Modeling and Control of Vibration in Mechanical Systems (Automation and Control Engineering)

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Modeling and Control of Vibration in Mechanical Systems

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AUTOMATION AND CONTROL ENGINEERING A Series of Reference Books and Textbooks Series Editors FRANK L. LEWIS, PH.D., FELLOW IEEE, FELLOW IFAC

SHUZHI SAM GE, PH.D., FELLOW IEEE

Professor Automation and Robotics Research Institute The University of Texas at Arlington

Professor Interactive Digital Media Institute The National University of Singapore

Modeling and Control of Vibration in Mechanical Systems, Chungling Du and Lihua Xie Analysis and Synthesis of Fuzzy Control Systems: A Model-Based Approach, Gang Feng Lyapunov-Based Control of Robotic Systems, Aman Behal, Warren Dixon, Darren M. Dawson, and Bin Xian System Modeling and Control with Resource-Oriented Petri Nets, Naiqi Wu and MengChu Zhou Sliding Mode Control in Electro-Mechanical Systems, Second Edition, Vadim Utkin, Jürgen Guldner, and Jingxin Shi Optimal Control: Weakly Coupled Systems and Applications, Zoran Gajic´, Myo-Taeg Lim, Dobrila Skataric´, Wu-Chung Su, and Vojislav Kecman Intelligent Systems: Modeling, Optimization, and Control, Yung C. Shin and Chengying Xu Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory, Second Edition, Frank L. Lewis; Lihua Xie and Dan Popa Feedback Control of Dynamic Bipedal Robot Locomotion, Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, and Benjamin Morris Intelligent Freight Transportation, edited by Petros A. Ioannou Modeling and Control of Complex Systems, edited by Petros A. Ioannou and Andreas Pitsillides Wireless Ad Hoc and Sensor Networks: Protocols, Performance, and Control, Jagannathan Sarangapani Stochastic Hybrid Systems, edited by Christos G. Cassandras and John Lygeros Hard Disk Drive: Mechatronics and Control, Abdullah Al Mamun, Guo Xiao Guo, and Chao Bi Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications, edited by Shuzhi Sam Ge and Frank L. Lewis Neural Network Control of Nonlinear Discrete-Time Systems, Jagannathan Sarangapani

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Modeling and Control of Vibration in Mechanical Systems

Chunling Du Data Storage Institute Singapore

Lihua Xie Nanyang Technological University Singapore

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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MATLAB® and Simulink® are trademarks of The MathWorks, Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text of exercises in this book. This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4398-1798-8 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents

Preface

xi

List of Tables

xiii

List of Figures

xv

Symbols and Acronyms

xxiii

1 Mechanical Systems and Vibration 1.1 Magnetic recording system . . . . . . . . . . . . . . . . . . . . 1.2 Stewart platform . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Vibration sources and descriptions . . . . . . . . . . . . . . . . 1.4 Types of vibration . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Free and forced vibration . . . . . . . . . . . . . . . . . 1.4.2 Damped and undamped vibration . . . . . . . . . . . . 1.4.3 Linear and nonlinear vibration . . . . . . . . . . . . . . 1.4.4 Deterministic and random vibration . . . . . . . . . . . 1.4.5 Periodic and nonperiodic vibration . . . . . . . . . . . . 1.4.6 Broad-band and narrow-band vibration . . . . . . . . . 1.5 Random vibration . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Random process . . . . . . . . . . . . . . . . . . . . . 1.5.2 Stationary random process . . . . . . . . . . . . . . . . 1.5.3 Gaussian random process . . . . . . . . . . . . . . . . . 1.6 Vibration analysis . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Fourier transform and spectrum analysis . . . . . . . . . 1.6.2 Relationship between the Fourier and Laplace transforms 1.6.3 Spectral analysis . . . . . . . . . . . . . . . . . . . . .

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1 1 2 4 5 6 6 6 6 7 8 11 11 12 12 13 13 14 14

2 Modeling of Disk Drive System and Its Vibration 2.1 Introduction . . . . . . . . . . . . . . . . . 2.2 System description . . . . . . . . . . . . . 2.3 System modeling . . . . . . . . . . . . . . 2.3.1 Modeling of a VCM actuator . . . . 2.3.2 Modeling of friction . . . . . . . . 2.3.3 Modeling of a PZT microactuator . 2.3.4 An example . . . . . . . . . . . . . 2.4 Vibration modeling . . . . . . . . . . . . .

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17 17 17 19 19 23 29 30 39

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Modeling and Control of Vibration in Mechanical Systems

2.5

2.4.1 Spectrum-based vibration modeling . . . . . . . . . . . . 2.4.2 Adaptive modeling of disturbance . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Modeling of Stewart Platform 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 System description and governing equations 3.3 Modeling using adaptive filtering approach . 3.3.1 Adaptive filtering theory . . . . . . 3.3.2 Modeling of a Stewart platform . . 3.4 Conclusion . . . . . . . . . . . . . . . . .

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4 Classical Vibration Control 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.2 Passive control . . . . . . . . . . . . . . . . . . . . 4.2.1 Isolators . . . . . . . . . . . . . . . . . . . 4.2.2 Absorbers . . . . . . . . . . . . . . . . . . 4.2.3 Resonators . . . . . . . . . . . . . . . . . 4.2.4 Suspension . . . . . . . . . . . . . . . . . 4.2.5 An application example − Disk vibration stacked disks . . . . . . . . . . . . . . . . 4.3 Self-adapting systems . . . . . . . . . . . . . . . . 4.4 Active vibration control . . . . . . . . . . . . . . . 4.4.1 Actuators . . . . . . . . . . . . . . . . . . 4.4.2 Active systems . . . . . . . . . . . . . . . 4.4.3 Control strategy . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . .

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5 Introduction to Optimal and Robust Control 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 5.2 H2 and H∞ norms . . . . . . . . . . . . . . . . . . 5.2.1 H2 norm . . . . . . . . . . . . . . . . . . . 5.2.2 H∞ norm . . . . . . . . . . . . . . . . . . . 5.3 H2 optimal control . . . . . . . . . . . . . . . . . . 5.3.1 Continuous-time case . . . . . . . . . . . . . 5.3.2 Discrete-time case . . . . . . . . . . . . . . 5.4 H∞ control . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Continuous-time case . . . . . . . . . . . . . 5.4.2 Discrete-time case . . . . . . . . . . . . . . 5.5 Robust control . . . . . . . . . . . . . . . . . . . . . 5.6 Controller parametrization . . . . . . . . . . . . . . 5.7 Performance limitation . . . . . . . . . . . . . . . . 5.7.1 Bode integral constraint . . . . . . . . . . . 5.7.2 Relationship between system gain and phase 5.7.3 Sampling . . . . . . . . . . . . . . . . . . .

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39 43 48 53 53 53 55 55 58 62 63 63 63 63 64 64 65 66 82 83 84 84 86 87 89 89 89 89 91 92 92 94 96 96 99 101 104 108 108 111 111

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Table of Contents 5.8

Conclusion

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6 Mixed H2 /H∞ Control Design for Vibration Rejection 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Mixed H2 /H∞ control problem . . . . . . . . . . 6.3 Method 1: slack variable approach . . . . . . . . . 6.4 Method 2: an improved slack variable approach . . 6.5 Application in servo loop design for hard disk drives 6.5.1 Problem formulation . . . . . . . . . . . . 6.5.2 Design results . . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . .

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115 115 115 116 117 123 123 128 131

7 Low-Hump Sensitivity Control Design for Hard Disk Drive Systems 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Design in continuous-time domain . . . . . . . . . . . . . . . . 7.3.1 H∞ loop shaping for low-hump sensitivity functions . . 7.3.2 Application examples . . . . . . . . . . . . . . . . . . . 7.3.3 Implementation on a hard disk drive . . . . . . . . . . . 7.4 Design in discrete-time domain . . . . . . . . . . . . . . . . . . 7.4.1 Synthesis method for low-hump sensitivity function . . . 7.4.2 An application example . . . . . . . . . . . . . . . . . . 7.4.3 Implementation on a hard disk drive . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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133 133 133 137 137 141 148 152 152 153 158 158

8 Generalized KYP Lemma-Based Loop Shaping Control Design 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem description . . . . . . . . . . . . . . . . . . . . . 8.3 Generalized KYP lemma-based control design method . . 8.4 Peak filter . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Conventional peak filter . . . . . . . . . . . . . . 8.4.2 Phase lead peak filter . . . . . . . . . . . . . . . . 8.4.3 Group peak filter . . . . . . . . . . . . . . . . . . 8.5 Application in high frequency vibration rejection . . . . . 8.6 Application in mid-frequency vibration rejection . . . . . . 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .

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161 161 162 163 166 166 168 169 169 177 178

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9 Combined H2 and KYP Lemma-Based Control Design 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Controller design for specific disturbance rejection and overall error minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Q parametrization to meet specific specifications . . . . . 9.3.2 Q parametrization to minimize H2 performance . . . . . 9.3.3 Design steps . . . . . . . . . . . . . . . . . . . . . . . .

183 183 184 185 185 187 188

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Modeling and Control of Vibration in Mechanical Systems 9.4

9.5

Simulation and implementation results . . . . . . . . . . . . . . . 9.4.1 System models . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Rejection of specific disturbance and H2 performance minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Rejection of two disturbances with H2 performance minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 190 193 194

10 Blending Control for Multi-Frequency Disturbance Rejection 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Control blending . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 State feedback control blending . . . . . . . . . . . . . . 10.2.2 Output feedback control blending . . . . . . . . . . . . . 10.3 Control blending application in multi-frequency disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . 10.3.2 Controller design via the control blending technique . . . 10.4 Simulation and experimental results . . . . . . . . . . . . . . . . 10.4.1 Rejecting high-frequency disturbances . . . . . . . . . . . 10.4.2 Rejecting a combined mid and high frequency disturbance 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 197 199 200 203 203 205 207 207 211 213

11 H∞ -Based Design for Disturbance Observer 11.1 Introduction . . . . . . . . . . . . . . . 11.2 Conventional disturbance observer . . . 11.3 A general form of disturbance observer . 11.4 Application results . . . . . . . . . . . 11.5 Conclusion . . . . . . . . . . . . . . .

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12 Two-Dimensional H2 Control for Error Minimization 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 2-D stabilization control . . . . . . . . . . . . . . . . . . . 12.3 2-D H2 control . . . . . . . . . . . . . . . . . . . . . . . 12.4 SSTW process and modeling . . . . . . . . . . . . . . . . 12.4.1 SSTW servo loop . . . . . . . . . . . . . . . . . . 12.4.2 Two-dimensional model . . . . . . . . . . . . . . 12.5 Feedforward compensation method . . . . . . . . . . . . . 12.6 2-D control formulation for SSTW . . . . . . . . . . . . . 12.7 2-D stabilization control for error propagation containment 12.7.1 Simulation results . . . . . . . . . . . . . . . . . . 12.8 2-D H2 control for error minimization . . . . . . . . . . . 12.8.1 Simulation results . . . . . . . . . . . . . . . . . . 12.8.2 Experimental results . . . . . . . . . . . . . . . . 12.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .

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ix

Table of Contents 13 Nonlinearity Compensation and Nonlinear Control 13.1 Introduction . . . . . . . . . . . . . . . . . . . 13.2 Nonlinearity compensation . . . . . . . . . . . 13.3 Nonlinear control . . . . . . . . . . . . . . . . 13.3.1 Design of a composite control law . . . 13.3.2 Experimental results in hard disk drives 13.4 Conclusion . . . . . . . . . . . . . . . . . . .

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251 251 251 252 256 257 259

14 Quantization Effect on Vibration Rejection and Its Compensation 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Description of control system with quantizer . . . . . . . . . . 14.3 Quantization effect on error rejection . . . . . . . . . . . . . . 14.3.1 Quantizer frequency response measurement . . . . . . 14.3.2 Quantization effect on error rejection . . . . . . . . . 14.4 Compensation of quantization effect on error rejection . . . . . 14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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261 261 261 266 266 266 269 272

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15 Adaptive Filtering Algorithms for Active Vibration Control 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Adaptive feedforward algorithm . . . . . . . . . . . . . . . . . . 15.3 Adaptive feedback algorithm . . . . . . . . . . . . . . . . . . . . 15.4 Comparison between feedforward and feedback controls . . . . . 15.5 Application in Stewart platform . . . . . . . . . . . . . . . . . . . 15.5.1 Multi-channel adaptive feedback AVC system . . . . . . . 15.5.2 Multi-channel adaptive feedback algorithm for hexapod platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Simulation and implementation . . . . . . . . . . . . . . 15.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 275 275 277 280 280 280 281 284 290

References

293

Index

305

Preface

This book is primarily intended for researchers and engineering practitioners in systems and control, especially those engaged in the area of modeling and control of vibrations in mechanical structures and systems. The book aims at empowering readers with a clear understanding of characteristics of various vibrations, their effects on system stability and performance, and techniques for rejecting vibrations of different frequency ranges and their limitations. Special attention is given to recently developed vibration modeling and control techniques in high precision systems. Many real-world examples are given to demonstrate the modeling and control techniques. Vibration exists in a wide spectra of engineering systems such as hard disk drives, automotives, aerospace and aeronautic systems, manufacturing systems, etc. Vibration is undesirable in most engineering applications, lowering system performance, wasting energy and creating unwanted noise. Although the problem of vibration control has been studied for a long time, it remains and indeed becomes more challenging in many applications such as precision engineering and hard disk drives, where an extremely high positioning accuracy is required. Therefore, vibration control has drawn more intensive efforts from researchers and engineering practitioners in recent years. It is our intention in this book to present to readers some of the recent developments in this field. The book presents the latest results in vibration modeling and advanced control design for vibration attenuation in mechanical actuation systems to achieve high precision positioning performance. It focuses on vibration and disturbance rejections using recently developed control techniques for high precision positioning, and demonstration of the benefits gained from the applications of these techniques. The theoretical developments and principles of control design are elaborated in detail so that the reader can apply the techniques developed to obtain solutions with the help of MATLABr . Examples are presented throughout the book so that the subject can be better understood. A number of simulation and experimental results with comprehensive evaluations are provided in each chapter, except Chapters 1, 4, and 5, which are dedicated to the review of related background knowledge. The book summarizes a collective research effort which we have had the pleasure to contribute to. Many results reported in the book are due to the collaboration with Guoxiao Guo from Western Digital Corporation, Jianliang Zhang and Jul Nee Teoh from Data Storage Institute (DSI) of Singapore, Youyi Wang from Nanyang Technological University (NTU), and Frank Lewis from the University of Texas at Arlington. The research work contained in this book was mainly performed at DSI and the School of Electrical and Electronic Engineering (EEE) of NTU, Singapore.

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Modeling and Control of Vibration in Mechanical Systems

Algorithms applied in magnetic recording systems were implemented at DSI and those in the Stewart platform at the School of EEE, NTU. We would like to express our sincere appreciation to DSI for its supportive environment and vibrant research atmosphere. We are also sincerely grateful to Dr. Ong Eng Hong and the colleagues in Mechatronics and Recording Channel Division of DSI, and EEE, NTU for their support. Lihua Xie Chunling Du

MATLABr is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, Tel: 508 647 7000, Fax: 508-647-7001, E-mail: info@mathworks. com, Web: www.mathworks.com.

List of Tables

2.1

σ values of the modeling error (dˆ1 − d1 ) for different p and Γ . . .

4.1

% reduction of σ values of PES, RRO and NRRO and disk vibration amplitude with stacked disks compared with single disk . . . . . . .

49 81

6.1

Control performance comparison . . . . . . . . . . . . . . . . . . . 131

7.1

Control performance comparison. . . . . . . . . . . . . . . . . . . 146

9.1

Comparison of performance specifications . . . . . . . . . . . . . . 191

14.1 Quantization and friction effect . . . . . . . . . . . . . . . . . . . 272

xiii

List of Figures

1.1 1.2 1.3 1.4 1.5 1.6 1.7

The servo control loop of a hard disk drive. Hexapod from Micromega Dynamics. . . . Zoomed-in view of the hexapod. . . . . . . An example of random vibration. . . . . . . Spectrum density of broad-band vibration. . Spectrum density of narrow-band vibration. Histogram of signal x. . . . . . . . . . . .

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2 3 4 7 9 10 13

2.1 2.2 2.3

A read back signal of embedded servo. . . . . . . . . . . . . . . . . Frequency responses of a second order transfer function. . . . . . . Measured VCM Bode plots (straight lines: pure double integrator k/s2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplicative uncertainty of a VCM. . . . . . . . . . . . . . . . . . The operator zr versus x. . . . . . . . . . . . . . . . . . . . . . . . Friction f versus actuator displacement x. . . . . . . . . . . . . . . A PZT actuated suspension. . . . . . . . . . . . . . . . . . . . . . Equivalent spring mass system of PZT microactuator. . . . . . . . . A typical frequency response of the PZT microactuator. . . . . . . . An opened 1.8-inch hard disk drive. . . . . . . . . . . . . . . . . . Measured and modeled frequency responses of the VCM actuation system (LDV range 0.5 µm/V). . . . . . . . . . . . . . . . . . . . . Closed control loop of a disk drive with a VCM actuator for friction measurement via LDV. . . . . . . . . . . . . . . . . . . . . . . . . Control signal u versus displacement x. . . . . . . . . . . . . . . . VCM actuator modeling with friction nonlinearity model F (x). . . Measured and modeled friction and error. . . . . . . . . . . . . . . Actuator frequency response for sinusoidal reference with amplitude of 1 and 3 V, respectively. . . . . . . . . . . . . . . . . . . . . . . . Actuator frequency response for sinusoidal reference with amplitude of 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preload and two-slope model for friction modeling. . . . . . . . . . Plant input voltage u versus displacement x. . . . . . . . . . . . . . Closed-loop control system disturbances d1 , d2 and noise n. . . . . Closed-loop control system with disturbance and noise models. . . . Sensitivity function S(z). . . . . . . . . . . . . . . . . . . . . . . . PES NRRO spectrum. . . . . . . . . . . . . . . . . . . . . . . . . .

20 21

2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23

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. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

22 24 26 28 28 29 30 31 32 33 34 34 36 37 37 38 38 39 40 41 42

xv

xvi

Modeling and Control of Vibration in Mechanical Systems 2.24 2.25 2.26 2.27

Three-layer RBF neural network. . . . . . . . . . . . . . . . . . . . Original disturbance d1 and the modeled dˆ1 . . . . . . . . . . . . . . Power spectrum of dˆ1 . . . . . . . . . . . . . . . . . . . . . . . . . NRRO power spectrum from measurement and disturbance models, i.e., e = −P (s) · dˆ1 − dˆ2 + n ˆ. . . . . . . . . . . . . . . . . . . . . 2.28 Modeling error (d1 − dˆ1 ) for different Γ with p = 1. . . . . . . . .

44 49 50

3.1 3.2 3.3 3.4 3.5 3.6

Single-axis system using piezoelectric stiff actuator. Linear discrete time adaptive filter. . . . . . . . . . Block diagram of the LMS adaptive filter. . . . . . System identification using LMS adaptive filter. . . Frequency responses of a PZT actuator. . . . . . . Estimated and experimental frequency responses. .

. . . . . .

54 55 57 59 60 61

4.1 4.2

67

4.14

Disk and spindle motor assembly of the spin stand. . . . . . . . . . Comparison of single-disk and dual-disk axial vibrations measured via LDV at 7200 RPM. . . . . . . . . . . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk RRO power spectrum at 7200 RPM (23% improvement of σ value). . . . . . . . . . . . . . Comparison of single-disk and dual-disk NRRO power spectrum at 7200 RPM (18% reduction of σ value). . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk PES in time domain at 7200 RPM (21% reduction of σ value). . . . . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk axial vibrations measured via LDV at 8400 RPM. . . . . . . . . . . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk RRO power spectrum at 8400 RPM (41% improvement of σ value). . . . . . . . . . . . . . Comparison of single-disk and dual-disk NRRO power spectrum at 8400 RPM (28% reduction of σ value). . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk PES in time domain at 8400 RPM (38% reduction of σ value). . . . . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk axial vibrations measured via LDV at 10200 RPM. . . . . . . . . . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk RRO power spectrum at 10200 RPM (the 3rd and 5th harmonics reduced significantly, 33% improvement of σ value). . . . . . . . . . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk NRRO power spectrum at 10200 RPM (27% reduction of σ value). . . . . . . . . . . . . . . . Comparison of single-disk and dual-disk PES in time domain at 10200 RPM (32% reduction of σ value). . . . . . . . . . . . . . . . . . . Generic feedback control system. . . . . . . . . . . . . . . . . . . .

