convertingone multi-objective controller design into multiple single-objective ones. Inaddition, this structure allows the existing CNC machine tool to augment theequipped tracking controller with a vibration suppression controller.• For CNC machine tool drives with large range of dynamic variations and highprecision requirements, switching LPV control techniques have been appliedto tracking and vibration suppression. These techniques can advance the per-formance of the machining process at the expense of controller complexity.• To avoid any poor transient performance associated with the switching ofcontrollers, a new smooth switching LPV controller has been devised. Thiscontroller ensures a smooth transition of switching controllers, and maintainssmall tracking errors at the controller switching instants.1579.3 Future WorkSome of the possible future research directions are discussed in the following.• In the smooth switching controller design method in Chapter 6, it is assumedthat the controller structure order is the same as the plant structure order.Assigning an arbitrary order of the smooth switching controller can simplifythe design and implementation of the controller, especially for a system witha high model order and/or with more than two gain-scheduling parameters.• It is expected that we can take full advantage of the switching gain-scheduledcontroller in its application to large industrial machine tool systems and multi-axis SKM and PKM mechanisms which are non-linear functions of the work-piece location and inertia. In Chapter 7, the smooth switching controllershows successful performance in SKM and PKM control. As a future work,this method can be implemented to control more complicated industrial sys-tems with higher DOFs.• In Chapter 8, the LPV controller needed to be extended to cover the whole 3Dspace. In the design stage, we noticed that designing only one LPV controllerfor the whole 3D space is computationally very complicated, or even infeasible.Therefore, as future work, one should design a switching LPV controller inwhich each local LPV controller covers a smaller subregion of the machinetool work-space.• The LPV contouring control should be extended for four and five-axis CNCmachine tool system. 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In IEEEInternational Conference on Mechatronics and Automation, pages 2194–2199,2007.174Appendix AHybrid Method in Ball-ScrewDrive Systems ModelingA.1 Wave Equations with Boundary ConditionsThe axial and torsional vibration of the distributed screw can be obtained by writingsecond order wave differential equation as:Es∂u2s∂x2(x, t) + ρsω2us(x, t) = 0, (A.1)Gs∂θ2s∂x2(x, t) + ρsω2θs(x, t) = 0, (A.2)where us(x, t) and θs(x, t) are axial displacement and torsional angle of the screwas a function of time and position. Also Es, Gs, ρs are density, Young and shearmodule of the screw, respectively.The screw axial boundary conditions can be obtained based on the matchingof the axial force in the screw with the axial of the thrust bearing in the side near175to the motor, and no axial force at the other end of the screw asaxial mode at x = 0, EsAs∂us∂x(0, t) = kbus(0, t), (A.3)axial mode at x = ls, EsAs∂us∂x(ls, t) = 0, (A.4)where As is the average cross section of the screw and ls is the screw length. Similarto the axial mode, the torsional boundary conditions can be obtained based onmatching of input torque in the coupling and twisting moment in the screw at x = 0and no moment at x = ls as:Torsional mode at x = 0, GsJs∂θs∂x(0, t) = kc (θ(0, t)− θm(t)) , (A.5)Torsional mode at x = ls,∂θs∂x(ls, t) = 0, (A.6)where Js is average second polar inertia of the screw cross section. In the nutlocation, the force and torque generated in the nut location can be estimated asdifference force and torque before and after nut due to axial and torsional force andmoment. By matching this force and moment with twisting force and moment ofsprings, we obtain the nut boundary condition as:axial at x = xc, EsAs[0−∂us∂x(x−c , t)] = kn(un(t)− us(xc, t)), (A.7)torsional at x = xc, GsJs[0−∂θs∂x(x−c , t)] = r2gkn(θc(t)− θs(xc, t)), (A.8)whereun(t) := uc(t)− rgθs(xc, t). (A.9)A.2 Motor and Table Equations of MotionAccording to the nut free-body diagram in 3.4, the equation of motion of the motorshaft can be written asJmθ¨m(t) + cmθ˙m(t) + kcθm(t) = τm(t) + kcθs(0, t), (A.10)176where Jm is motor armature moment of inertia, cm is the motor viscous damp-ing, and kc (θs(0, t)− θm(t)) is the torsional torque generated by coupling stiffnesskc. In a similar way, by considering the axial force from the nut to the table,kn (u(xc, t)− un(t)), and substitute un from (A.9), we can write the table equationof motion as:mcu¨c(t) + ccu˙c(t) + knuc(t) = fc(t) + kn (us(xc, t) + rgθs(xc, t)) , (A.11)where mc and cc are summation of workpiece and table mass, and the table viscousdamping, respectively.A.3 Solving the Differential Equation ProblemIn order to solve the wave equations (A.1-A.2) which is coupled with the motor andtable differential equation (A.10-A.11), with considering the boundary conditions(A.3-A.7), Varasani and Nyfe [99], proposed quasi-static (slow-varying) deformationof torsional and axial deformation. In the low frequency range where ωls √Es/ρsand ωls √Gs/ρs, it can be assumed that axial and torsional waves do not prop-agate in the screw. Hence, us and θs are approximated linearly as a function ofx:us(x, t) =(1− x/xc)us(0, t) + (x/xc)us(xc, t), x < xc,0 + us(xc, t), x > xc,(A.12)θs(x, t) =(1− x/xc)θs(0, t) + (x/xc)θs(xc, t), x < xc,0 + θs(xc, t), x > xc,(A.13)177me(xc) :=ρsAsxc3[(kekb)2+k2ekbk1+(kek1)2]+ ρsAs(ls − xc)(kek1)2,J1(xc) := ρsJsls(1−r2gkektc)2xc3ls1 +(1−r2gkekc)(1−r2gktktc)−1+(1−r2gktkc)2(1−ktr2gktc)−2+(1−xcls),J2(xc) := ρsJsls(r2gkektc)2 [xc3ls(1 +ktckc+k2tck2c)+(1−xcls)],J12(xc) :=r2gρsJslskektc{xcls[16+ktckc(12−r2gkt3kc)−2xc3ls]+(1−2xc3ls)(1−r2gkektc))},ktc(xc) :=(1kc+xcGsJs)−1,k1(xc) :=(xcEsAs+1kb)−1.(A.14)178

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