5.1 5.2

Configuration of standard optimal control. . . . . . . . . . . . . . . 94 A closed-loop system with uncertainty. . . . . . . . . . . . . . . . . 102

4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

4.12 4.13

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

51 52

69 70 71 72 73 74 75 76 77

78 79 80 86

xvii

List of Figures 5.3 5.4 5.5 5.6 5.7 5.8 6.1 6.2 6.3 6.4 6.5 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24

A closed-loop system with additive uncertainty for robust stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A closed-loop system with multiplicative uncertainty for robust stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control system structure for Youla parametrization. . . . . . . . . . Sensitivity function for continuous-time system. . . . . . . . . . . . Sensitivity function for discrete-time system. . . . . . . . . . . . . Sensitivity function in discrete-time domain. . . . . . . . . . . . . . Mixed H2 /H∞ control scheme for HDD servo bance models. . . . . . . . . . . . . . . . . . . . Frequency responses of the VCM actuator. . . . . Multiplicative uncertainty of the VCM actuator. . Frequency response of sensitivity functions. . . . Frequency response of sensitivity functions. . . .

loop with distur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 107 110 110 113 123 125 126 129 130

Parallel structure of a dual-stage actuation system with disturbances and noise injected. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Power spectrum of PES nonrepeatable runout in open loop. . . . . . 135 Decoupled structure of dual-stage actuation systems. . . . . . . . . 136 Structure of H∞ loop shaping. . . . . . . . . . . . . . . . . . . . . 136 Frequency responses of Sv (s) and Sm (s). . . . . . . . . . . . . . . 139 Frequency responses of Pv (s)Cv (s) (solid line) and Pm (s)Cm (s)(dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Frequency response of VCM actuator P (s). . . . . . . . . . . . . . 141 Frequency response of VCM controller Cv (s). . . . . . . . . . . . . 142 Frequency response of microactuator controller Cm (s). . . . . . . . 143 Sensitivity function Sm (s) (solid) and its weighting function inverse (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Open loop frequency response of the dual-stage system. . . . . . . . 144 Sensitivity and complementary sensitivity functions. . . . . . . . . 145 Microactuator frequency response. . . . . . . . . . . . . . . . . . . 146 Frequency response of microactuator controller Cm (s). . . . . . . . 147 Open loop frequency response of the dual-stage system. . . . . . . . 147 Sensitivity and complementary sensitivity functions. . . . . . . . . 148 Experimental structure. . . . . . . . . . . . . . . . . . . . . . . . . 149 Sensitivity function of the dual-stage system (smooth line: simulation result; rough line: testing result; dotted line: PID design). . . . 150 Open loop frequency response of the dual-stage system (smooth line: simulation result; rough line: testing result.) . . . . . . . . . . . . . 150 Step response of the dual-stage system. . . . . . . . . . . . . . . . 151 3σ of PES NRRO versus frequencies. . . . . . . . . . . . . . . . . 151 VCM controller Cv (z). . . . . . . . . . . . . . . . . . . . . . . . . 155 Microactuator controller Cm (z). . . . . . . . . . . . . . . . . . . . 155 Sensitivity function of the dual-stage system. . . . . . . . . . . . . 156

xviii

Modeling and Control of Vibration in Mechanical Systems

7.25 Sensitivity function. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.26 Sensitivity function of the dual-stage system. . . . . . . . . . . . . 7.27 Frequency responses of Pv (z)Cv (z) (solid curve) and Pm (z)Cm (z) (dashed curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.28 Sensitivity function of the dual-stage system. . . . . . . . . . . . . 7.29 Open loop frequency responses of the dual-stage system. . . . . . . 7.30 Step response of the dual-stage system. . . . . . . . . . . . . . . . 7.31 3σ value of PES NRRO versus frequency. . . . . . . . . . . . . . .

156 157

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

163 167 167 170 170 171 172

8.9 8.10 8.11 8.12 8.13 8.14

8.15 8.16

8.17 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Q parameterization for control design. . . . . . . . . . . . . . . . . Peak filter F in the nominal feedback loop. . . . . . . . . . . . . . Peak filter in the frequency domain. . . . . . . . . . . . . . . . . . Sensitivity functions before and after group peak filtering activated. PZT microactuator attached to VCM actuator arm. . . . . . . . . . Power spectrum of the position error before servo control. . . . . . PZT micro actuator frequency response. . . . . . . . . . . . . . . . Sensitivity functions before and after the KYP lemma-based design: simulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-loop Bode plot before and after the KYP lemma-based design. Structure of experimental setup. . . . . . . . . . . . . . . . . . . . Sensitivity functions before and after the KYP lemma-based design: experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . σ value of PES versus frequency. . . . . . . . . . . . . . . . . . . . Frequency response of the PZT microactuator. . . . . . . . . . . . . PES NRRO power spectrum calculated from measured PES signal without servo control, reflecting the vibration distribution of the system (3σ = 21 nm including the noise 3σ = 15.2 nm). . . . . . . . . Comparison of sensitivity functions. . . . . . . . . . . . . . . . . . Open loop frequency responses (PLPF (GM: 6 dB, PM: 50 deg., Bandwidth 1.4kHz)); KYP(GM: 6 dB, PM: 34 deg., Bandwidth: 1.7 kHz))). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NRRO power spectrum with PLPF and KYP (50% reduction before 1 kHz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H2 control scheme with Q parametrization for controller design. . . Frequency response of a PZT microactuator. . . . . . . . . . . . . . Open-loop frequency responses. . . . . . . . . . . . . . . . . . . . Designed sensitivity functions. . . . . . . . . . . . . . . . . . . . . Comparison of sensitivity functions obtained from experiment. . . . NRRO power spectrum with KYP Lemma-based controller with and without H2 minimization. . . . . . . . . . . . . . . . . . . . . . . PES NRRO spectrum without servo control. . . . . . . . . . . . . . Open-loop frequency response. . . . . . . . . . . . . . . . . . . . . Resultant sensitivity function (Solid line: with Spec. (i), (ii) and (iii); Dashed line: with Spec. (i) and (ii)). . . . . . . . . . . . . . . . . .

157 159 159 160 160

174 175 175 176 176 179

180 181

181 182 184 190 191 192 192 193 194 195 195

List of Figures

xix

9.10 Resultant sensitivity function with all the three requirements fulfilled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 9.11 NRRO power spectrum with rejection of two specific disturbances at 0.65 and 2 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.1 10.2 10.3 10.4 10.5

Blending control scheme. . . . . . . . . . . . . . . . . . . . . . . . Control loop with injected disturbances at different frequencies. . . Control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency response of the VCM actuator. . . . . . . . . . . . . . . Open-loop frequency response with disturbance rejection at 4 and 8 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Simulated sensitivity function with disturbance rejection at 4 and 8 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Measured (solid curve) sensitivity function with disturbance rejection at 4 and 8 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Open-loop with disturbance rejection at 3, 6.5, and 10 kHz. . . . . . 10.9 Sensitivity function with disturbance rejection at 3, 6.5, and 10 kHz. 10.10 Open-loop with disturbance rejections at 0.65 and 2 kHz. . . . . . . 10.11 Sensitivity function with disturbance rejections at 0.65 and 2 kHz. . 11.1 Block diagram of the control loop with a conventional disturbance observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Block diagram of the control loop with a general disturbance observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Frequency response of the designed Q(z). . . . . . . . . . . . . . . 11.4 The sensitivity functions without and with the general disturbance observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 The sensitivity function comparison with the general and the conventional disturbance observers. . . . . . . . . . . . . . . . . . . . 11.6 Disturbance d1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Error signal e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Measured sensitivity functions without and with the general disturbance observer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Comparison of TEQ−OL about the general disturbance observer. . . 11.10 Comparison of TEQ−OL about conventional disturbance observer. . 12.1 SSTW process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 SSTW servo loop with disturbances and noise models. . . . . . . . 12.3 PES NRRO and its σ values versus track number during propagation. (The time sequence and the σ value increase with the track number.) 12.4 SSTW servo loop modeling in two dimensions. . . . . . . . . . . . 12.5 SSTW servo loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Frequency response of a VCM actuator. . . . . . . . . . . . . . . . 12.7 Frequency response of the closed-loop transfer function with the PD controller, PID controller, and H2 controller. . . . . . . . . . . . .

198 204 204 207 208 209 210 210 211 212 213

216 218 222 223 223 224 224 225 225 226 231 232 233 234 236 237 238

xx

Modeling and Control of Vibration in Mechanical Systems 12.8 |Φ| versus frequency. . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 3σ of PES NRRO. . . . . . . . . . . . . . . . . . . . . . . . . . . 12.10 Frequency response of the H2 controller. . . . . . . . . . . . . . . 12.11 Frequency response of the open-loop system with the H2 controller. 12.12 Comparison of sensitivity functions. . . . . . . . . . . . . . . . . . 12.13 σ value of PES NRRO versus track number. . . . . . . . . . . . . . 12.14 2-D controller for SSTW servo loop. . . . . . . . . . . . . . . . . 12.15 σ of PES NRRO versus track number. . . . . . . . . . . . . . . . . 12.16 Open-loop frequency response with stabilization controller. . . . . 12.17 Sensitivity function with stabilization controller (Ac2 , Bc2 , Cc , Dc ). 12.18 Frequency response of controller (Ac2 , Bc2 , Cc, Dc ). . . . . . . . . 12.19 Open-loop frequency response with controller (Ac2 , Bc2 , Cc, Dc ). . 12.20 Sensitivity function with controller (Ac2 , Bc2 , Cc , Dc ). . . . . . . 12.21 Step response (Channel 1/2/3: Reference/Output/Control signal). .

239 240 241 241 242 243 245 246 246 247 248 249 249 250

13.1 Friction compensation for the actuation system. . . . . . . . . . . . 13.2 Input u versus displacement x with and without compensation. . . . 13.3 Actuator frequency responses with and without friction compensation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Actuator frequency responses with friction compensation for different displacements in voltage with 0.5µm/V (Straight smooth lines: the pure double integrator). . . . . . . . . . . . . . . . . . . . . . . 13.5 Actuator frequency responses with friction compensation for different displacements in voltage with 0.5 µm/V (Straight smooth lines: the pure double integrator). . . . . . . . . . . . . . . . . . . . . . . 13.6 Sensitivity functions with and without friction compensation. . . . . 13.7 Control structure of a plant P (s) with Youla parametrization approach and adaptive nonlinear compensation. . . . . . . . . . . . . 13.8 Comparison of error rejection frequency response without and with uN of different p and Γ. . . . . . . . . . . . . . . . . . . . . . . . . 13.9 NRRO power spectrum with KYP lemma-based linear control and nonlinear compensation (80% reduction before 400 Hz). . . . . . .

252 253

14.1 Experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Frequency response of the VCM actuator measured by injecting a swept sine wave of 5mV amplitude. . . . . . . . . . . . . . . . . . 14.3 The servo loop in experiment. . . . . . . . . . . . . . . . . . . . . 14.4 Frequency response of the controller C(z). . . . . . . . . . . . . . 14.5 Frequency response of the sensitivity function S(z) with different reference levels (i.e., actuator moving ranges are different). . . . . . 14.6 Frequency response of the quantizer before and after compensation (bit number n = 6). . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Frequency response of the quantizer with compensation (bit number n = 8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253

254

254 255 256 258 259 263 263 264 264 265 267 267

List of Figures

xxi

14.8 Frequency response of the quantizer with compensation (bit number n = 10). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9 Measured sensitivity function SQ (z) with different bits n. . . . . . . 14.10 Sensitivity function |SQ (f)| with f = 10 Hz versus bit n. . . . . . 14.11 Compensation scheme of quantization effect. . . . . . . . . . . . . 14.12 Choosing the threshold δ and the scaling factor a. . . . . . . . . . . 14.13 Sensitivity function with quantization compensation. . . . . . . . .

268 268 269 270 270 271

15.1 Block diagram of FXLMS algorithm. . . . . . . . . . . . . . . . . 15.2 Filtered-X LMS adaptive feedback algorithm. . . . . . . . . . . . . 15.3 Adaptive inverse control scheme. . . . . . . . . . . . . . . . . . . 15.4 Block diagram of 2 × 2 adaptive feedback algorithm. . . . . . . . . 15.5 Block diagram of 6 × 1 FXLMS adaptive feedback control system. 15.6 General layout of the experimental setup. . . . . . . . . . . . . . . 15.7 60 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 15.8 210 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 15.9 240 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 15.10 270 Hz error signal in dB unit. . . . . . . . . . . . . . . . . . . . . 15.11 Simulink diagram of automatic gain control. . . . . . . . . . . . . 15.12 180 Hz error signal without automatic gain control. . . . . . . . . . 15.13 180 Hz error signal with automatic gain control. . . . . . . . . . .

276 279 279 282 283 286 287 287 288 288 291 292 292

Symbols and Acronyms

Rn :

n-dimensional real Euclidean space

Rn×m :

set of n × m real matrices

In :

n × n identity matrix

(A, B, C, D):   A B1 B2  C1 D11 D12 : C2 D21 D22

state-space representation of a system

compact representation of system: x(k + 1) = Ax(k) + B1 w(k) + B2 u(k) z(k) = C1 x(k) + D11 w(k) + D12 u(k) y(k) = C2 x(k) + D21 w(k) + D22 u(k)

diag{A1 , A2 , · · · , An }: block diagonal matrix with Aj ( not necessarily square), j = 1, 2, · · · , n, on the diagonal XT :

transpose of matrix X

X∗:

complex conjugate transpose of matrix X

P ≥ 0:

symmetric positive semidefinite matrix P ∈ Rn×n

P > 0:

symmetric positive definite matrix P ∈ Rn×n

P ≥ Q:

P − Q ≥ 0 for symmetric P, Q ∈ Rn×n

P > Q:

P − Q > 0 for symmetric P, Q ∈ Rn×n

σ ¯ (X):

largest singular value of X

Trace(X):

trace of X

k · k:

Euclidean vector norm

kwk2 :

ℓ2 -norm of a signal {w(k)}, i.e.,

s

∞ P

k=0

2

kw(k))k .

xxiii

xxiv

Modeling and Control of Vibration in Mechanical Systems ℓ2 {[0, ∞)}: space of square summable sequences on {[0, ∞)}. The signal {w(k)} is said to be from ℓ2 {[0, ∞)} or simply ℓ2 if kwk2 < ∞. kGk2 :

H2 norm of transfer function G

kGk∞:

H∞ norm of transfer function G

Re( ):

the real part of a complex number

Im( ):

the imaginary part of a complex number

ρ( ):

spectral radius

AGC :

automatic gain control

AV C :

active vibration control

deg:

degree

det:

determinant

DSA:

Dynamic Signal Analyzer

FFT:

fast Fourier transform

F XLM S:

filtered-X LMS

HDD:

hard disk drive

LDV :

Laser Doppler Vibrometer

LF T :

Linear fractional transformation

LM I:

linear matrix inequality

LM S:

least mean square

LQG:

linear quadratic Gaussian

LT R:

loop transfer recovery

M EM S:

micro electro-mechanical system

M SE:

mean square error

N RRO:

nonrepeatable runout

P ES:

position error signal

xxv

Symbols and Acronyms P ID :

proportional-integral-derivative

P LP F :

phase lead peak filter

P ZT :

lead zirconate titanate/piezoelectric

RBF :

radial basis function

RM S:

root mean square

RP M :

rotations per minute

RRO:

repeatable runout

SST W :

self-servo track writing

ST W :

servo track writing

T M R:

track misregistration

V CM :

voice coil motor

1 Mechanical Systems and Vibration

When studying mechanical systems, we have to include the subject of dynamics and vibration. Dynamics is a branch of mechanics that deals with motion and its effect on a body. Unlike statics, which deals with bodies at rest, dynamics takes into account the effect of velocities and accelerations on the forces acting on bodies. Vibration is regarded as a branch of dynamics, since forces and masses are taken into account in vibration analysis. Thus it is natural that both dynamics and vibrations of mechanical systems are studied in this book. A vibration may be a signal, force, or temperature variation that affects the response of a system in an unacceptable manner. If our analysis of system response to a vibration shows that it regularly affects the system performance in an unacceptable manner, we need to alter or control the vibration response and bring the response within acceptable levels by adding appropriate forces, called control forces which are functions of system response such as displacement. This may require a design by using the value of the system response to generate additional forces according to certain rules or laws such that the modified response behaves according to desired performance and within certain bounds. This results in a closed-loop system that incorporates feedback controls. The purpose of designing a system with feedback force is to minimize unwanted behavior. Examples include magnetic recording systems, Stewart platforms, positioning stages [27], the atomic force microscope (AFM) [28], industry robots [29], as well as some automotive systems [34]. The magnetic recording system and Stewart platform are examples to be presented next to help in fixing these ideas more firmly. With such a grounding, more advanced problems become accessible.

1.1

Magnetic recording system

Figure 1.1 shows a servo control loop of a hard disk drive (HDD) with a voice coil motor (VCM) and a piezoelectric (PZT) actuated servo system. It consists of a stack of flat rotating disks with positioning information or servo information embedded in their surfaces. The servo information is used to position the magnetic heads on the disk surfaces. Position measurement of the magnetic heads is achieved by means of analyzing the position error signal (PES) calculated from the read back signal. To

1

2

Modeling and Control of Vibration in Mechanical Systems

have a disk drive with a high storage capacity, the head positioning error with respect to the target track center needs to be as small as possible. The error is mainly due to, (1) torque disturbances from spindle motor, (2) actuator pivot friction, (3) airflow-induced non-repeatable disk, suspension and slider vibrations, (4) mechanical resonance vibration, and (5) head sensing and electronic noises, media noise, and quantization noises. Hence how to deal with the variety of the disturbances is critical to the head positioning accuracy, and subsequently the track density for a high capacity disk drive.

FIGURE 1.1 The servo control loop of a hard disk drive.

1.2

Stewart platform

The six-leg parallel linkage mechanism known as the “Stewart platform” was discovered as early as in 1965 [30]. It is designed according to a cubic configuration, consisting of two triangular parallel plates connected to each other by six active legs orthogonal to each other. Each leg is equipped with a voice coil actuator, a force sensor and two flexible joints. The closed kinematical linkage structure of a Stewart platform has major advantages over any serial link robots: great rigidity, high force to weight ratio, six degrees of freedom (DOF), etc. [31] The Stewart platform is

Mechanical Systems and Vibration

3

widely used as space born structures, as well as a high precision pointing device and vibration isolator. Stewart platforms can be divided into two main classes according to the stiffness of the legs: stiff type and soft type. For the soft design, each leg essentially acts as an axial springing parallel with a voice coil actuator, while the stiff design involves piezoelectric or magneto restrictive legs whose extensions can be controlled [32]. The Stewart platform has been widely used in active vibration control. It has an important property for vibration control application: forces transmitted between the mobile plate and the base plate are totally axial forces of actuators. This implies that if the axial forces can be measured and eliminated, the vibration created by these forces can thus be eliminated. Thus the Stewart platform has become one of the most popular approaches for 6-DOF active vibration control in precision systems due to its attractive properties.

FIGURE 1.2 Hexapod from Micromega Dynamics.

The Stewart platform (Hexapod) from Micromega-Dynamics used as a vibration isolation device is shown in Figure 1.2. The hexapod has a cubic architecture and consists of two parallel plates connected to each other by six active legs. The plates are made of aluminum with a thickness of 20 mm and diameter of 250 mm, with the weight of the mobile plate at 1 kg. Each leg of the active interface consists of a linear piezoelectric actuator, a collocated force sensor, and flexible tips to connect the two end plates. Flexible tips are used in order to avoid the problem of friction and backlash, which comes with the use of spherical joints. The hexapod can be used to

4

Modeling and Control of Vibration in Mechanical Systems

FIGURE 1.3 Zoomed-in view of the hexapod.

actively increase the structural damping of flexible systems attached to it. Figure 1.3 shows the zoomed-in view of the inside of the hexapod, revealing the arrangement of the six collocated sensor-actuator legs. The wires shown in Figure 1.2 are the outputs of the force sensors (each for one sensor) and inputs to the piezoelectric actuators (each for one actuator), respectively. Each of the legs in the Stewart platform consists of a PZT force sensor and an amplified PZT actuator. They form a collocated sensor-actuator pair configuration. If an actuator and a sensor are collocated, the associated signals for the actuator and sensor are power conjugated, i.e., the product of the actuated velocity and the measured force represents the power that is extracted from the mechanical structure. A collocated actuator-sensor pair thus enables the control of power that is supplied to the mechanical structure. Collocated actuator-sensor pairs are suitable in active vibration control applications in the sense that they guarantee damping and stability robustness if designed properly [33]. In this book, the modeling and vibration controls for the magnetic recording system and the Stewart platform will be detailed.

1.3

Vibration sources and descriptions

Any oscillatory motion of bodies that repeatedly appears is called vibration or oscillation. There are usually forces associated with vibrations. They can be induced by various types of excitation. Some of them are: fluid flow; rotating unbalanced machinery; structure flexible modes; electrical torque; reciprocating machinery; motion induced in vehicles traveling over uneven surfaces; and ground motion caused by earthquakes. Flow induced vibration is generated by the forces exerted on an object by fluid motion. Such situations can be complicated by the fact that the motion of the vibrat-

Mechanical Systems and Vibration

5

ing object can alter the fluid flow conditions, thus changing the fluid forces. Another complicating factor is the mass of the fluid, which increases the effective mass of the system. Examples of vibration caused by fluid motion include: wave action on structure; vortex-induced vibration such as vibration of transmission cables, underwater cables used for towing and structural support, and cooling towers and chimneys; vibration caused by internal flows, such as air flow in hard disk drives, flow through pipes and hoses having bends; structural vibration caused by fluctuating aerodynamic forces such as turbulence [1][56]. In certain situations, the steady-state excitation force due to fluid motion is sinusoidal with an amplitude proportional to the square of the forcing frequency. A model of the system undergoing such excitation is m¨ x + cx˙ + kx = F0 ω2 sinωt,

(1.1)

where m is the system mass, x is the system response, ω is the forcing frequency, k and c are respectively stiffness and damping, and F0 is the force coefficient. Further analysis of system response to flow fluid motion is quite complicated and requires detailed consideration of fluid mechanics of the system. Some oscillatory systems have simple harmonic motion of the form y(t) = Bsin(ωt) + Ccos(ωt),

(1.2)

where y(t) is the displacement of a mass, and is equivalent to y(t) = Asin(ωt + φ), p A = B2 + C 2 , C B cos(φ) = , sin(φ) = . A A There occur some oscillations having exponential amplitude as follows: y(t) = Aert sin(ωt + φ),

(1.3) (1.4) (1.5)

(1.6)

where oscillation amplitude decays exponentially when r < 0, and grows indefinitely when r > 0. Many types of motion cannot be easily represented by simple functions because they are essentially random. Examples include air turbulence to arm and suspension in hard disk drives, ground motion due to an earthquake, and base motion that occurs when a vehicle travels over an uneven surface. It is possible, however, to characterize them by means of statistical averages and spectrum plots, in which Fourier analysis is used to identify the major frequency components in the vibration and they will be described in more detail in the later part of the chapter.

1.4

Types of vibration

There are several ways to categorize vibrations. Basically, they can be calssified as follows.

6

1.4.1

Modeling and Control of Vibration in Mechanical Systems

Free and forced vibration

If a system vibrates on its own after an initial disturbance and no external force acts on it, the ensuing vibration is known as free vibration. A direct example of free vibration is the oscillation of a simple pendulum. If a system vibration is due to an external force, the arising vibration is known as forced vibration. The oscillation in machines such as diesel engines that results from an external force is an example of forced vibration. If the frequency of the external force coincides with one of the natural frequencies of the system, the phenomenon known as resonance occurs, and the system undergoes oscillation. The occurrence of that resonance causing large oscillation may lead to failures of some structures such as buildings, bridges, turbines, and airplane wings.

1.4.2

Damped and undamped vibration

If during oscillation there is no energy lost or dissipated in friction or other resistance, the vibration is known as undamped vibration. On the other hand, if there is energy lost during oscillation, it is called damped vibration. When analyzing vibration near resonance in physical systems, consideration of damping becomes extremely important.

1.4.3

Linear and nonlinear vibration

If all the basic components in a vibratory system such as spring, mass and damper behave linearly, the resulting vibration is classified as linear vibration. On the other hand, if any of the basic components behaves nonlinearly, the vibration is categorized as nonlinear vibration. Linear and nonlinear differential equations are used to govern the behaviors of linear and nonlinear vibratory systems, respectively. If a vibration is linear, the principle of linear systems such as superposition holds, and there are well developed mathematical tools for analysis. As for nonlinear vibration, the superposition principle is not valid, and techniques of analysis are more complicated and less well known. Since all vibratory systems tend to behave nonlinearly with respect to amplitude level of oscillation, some knowledge of nonlinear vibration is desirable in dealing with practical vibratory systems. As is known, a describing function is one approximation method used to analyze nonlinear vibratory systems.

1.4.4

Deterministic and random vibration

A vibration is known as deterministic vibration if it results from an excitation with value or amplitude known at any given time. In some cases, the excitation acting on a vibratory system is nondeterministic or random, and the value of the excitation at a given time cannot be predicted. In these cases, a large amount of excitation data collected may exhibit some statistical regularity. Statistical methods can be used for analysis, as it is possible to estimate averages such as the mean and variance values of the random excitation. Examples of random excitations are air flow inside hard

7

Mechanical Systems and Vibration

disk drives, road roughness and ground motion during earthquakes. If the excitation is random, the induced vibration is called random vibration, such as that shown in Figure 1.4. It can be described in terms of statistical quantities.

0.8

0.6

0.4

Vibration signal

0.2

0

−0.2

−0.4

−0.6

−0.8 0

0.01

0.02

0.03

0.04

0.05 Time(sec)

0.06

0.07

0.08

0.09

0.1

FIGURE 1.4 An example of random vibration.

1.4.5

Periodic and nonperiodic vibration

Periodic vibration can be represented by Fourier series as superposition of harmonic components of various frequencies. That is, if x(t) is a periodic function with period τ , its Fourier series representation is given by a0 x(t) = + a1 cos(ωt) + a2 cos(2ωt) + · · · + b1 sin(ωt) + b2 sin(2ωt) + ... 2 ∞ a0 X = + (an cos(nωt) + bn sin(nωt)), (1.7) 2 n=1 where ω = 2π/τ is the fundamental frequency, and a0 , an , bn are constant coefficients given by

a0 =

ω π

Z

0

2π/ω

x(t)dt =

2 τ

Z

0

τ

x(t)dt,

(1.8)

8

Modeling and Control of Vibration in Mechanical Systems Z ω 2π/ω 2 τ an = x(t)cos(nωt)dt = x(t)cos(nωt)dt, (1.9) π 0 τ 0 Z Z 2 τ ω 2π/ω x(t)sin(nωt)dt = x(t)sin(nωt)dt. (1.10) bn = π 0 τ 0 Z

Let a0 , 2 cn = (a2n + b2n )1/2 . c0 =

(1.11) (1.12)

The root mean square (RMS) value of the periodic function x(t) can be determined as v u∞ uX RM S(x(t)) = t |cn |2 . (1.13) n=0

Thus the mean square value of x(t) is given by the sum of the squares of the absolute values of the Fourier coefficients. Equation (1.13) is known as Parseval’s formula for periodic functions. Although the series in (1.7) is an infinite sum, most periodic functions can be approximated by only a few harmonic functions. A nonperiodic vibration x(t) can be represented by the integral Fourier transform pair:

x(t) =

1 2π

Z

∞

X(ω)eiωt dω

(1.14)

−∞

and X(ω) =

Z

∞

x(t)e−iωt dt.

(1.15)

−∞

The RMS value of the nonperiodic function x(t) can be determined as sZ ∞ |X(ω)|2 RM S(x(t)) = dω. 2πτ −∞

(1.16)

Equation (1.16) is known as Parseval’s formula for nonperiodic functions.

1.4.6

Broad-band and narrow-band vibration

A broad-band vibration is a stationary random process whose spectral density function has significant values over a range or band of frequencies which is approximately of the same order of magnitude as the center frequency of the band. The density in

9

Mechanical Systems and Vibration

Figure 1.5 describes a broad-band vibration, which is composed of components containing frequencies over a wide or broad frequency range. A narrow-band random vibration is a stationary vibration whose spectral density function has significant values only in a range of frequency whose width is small compared to the magnitude of the center frequency. Figure 1.6 shows a vibration containing frequencies over a narrow band.

16

14

12

Magnitude(dB)

10

8

6

4

2

0 1 10

2

3

10

10

4

10

Frequency(Hz)

FIGURE 1.5 Spectrum density of broad-band vibration.

A random process whose power spectral density is constant over a frequency range is called white noise. It is called ideal white noise if the band of frequencies is infinitely wide. An ideal white noise is physically unrealizable, since the variance of such a random process would be infinite because the area under the spectrum would be infinite. It is called band-limited white noise if the band of frequencies has finite cut-off frequencies ω1 and ω2 . The variance of a band-limited white noise is given by the total area under the spectrum, namely, 2S0 (ω2 − ω1 ), where S0 denotes the constant value of the spectral density.

10

Modeling and Control of Vibration in Mechanical Systems

7

6

Magnitude(dB)

5

4

3

2

1

0 2 10

3

10 Frequency(Hz)

FIGURE 1.6 Spectrum density of narrow-band vibration.

11

Mechanical Systems and Vibration

1.5 1.5.1

Random vibration Random process

In contrast to deterministic excitations, in many applications the inputs are not well known and can be described only in terms of statistical measures such as mean value, variance and standard deviation. Mean value The average is called the mean or expected value, often represented by µ. For discrete values, such as those produced by digital data acquisition, the mean is defined as n

E(x) =

1X xj . n j=1

(1.17)

For a continuous function x(t), the mean is 1 E[x(Td )] = Td

Z

Td

x(t)dt

(1.18)

0

where the time duration of the data sample is Td . Variance Two signals may have the same mean value but one may fluctuate with greater amplitude about the mean. So we also need a measure of the range of fluctuation. Simply specifying the minimum and maximum values is insufficient because a single large fluctuation (above or below the mean) can be misleading. A measure that indicates the spread about the mean is the variance, which is defined as the average value of the square of the difference between the signal and its mean. The variance is calculated for a discrete signal as follows. n

var(x) = σ 2 =

1X (xj − µ)2 , n j=1

where σ 2 is the variance and σ is called the standard deviation. For a signal continuous in time, the variance is calculated from Z Td 1 [x(t) − µ]2 dt. var(x, Td ) = Td 0

(1.19)

(1.20)

The mean-square p value of x is the expected value of x2 and is denoted E(x2 ). The RMS value of x is E(x2 ). The relation between the variance, the mean square, and the mean is as follows. σ 2 = E(x2 ) − [E(x)]2.

(1.21)

12

Modeling and Control of Vibration in Mechanical Systems

If the mean of x is zero, the standard deviation σx of x is calculated from

σx =

p E(x2 ),

(1.22)

and is thus the same as the RMS value. A random input will generate a random response, and the output signal given by its sensor will appear to be random. Random signals, like the example shown in Figure 1.4, have no apparent pattern and never repeat. Another characteristic of a random signal is that it is impossible to predict what the signal will be in the future, even if we have the past values of the signal.

1.5.2

Stationary random process

A special case of a random process is one that is stationary, which means that its statistical properties (such as its mean and variance) are time-independent. If x(t) is stationary, its mean and covariance will be independent of t: E[x(t)] = E[x(t + τ )] = µ,

(1.23)

E[x(t)x(t + τ )] = σxx(τ ).

(1.24)

and

1.5.3

Gaussian random process

A Gaussian or normal random process has a number of remarkable properties that permit the computation of random vibration characteristics in a simple manner. The probability density function of a Gaussian process x(t) is given by

p(x) = √

x 2 1 x−¯ 1 e− 2 ( σx ) , 2πσx

(1.25)

where x ¯ and σx denote the mean value and standard deviation of x. The mean x¯ and standard deviation σx of x(t) vary with t for a nonstationary process but are constants for a stationary process. A very important property of a Gaussian process is that the form of its probability distribution is invariant with respect to linear operations. This means that if the excitation of a linear system is a Gaussian process, the steady-state response is generally a different random process, but still a normal one. The only changes are that the magnitude of the mean and standard deviation of the response are different from those of the excitation. The graph of a Gaussian probability density function has a bell-shaped envelope as seen in Figure 1.7, and is symmetric about the mean value; its spread is governed by the value of the standard deviation.

13

Mechanical Systems and Vibration 180

160

140

Probability density

120

100

80

60

40

20

0 −4

−3

−2

−1

0 Signal x

1

2

3

4

FIGURE 1.7 Histogram of signal x.

1.6 1.6.1

Vibration analysis Fourier transform and spectrum analysis

We know that a periodic signal can be expressed as a Fourier series of harmonic functions. The Fourier series coefficients, when plotted versus frequency, gives a plot of the spectrum of the signal. The spectrum graphically displays the frequency content of the signal. A nonperiodic function is expressed with the following Fourier transform pair:

x(t) =

Z

∞

a(ω)cos(ωt)dω +

−∞

Z

∞

b(ω)sin(ωt)dω,

(1.26)

−∞

where Z ∞ 1 x(t)cosωtdt, 2π −∞ Z ∞ 1 x(t)sinωtdt. b(ω) = 2π −∞

a(ω) =

(1.27) (1.28)

14

Modeling and Control of Vibration in Mechanical Systems The Fourier transform of x(t) is X(ω) = a(ω) − ib(ω).

(1.29)

An equivalent form of the Fourier transform is given by

x(t) =

Z

∞

1 2π

Z

X(ω)eiωt dω,

(1.30)

−∞

and

X(ω) =

∞

x(t)e−iωt dt.

(1.31)

−∞

The spectrum of a nonperiodic signal is the magnitude of its Fourier transform, that is, X(ω). Equation (1.31) implies that the transform is symmetric about ω = 0, that is X(−ω) = X(ω).

(1.32)

The transform of a time-shifted signal x(t − d) is X(ω)e−iωd and its spectrum is the same as the spectrum of x(t), while the time shift d affects only the phase angle of the transform.

1.6.2

Relationship between the Fourier and Laplace transforms

The Fourier transform X(ω) of x(t) is related to the Laplace transform X(s) as follows.

X(ω) =

1.6.3

1 X(s)|s=iω . 2π

(1.33)

Spectral analysis

The spectral density, or power spectral density, Sxx (ω), is one of the most useful functions in vibration testing. It is defined as Z ∞ 1 Rxx(τ )e−iωτ dτ, (1.34) Sxx (ω) = 2π −∞ where Rxx(τ ) = lim

T →∞

1 T

Z

T /2

x(t)x(t + τ )dt

−T /2

is the autocorrelation function and T is the time duration.

(1.35)

15

Mechanical Systems and Vibration Rxx(τ ) can be written as the inverse Fourier transform of Sxx (ω):

Rxx(τ ) =

Z

∞

Sxx (ω)eiωτ dω.

(1.36)

−∞

The cross-spectral density is written as Z ∞ 1 Sxy (ω) = Rxy (τ )e−iωτ dτ, 2π −∞

(1.37)

where 1 T →∞ T

Rxy (τ ) = lim

Z

T /2

x(t)y(t + τ )dt

(1.38)

−T /2

is the cross-correlation function. A very useful relation is that

Rxx(0) =

Z

∞

Sxx (ω)dω = E(x2 ),

(1.39)

−∞

which means that the mean-square value can be computed from the spectral density.

2 Modeling of Disk Drive System and Its Vibration

2.1

Introduction

Modeling plays an important role in any control system analysis and design. A physical system has to be modeled in order to design a control system. Physical systems are more or less nonlinear and may vary with time. Researchers have made many attempts to deal with a physical system: approximating it with a linear model at an operating point, finding a control strategy that is robust and adaptable to the changes in the physical system. The system modeling in this chapter is discussed from the first principle of actuators. System dynamics measurement in the frequency domain is used to determine the parameters of a model in the form of transfer function. To have a more realistic model, uncertainties, especially due to high frequency unmodeled dynamics, have to be involved inherently. Moreover, nonlinearity induced by friction is measured under the condition that the actuator is controlled to avoid unsteady signal measurement. The hysteresis of friction versus actuator position is then obtained from the measurement in closed-loop. An operator based modeling approach is adopted to model the hysteresis, and an optimal model is obtained by minimizing the energy gain between the position and the modeling error. In this chapter, vibration modeling is based primarily on the spectral decomposition of the error signal measured in a closed-loop system. A decoupling procedure is proposed which leads to approximate disturbance and noise models. Particularly, low-frequency disturbances are modeled as the output of an adaptive nonlinear mechanism with the error signal as the input.

2.2

System description

A hard disk drive as shown in Figure 1.1 includes five major parts: baseplate and cover, spindle and motor assembly, actuator assembly, disk, head/suspension assembly, and electronics card.

17

18

Modeling and Control of Vibration in Mechanical Systems

The spindle and motor assembly includes disk clamps to clamp disks. The actuator assembly contains an actuator driven by voice-coil motor (VCM) and mounted via ball-bearing at each end of a pivot shaft, flex cable carrying head and VCM leads, and arms to support suspension/head extension between the disks. In the head/suspension assembly, an airbearing surface is created on surface next to the rotating disk, the slider carrying heads flies on top of the disk surface, and a gimbal attaches the slider to the suspension. The electronics card involves drivers for the spindle motor and VCM, read/write (R/W) electronics, a servo demodulator, and micro processors for servo control and control of interface to host computer. The actuator servo channel consists of a demodulator producing position information from the servo burst read from the disk during seeking and following; a servo controller to control the position of the R/W head during reading, writing and seeking; spindle control to keep the spindle rotating at a specific speed with a minimum speed fluctuation and a power driver to drive the spindle motor and the VCM actuator. The servo or position information is used to position the magnetic head on the disk surfaces. Position measurement of the magnetic head is achieved by means of analyzing the position error signal (PES) calculated from the read back signal. The head-positioning servomechanism is a control system that positions the R/W head from one track to another in minimum time, and repositions the R/W head over a desired track with minimum statistical deviation from the track center. A settling controller is used in between the above seeking and following modes. The seek time is a measure of how fast the disk drive actuators can move the R/W head to a desired location. The seek time is limited by the actuator behavior, acceleration current level and the control algorithm. The major requirement in the seeking process is fast and smooth seeking with small or even no overshoot. Dual-stage actuation with a VCM as primary actuator and a microactuator as secondary actuator works as one way to achieve fast seeking and settling due to higher bandwidth. Once the actuator is regulating the position of the R/W head at the desired track, the smaller the head position deviates from the desired track center, the closer the tracks can be put together and the higher the track density becomes. In this stage, the servo performance is limited by mechanical factors in the actuator, disk platter, spindle motor, etc. An improved mechanical design is supposed to present less disturbance, causing less off-track. On the other hand, a good closed-loop servo system is expected to reject the disturbances. The error transfer function must be well designed to yield a sufficiently small closed-loop non-repeatable runout. This typically requires a satisfactory servo loop based on the disturbance spectrum. Generally, it demands a high servo bandwidth, a high 0-dB crossover frequency and a low hump of error rejection transfer function. A secondary microactuator activated together with the VCM primary actuator is generally used to produce a higher bandwidth closed-loop system. In a disk drive, the positioning information or servo information (“servo bursts”) is embedded in each disk surface. Servo bursts are conventionally written by costly dedicated servo writing equipment external to the disk drive, which uses a laserguided push-pin mechanism to position the write head on the disk surface until the servo burst information is written on the disk completely [53] [54]. The defects such

Modeling of Disk Drive System and Its Vibration

19

as non-circularity caused by the spindle motor vibration in the servo bursts will make servo tracks more difficult to follow in disk drives. The demand of big storage capacity in hard disk drives requires high track density. More accurate placement of servo bursts is thus required accordingly. As technologies such as servo mechanism and head and media technology advance, the capability of writing and reading narrower tracks is improved, resulting in an increased track density. Conventional servo writers require a clean room environment because the disk and head will be exposed to the environment to allow access to the external head and actuator. A self-servo track writer regenerates timing and radial information from previously written tracks using the existing R/W head [135]. The external equipment is no longer needed in servo pattern writing and thus no open access is required for the head disk assembly and servo track writing does not have to be carried out in a clean room environment. However, the self-servo track writing creates the radial error propagation problem, which will hinder the whole process of servo track writing if not properly solved [140]. In an embedded servo, data and servo information are written on all tracks in an interleaved manner. All tracks are divided into a fixed number of radial sectors. Each sector starts with a servo pattern followed by user data information. The pattern is repeated for each sector. The fields in an embedded servo pattern include Preamble, AGC field, sector or index mark, track address in gray code and servo bursts. The position feedback information for the disk drive servo mechanism consists of two components: gray code and position error signal. The gray code track number provides coarse position information. It determines the absolute position of the read/write head during seeking and tracking. Position error signal, or PES, is the relative displacement of the R/W head from the track center. When PES and gray code are combined together, the position of the read/write head is obtained. Figure 2.1 shows one read back signal of the embedded servo. The A, B, C, D bursts give fine position error quantifying the amount of track misregistration, i.e., the deviation of the R/W head from the center of the track.

2.3 2.3.1

System modeling Modeling of a VCM actuator

A linear VCM actuator moves in and out along a disk radius in one direction. It contains a coil which is rigidly attached to the structure to be moved and suspended in a magnetic field created by permanent magnets. When a current passes through the coil, a force is produced which accelerates the actuator radially inward or outward, depending on the direction of the current. The produced force is a function of the current ic . Approximately, fm = kt ic ,

(2.1)

20

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.1 A read back signal of embedded servo. where kt is a linearized nominal value called torque constant. The resonance of the actuator is mainly due to the flexibility of the pivot bearing, arm, suspension, etc. When the bandwidth of a control loop is very low and the resonance may not be a limiting factor to the control design, the actuator model can be considered as the simplified and rigid one which is a double integrator with transfer function k/s2 , i.e., k u, s2

(2.2)

y˙ = ky v,

(2.3)

v˙ = kv u,

(2.4)

y= or

where u is the input to the actuator, y and v are the displacement and the velocity of the read/write head, ky is the position measurement gain, and kv = kt /m with the actuator mass m. With higher bandwidth, the actuator resonances have to be considered in the control design, since the flexible resonance modes will reduce the system stability and affect control performance if ignored. Then the actuator model becomes y=

kv ky Pr (s)u, s2

(2.5)

which includes the resonance model Pr (s). Let ωn = 2πfn correspond to a single resonance frequency fn , and ξn be the associated damping coefficient. A second order transfer function can be used to represent the resonance, i.e.,

Pr (s) =

ωn2 . s2 + 2ξn ωn s + ωn2

(2.6)

Modeling of Disk Drive System and Its Vibration

21

Magnitude(dB)

Different ξn gives different frequency responses, shown in Figure 2.2. The peak of magnitude is higher when ξn decreases.

0

Phase(deg)

0

Frequency(Hz)

FIGURE 2.2 Frequency responses of a second order transfer function.

Other forms of Pr (s) include Pr (s) =

s2

b1 ωn s + b0 ωn2 + 2ξn ωn s + ωn2

(2.7)

and Pr (s) =

b2 s2 + b1 ωn s + b0 ωn2 s2 + 2ξn ωn s + ωn2

(2.8)

with zeros included to facilitate a phase lift which is usually associated with resonance modes. For lightly damped resonance, 0.005 ≤ ξn ≤ 0.05 is typical.

22

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.3 Measured VCM Bode plots (straight lines: pure double integrator k/s2 ).

Modeling of Disk Drive System and Its Vibration

23

Figure 2.3 shows the measured frequency response of a VCM actuator system. By curve fitting each resonant mode, one can obtain the parameters of the transfer function. The deviation from the double integrator model in low frequencies in Figure 2.3 is due to the pivot friction and other nonlinearities. The nonlinearity modeling will be discussed in detail in the next section. Consider the resonance model in (2.7) ((2.6) and (2.8) can be handled similarly). y˙ = ky v, v˙ = kv

(2.9) b0 ωn2

s2

b1 ωn s + u. + 2ξn ωn s + ωn2

(2.10)

Define x1 =

sωn ωn2 u, x2 = 2 u, s2 + 2ξn ωn s + ωn2 s + 2ξn ωn s + ωn2

(2.11)

i.e., x˙ 1 = ωn x2 . Then, we have a state-space description        y 0 0 ky 0 0 y˙  v˙   0 0 kv b0 kv b1   v   0     +   u.  =  x˙ 1   0 0 0 ωn   x 1   0  x2 ωn 0 0 −ωn −2ξn ωn x˙ 2

(2.12)

(2.13)

More resonance modes can be modeled similarly. Assume that nonlinearities such as friction and bias force and resonances are properly compensated, the actuator model is a double integrator model. It is written as        y˙ 0 ky y 0 = + u. (2.14) v˙ 0 0 v kv On the other hand, to make the mathematical models more realistic, we need to consider uncertainties inherent in the plant model. Note that it is difficult to involve all high-frequency dynamics in a model-based servo control design. In this case, an uncertainty is introduced in the plant model. For instance, a model with multiplicative uncertainty can be given by P = (1 +∆)Pn . Figure 2.4 shows the multiplicative uncertainty ∆ of a VCM. To meet the robustness requirement against this unmodeled high frequency dynamics, we need to properly handle the uncertain system so that it can perform under stricter margins.

2.3.2

Modeling of friction

In this section, we focus on friction modeling. There are basically two kinds of methodologies for friction control in the literature: Model-based friction compensation and non-model based friction control. Friction models for the former can be

24

Modeling and Control of Vibration in Mechanical Systems

20

10

0

Magnitude(dB)

−10

−20

−30

−40

−50

−60

2

10

3

10

Frequency(Hz)

FIGURE 2.4 Multiplicative uncertainty of a VCM.

4

10

25

Modeling of Disk Drive System and Its Vibration

roughly classified into two categories: static models and dynamic models. There are various static models for friction, for example, the Coulomb model, the viscous model, and friction models with the Stribeck effect, etc. However, a static friction model cannot capture observed friction phenomena like hysteresis, position dependence, and variations in breakaway forces. Therefore, a friction model involving dynamics is necessary to describe friction phenomena accurately. A relatively new dynamic friction model proposed in [43] combines the Dahl stiction behavior with arbitrary steady state friction characteristics, which is able to include the Stribeck effect. A nonlinear friction observer is then required for position control because the involved interim state is not measurable and has to be observed in order to estimate the friction force. Later, to overcome the limitations of the above model, an integrated model is proposed in [44], which is used in [41] for VCM pivot friction modeling in HDDs. The resultant friction model needs to be iteratively improved and verified using the measured and the simulated responses. Another dynamic model used in VCM pivot friction modeling is the preload and two-slope model, which is detailed in [40, 42] respectively in the frequency domain and the time domain. However, although the time-domain approach provides a good match between the time domain response of the model and the data collected, it cannot guarantee a good match in the frequency domain, and vice versa. The non-model based approaches include the neural network method [46][47] and the disturbance observer method [45]. The neural network method does not require full knowledge of the nonlinearity model, but its implementation in real disk drives seems difficult because of slow convergence [48]. In [45], a novel method for the cancelation of pivot nonlinearities is proposed and it consists of an accelerometer and a disturbance observer. The accelerometer is employed to linearize the dynamics from the desired input signal to carriage angular acceleration, and the observer estimates the nonlinear disturbances due to pivot friction for disturbance cancelation. In this section, a mathematical model will be developed to closely describe the friction hysteresis behavior. Among existing hysteresis models in the literature, the Prandtl model [50] is less complex and more attractive in real-time applications. The elementary operator in the Prandtl hysteresis model is a rate-independent backlash or linear play operator, defined by pr (π0 , x(t)), where x(t) is the actuator response and π0 ∈ R is usually initialized to 0. Hysteresis nonlinearity can be modeled by a linearly weighted superposition of many backlash operators with different threshold r > 0 and weight values wb , i.e., Z ∞ Fh (x(t)) = wb (r)pr [π0 , x(t)]dr, (2.15) 0

where the weight wb defines the ratio of the backlash operator, as seen in Figure 2.5. In order to have an accurate mathematical model for the hysteresis, the creep model proposed in [49] is also incorporated. Hence we consider the operator model given by Z ∞ Z ∞ wc (λ)lλ [ξ0 , x(t)]dλ, wb (r)pr [π0 , x(t)]dr + F (x(t)) = ax(t) + 0

0

26

Modeling and Control of Vibration in Mechanical Systems (2.16)

where t ∈ [0, T ], a, wb (r) and wc (λ) are parameters to be determined, pr and lλ are the elementary hysteresis and linear creep operators, and are defined as follows. The elementary hysteresis operator pr with threshold r is defined as the solution operator pr [π0 , x(t)] = zr (t) of the rate independent hybrid differential equation  ˙ if x(t) = zr (t) − r  x(t), z˙r (t) = 0, if zr (t) − r < x(t) < zr (t) + r  x(t), ˙ if x(t) = zr (t) + r

with the initial value

zr (0) = max{x(0) − r, min{x(0) + r, π0 (r)}}.

(2.17)

FIGURE 2.5 The operator zr versus x. Define the linear creep operator lλ with λ > 0 as the solution operator lλ [ξ0 , x(t)] = zλ (t) of the differential equation 1 z˙λ (t) + zλ (t) = x(t) λ with the initial value equation zλ (0) = ξ0 (λ).

(2.18)

27

Modeling of Disk Drive System and Its Vibration

The explicit integral formula for the linear creep operator lλ is as follows. Z t lλ [ξ0 , x(t)] = e−λt ξ0 (λ) + λ eλ(τ−t) x(τ )dτ. (2.19) 0

For numerical implementation of the operator-based modeling, a discrete-time model F (x(k)) of the operator F (x(t)) in (2.16) is developed as follows. F (x(k)) = ax(k) +

n X

wbi pri [π0 , x(k)] +

i=1

m X

wcj lλj [ξ0 , x(k)],

(2.20)

j=1

where 1) the output sequence of the discrete hysteresis operator is calculated by pri [π0 , x(k)] = zri (k),

(2.21)

  x(k) + ri , if zri (k − 1) − ri ≥ x(k) zri (k) = zri (k − 1), if zri (k − 1) − ri < x(k) < zri (k − 1) + ri  x(k) − ri , if zri (k − 1) + ri ≤ x(k)

with the initial value zri (0) = max{x(0) − ri , min{x(0) + ri , π0 (ri )}}; 2) the discrete counterpart to the continuous elementary creep operator is given by lλj [ξ0 , x(k)] = zλj (k)

(2.22)

zλj (k + 1) = e−λj Ts · zλj (k) + (1 − e−λj Ts ) · x(k)

(2.23)

with

and the initial value zλj (0) = ξ0 (λj ). Consider the hysteresis curve of the friction f versus the displacement x in Figure 2.6. Let fe = F (x(k)) be the approximated friction, then the approximation error e = f − fe . We define the energy gain between the actuator position and the error as v u PL u eT (k)e(k) kTex k∞ = t PLk=1 , (2.24) T k=1 x (k)x(k) where L is the number of data points. Denote wb = (wb1 , wb2 , · · · , wbn ), wc = (wc1 , wc2 , · · · , wcm ), and Λ = (λ(1), λ(2), · · ·, λ(m)). We aim to find optimal parameters a, wb , wc , and Λ in (2.20) so that (2.24) is minimized, and thus a model (2.20) can be obtained to approximate the friction f with the displacement x as the input. Note that kTexk∞ is a function of a, wb , wc , and Λ, and is denoted as kTex k∞ = ℓ (a, wb , wc , Λ) .

(2.25)

The MATLAB function fminsearch can be used to minimize ℓ (a, wb , wc , Λ) with respect to (a, wb , wc , Λ).

28

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.6 Friction f versus actuator displacement x.

FIGURE 2.7 A PZT actuated suspension.

29

Modeling of Disk Drive System and Its Vibration

2.3.3

Modeling of a PZT microactuator

A piezoelectric-based microactuator located on the suspension as shown in Figure 2.7 is considered in this section. The mechanical operation of the microactuator can be understood via an equivalent spring-mass system. The compliance of the base plate is simplified as a single spring Kb , and the compliance of the flex hinge elements is simplified as a single rotational spring Kr .

FIGURE 2.8 Equivalent spring mass system of PZT microactuator.

An important point for PZT microactuator modeling is that the PZT element acts in series with the base plate springs. Thus the displacement of the PZT element results in displacements of the springs. The PZT and the base plate with spring constants Km and Kb can be equivalent to a single spring with constant KT =

1 Km

2 +

1 Kb

.

(2.26)

The model is derived by applying forces at the interface of the piezo element and the base plate spring and by summing moments about the pivot point. The free expansion of the piezo element is expressed as θf =

Lm dexpV , cl1

(2.27)

30

Modeling and Control of Vibration in Mechanical Systems

where Lm is the piezo length, dexp is the piezo expansion coefficient, V is the voltage, c is the piezo thickness, and l1 is the length as indicated in Figure 2.8. The following second order differential equation can be derived to capture the dynamic behavior of the microactutor:

K

d2 θ dθ KT Lm dexpl1 +C + (Kr + KT l12 )θ = V, dt dt c

(2.28)

where K is the torsional inertia, Kr is the torsional spring rate. A typical frequency response of the PZT microactuator from voltage input to position output is shown in Figure 2.9.

Magnitude(dB)

0

−10

−20

−30 1 10

2

3

10

10

4

10

Phase(deg)

0 −40 −80 −120

−160 −200 1 10

2

3

10

10

4

10

Frequency(Hz)

FIGURE 2.9 A typical frequency response of the PZT microactuator.

2.3.4 2.3.4.1

An example Dynamics modeling

The hard disk drive under consideration is shown in Figure 2.10. It includes the VCM actuator mounted with the arm and the suspension/head. The actuator is driven by a driver which converts voltage differences into current differences linearly. The actuator dynamics measurement is taken using a Laser Doppler Vibrometer (LDV) and a Dynamic Signal Analyzer (DSA). The displacement y is measured via

Modeling of Disk Drive System and Its Vibration

31

FIGURE 2.10 An opened 1.8-inch hard disk drive. the LDV, and the frequency response is measured by using the DSA to generate a swept sine signal to excite the actuator. The measured frequency response is shown in Figure 2.11, where the modeled one is plotted from the following transfer function P (s) obtained by curve fitting to the measured frequency responses. P (s) = KPf (s)Pres (s), K = 5.3290 × 1017 , 1 , Pf (s) = 2 s + 2 × 0.25 × 2π120s + (2π120)2 s2 + 1081s + 7.3 × 108 Pres (s) = 2 . (s + 1056s + 6.964 × 108 )(s2 + 6032s + 2.527 × 109 )

(2.29)

The resonance of Pf (s) at 120 Hz is due to the nonlinearity of actuator pivot friction. When the friction nonlinearity is neglected, it is replaced with the pure double integrators model, i.e., P (s) =

K Pres (s), K = 4.8209 × 1017 , s2

(2.30)

which is plotted as the dotted curves in Figure 2.11. However, the friction in the actuator pivot [39] [40] is known to limit the low frequency gain of the control loop. Translated to the error rejection function or sensitivity function, it lifts the magnitude

32

Modeling and Control of Vibration in Mechanical Systems

100 Measured Modeled

80

Magnitude(dB)

60 40 20 0 −20 −40 1 10

10

2

3

10

10

4

200

Phase(deg)

0

−200

−400

−600 1 10

10

2

10

3

10

4

Frequency(Hz)

FIGURE 2.11 Measured and modeled frequency responses of the VCM actuation system (LDV range 0.5 µm/V).

Modeling of Disk Drive System and Its Vibration

33

of the sensitivity function at low frequencies, and thus reduces the ability of the control loop to reject low-frequency vibrations and affects the positioning accuracy. Therefore, it is necessary to compensate the friction impact. 2.3.4.2

Friction measurement and modeling

Due to the fluctuation of the head when the disk is rotating (rotational speed is 4200 RPM), it is difficult to have a steady displacement signal of the head. Thus the friction measurement is carried out under the closed loop control as shown in Figure 2.12. The controller C(z) is a PID controller combined with a notch filter and is expressed in (2.31). The sampling time Ts is 83.3 ms. With the controller, the openloop 0 dB crossover frequency is 945 Hz, the gain margin is 8.7 dB, and the phase margin is 49 deg. z−1 Ts 0.9023z 2 + 0.9467z + 0.7242 + ki )× , Ts z z−1 z 2 + 0.929z + 0.6442 kc = 0.0625, kp = 0.8, kd = 400 × 10−6 , ki = 400. (2.31)

C(z) = kc(kp + kd

FIGURE 2.12 Closed control loop of a disk drive with a VCM actuator for friction measurement via LDV. A 10 Hz sinusoidal signal with increasing amplitude of 0.5, 1, and 3 V is respectively used as the reference signal in Figure 2.12. The control signal u and displacement x are measured, and shown in Figure 2.13. The VCM actuator model with consideration of nonlinearity F (x) is shown in Figure 2.14, which includes two pure integrators, the resonance modes Pres (s) and the gain K given in (2.30). With the measured u and x, ua can be obtained from x, and thus f = u − ua. The relation between x and f can be obtained and shown as hysteresis curves in Figure 2.6.

34

Modeling and Control of Vibration in Mechanical Systems

−3

2

x 10

1.5

Control signal u (V)

1 0.5 0 −0.5 −1 −1.5 −2 −4

−3

−2

−1 0 1 Displacement (V, 0.5 µm/V)

2

FIGURE 2.13 Control signal u versus displacement x.

FIGURE 2.14 VCM actuator modeling with friction nonlinearity model F (x).

3

4

Modeling of Disk Drive System and Its Vibration

35

In what follows, we shall find F (x) to model the relationship between f and x. The above mentioned operator based method to approximate hysteresis will be applied to model the hysteresis of f and x as shown in Figure 2.6. ri in (2.21) can be chosen as the amplitude of x. Since with the chosen peak-topeak values of x = 0.5, 1 and 3V, the ri are respectively r1 = 0.25; r2 = 0.5; r3 = 1.5.

(2.32)

With m = 3, n = 3 and the initial values π0 = 0, ξ0 = 0, after 2000 iterations, a minimum error e is achieved and the optimal parameters are obtained as a = 9.0024, wb1 = −1.5783, wb2 = 0.1667, wb3 = −0.0655,

wc1 = −0.1431, wc2 = −7.3341, wc3 = 0.4403, which gives the minimal kTex k∞ = 0.08. With these parameters, fe = F (x(k)) can be calculated from (2.20). The time traces of fe and f are compared in Figure 2.15. It is observed that the time trace from the model (2.20) can closely track f, and the error f −fe is small. The modeled hysteresis from x to fe is drawn and compared with the measured one in Figure 2.6. It is seen that the modeled hysteresis and the measured one are close to each other. If the creep term in the model (2.20) is removed, i.e., wcj = 0, j = 1, · · · , m, the modeling accuracy decreases. Indeed, in this case, the minimal kTexk∞ = 0.1293 > 0.08. Figures 2.16 and 2.17 show that the quality of the agreement in the frequency domain between 70 Hz and 150 Hz decreases with lower excitation amplitude especially for the 0.5V case. Note that the operator model fe = F (x(k)) describes the hysteretic characteristics in Figure 2.6 as a mapping between the actuator position x and the friction force f. It turns out that the model makes it possible to approximate hysteretic transfer characteristics without modeling the underlying physics. This is different from the friction models such as the preload and two-slope model in [40][42]. Compared to the model in [40] [42], an advantage of our operator based model is that the frequency response of the actuator can also fit well to the measured one, as shown in Figures 2.16 and 2.17 . For comparison, we also apply the preload and two slope model [40] for the hysteresis, as shown in Figure 2.18. The preload model for velocity v and the two-slope model for position x are given by fv = kv v + ks sgn(v), and fx =



kax, |x| ≤ sx kb x + (ka − kb )sx , |x| > sx .

(2.33)

36

Modeling and Control of Vibration in Mechanical Systems Using the data with the amplitude of x(t) of 0.25V, we obtain that

ks = 1.85e − 4, kv = 0.48, ka = 0.0145, kb = 0.0014, sx = 0.009.

Using this model, a comparison with the measured data in the time domain is shown in Figure 2.19 for the amplitude of x(t) of 0.5 and 1 V, respectively. It can be observed that the plant input u versus position x fits reasonably well to the measured results in the time domain. However, in the frequency domain, the magnitude response for the case of the amplitude of x(t) of 0.5V, plotted as the dotted curve in Figure 2.17 deviates significantly from the measurement results.

−3

2.5

x 10

From measurement (f) From modeling(fe) Error (f−fe)

2

1.5

Friction and error

1

0.5

0

−0.5

−1

−1.5

−2

−2.5

0

0.05

0.1

0.15

0.2

0.25

0.3

Time(sec)

FIGURE 2.15 Measured and modeled friction and error.

0.35

0.4

0.45

0.5

Modeling of Disk Drive System and Its Vibration

37

FIGURE 2.16 Actuator frequency response for sinusoidal reference with amplitude of 1 and 3 V, respectively.

FIGURE 2.17 Actuator frequency response for sinusoidal reference with amplitude of 0.5 V.

38

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.18 Preload and two-slope model for friction modeling.

−3

1

x 10

0.8

Measured Modeled

0.6

Input voltage u

0.4 1V 0.2 0.5V 0 −0.2 −0.4 −0.6 −0.8 −1 −0.8

−0.6

−0.4

−0.2 0 Displacement (µm)

FIGURE 2.19 Plant input voltage u versus displacement x.

0.2

0.4

0.6

Modeling of Disk Drive System and Its Vibration

2.4

39

Vibration modeling

Vibration in disk drives causes the deviation of the R/W head positioning from the desired track center. It is the combination of the repeatable runout which is synchronous with the spindle revolution and the nonrepeatable runout. Figure 2.20 shows a simplified block-diagram of disk drive servo loop. y is the position of the R/W head and e is the position error signal. The signal d1 represents all the torque disturbances to the system. Such disturbances include any torque due to air-turbulence force to the actuator, the suspension and the slider. The effects of the torque disturbances are dominant at frequencies that are relatively low when compared to the servo bandwidth. The signal d2 represents disturbances that are due to non-repeatable motions of the disk and motor, suspension and slider vibrations, which directly add to the relative position of the R/W head and the servo track. The noise signal n includes media and head sensing noises and also represents the effects of the PES demodulation noise which includes actual electrical noise and A/D quantization noise. The noise signal n is thus reasonably modeled as a broad-band white noise.

2.4.1

Spectrum-based vibration modeling

FIGURE 2.20 Closed-loop control system disturbances d1 , d2 and noise n.

Since a controlled closed-loop can provide steady signals, the vibration source analysis is based on the signal that is collected from the closed-loop system. From Figure 2.20, e(k) = −P (z)S(z)d1 (k) − S(z)d2 (k) + S(z)n(k),

(2.34)

40

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.21 Closed-loop control system with disturbance and noise models. where P (z) is the transfer function of the discretized plant model P (s) and the sensitivity function or error rejection function is given by S(z) =

1 . 1 + P (z)C(z)

(2.35)

Figure 2.22 shows the sensitivity function S(z) of a closed-loop control system. Assume that d1 , d2 , and n are uncorrelated. The power spectrum denoted by Se of the error signal e is given by Se = |P (z)S(z)|2 |d1 (k)|2 + |S(z)|2 |d2 (k)|2 + |S(z)|2 |n(k)|2.

(2.36)

Figure 2.23 shows the spectrum of NRRO component in the error signal e. Two humps are obviously observed in the baseline curve. One is in the frequency range lower than 100 Hz, the other one is after 500 Hz. Considering (2.36) and Figure 2.22, the second hump is caused by S(z) through |S(z)|2 |n(k)|2 , and the first hump is due to d1 through P (z)S(z) with a hump in a lower frequency range. Hence the disturbance and noise modeling can be carried out as follows. In Figure 2.21, models D1 (s), D2 (s) and N (s) are used to describe d1 (s), d2 (s) and n(s) respectively, and wi (i = 1, 2, 3) are independent white noises with variance 1. Eq. (2.36) becomes Se = |P (z)S(z)|2 |D1 (z)|2 + |S(z)|2 |D2 (z)|2 + |S(z)|2 |N (z)|2 ,

(2.37)

where D1 (z), D2 (z) and N (z) are the discrete forms of D1 (s), D2 (s) and N (s), respectively. D1 (z) and N (z) can be determined by fitting weighted versions of P (z)S(z) and S(z) to the baseline curve of the spectrum and the spikes are considered as the effect of D2 (z). Hence the steps to obtain D1 , N and D2 are

Modeling of Disk Drive System and Its Vibration

FIGURE 2.22 Sensitivity function S(z).

41

42

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.23 PES NRRO spectrum.

Step 1). Find Sb (j),

Sb (j) =

jq

min

i=1+(j−1)q

Se (i), j = 1, 2, · · · , L/q,

(2.38)

where L is the length of Se and q is as small as possible. Step 2). Compute D1 (z), |D1 (z)|2 = WL (z)Sb /|P (z)S(z)|2 ,

(2.39)

where WL is a low-pass weighting function used to select Sb in low frequency range. Step 3). Compute N (z), |N (z)|2 = WH Sb /|S(z)|2 ,

(2.40)

where WH is the high-pass weighting function to select Sb in high frequency range. Step 4). Till now, the baseline curve Sb can be fit well by the identified D1 and N . The remaining part of the spectrum is then regarded as D2 . Thus, |D(z)|2 = {Se − [|P (z)S(z)|2 |D1 (z)|2 + |S(z)|2 |N (z)|2 ]}/|S(z)|2 .

(2.41)

Modeling of Disk Drive System and Its Vibration

43

D1 (s), D2 (s) and N (s) are then obtained as follows from D1 (z), D2 (z) and N (z) using the bilinear approximation method [81]. 1.3916 × 10−5 (s + 575.8)(s + 575.6)(s2 + 0.04389s + 161.6) , (2.42) (s2 + 315.5s + 8.178 × 104 )(s2 + 315.4s + 8.178 × 104 ) 0.701588(s + 1.271 × 104 )2 (s2 + 6.125 × 10−5 s + 4.373 × 108 ) D2 (s) = , (s + 708.4)2(s2 + 0.0001317s + 4.376 × 108 ) (2.43) 1.1695(s + 1.431 × 104 )(s + 766.2)(s2 + 8609s + 4.672 × 107 ) N (s) = . (2.44) (s + 4603)(s + 1538)(s2 + 4451s + 1.507 × 107 )

D1 (s) =

With the disturbance and noise models, the feedback controller C(z) can be optimized to minimize the error due to w = [w1 w2 w3 ]T by using the H2 optimal control method or other advanced control methods.

2.4.2 2.4.2.1

Adaptive modeling of disturbance Neural network approximation

A neural network usually consists of a large number of simple processing elements called nodes. The nodes are interconnected by weighted links with weight parameters adjustable. The different arrangement of the nodes and the interconnections defines various architectures of neural networks [73][74], which are suitable to different kinds of applications. In control engineering, a multi-layer neural network is usually used to generate the mapping from input to output since it can approximate any function under mild assumption with any desired accuracy. The function approximation is defined as follows. Definition 2.1 [75] If f(x) : Rn → Rm is a continuous vector function defined on a compact set Ω, and any y(W, x) : Rt × Rn → Rm is an approximating function that depends continuously on W and x, then, the approximation problem is to determine the optimal W denoted by W⋆ , for some index d such that d(y(W⋆ , x), f(x)) ≤ ε,

(2.45)

for an acceptable small ε. There are a number of neural networks studied for function approximation such as multi-layer perceptron networks, radial basis function (RBF) networks, and higher order neural networks. The RBF network is suitable for online nonlinear adaptive modeling and control, because it is a linearly parameterized network, has spatially localized learning capability and thus has better memory during learning, and exhibits a fast initial rate of learning convergence. The three-layer neural network shown in Figure 2.24 is a RBF network, where x ∈ Rn , y ∈ Rm , and s ∈ Rp are respectively the input, the output, and the activation function vectors, and wij is the second to the third layer interconnection

44

Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.24 Three-layer RBF neural network.

45

Modeling of Disk Drive System and Its Vibration weights. The output yi is given by yi =

p X j=1

wij sj (kx − cj k2 ), i = 1, 2, · · · , m,

(2.46)

or equivalently in a matrix form, y(x) = WT s(x),

(2.47)

where p is the number of nodes, x = [x1 x2 · · · xn ]T ∈ Rn is the network input vector, k • k denotes the Euclidean norm, cj ∈ Rn is the center vector, and WT = [wij ]T . The approximation of a general function f(x) : Rn → Rm can then be expressed as f(x) = WT s(x) + ε(x)

(2.48)

where ε(x) ∈ Rm is the reconstruction error vector. Several functions such as Gaussian, Hardy’s multiquadric and inverse Hardy’s multiquadric functions have been used as activation functions. They are separately written as −(x − cj )T (x − cj ) ], σj2

(2.49)

σj2 + (x − cj )T (x − cj ),

(2.50)

sj (x) = exp[

sj (x) = and

q

sj (x) = q

1 σj2

+ (x − cj )T (x − cj )

.

(2.51)

For a mechanical system with dynamics given as a function of position, velocity and acceleration, a dynamic neural network is needed in order to fully emulate the dynamics. In a dynamic neural network, dynamic variables such as velocity and acceleration are involved in the input x. In discrete-time neural networks, past information such as x(k − 1) and x(k − 2) is used as input. 2.4.2.2

Disturbance modeling

In some mechanical motion systems, disturbances in frequency range lower than a few hundred Hz are quite dominant and they may be due to torque disturbances, nonlinear or unknown vibration sources. Here we use d1 to represent these disturbances. This section aims to model the low-frequency disturbance d1 with an

46

Modeling and Control of Vibration in Mechanical Systems

adaptive nonlinear strategy based on the measured error signal e. In the open-loop without controller C(z), e = −P (s) · d1 − d2 + n.

(2.52)

The disturbance d1 is modeled as dˆ1 = f(Φ(e(k))),

(2.53)

where f(·) is an unknown function of the bounded vector-valued function Φ(e(k)) with e(k) = [e(k) e(k − 1) · · · e(k − l)]T ,

(2.54)

where l is to be determined. Thus (2.53) is dependent on some time history of the measured signal e. For brevity, f(Φ(e(k))) is denoted as f(Φk ). As stated in the previous section, the unknown function f can be approximated using neural networks. Here we select a Gaussian RBF based neural network. The Gaussian RBF network has some attractive properties and thus has been widely used in nonlinear control and signal approximation. The properties are, (1) it is bounded and strictly positive, and (2) it possesses a localized response. Based on the Gaussian RBF network, the low frequency disturbances can be approximated as dˆ1 = f(Φk ) = W T (k)s(Φk ) + ε,

(2.55)

where ε is the modeling error, s(Φk ) ∈ Rp is a basis vector function with a suitable number of nodes p. The radial basis function s(Φk ) is written as −

(e(k)−ce )2 i 2 σe i

−

(e(k)−e(k−1)−c∆e )2 i σ2 ∆yi

si (Φk ) = e ·e s(Φk ) = [s1 (Φk ) s2 (Φk ) ... sp (Φk )]T ,

, (2.56)

where σei and cei (σ∆ei and c∆ei ) are the variance and center position of the measurement e (velocity e). ˙ The weight vector W (k) in (2.55) can be calculated iteratively according to the following weight update law: W (k + 1) = (1 − δ)W (k) − Γs(Φk )e(k)

(2.57)

with the adaptation gain matrix Γ ∈ Rp×p being diagonal and satisfying Γ > 0 and the forgetting factor 1 > δ > 0. The forgetting factor δ is to ensure the boundedness of W (k) when the system is subjected to bounded disturbances. The speed of learning rate is related to the chosen Γ matrix. Next, the power spectrum in Figure 2.27 of a position error signal e will be used to verify the adaptive modeling scheme. It is noted that the contribution of disturbances and noise to e is described as in (2.52).

Modeling of Disk Drive System and Its Vibration

47

A low frequency signal with the spectrum as in the low frequency range in Figure 2.27 is generated and injected as disturbance d1 in Figure 2.21. The disturbance d1 affects the error signal or the output e through the plant P (s), thus the nonlinear model from e to dˆ1 aims to generate a cancelation signal of the disturbance d1 . Since the low frequency disturbances due to torque and bias are generally nonlinear, the model from e to d1 is chosen to be nonlinear. The verification is implemented with the sampling period Ts = 1/45000sec. For p = 1 with zero center positions cy1 and c∆y1 , after some trials, it was found that 2 when σy21 = σ∆y = 10, δ = 0.5, and 1 Γ = Ts · 106 · 220, p = 1, l = 1,

(2.58) (2.59)

the time trace dˆ1 calculated from (2.55) gives the best approximation of the disturbance d1 . The effect of different p and Γ on the modeling accuracy will be evaluated later. Figure 2.25 shows the simulated time trace comparison of d1 and dˆ1 from the nonlinear model (2.55). It is observed that the time trace from the model (2.55) can give a close tracking of the original one. The spectrum of dˆ1 is seen in Figure 2.26. Moreover, from the spectrum in Figure 2.27 with the component of disturbance P (s) · dˆ1 removed and considering that noise n is a white noise, the disturbance d2 and the noise n are represented approximately by dˆ2 = D2 (s)w 2 ,

(2.60)

D2 (s) = 0.0019(s2 + 3329s + 1.695 × 107 )(s2 + 3340s + 5.61 × 108 ) , (s2 + 245s + 1.668 × 107 )(s2 + 477.5s + 5.701 × 108 ) and n ˆ = N (s)w3 = 0.005 w3 .

(2.61)

The NRRO spectrum obtained by combining the nonlinear and linear models is compared with the measured one in Figure 2.27. It is found that the adaptive nonlinear modeling method can indeed be used to model the disturbance d1 that is dominant in low frequency range. Let the modeling error de = d1 − dˆ1 . To evaluate the effects of different values of p on the modeling error, two more cases with p = 5, 9 and the following parameters for Γ are investigated. Γ = Ts · 106 · diag{33, 66, 220, 66, 33}, p = 5; Γ = Ts · 106 · diag{38.5, 88, 88, 38.5, 220, 38.5, 88, 88, 38.5}, p = 9.

(2.62) (2.63)

The σ value of the modeling error e can be seen in Table 2.1. With p = 1, 5, 9, σ increases from 4.43 to 4.60, which means higher p may not give a better result

48

Modeling and Control of Vibration in Mechanical Systems

in this practical application. Compared with p, Γ influences the modeling accuracy more significantly. For p = 1, with Γ being changed to 10% of the value in (2.58), the σ value of the error e increases from 4.43 to 11. The time sequence comparison of d1 − dˆ1 is shown in Figure 2.28, where the difference is clearly seen. The other two cases are similar.

2.5

Conclusion

In this chapter the models of a VCM actuator and a PZT microactuator have been derived on the basis of their physical operations. The models have been verified with the measured frequency responses from voltage input to displacement output. To have a complete model of an actuation system, this chapter has also addressed the modeling of uncertainties in high frequency and nonlinearities such as actuator pivot friction. A Prandtl operator based model has been used to model the friction nonlinearity, and the optimal model parameters have been obtained by minimizing the energy gain between the actuator position and the modeling error. It turns out that the derived model matches well the measured model not only in the time domain, but also in the frequency domain. The developed vibration and noise modeling approaches are based on the error signal power spectrum of the closed-loop system. In particular, for low frequency disturbance modeling, an adaptive nonlinear scheme with system measurement as input has been applied to approximate the original disturbance for real-time compensation.

49

Modeling of Disk Drive System and Its Vibration TABLE 2.1

σ values of the modeling error (dˆ1 − d1 ) for different p and Γ

p 1 5 9 σ values of (dˆ1 − d1 ) for Γ and 0.1Γ.(×10−4) 4.43/11 4.55/11 4.60/10 −3

5

x 10

Original disturbance and the modeled (µ m)

Original disturbance d1 d1 from nonlinear modeling

4 3 2 1 0 −1 −2 −3 −4

0

0.05

0.1

0.15 Time(sec)

FIGURE 2.25 Original disturbance d1 and the modeled dˆ1 .

0.2

0.25

50

Modeling and Control of Vibration in Mechanical Systems

−3

4

x 10

3.5

NRRO magnitude(µm)

3

2.5

2

1.5

1

0.5

0

0

1000

FIGURE 2.26 Power spectrum of dˆ1 .

2000

3000 Frequency(Hz)

4000

5000

6000

51

Modeling of Disk Drive System and Its Vibration

−3

x 10

Modeled Measured

4

NRRO magnitude(µm)

3.5 3 2.5 2 1.5 1 0.5 0

0

1000 2000 Frequency(Hz)

3000

4000

5000

6000

FIGURE 2.27 NRRO power spectrum from measurement and disturbance models, i.e., e = −P (s)· dˆ1 − dˆ2 + n ˆ.

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Modeling and Control of Vibration in Mechanical Systems

−3

4

x 10

Γ 0.1Γ

3

Modeling error (µm)

2 1 0 −1 −2 −3 −4

0

0.05

0.1

0.15 Time(sec)

FIGURE 2.28 Modeling error (d1 − dˆ1 ) for different Γ with p = 1.

0.2

0.25

3 Modeling of Stewart Platform

3.1

Introduction

An adaptive control system is able to adapt to the changes of a physical system. In order to obtain more accurate models for physical systems, adaptive identification algorithms can be used. Indeed, adaptive filtering is often adopted in control and signal processing fields, for the modeling of an unknown system. In this chapter, the governing motion equation of a piezoelectric Stewart platform will be obtained first, followed by the derivation of a transfer function from the actuator force to the sensor output based on frequency response data. An adaptive filtering approach will then be introduced and subsequently used to model the platform and verify the transfer function obtained.

3.2

System description and governing equations

The study of active damping of flexible structures has been induced by the requirement of structural stability. Most solutions to the active damping problem rely on the integration of smart actuators and sensors in the structure itself. Referring to the discussion in Chapter 1, each leg of the hexapod Stewart platform consists of an amplified piezoelectric actuator, a force sensor and two flexible joints. This piezoelectric Stewart platform can be used as a precision pointing device, a vibration isolation mount, or an active damping mount. Shown in Figure 1.2 and Figure 1.3, the hexapod consists of two parallel plates connected to each other by six active legs. The legs are mounted in such a way to achieve the geometry of cubic configuration. Each active leg consists of a force sensor, an amplified piezoelectric actuator, and two flexible joints. Figure 3.1 shows an equivalent system of each leg connecting two rigid bodies: the disturbance source m and the sensitive payload M that are connected by a force sensor and a piezoelectric actuator represented by its elongation δ and spring stiffness k. The Laplace form of the governing equation of the motion in this system is

53

54

Modeling and Control of Vibration in Mechanical Systems

FIGURE 3.1 Single-axis system using piezoelectric stiff actuator.

M s2 xp = −ms2 xd = k(xd − xa ) = y

(3.1)

δ = xp − xa .

(3.2)

and

The transfer function between the extension δ of the piezo stack and the sensor output y is M ms2 y =k . δ M ms2 + k(M + m)

(3.3)

Consider the piezoelectric hexapod integrated in a structure. Let M and K be the mass and stiffness matrices of the global passive system, i.e., structure and hexapod. The dynamic equation governing the system is written as M s2 x + Kx = Bf,

(3.4)

where B is the force Jacobian matrix, x represents payload frame displacement, and f = kδ represents the equivalent piezoelectric force in the leg. A force sensor is located in each leg of the hexapod and collocated with the actuator. The corresponding sensor output is y = k(q − δ),

(3.5)

where y is the force sensor output and q is the leg extension from the equilibrium position. Taking into account the relationship between the leg extension and the payload frame displacement q = B T x,

(3.6)

55

Modeling of Stewart Platform the sensor output equation becomes y = k(B T x − δ).

(3.7)

The sensor output y is to be used as the input to the controller. Next an adaptive filtering approach will be used to model the hexapod Stewart platform and verify the dynamics equation (3.3).

3.3 3.3.1

Modeling using adaptive filtering approach Adaptive filtering theory

LMS algorithm

FIGURE 3.2 Linear discrete time adaptive filter.

In Figure 3.2, the statistically stationary time sequence of input signal, x(0), x(1), · · ·, is applied to the linear discrete time filter whose coefficients are W 0, W 1, · · ·. The filter output at time k, y(k), is to be as close as possible to the desired response, d(k). The difference between y(k) and d(k) is defined as the estimation error, e(k) that is then applied back to the filter to adjust the filter weights so as to minimize the estimation error in the statistical sense. y(k) = W T (k)X(k), where W T (k) = [W 0 W 1 W 2 · · ·], and X T (k) = [x(k) x(k − 1) x(k − 2) · · ·]. In most practical instances the adaptive process aims to minimize the mean-square value, or average power of the error signal [37]. Optimization under this criterion is

56

Modeling and Control of Vibration in Mechanical Systems

most effective in statistical sense. So the performance index of the adaptive filter is determined by the mean-squared error (MSE) criterion in which the cost function is defined as follows. ξ(k) = E[e2 (k)],

(3.8)

where E[ ] denotes the expected value.

e(k) = d(k) − y(k) = d(k) − W T (k)X(k).

(3.9)

ξ(k) = E[(d(k) − W T (k)X(k))2 ] = E[d2(k)] − 2W T (k)E[d(k)X(k)] + W T (k)E[X(k)X T (k)]W (k) = Rdd − 2W T (k)Rdx + W T (k)Rxx W (k), (3.10) where Rdx is the cross correlation function between d(k) and x(k), Rdd and Rxx are respectively the autocorrelation functions of d(k) and x(k). In many practical applications, the statistics of d(k) and x(k) are unknown; the exact knowledge of gradient vector is not available. Thus the method of steepest descent [35] cannot be implemented in practical situations. Therefore another method called the Least Mean Square (LMS) algorithm was introduced by Bernard Widrow [37]. In the LMS algorithm, instantaneous squared error is used as an estimation of the mean squared error. The mathematical derivation of the LMS algorithm is as follows. ˆ = e2 (k), ξ(k) ˆ = ∇e2 (k) = 2[∇e(k)]e(k), ∇ξ(k) e(k) = d(k) − W T (k)X(k), ∂e(k) ∇e(k) = = −X(k), ∂W (k) ˆ = −2X(k)e(k). ∇ξ(k)

(3.11)

The updating equation of the adaptive filter coefficient: W (k + 1) = W (k) −

µ ˆ ∇ξ(k), 2

becomes W (k + 1) = W (k) + µX(k)e(k).

(3.12)

57

Modeling of Stewart Platform

FIGURE 3.3 Block diagram of the LMS adaptive filter. Equation (3.12) is the well-known LMS algorithm that is suitable for practical signal processing applications because of its simplicity and the availability of the instantaneous error in real time. The block diagram of the LMS algorithm is illustrated in Figure 3.3. Normalized LMS algorithm The LMS adaptation process is very much dependant on the step size µ and the reference signal power. The step size determines the system convergence rate and stability [36][37]. The maximum stable step size is inversely proportional to the filter order L and the power of the reference signal x(k). A technique used to optimize the convergence speed, independent of the reference signal power, is known as the normalized LMS algorithm (NLMS). The NLMS algorithm is given as follows. W (k + 1) = W (k) + µ(k)X(k)e(k),

(3.13)

where each variable is identical to the one for the LMS algorithm except for µ(k) that is an adaptive step size computed as µ(k) =

α , ˆ LPx(k)

(3.14)

where α is a normalized step size that satisfies 0 < α < 2, L is the filter length, and Pˆx (k) is the estimated power of the reference signal x(k). The estimation of Pˆx (k) can be done using the mean square value of the reference signal as follows. X T (k)X(k) . Pˆx (k) = L

(3.15)

58

Modeling and Control of Vibration in Mechanical Systems

Then Equation (3.14) reduces to µ(k) =

α . X T (k)X(k)

(3.16)

This step size is the most widely used for the NLMS algorithm. A variant of the NLMS algorithm uses a small constant ε as follows. µ(k) =

α . ε + X T (k)X(k)

(3.17)

This constant ensures that the step size does not tend to infinity in the case of a zero input signal. The NLMS algorithm guarantees an attenuation level of γ ≤ 1, where γ is the induced-norm from noise or signal inputs to the filtering error. Therefore, a salient feature of this algorithm is that it lowers the influence of the input signal on the noise amplification effect, especially when the input signal x(k) is large.

3.3.2

Modeling of a Stewart platform

Adaptive identification is a technique that uses an adaptive filter to model an unknown system. An adaptive identification method can be applied either online along with the vibration control or offline prior to the vibration control. In an online identification system, the number of adaptive filters required for an adaptive control system will be increased. Furthermore, convergence of the adaptive filter used in the identification section of the system can be affected by a large amount of primary vibration signal [36]. Therefore an offline identification technique will be applied for the system modeling. The block diagram of the adaptive identification is illustrated in Figure 3.4. P (z) is the transfer function of the system to be identified. W (z) is a digital filter used to model P (z) based on the LMS error minimization algorithm. Both systems P (z) and W (z) are excited by a band limited white noise. The difference between the two outputs d(k) and y(k) is fed back into the LMS algorithm as error signal e(k). The LMS algorithm will adaptively adjust the coefficients of filter W (z) to minimize e(k) based on the least MSE criterion. When the error signal reaches the minimum level, the filter W (z) represents a model of P (z). The LMS adaptive filter approach is applied for the modeling of the Stewart platform. A Simulink program is developed for offline identification of the platform. A 16th order LMS adaptive filter with adaptation step size of 0.01 is chosen. A band-limited white noise is used as the training signal. Sampling frequency of the white noise generator is set at 1 kHz so that the response of the PZT actuator will be confined to the bandwidth of 500 Hz (half of sampling frequency). But the sampling frequency of the adaptive filter is set to 10 kHz to give enough time for adaptive filter to compute the filter weights within each sample of the training signal. The Simulink program for the identification is downloaded into a dSPACE real time interface board DS1104. Identification process for each PZT actuator is performed alternatively. Filter tap values are recorded through the dSPACE manager

Modeling of Stewart Platform

59

FIGURE 3.4 System identification using LMS adaptive filter.

software during the identification process. From the results of the identification of six PZT actuators of the Stewart platform in Figure 1.2, one of the resonance frequencies of the smart structure is found to be around 240 Hz. Figure 3.5 is the phase and magnitude responses of one of the six PZT actuators. The result shows that there is a resonance peak at about 240 Hz. The phase response between 60 Hz and 200 Hz is approximately linear. Experimental results of the active vibration control system also indicate that the frequency region that can attenuate the vibration signal is between 60 Hz and 220 Hz. By applying a frequency domain modeling approach, an approximate model in terms of a transfer function for each leg can be obtained. Figure 3.6 shows the frequency responses of an PZT actuator, including the estimated and the experimental ones. It is clearly shown that the estimated transfer function is well fitted to the real one. It is also noted that the shape of the frequency responses in Figure 3.6 agrees with that of the model in (3.3). The transfer function of the PZT actuator is obtained as:

P (s) = −190.9s4 − 1.499 × 104 s3 − 5.703 × 108 s2 − 2.507 × 1010 s − 5.624 × 1011 . s5 + 3232s4 + 5.378 × 106 s3 + 1.621 × 1010 s2 + 6.82 × 1012 s + 1.943 × 1016 (3.18)

60

Modeling and Control of Vibration in Mechanical Systems

FIGURE 3.5 Frequency responses of a PZT actuator.

Modeling of Stewart Platform

FIGURE 3.6 Estimated and experimental frequency responses.

61

62

3.4

Modeling and Control of Vibration in Mechanical Systems

Conclusion

This chapter has studied the modeling of an active piezoelectric Stewart platform. The LMS adaptive filtering approach has been adopted for the system identification. The model obtained via the adaptive identification or the experimental testing has shown consistency with the governing motion equation.

4 Classical Vibration Control

4.1

Introduction

The presence of vibration often leads to undesirable effects such as structural or mechanical failure, frequent and costly maintenance of machines, worsening positioning performance, and human pain and discomfort. Vibrations can sometimes be eliminated theoretically. However, due to the high manufacturing cost that may be involved in eliminating vibration, reduction of vibration is preferred so as to achieve a compromise between an acceptable amount of vibration and a reasonable manufacturing cost. Various classical techniques of vibration control for the purpose of vibration reduction have been presented, such as balancing of rotating and reciprocating machine, control of natural frequency, damping and stiffness modification, isolators, and absorbers. Some of the techniques will be introduced briefly in this chapter. An active vibration control is required for a system if it needs an external power to perform its function. Examples of some active vibration control include, (1) using hybrid mass dampers to apply a control force to a movable mass so as to reduce building sway caused by wind and seismic waves, (2) reducing aircraft cabin noise by attenuating the vibration of the large panels of thin metal that form the cabin walls, (3) damping out vibrations using piezoelectric devices installed on the trailing edge of helicopter blades, etc.

4.2 4.2.1

Passive control Isolators

Vibration isolation methods are used to reduce the undesired effects of vibration. It involves the insertion of a resilient member called an isolator between the vibrating mass and the source of vibration so that a reduction in the dynamic response of the system is achieved under specified conditions of vibration excitation [4]. An isolation system is said to be active or passive depending on whether or not external power is required for the isolator to perform its function. A passive isolator consists

63

64

Modeling and Control of Vibration in Mechanical Systems

of a resilient member, i.e., stiffness, and an energy dissipator, i.e., damping. Typical examples of passive isolators include metal springs, cork, pneumatic springs, and rubber springs.

4.2.2

Absorbers

If a system is acted upon by a force whose excitation frequency nearly coincides with its natural frequency, the vibration of the system can be reduced by using a dynamic vibration absorber, which is simply a spring-mass system. We consider an auxiliary mass m2 attached to a machine of mass m1 through a spring of stiffness k2 . The motion equations of the masses m1 and m2 are written as m1 x¨1 + k1 x1 + k2 (x1 − x2 ) = F0 sinωt, m2 x¨2 + k2 (x2 − x1 ) = 0,

(4.1) (4.2)

where x1 and x2 represent displacements of m1 and m2 , respectively. The steadystate amplitudes of the mass m1 and m2 are given by (k2 − m2 ω2 )F0 , (k1 + k2 − m1 ω2 )(k2 − m2 ω2 ) − k22 k2 F0 . X2 = (k1 + k2 − m1 ω2 )(k2 − m2 ω2 ) − k22 X1 =

(4.3) (4.4)

Equation (4.3) implies that if k2 , m2 the amplitude X1 of the machine m1 will be zero. Consider the machine operating near its resonance ω2 ≃ k1 /m1 before the addition of the dynamic vibration absorber. If the absorber is designed such that ω2 =

ω2 =

k2 k1 = , m2 m1

(4.5)

the vibration amplitude of the machine, while operating at its original resonant frequency, will be zero. The above mentioned dynamic vibration absorber removes the original resonance peak in the response of the machine, but introduces two new resonance peaks. If it is necessary to reduce the amplitude of vibration of the machine over a range of frequencies, a damped dynamic vibration absorber can be used [5].

4.2.3

Resonators

Resonators are used in certain cases where it is easier and more efficient to locate anti-vibration systems near the vibration source. The main idea is to create another source of vibrations which will cancel out the original vibration if it is correctly tuned. The principle is to use the kinetic energy in the resonant mass system, where

Classical Vibration Control

65

the mass is coupled to the vibrating source either by kinematic coupling or via a very flexible link [8]. For example, in a rotating structure the stator is often excited by the rotor via inertial effects such as imbalance or other effects such as the aerodynamic asymmetry of the blades or flaps. Positioning a resonator on the rotor is to generate loads that oppose the excitation loads. In the case of a helicopter, the resonant mass is placed on the axis of the rotor hub. It is supported by three springs which allow it to vibrate in a plane perpendicular to the axis of rotation. The motion of the counterweight is mainly in-plane motion so that the system will generate in-plane loads to eliminate the in-plane vibration.

4.2.4

Suspension

Suspension acts as the link between two structures that is designed in order to isolate one of the structures from the other. This is a concept for structure isolation that is achieved by canceling the variable loads transmitted to the support structure. The characteristics of the link are determined by analyzing the dynamic behavior and vibrations without modifying its main function in static characteristics. In practice, the link characteristics can be modified by varying its stiffness or appropriate positioning of the system natural frequencies and its damping in terms of energy transfer. An additional technique consists of introducing a flapping mass whose inertial effects will neutralize the excitation inputs. Next we will briefly introduce stiffness modification and damping modification. Stiffness modification The dynamic behavior of structures results from the exchange and dissipation of energy. Dynamic forces transfer their energy to the structure, which then responds via several mechanisms, such as bending or extension. Dynamic behavior can be modeled in several ways. The best known is Newton’s second law of motion. If the external force is static or quasi-static, structural stiffness forces develop to create an equilibrium. External dynamic forces are balanced in a more complex way with inertial and damping forces. Stiffness, often schematically and conceptually represented by a spring, denotes the capacity of a system to store strain energy. The stiffness force follows Hooke’s law, Fs = ks ∆x, where the stiffness constant ks is expressed in the unit of force per unit length. This is a linear model where the spring displacement is measured from the rest length. More complicated laws exist, for example, the nonlinear relation F = k(x)∆xa, where the parameter a would depend on the particular material being modeled, and the stiffness parameter k(x) is a function of how much the spring has been elongated. Damping modification Damping defines the ability of a structure to dissipate energy. For an oscillatory system, damping is a measure of how much energy is dissipated by the system during an oscillation cycle. For example, structural connections between components add damping to a structure.

66

Modeling and Control of Vibration in Mechanical Systems

Most systems possess damping to some extent, which is helpful in vibration control. If a system undergoes a forced vibration, its response or amplitude of vibration near resonance tends to become large if there is no damping. The presence of damping always limits the amplitude of vibration. Damping can be modified in the system to control its response, by the use of structural materials having high internal damping, such as laminated or sandwiched materials. An example is the use of viscoelastic materials. We know that the response amplitude of a system at resonance ω = ωn under harmonic excitation F (t) = F0 eiωt is given by F0 F0 = , kη αEη

(4.6)

where η is the energy loss factor, and the stiffness k is proportional to the Young’s modulus, i.e., k = αE with a constant α. When viscoelastic materials are used for vibration control, they are subjected to shear or direct strains. A simplest arrangement is that a layer of the viscoelastic material is attached to an elastic one. Another arrangement is that a viscoelastic layer is sandwiched between the elastic layers. The material with the largest loss factor will be subjected to the smallest stress, while the stress is proportional to the displacement. Hence viscoelastic materials having large loss factor are used to provide internal damping for vibration control. Damping tapes, consisting of thin metal foil covered with a viscoelastic adhesive, are used on vibrating structures. Another application example is presented as follows with detailed discussion and evaluation [14].

4.2.5

An application example − Disk vibration reduction via stacked disks

To support higher track per inch (TPI) density hard disk drive, the positioning accuracy of test equipment such as spin stand and servo track writer has to be increased. The Spin stand is commonly-used equipment for magnetic recording component testing [10]. A servo track writer (STW) [25] is used to pattern the recording media with servo information for the HDD servo system. In both cases, spindle motor and disk vibrations [23] affect the positioning accuracy and limit how close adjacent tracks can be placed together, and thus restrict the TPI number that can be achievable. In [11], damped laminated disks were used to reduce the amplitude of rocking modes by sandwiching a viscoelastic layer in between two aluminum layers to increase disk damping. Reference [24] investigated the effect of suppressing resonance amplitude of disk vibrations by applying a squeeze film damping. In addition to the laminated disks and the squeezed air bearing plate methods, in this section we discuss an approach with minimal mechanical alternation. That is to stack more than one normal recording media together for reading and writing on the disk surfaces. The Guzik spin stand shown in Figure 4.1 is used to evaluate the effectiveness of dual-disk stack in position accuracy improvement. The disk vibrations in the axial

Classical Vibration Control

67

direction are also measured at the outer diameter (OD) region of the disk via LDV. In the experiment, the spindle motor is spinning at 7200, 8400 and 10200 RPM respectively, and the disk is an aluminum disk of 3.5-inch in diameter and 0.8 mm in thickness. Two such disks are stacked together and mounted on the spindle motor of the spin stand. The experimental process is described as follows. Step 1: Stack two disks together and mount on the spin stand; Step 2: Spin the motor and write servo information on the top disk surface; Step 3: Read back the written information and obtain the PES. To write servo information on the disk surface in between the two disks, the spindle motor is stopped and the written surface of the disk is flipped over, following which, steps 1, 2 and 3 are repeated.

FIGURE 4.1 Disk and spindle motor assembly of the spin stand.

The disk vibration in the axial direction of the single disk and the dual-disk stack are measured via LDV. The position error signals are also collected for both the single disk and the dual-disk stack cases. To evaluate the improvement of position accuracy, 20 traces of the position error signal are collected and denoted by P ESi

68

Modeling and Control of Vibration in Mechanical Systems

(i = 1, · · · , N ). N = 20. The repeatable runout (RRO) and the nonrepeatable runout (NRRO) are given by

RRO =

N 1 X P ESi , N RROi = P ESi − RRO. N

(4.7)

i=1

A. Positioning accuracy improvement at 7200 RPM Tested disk vibration results of the dual-disk and the single disk are shown in Figure 4.2. Besides the harmonics with respect to the spindle speed, the disk vibration modes can be seen clearly as denoted by T1-T9. It is found that the frequencies of the disk vibration modes T1-T9 are shifted and their amplitudes are apparently reduced. The vibration modes are all shifted by 40 Hz to higher frequencies. The power spectra of the RRO and NRRO, computed from the collected PES data, are shown in Figure 4.3 and Figure 4.4. As seen in Figure 4.3, except for the 1st harmonic, other dominant harmonics before the 11th harmonic are all reduced. In Figure 4.4, we can observe that almost all the reduced disk vibration modes T1-T9 in Figure 4.2 are reflected in NRRO. The other peaks indicated by S1-S4 can be identified to be caused by slider vibrations. They are both lowered significantly, which may be due to better slider-disk interaction. It is noted that the peak S2 for the single disk case is shifted to S3 in the dual-disk case. This randomly happens and the writing process at a different time may cause such a shifting. This phenomenon can be seen in other testing results which will be shown later. The standard deviations (or σ values) of RRO, NRRO and PES are obtained. It turns out that the σ value of RRO is reduced by 23%, NRRO by 18%, and the PES σ value is reduced by 21%. Figure 4.5 shows the time sequences of PES in both the cases and the amplitude reduction is seen apparently. B. Positioning accuracy improvement at 8400 RPM The disk or the dual-disk stack is rotating at 8400 RPM, and the disk vibrations in the axial direction are shown in Figure 4.6. It can be seen that the amplitudes of the vibration modes are all reduced with shifts in frequencies, as indicated by T1-T6. The power spectra of the RRO and NRRO are calculated from the tested PES and shown in Figure 4.7 and Figure 4.8. Figure 4.7 shows that almost all the first seven harmonics are decreased and as a result, the σ value of RRO is reduced by 41%. In Figure 4.8, the corresponding T1-T6 in Figure 4.6 can be found and they are all suppressed in the case of the dual-disk stack. Notice that S1 is reduced significantly, but can not be traced in the tested disk vibrations in Figure 4.6. This further verifies that S1 is caused by the disk-slider interaction. The corresponding S4 cannot be found in this case, while T1 appears very close to S4. The resultant σ value of NRRO is improved by 28%, and PES by 38%. Figure 4.9 shows the comparison of single-disk and dual-disk PES in time domain, and the amplitude of PES in the case of the dual-disk stack approach is much lower as compared to conventional single disk approach.

69

Classical Vibration Control

−4.5 2−disk 1−disk −5

Dsplacement amplitude(log10(m))

−5.5

−6

−6.5

−7 T1

−7.5

T2 T3

T4 T5

T6 T7

T8 T9

−8

−8.5

−9

0

500

1000

1500

2000

2500

3000

Frequency(Hz)

FIGURE 4.2 Comparison of single-disk and dual-disk axial vibrations measured via LDV at 7200 RPM.

70

Modeling and Control of Vibration in Mechanical Systems

1−disk 2−disk

FFT(RRO)(Vrms)

0.1

0.05

0

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency(Hz)

FIGURE 4.3 Comparison of single-disk and dual-disk RRO power spectrum at 7200 RPM (23% improvement of σ value).

71

Classical Vibration Control

0.1 2−disk 1−disk

S1 0.09

0.08

FFT(NRRO)(Vrms)

0.07

0.06

0.05

S4

0.04 S3 S2

0.03

T3 T7

0.02 T1

T2

T6 T4 T9 T8

0.01

0

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency(Hz)

FIGURE 4.4 Comparison of single-disk and dual-disk NRRO power spectrum at 7200 RPM (18% reduction of σ value).

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FIGURE 4.5 Comparison of single-disk and dual-disk PES in time domain at 7200 RPM (21% reduction of σ value).

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−5 2−disk 1−disk −5.5

Dsplacement amplitude(log10(m))

−6

−6.5

−7

T1

−7.5

T2 −8

T3 T4 T5

T6

−8.5

−9

0

500

1000

1500

2000

2500

3000

Frequency(Hz)

FIGURE 4.6 Comparison of single-disk and dual-disk axial vibrations measured via LDV at 8400 RPM.

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1−disk 2−disk

FFT(RRO)(Vrms)

0.1

0.05

0

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency(Hz)

FIGURE 4.7 Comparison of single-disk and dual-disk RRO power spectrum at 8400 RPM (41% improvement of σ value).

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0.1 1−disk 2−disk

S1 0.09

0.08

FFT(NRRO)(Vrms)

0.07

0.06

0.05 T1 0.04

0.03 T3 T2

0.02

T4

S2(S3) T6

T5 0.01

0

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency(Hz)

FIGURE 4.8 Comparison of single-disk and dual-disk NRRO power spectrum at 8400 RPM (28% reduction of σ value).

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 4.9 Comparison of single-disk and dual-disk PES in time domain at 8400 RPM (38% reduction of σ value).

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C. Positioning accuracy improvement at 10200 RPM A higher rotational speed of 10200 RPM is performed on both the dual-disk stack and single disk cases. Figure 4.10 shows the obtained disk axial vibration by LDV. The modes T1-T6 are quite obvious with the amplitude reduced and the frequencies shifted higher. Figure 4.11 and Figure 4.12 show the power spectrum of RRO and NRRO. The 3rd and 5th harmonics are significantly reduced as seen in Figure 4.11. Figure 4.12 reflects the corresponding disk vibration modes T1-T6 in Figure 4.10. As for the slider related vibrations, S1 is reduced significantly again and S2 and S3 are shifted, while S4 cannot be seen at all in Figure 4.12. As a result, the σ value of RRO is improved by 33%, NRRO by 27%, and PES by 32%, compared with that of the single disk approach. Figure 4.13 shows the comparison of PES in time domain, and it is observed that the amplitude of PES in the case of the dual-disk stack is decreased.

−4.5 2−disk 1−disk −5

Dsplacement amplitude(log10(m))

−5.5

−6

−6.5

−7 T1 −7.5 T2

T4

−8 T3

T6 T5

−8.5

−9

0

500

1000

1500

2000

2500

3000

Frequency(Hz)

FIGURE 4.10 Comparison of single-disk and dual-disk axial vibrations measured via LDV at 10200 RPM.

The reduction of disk vibration amplitude is evaluated from Figures 4.2, 4.6 and 4.10 and tabulated in Table 4.1 for different rotational speeds. The improvements of

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1−disk 2−disk 0.2

FFT(RRO)(Vrms)

0.15

0.1

0.05

0

0

500

1000

1500

2000

2500

3000

3500

4000

Frequency(Hz)

FIGURE 4.11 Comparison of single-disk and dual-disk RRO power spectrum at 10200 RPM (the 3rd and 5th harmonics reduced significantly, 33% improvement of σ value).

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0.1 1−disk 2−disk

S1

0.09

0.08

FFT(NRRO)(Vrms)

0.07

0.06

0.05

0.04

0.03

T1 0.02

S2

T2 T3

T4

S3

T5 T6

0.01

0 0

500

1000

1500

2000

2500

3000

3500

4000

Frequency(Hz)

FIGURE 4.12 Comparison of single-disk and dual-disk NRRO power spectrum at 10200 RPM (27% reduction of σ value).

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 4.13 Comparison of single-disk and dual-disk PES in time domain at 10200 RPM (32% reduction of σ value).

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% reduction of σ values of PES, RRO and NRRO and disk vibration amplitude with stacked disks compared with single disk Speed(RPM) 7200 8400 10200

RRO NRRO PES 23 18 21 41 28 38 33 27 32

Disk vibration amplitude 24 45 35

the σ values of RRO, NRRO and PES are also summarized in Table 4.1. Remarkable improvement is achieved for every rotational speed. It can be seen clearly from the table that the best improvement is obtained when operating at 8400 RPM. The second best result obtained is performed at 10200 RPM, where the dual-disk stack approach leads to an improvement of 32% in positioning accuracy. For all the cases studied, we have observed a remarkably reduced amplitude of the disk vibration and its reflection in positioning accuracy improvement via the dualdisk stack. This indicates that the stacked disk is a simple way of reducing the disk vibrations in spin stand tests, particularly when a single disk surface is accessed. In Figures 4.2, 4.6, and 4.10, we have also observed the frequency shifting of disk vibration modes. Theoretically the maximum amplitude of disk vibration is given by [22]

V ∝

ρR4 ω2 , t3 Ed

(4.8)

where R is the disk outer radius, ρ is the gas density, ω is the angular speed, t is the disk thickness, E is the Young’s modulus of the disk substrate, and d is the disk substrate damping. Equation (4.8) indicates that the disk vibration is inversely proportional to t3 . In the dual-disk case, the disk stack thickness ˜t = 2t. Hence assuming that the change of E is negligible, the amplitude of disk vibration ρR4 ω2 V˜ ∝ , 8t3 E d˜

(4.9)

which means that V˜ < V when the damping d˜ > 81 d. The experimental results in Figures 4.2, 4.6 and 4.10 verified the amplitude reduction of the disk vibration. On the other hand, it is known that the natural frequency of disk vibration mode is defined by [23] fmn =

 1/2 λ2mn Et3 , 2πR2 12γ(1 − ν 2 )

(4.10)

where fmn is the natural frequency of the (m, n) mode, γ is the mass per unit area of the disk, ν is the Poisson’s ratio, and λmn is the dimensionless frequency parameter which is generally a function of the boundary conditions on the plate, the ratio of the

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inner to outer diameter, and Poisson’s ratio. Denoting Ω as the rotational speed of the disk, the frequency of the disk vibration mode is given by [23] f = fmn ± nΩ.

(4.11)

Assuming that the change of ν is also negligible, and considering that ˜t = 2t, 1/2 ˜2  √ λ Et3 f˜mn = 2 2 mn2 , 2πR 12γ(1 − ν 2 )

(4.12)

which implies some shifting of the disk vibration modes, as can be seen in the experimental results in Figures 4.2, 4.6 and 4.10. In addition to the doubled thickness, the mechanical properties such as d and λmn [22][23] are certain to change correspondingly after stacking two disks. Further research for details on how the stacked disks change these properties would be significant. To summarize, the proposed process is indeed an approach to reduce the disk vibration on both disk surfaces for reading and writing, such as multi-disk servo track writers [25]. Thus such a disk vibration reduction approach can be used as an alternative or complementary technique to the air shroud design [24] and the active control approaches to further improve positioning accuracy.

4.3

Self-adapting systems

In certain applications, system dynamic parameters vary with time. For example, the mass of a system like a car, airplane or helicopter decreases as it consumes fuel. Distinct flight conditions such as flight level, maneuvers, or landing produce different types of excitation. When the initial characteristics of the system no longer meet the requirements of the system’s working conditions, the system characteristics therefore need to adapt to the parameter variations of the excitation and the system itself. Two ways are possible: one is to change the stiffness of the structure; the other is to add moving masses so that their inertial effects counteract the effects of the excitation variation. Some self-adapting systems in use are briefly introduced as follows [8]. Self-tuning suspension Self-tuning suspension systems are equipped suitably for slow variation of the system characteristics. One method, for example, used in self-tuning systems involves analyzing the system’s vibration frequency and its timewise variation. The tuning is usually performed by an actuator. After the system status is measured, the tuning system will modify the feature of the system, such as stiffness, damping and position of a mass. The self-tuning system is tuned by a designed control algorithm. Simulations using the control algorithm are required to identify the setting parameters of the

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algorithm and to verify the reliability of the system. Subsequently, the algorithm must be validated by experimental tests on the real structure. The car suspension is an example of self-tuning suspensions. It is able to adapt to various operating configurations depending on the type of driving, the driving conditions, or the desired comfort. One method to achieve this is to modify the suspension stiffness or its damping using various techniques. Self-adjusting absorbers The mechanical resonator, which is basically a mass-spring system, is used to control certain vibration sources. It is placed where the vibrations are to be reduced. The principle is that the resonance frequency of the mass-spring system as a resonator is adjusted with the excitation frequency in operation, and as the mass vibrates, the vibratory level is decreased. In certain structures, the excitation frequency evolves with time gradually. These changes take place over a long period, compared to the excitation frequency. As a result, it is necessary to create a system whose resonance frequency would adjust automatically to the variations of the excitation frequency. For a 20 Hz excitation frequency, this variation usually takes place between 1 and 3 seconds. This kind of system actually works as the case of self-adjusting absorbers. Self-adapting resonator Self-adapting resonators are capable of adjusting to changes in the excitation frequency. The hub resonator used in a helicopter is an example. It consists of a flapping mass that vibrates in a plane perpendicular to the rotation axis of the rotor. The mobile flapping mass is supported by three flexible elements indexed at 120◦, and it slides along the rotor axis. The stiffness of the three elements is designed so that the resulting anti-resonance corresponds to the excitation frequency. The required position of the mobile mass is determined by control algorithms through an actuator and the mobile mass is moved by the actuator. The z position of the mass makes the variation of the inertia of the assembly and thus the anti-resonance frequency varies accordingly.

4.4

Active vibration control

The self-adapting systems presented previously will not be sufficient in the cases where the source characteristics vary too fast for the involved algorithms or the required level of performance is too high. Active methods should be used to decrease the vibrations. A vibration control system is called active if it uses external power to perform its function. It is comprised of a servomechanism with an actuator, a sensor, and a microprocessor-based system. The actuator applies a force to the mass whose vibration is to be reduced. The sensor measures the motion of the mass in terms of displacement, velocity, or acceleration, depending on the application. The micropro-

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cessor based system consists of analog-to-digital converters to process sensor inputs and digital-to-analog converters to convert the microprocessor’s output command into an input signal to the actuator. The control logic, called the control algorithm, programmed in the computer uses sensor measurement to decide how much force the actuator should apply.

4.4.1

Actuators

Piezoelectric materials can convert electrical current into motion, and vice versa. They change shape when an electric current passes through them, and they generate an electric signal when they flex. Thus they can be used as actuators to create force or motion, and as sensors to sense motion. The materials used for high-precision actuation include electrostrictive and magnetostrictive materials, which are similar to piezoelectric materials. These are ferromagnetic materials that expand or contract when subjected to an electric or a magnetic field. The previously mentioned voice coil motor (VCM) actuator in hard disk drives is a linear actuator moving in one direction. Because of its similarity to a loudspeaker, it is referred to as a voice coil motor. A VCM actuator has a coil of wire rigidly attached to the structure and suspended in a permanent magnetic field. It is driven when a force is produced to accelerate it radially as a current is passed through the coil. Some actuators cannot provide enough force for larger applications such as vehicle suspension and control of building motions. In building application, hydraulic cylinders are usually used. With a working hydraulic pressure of commonly 2000 psi, a cylinder containing a piston whose area is only 1 square inch will generate 1 ton of force. Active vehicle suspensions use hydraulic devices, electric motors, and magneto-rheological fluid dampers.

4.4.2

Active systems

An active system is applied to decrease vibrations by introducing dynamic loads locally in the structure [8]. The dynamic loads are controlled by a processor in order to minimize the vibratory level. The technology of generating loads is fundamental in control strategy. Active suspension Several active principles are used for suspensions to isolate one structure from another. One of them is internal load control to modify the distribution of internal loads in the structure. Its principle is to inject a set of dynamic loads into the structure in order to minimize its vibratory response. The loads depend on the vibratory condition of the structure, which is sensed with the help of a set of accelerometers or strain gauges. Hydraulic or electro-dynamic actuators suitably located on the structure are used to inject the loads. The role of the actuators is to modify the distribution of vibratory energy for different modes to minimize the structure vibrations, instead of dispersing the vibratory energy of the structure.

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Active resonator An active resonator works by displacing a mass with an actuator. It uses the dynamic amplification of the mass to generate high loads using minimal energy. Different types of actuator technologies such as hydraulic, electromagnetic, or piezoelectric actuators, have been used to match the fields of application in terms of forces and frequencies. We take an electromagnetic actuator as an example. The type of single stage resonator uses the principle of single stage mechanical systems with displacement operated by an electromagnetic force. The motion equation of the mass M of a permanent magnet for small motion around its static position is given by

M x¨1 + c1 x˙ 1 + k1 x1 = FV (t).

(4.13)

With FV (t) = F0 cos(ωt), the Laplace transform of the displacement produced by the mass is

X1 (s) =

M s2

1 F0 , + c1 s + k 1

(4.14)

where k1 and c1 are respectively the stiffness and damping of the mass. The load transmitted to the structure equals to FT (t) = c1 x˙ 1 − k1 x1 − FV (t).

(4.15)

The mechanical parameters such as mass and stiffness are designed so that the magnetic loads remain small and compatible with an acceptable energy consumption. The natural frequency of the mass-spring system must be designed as close as possible to the frequency of the vibration that needs to be controlled. A two-stage electromagnetic resonator uses two masses called stages [8]. The control force FV is introduced between the two masses by an electromagnetic load. It is produced from a voltage V that is fed to a power amplifier to generate a magnetic force via the current in the coil. The role of the amplifier is to ensure the law between the generated force FV and the control voltage V is linear. The function of control consists of tuning the system parameters such as mass and stiffness so that at the control frequency, the force generated on the structure is amplified with respect to the force FV , while maintaining a small relative displacement of the two stages. The motion equations of the two masses are written as M1 x¨1 + c1 (x˙ 1 − x˙ 2 ) + k1 (x1 − x2 ) = FV (t), (4.16) M2 x¨2 + c1 (x˙ 2 − x˙ 1 ) + c2 x˙ 2 + k1 (x2 − x1 ) + k2 x2 = −FV (t), (4.17) with the primary stage M2 and the secondary stage M1 . The load transmitted to the structure is FT = c2 x˙ 2 + k2 x2 .

(4.18)

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4.4.3

Modeling and Control of Vibration in Mechanical Systems

Control strategy

When the system response is not acceptable, the value of the response is used to generate additional forces according to rules or laws such that the modified response behaves according to the design and within certain bounds. This results in a closedloop system that incorporates feedback control, where the response is evaluated by a sensor and is fed back to an actuator that generates a force or motion. The purpose of designing a system with feedback force is to minimize unwanted behaviors and eliminate the effect of vibration on system performance in a desired manner. Feedback control provides a mechanism for tailoring system behavior to specific standards and needs. The block diagram depicting feedback control is in Figure 4.14. One can see that the signal from a block goes concurrently to other blocks or a summing point. The summing point indicates that the input and output signals are compared. The difference is an error that generates a control action. The disturbance or forcing is also shown being input directly to the plant or being injected to the output of the plant.

FIGURE 4.14 Generic feedback control system.

In order to properly design a feedback control system, performance must be defined in terms of system specifications. Standard performance measures are usually defined in terms of the step response. The general design objectives are the speed of response, stability, and accuracy or allowable error. The first objective implies that it is desirable for a control system to respond to a reference command rapidly. Stability is the prerequisite of any active control system design. The allowable error represents how close to the desired response the control force must bring the structure and how much the control system can reject vibrations. The controller produces control action or signal by using the comparison signal of plant output with the desired reference value. There are a number of control

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methods used by people to design control actions. Classical control is represented by the well-known proportional-integral-derivative (PID) control. Advanced control techniques include robust and optimal controls, which will be introduced briefly in the next chapter. An adaptive control system is one that can change or adapt either its gain values or even its control algorithm to accommodate changing conditions. In mathematics how the control gains or the control algorithm should adapt to changing conditions is difficult. There are thus fewer practical applications of adaptive controls.

4.5

Conclusion

Various classical techniques of vibration control were discussed in this chapter and some typical techniques such as using isolators, absorbers, a resonator, and suspension were introduced individually. Specifically, disk vibration reduction via stacked disks as an application example was presented with detailed discussion and evaluation. Subsequently, active vibration control was introduced with actuators, active systems, and control strategies.

5 Introduction to Optimal and Robust Control

5.1

Introduction

This chapter intends to briefly review the optimal and robust control. H2 and H∞ norms and their calculation are first introduced since they are two commonly used measures of performance specifications of a control system. What follow are the H2 and H∞ control problems and controller design via a linear matrix inequality approach. Robust control of systems with multiplicative uncertainty and additive uncertainty, and the parametrization of all stable controllers, i.e., the so-called Youla parametrization, will be investigated. Moreover, performance limitation of a feedback control system is discussed, which is necessary to help understand the vibration control schemes in the later chapters.

5.2 5.2.1

H2 and H∞ norms H2 norm

The H2 norm of a matrix transfer function G(s) analytic in Re(s) > 0 (open righthalf plane) is defined as s Z +∞ 1 T race[G∗ (σ + jω)G(σ + jω)]dω}, (5.1) kGk2 := sup{ σ>0 2π −∞ or equivalently [2] kGk2 =

s

1 2π

Z

+∞

T race[G∗ (jω)G(jω)]dω.

(5.2)

−∞

Although kGk2 can be computed from its definition, there are some simple alternatives taking advantage of a state-space representation of G(s). LEMMA 5.1 [6] Consider a system G(s) with a state-space representation (A, B, C, D). If

89

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A is stable and D = 0, then we have kGk22 = T race(B T Y2 B) = T race(CX2 C T )

(5.3)

where X2 and Y2 are the controllability and observability Gramians that can be obtained from the following Lyapunov equations: AX2 + X2 AT + BB T = 0, T

T

A Y2 + Y2 A + C C = 0.

(5.4) (5.5)

We also consider a discrete-time linear time-invariant system G(z) with the following state-space representation: x(k + 1) = Ax(k) + Bw(k), z(k) = Cx(k) + Dw(k),

(5.6) (5.7)

where x ∈ Rnx is the state, z ∈ Rnz is the controlled output, w ∈ Rnw is the disturbance input. Let Tzw denote the transfer function from the input w to the output z. Then the H2 norm is defined as s Z π 1 ∗ (ejω )T kTzw k2 = T race[ Tzw (5.8) zw (ejω )dω]. 2π −π By Parseval’s theorem, kTzw k2 can equivalently be defined as v u ∞ X u kTzw k2 = tT race[ g(k)gT (k)],

(5.9)

k=0

where g(k) is the impulse response of Tzw . Let the input w to the system be a wide-band stationary stochastic process. The H2 norm of Tzw can also be interpreted as the RMS value of the output z(k) when the system is subject to a white noise having zero mean and unit variance. That is q (5.10) kTzw k2 = E[zT z]. The H2 norm for the discrete-time system Tzw can be computed as q q kTzw k2 = T race(DT D + B T Y2 B) = T race(DDT + CX2 C T ), (5.11)

where Y2 and X2 are the reachability and observability Gramians that can be obtained from the following Lyapunov equations: AY2 AT − Y2 + BB T = 0, AT X2 A − X2 + C T C = 0.

(5.12) (5.13)

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The following theorem presents an alternative LMI condition for bounding the H2 norm of the discrete-time system Tzw . LEMMA 5.2 Consider a discrete-time transfer function Tzw of realization (A, B, C, D). Given a scalar µ > 0, kTzw k22 < µ if and only if there exist X2 = X2T and Y2 = Y2T such that T race(Π) < µ and   Π CX2 D  X2 C T X2 0  > 0, (5.14) DT 0 I   Y2 AY2 B  Y2 AT Y2 0  > 0. (5.15) BT 0 I Observe that (5.14) and (5.15) are linear in X2 and Y2 , and hence can be solved by employing the LMI Tool [69] in MATLAB. The H2 norm of the system can be computed by minimizing µ using the function mincx in MATLAB Optimization Toolbox.

5.2.2

H∞ norm

The H∞ norm of a matrix transfer function G(s) that is analytic and bounded in the open right-half plane is defined as [3] kGk∞ :=

kzk2 sup σ ¯ [G(s)] = sup σ ¯ [G(jω)] = sup , kwk ω∈R w= 6 0 2 Re(s)>0

(5.16)

where w and z are respectively the input and output of G(s). A control engineering interpretation of the infinity norm of a scalar transfer function G(s) is the distance in the complex plane from the origin to the farthest point on the Nyquist plot of G, and it also appears as the peak value on the magnitude plot of |G(jω)|. Hence the H∞ norm of a transfer function can, in principle, be obtained graphically. In general, the H∞ norm of a stable matrix transfer function can be read directly from its singular value plots. The H∞ norm can also be computed in state-space. LEMMA 5.3 [6] Let γ > 0 and G(s) : (A, B, C, D) with A stable. Then kG(s)k∞ < γ if and only if σ ¯ (D) < γ and the Hamiltonian matrix H has no eigenvalues on the imaginary axis, where H :=



A + BR−1 DT C −C T (I + DR−1 DT )C

 BR−1 B T , − (A + BR−1 DT C)T

(5.17)

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and R = γ 2 I − DT D. The following so-called Bounded Real Lemma gives a matrix inequality condition for the system G(s) to have a pre-specified level of H∞ norm. LEMMA 5.4 Continuous-time Bounded Real Lemma Consider a continuous-time transfer function G(s) of realization G(s) = C(sI − A)−1 B + D, where A is a stable matrix. Given a scalar λ > 0, the H∞ norm kGk∞ < λ if and only if there exists X = X T > 0 such that  T  A X + XA XB CT  BT X − λI DT  < 0. (5.18) C D − λI In the discrete-time case, the H∞ norm of a stable transfer function matrix Tzw (z) with realization (A, B, C, D) is defined as kTzw k∞ =

kzk2 sup σ¯ [Tzw (ejω )] = sup , kwk 2 w= 6 0 ω∈[0,2π]

(5.19)

where w and z are respectively the input and output of Tzw . The following discrete-time Bounded Real Lemma provides a linear matrix inequality condition for the system Tzw to have kTzw k∞ < γ. LEMMA 5.5 Discrete-time Bounded Real Lemma Consider a discrete-time and a stable transfer function matrix Tzw (z) with a state-space realization (A, B, C, D). Then, for some given λ > 0, kTzw k∞ < λ if and only if there exists X = X T > 0 such that  T  A XA − X AT XB CT  B T XA B T XB − λ2 I DT  < 0. (5.20) C D −I

5.3 5.3.1

H2 optimal control Continuous-time case

We consider the closed-loop system described by the block diagram in Figure 5.1. The continuous-time linear time-invariant plant P (s) is described by the state-space equations: x(t) ˙ = Ax(t) + B1 w(t) + B2 u(t),

(5.21)

Introduction to Optimal and Robust Control z(t) = C1 x(t) + D11 w(t) + D12 u(t), y(t) = C2 x(t) + D21 w(t) + D22 u(t),

93 (5.22) (5.23)

where x ∈ Rnx is the state, y ∈ Rny is the measurement output, z ∈ Rnz is the controlled output, w ∈ Rnw is the disturbance input, u ∈ Rnu is the control input, and A, B1 , B2 , C1 , D11 , D12 , C2 , and D21 are of appropriate dimensions. We assume D22 = 0 without loss of generality [89]. Introduce the following dynamic output feedback controller C(s): x˙ c(t) = Ac xc(t) + Bc y(t),

(5.24)

u(t) = Ccxc (t) + Dc y(t).

(5.25)

Denote ξ = [xT xTc ]T . From (5.21)−(5.23) and (5.24)−(5.25), the closed-loop system is given by ˙ = ξ(t) z(t) =

¯ ¯ Aξ(t) + Bw(t), ¯ ¯ Cξ(t) + Dw(t),

(5.26) (5.27)

where     A + B2 Dc C2 B2 Cc B2 Dc D21 + B1 ¯ ¯ A= , B= , Bc C2 Ac Bc D21 ¯ = D12 Dc D21 + D11 . C¯ = [C1 + D12 Dc C2 D12 Cc ] , D

(5.28) (5.29)

The continuous-time H2 control problem is to find a proper, real rational controller C(s) that stabilizes P internally and minimizes the H2 norm of the closed-loop transfer function matrix Tzw from w to z in (5.26)−(5.27). Assume that the system (5.21)−(5.23) satisfies the following conditions: Assumption 5.1 (1). D12 is of full column rank; (2). The subsystem (A, B2 , C1 , D12 ) has no invariant zeros on the imaginary axis; (3). D21 is of full row rank; (4). The subsystem (A, B1 , C2 , D21 ) has no invariant zeros on the imaginary axis. Let X2 ≥ 0 and Y2 ≥ 0 be the solutions of the following Riccati equations: T AT X2 + X2 A − (X2 B2 + C1T D12 )(D12 D12 )−1 (X2 B2 + C1T D12 )T T +C1 C1 = 0, (5.30) T T T T −1 T T T Y2 A + AY2 − (Y2 C2 + B1 D21 )(D21 D21 ) (Y2 C2 + B1 D21 )

+B1 B1T = 0.

(5.31)

According to the H2 optimal control theory, an H2 optimal controller can be obtained as [7] Ac = A + B2 F + KC2 , Bc = −K, Cc = F, Dc = 0,

(5.32)

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FIGURE 5.1 Configuration of standard optimal control. where T T F = −(D12 D12 )−1 (D12 C1 + B2T X2 ), T T −1 K = −(Y2 C2T + B1 D21 )(D21 D21 ) .

(5.33)

The minimal H2 norm of the transfer function Tzw is given by q kTzw k2 = T race(B1T X2 B1 ) + T race[(AT X2 + X2 A + C1T C1 )Y2 ].(5.34)

If the conditions (1)−(4) in Assumption 5.1 are not satisfied, the so-called perturbation method is applied [7] so that the above design method to find an appropriate controller is still applicable.

5.3.2

Discrete-time case

Consider the discrete-time linear time-invariant system P (z) with the following statespace representation: x(k + 1) = Ax(k) + B1 w(k) + B2 u(k), z(k) = C1 x(k) + D11 w(k) + D12 u(k), y(k) = C2 x(k) + D21 w(k) + D22 u(k),

(5.35) (5.36) (5.37)

where x ∈ Rnx is the state, y ∈ Rny is the measurement output, z ∈ Rnz is the controlled output, w ∈ Rnw is the disturbance input, u ∈ Rnu is the control input, and A, B1 , B2 , C1 , D11 , D12 , C2 , and D21 are of appropriate dimensions. D22 = 0 is also assumed.

Introduction to Optimal and Robust Control

95

Introduce the following dynamic output feedback controller C(z): xc (k + 1) = Ac xc(k) + Bc y(k), u(k) = Ccxc (k) + Dc y(k).

(5.38) (5.39)

Denote ξ = [xT xTc ]T . From (5.35)−(5.37) and (5.38)−(5.39), the closed-loop system is given by ¯ ¯ ξ(k + 1) = Aξ(k) + Bw(k), ¯ ¯ z(k) = Cξ(k) + Dw(k),

(5.40) (5.41)

where     A + B2 Dc C2 B2 Cc B2 Dc D21 + B1 ¯ ¯ A= , B= , Bc C2 Ac Bc D21 ¯ = D12 Dc D21 + D11 . C¯ = [C1 + D12 Dc C2 D12 Cc ] , D

(5.42) (5.43)

The discrete-time H2 control problem is to find a proper, real rational controller C(z) that stabilizes P (z) internally and minimizes the H2 norm of the transfer function matrix Tzw (z) from w to z of the closed-loop system (5.40)−(5.41). The counterpart of the Riccati equations (5.30)−(5.31) for discrete-time systems is as follows. T AT X2 A − (AT X2 B2 + C1T D12 )(D12 D12 + B2T X2 B2 )−1 (AT X2 B2 + C1T D12 )T T +C1 C1 = 0, (5.44) T T T T AY2 AT − (AY2 C2T + B1 D21 )(D21 D21 + C2 Y2 C2T )−1 (AY2 C2T + B1 D21 ) T +B1 B1 = 0. (5.45)

A discrete-time H2 optimal controller can then be obtained as (5.32). And the minimal H2 norm of the transfer function Tzw is given by q kTzw k2 = T race(B1T X2 B1 ) + T race[(AT X2 A + C1T C1 )Y2 ]. (5.46)

A parametrization of all H2 controllers is developed in terms of LMIs as in the following theorem which linearizes the H2 norm conditions (5.14)−(5.15) for synthesis. THEOREM 5.1 [90] Consider system (5.35)−(5.37). There exists a controller (5.38)−(5.39) such that kTzw k22 < µ if and only if the following linear matrix inequalities and equality admit a solution: T race(Π) < µ,

(5.47)

96 

and

Π ∗ ∗  P2  ∗   ∗   ∗ ∗

Modeling and Control of Vibration in Mechanical Systems  C1 X + D12 E C1 + D12 Dc C2 X + X T − P2 I + Z T − J  > 0, (5.48) ∗ Y +YT −H  J AX + B2 E A + B2 Dc C2 B1 + B2 Dc D21 H U Y A + W C2 Y B1 + W D21   ′ ′  > 0, ∗ X + X − P2 I +Z −J 0 (5.49)  T  ∗ ∗ Y +Y −H 0 ∗ ∗ ∗ I D11 + D12 Dc D21 = 0,

(5.50)

where ∗ denotes an entry that can be deduced from the symmetry of the matrix, the matrices X, E, Y , W , U , Dc , Z, J, and the symmetric matrices P2 , H and Π are the variables. A feasible H2 controller is obtained by choosing N1 and M1 nonsingular such that N1 M1 = Z − Y X and calculating Cc = (E − Dc C2 X)M1−1 , Bc = N1−1 (W − Y B2 Dc ),

Dc = Dc ,

(5.51) (5.52)

Ac = N1−1 [U − Y (A + B2 Dc C2 )X − N1 Bc C2 X − Y B2 Cc M1 ]M1−1 . (5.53)

5.4 5.4.1

H∞ control Continuous-time case

We consider the continuous-time system P described by (5.21)−(5.23), and the class of causal, linear, time-invariant and finite-dimensional controllers that internally stabilize P , or namely, all admissible controllers for P . Our aim is to find an admissible controller C such that the closed-loop system Tzw satisfies kTzw k∞ < γ.

(5.54)

Assumption 5.2 (1). The pair (A, B2 ) is stabilizable and the pair (A, C2 ) is detectable. T T (2). The matrices D12 and D21 satisfy D12 D12 = Inu and D21 D21 = Iny . (3).   A − jωI B2 rank = nx + nu , for all real ω. (5.55) C1 D12 (4). rank



 A − jωI B1 = nx + ny , for all real ω. C2 D21

(5.56)

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The assumption that (A, B2 , C2 ) is stabilizable and detectable is necessary and sufficient for the existence of admissible controllers. The full rank assumptions (3) and (4) are necessary for the existence of stabilizing solutions to the Riccati equations that are used to obtain the solution to the H∞ control problem. ˆ B, ˆ A˜ and C˜ as Define the matrices A, T T ˆB ˆ T = B1 (I − D21 Aˆ = A − B1 D21 C2 , B D21 )B1T , T T A˜ = A − B2 D12 C1 , C˜ T C˜ = C1T (I − D12 D12 )C1 .

(5.57) (5.58)

Suppose that Uy maps y to u and has a minimal realization (Au , Bu , Cu , Du ) satisfying det(I − D22 Du ) 6= 0 for a well-posed closed loop. Then a realization in terms of LFT is given by LF T (P, Uy ) =   A + B2 Du M C2 B2 (I + Du M D22 )Cu B1 + B2 Du M D21  , (5.59)  Bu M C2 Au + Bu M D22 Cu Bu M D21 C1 + D12 Du M C2 D12 (I + Du M D22 )Cu D11 + D12 Du M D21

with M = (I − D22 Du )−1 .

THEOREM 5.2 [71] Consider the system (5.21)−(5.23) satisfying Assumption 5.2. There exists an admissible C such that the closed-loop system (5.26)−(5.27) satisfies (5.54) if and only if 1. There is a solution X∞ ≥ 0 to the algebraic Riccati equation X∞ A˜ + A˜T X∞ − X∞ (B2 B2T − γ −2 B1 B1T )X∞ + C˜ T C˜ = 0,

(5.60)

such that A˜ − (B2 B2T − γ −2 B1 B1T )X∞ is asymptotically stable. 2. There is a solution Y∞ ≥ 0 to the algebraic Riccati equation ˆ ∞ + Y∞ AˆT − Y∞ (C2T C2 − γ −2 C1T C1 )Y∞ + B ˆB ˆ T = 0, AY

(5.61)

such that Aˆ − Y∞ (C2T C2 − γ −2 C1T C1 ) is asymptotically stable. 3. ρ(X∞ Y∞ ) < γ 2 . In the case when these conditions hold, C is an admissible controller satisfying (5.54) if and only if C is given by the LFT C = LF T (Ca , Uy ),

kUy k∞ < γ,

where Uy is a stable transfer function. The generator Ca is given by   Ak Bk1 Bk2 Ca =  Ck1 0 I  , Ck2 I 0

(5.62)

(5.63)

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where Ak = A + γ −2 B1 B1T X∞ − B2 F∞ − Bk1 C2z ,     T T T Bk1 Bk2 = B1 D21 + Z∞ C2z B2 + γ −2 Z∞ F∞ ,     Ck1 −F∞ = , Ck2 −C2z

T C2z = C2 + γ −2 D21 B1T X∞ , F∞ = D12 C1 + B2T X∞ , Z∞ = Y∞ (I − γ −2 X∞ Y∞ )−1 = (I − γ −2 Y∞ X∞ )−1 Y∞ .

(5.64) (5.65) (5.66) (5.67) (5.68)

A point given by Theorem 5.2 is that a solution to the H∞ generalized regulator problem exists if and only if there exist stabilizing, nonnegative definite solutions X∞ and Y∞ to the algebraic Riccati equations associated with the full information H∞ control problem and the H∞ estimation of C1 x such that the coupling condition ρ(X∞ Y∞ ) < γ 2 is satisfied. The optimal H∞ control problem is to find an internally stabilizing controller C(s) such that kTzw k∞ of the closed-loop system (5.26)−(5.27) is minimized. However, in practice it is often not necessary to design an optimal controller, and it is usually appropriate to obtain a controller that gives rise to an H∞ norm of the closed-loop system less than a prescribed value. More specifically, a suboptimal H∞ control problem is that given γ > 0, find an admissible controller C, if there is any, such that kTzw k∞ < γ. The following theorem gives a design method for a suboptimal H∞ output feedback controller. THEOREM 5.3 [90] Consider system (5.21)−(5.23). Given a scalar γ > 0, there exists an output feedback controller (5.24)−(5.25) such that kTzw k2∞ < γ if the following LMI admits a solution (E, W, U, Dc, X, Y ):    

AX + XAT + B2 E + (B2 E)T U T + A + B2 Dc C2 T ∗ A Y + Y A + W C2 + (W C2 )T , ∗ ∗ ∗ ∗  B1 + B2 W Dc D21 (C1 X + D12 E)T Y B1 + W D21 (C1 + D12 Dc C2 )T   < 0, (5.69) −γI (D11 + D12 Dc D21 )T  ∗ −γI   X I > 0. (5.70) I Y

In this case, a feasible H∞ controller is obtained from (5.51)−(5.53), where N1 M1 = I − Y X.

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5.4.2

99

Discrete-time case

Assume that the time-invariant discrete-time system (5.35)−(5.37) satisfies: Assumption 5.3 (1). (A, B2 , C2 ) is stabilizable and detectable. T T (2). D12 D12 > 0 and D21 D21 > 0. (3).   A − ejθ I B2 rank = nx + nu , for all θ ∈ (−π, π]. (5.71) C1 D12 (4). rank



 A − ejθ I B1 = nx + ny , for all θ ∈ (−π, π]. C2 D21

(5.72)

We seek a causal, linear, time-invariant and finite-dimensional controller C(z) such that the closed-loop system (5.40)−(5.41) is stable and kTzw k∞ < γ,

(5.73)

or equivalently, under zero initial conditions, kzk22 − γ 2 kwk22 ≤ −εkwk22 , for all w ∈ ℓ2 [0, ∞) and some ε > 0. Let     C1 B = B1 B2 , Cd = , 0   D11 D12 Dd = , Inw 0   Inz 0 Js = , 0 −γ 2 Inw   I 0 Jt = nw . 0 −γ 2 Inu

(5.74)

(5.75) (5.76) (5.77) (5.78)

With the assumptions (1)−(4), a causal, linear, finite-dimensional stabilizing controller that leads to kTzw k∞ < γ exists if and only if the following two conditions hold [71]. 1. There exists a solution to the Riccati equation X∞ = CdT Js Cd + AT X∞ A − W T R−1 W,

(5.79)

with  R1 RT2 = DdT Js Dd + B T X∞ B, R2 R3   W11 W = = DdT Js Cd + B T X∞ A, W21 ∈ Rnu ×nx , W21 R=



(5.80) (5.81)

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such that A − BR−1 W is asymptotically stable and X∞ ≥ 0,

(5.82)

R1 − RT2 R−1 3 R2

nw ×nw

< 0, R1 ∈ R

nu ×nu

, R3 ∈ R

.

(5.83)

Denote T −1 ∇ = R1 − RT2 R−1 3 R2 , W∇ = W11 − R2 R3 W21 ,

(5.84)

and let E1 be an nu × nu matrix such that E1T E1 = R3 ,

(5.85)

and E2 be an nw × nw matrix such that −2 E2T E2 = −γ −2 (R1 − RT2 R−1 ∇. 3 R2 ) = −γ

(5.86)

Define the system 

At Bt Ct Dt



A − B1 ∇−1 W∇ B1 E2−1 −1 −1 −1  = E1 R3 (W − R2 ∇ W∇ ) E1 R−1 3 R 2 E2 −1 −1 C2 − D21 ∇ W∇ D21 E2 

2. There exists a solution to the Riccati equation

 0 I. 0

Y∞ = Bt Jt BtT + At Y∞ ATt − Mt St−1 MtT ,

(5.87)

(5.88)

where  S1 S2 St = = , S2T S3   Mt = Bt Jt DtT + At Y∞ CtT = Mt1 Mt2 , Dt Jt DtT

+ Ct Y∞ CtT



(5.89) (5.90)

such that At − Mt St−1 Ct is asymptotically stable and Y∞ ≥ 0, S1 − S2 S3−1 S2T < 0.

Note that Ct in (5.87) is partitioned as     Ct1 E1 R−1 (W − R2 ∇−1 W∇ ) 3 Ct = = . Ct2 C2 − D21 ∇−1 W∇

(5.91) (5.92)

(5.93)

A controller that achieves the objective (5.73) is given by xc (k + 1) = At xc (k) + B2 u(k) + Mt2 S3−1 (y(k) − Ct2xc (k)), E1 u(k) = −Ct1 xc(k)1 − S2 S3−1 (y(k) − Ct2 xc (k)).

(5.94) (5.95)

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101

All controllers that achieve the objective (5.73) are generated by the LFT C = LF T (Ca , Uc), where Uc is a linear causal system such that kUc k∞ < γ , and the generator Ca is given by    I − B2 − Mt2 (X2T )−1 xc (k + 1)  0 E1 S2 (X2T )−1   u( k)  0 0 X2 η(k)    −1 xc (k) At 0 (Mt1 − Mt2 S3 S2 )(γ 2 X1T )−1   y(k)  , =  −Ct1 0 X1 (5.96) φ(k) −Ct2 I 0 with X2 X2T = S3 , X1 X1T = −γ −2 (S1 − S2 S3−1 S2T ).

(5.97) (5.98)

The following theorem gives one parametrization approach of all suboptimal discretetime H∞ output feedback controllers. THEOREM 5.4 [90] Consider system (5.35)−(5.37). Given a scalar γ > 0, there exists an output feedback controller (5.38)−(5.39) such that kTzw k2∞ < γ if the following LMI admits a solution:   P∞ J AX + B2 E A + B2 Dc C2 B1 + B2 Dc D21 0  ∗ H  U Y A + V C2 Y B1 + V D21 0   ′ T T T T   ∗ ∗ X + X ′ − P∞ I+Z −J 0 X C1 + E D12   T   ∗ ∗ ∗ Y +YT −H 0 C1T + C2T DcT D12   T T T T   ∗ ∗ ∗ ∗ I D11 + D21 Dc D12 ∗ ∗ ∗ ∗ ∗ γI > 0, (5.99) where the matrices X, E, Y , V , U , Dc , Z, J, and the symmetric matrices P∞ and H are the variables. A feasible H∞ controller is obtained from (5.51)(5.53).

5.5

Robust control

The H∞ norm is used to test robust stability of a nominally stable system under unstructured perturbations. The following so-called small gain theorem is the basis for robust stability analysis.

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THEOREM 5.5 Small Gain Theorem Consider a proper and stable transfer function matrix T (s). Suppose that a stable ∆(s) is connected from the output of T (s) to the input of T (s) as shown in Figure 5.2. Then the closed-loop system given in Figure 5.2 is internally stable if σ ¯ [∆(jω)]¯ σ [T (jω)] < 1, ∀ω ∈ R

[

∞.

(5.100)

FIGURE 5.2 A closed-loop system with uncertainty. The small gain condition is sufficient to guarantee internal stability of the closedloop system even if ∆ is nonlinear and time-varying. The small gain theorem tells us that an H∞ norm bound on T implies closed-loop stability in the presence of certain H∞ norm bounded system uncertainties. The H∞ norm bound implies a certain stability robustness. Note from (5.100) that the size of tolerable uncertainty varies inversely proportional to the H∞ norm bound of T , which means that the robustness increases as the H∞ norm bound decreases. In what follows, the small gain theorem will be used to test robust stability under model uncertainties. The modeling error ∆ is assumed to be stable and suitably scaled with weighting functions W1 and W2 , i.e., the uncertainty can be represented as W1 ∆W2 . Additive uncertainty We assume that the model uncertainty can be represented by an additive perturbation: Π = P + W1 ∆W2 ,

(5.101)

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which is shown in Figure 5.3.

FIGURE 5.3 A closed-loop system with additive uncertainty for robust stability analysis.

THEOREM 5.6 [70] Let Π = P + W1 ∆W2 , and C be a stabilizing controller for the nominal plant P . Then the closed-loop system is well-posed and internally stable for all k∆k∞ < 1 if and only if kW2 CSW1 k∞ ≤ 1, where S=

1 . 1 + PC

(5.102)

Similarly, the closed-loop system is stable for all stable ∆ with k∆k∞ ≤ 1 if and only if kW2 CSW1 k∞ < 1. Multiplicative uncertainty The system model is described by the following multiplicative perturbation: Π = (I + W1 ∆W2 )P

(5.103)

where W1 , W2 and ∆ are stable. Consider the feedback system shown in Figure 5.4. THEOREM 5.7 [70] Let Π = (I + W1 ∆W2 )P , C be a stabilizing controller for the nominal plant P , and T = 1 − S. Then (i) the closed-loop system is well-posed and internally stable for all stable ∆ with k∆k∞ < 1 if and only if kW2 T W1 k∞ ≤ 1. (ii) the closed-loop system is well-posed and internally stable for all stable ∆ with k∆k∞ ≤ 1 if kW2 T W1 k∞ < 1. (iii) the robust stability of the closed-loop system for all stable ∆ with k∆k∞ ≤ 1 does not necessarily imply kW2 T W1 k∞ < 1.

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FIGURE 5.4 A closed-loop system with multiplicative uncertainty for robust stability analysis. (iv) the closed-loop system is well-posed and internally stable for all stable ∆ with k∆k∞ ≤ 1 only if kW2 T W1 k∞ ≤ 1. (v) In addition, assume that neither P nor C has poles on the imaginary axis. Then the closed-loop system is well-posed and internally stable for all stable ∆ with k∆k∞ ≤ 1 if and only if kW2 T W1 k∞ < 1.

5.6

Controller parametrization

Consider the standard system block diagram in Figure 5.1 with 

 Ap B1 B2 P (s) =  C1 D11 D12  . C2 D21 D22

(5.104)

Suppose (Ap , B2 ) is stabilizable, (C2 , Ap ) is detectable, and D22 = 0. The problem discussed here is that given a plant P , parameterize all controllers C that internally stabilize P . The parametrization for all stabilizing controllers is known as Youla parametrization [68], as shown in Figure 5.5. The Youla parametrization starts with a nominal controller that is an estimated-state feedback. The estimated state feedback controller is given by u = −K xˆ,

(5.105)

where the state feedback gain K is some appropriate matrix and x ˆ is an estimate of the component of x, governed by the observer equation xˆ˙ = Ap xˆ + B2 u + L(y − C2 xˆ),

(5.106)

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105

where L is the estimator gain. The transfer function of the estimated-state feedback controller is thus Kn (s) = −K(sI − Ap + B2 K + LC2 )−1 L.

(5.107)

The nominal controller Kn (s) will stabilize P provided K and L are chosen such that Ap − B2 K and Ap − LC2 are stable. To augment the estimated state feedback controller, we inject v into u as shown in Figure 5.5, meaning that (5.105) is replaced by u = −K xˆ + v,

(5.108)

and therefore the signal v does not induce any observer error. For the signal e we take the output prediction error: e = y − C2 xˆ.

(5.109)

In Figure 5.5, the observer based controller applies output prediction error processed through a stable transfer function Q and added to the output of Kn . This augmentation is able to yield every controller that stabilizes the plant, which means every stabilizing controller can be realized as an observer based controller with some stable transfer function Q. Thus in the sequel we can form simple state-space equations for the parametrization of all controllers that stabilize the plant, and all closed-loop transfer matrices achieved by controllers that stabilize the plant. The state-space equations of the augmented controller are given by xˆ˙ = (Ap − B2 K − LC2 )ˆ x + Ly + B2 v,

u = −K xˆ + v, e = y − C2 x ˆ.

(5.110) (5.111) (5.112)

If Q has a state-space realization x˙ q = Aq xq + Bq e, v = Cq xq + Dq e,

(5.113) (5.114)

then a state-space realization of the observer based controller by eliminating e and v from the augmented controller equations (5.110)−(5.112) and the Q realization (5.113)−(5.114) is obtained as: xˆ˙ = (Ap − B2 K − LC2 − B2 Dq C2 )ˆ x + B2 Cq xq + (L + B2 Dq )y, x˙ q = −Bq C2 xˆ + Aq xq + Bq y, u = −(K + Dq C2 )ˆ x + Cq xq + Dq y,

(5.115) (5.116) (5.117)

or equivalently, C(s) = Cc (sI − Ac )−1 Bc + Dc ,

(5.118)

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where 

Ap − B2 K − LC2 − B2 Dq C2 Ac = −Bq C2   L + B2 Dq Bc = , Bq   Cc = −K − Dq C2 Cq ,

 B2 Cq , Aq

Dc = Dq .

(5.119) (5.120) (5.121) (5.122)

On the other hand, the state-space equations for the closed-loop system with only the augmented controller (5.110)−(5.112) are found as follows by eliminating u and y from (5.110)−(5.112) and the plant equations in (5.104). x˙ = Ap x − B2 K xˆ + B1 w + B2 v, xˆ˙ = LC2 x + (Ap − B2 K − LC2 )ˆ x + LD1 w + B2 v, z = C1 x − D12 K xˆ + D11 w + D12 v, e = C2 x − C2 xˆ + D21 w,

(5.123) (5.124) (5.125) (5.126)

which are equivalently written as 

 T11 (s) T12 (s) = CT (sI − AT )−1 BT + DT , T21 (s) 0

(5.127)

where

AT BT CT DT



 Ap −B2 L = , LC2 Ap − B2 K − LC2   B1 B2 = , LD1 B2   C1 − D12 K = , C2 − C2   D11 D12 = . D21 0

(5.128) (5.129) (5.130) (5.131)

It has been verified that by augmenting the stable transfer function Q, the closedloop transfer function from w to z is simply an affine function of Q and equals T11 + T12 QT21 .

Introduction to Optimal and Robust Control

FIGURE 5.5 Control system structure for Youla parametrization.

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5.7 5.7.1

Modeling and Control of Vibration in Mechanical Systems

Performance limitation Bode integral constraint

The block diagram in Figure 2.20 shows a typical closed-loop control system. The closed-loop transfer function from r to y is given by T =

PC . 1 + PC

(5.132)

The sensitivity function is also known as the disturbance rejection function or error rejection function, and given by S=

1 . 1 + PC

(5.133)

Note that S + T ≡ 1,

(5.134)

hence T is also commonly called the complementary sensitivity function. Denote Tyd2 the transfer function from d2 to y. In Figure 2.20 , note that S = Tyd2 = −Ted2 = Ten .

(5.135)

The sensitivity function is thus important, because it explains how disturbance d2 goes through the closed-loop system and shows up at the output y, or the error signal e. It is also important to understand how noise n will be filtered through the closedloop system. The Bode plot of sensitivity function in continuous-time domain is shown in Figure 5.6. It can be explained by the following Bode integral theorem. THEOREM 5.8 [67] Bode’s Integral Theorem for Continuous-time Systems For a stable, rational P and C with P (s)C(s) having at least 2-pole roll off, Z ∞ log|S|dω = 0. (5.136) 0

An implication of Theorem 5.8 is that if the system is made less sensitive to disturbance at some frequencies, it will be more sensitive at other frequencies. If the plant P or compensator is not stable, i.e., if P (s) and /or C(s) have a finite number of unstable poles pk , then (5.136) becomes Z

0

∞

log|S|dω = 2π

K X

k=1

Re(pk ) > 0,

(5.137)

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where K is the number of unstable poles. Equation (5.137) implies that any unstable poles in the system only make it worse, in that more of the disturbance would have to be amplified. Figure 5.7 means that for a discrete-time system, the main difference is the Nyquist frequency limits the frequency range we have to work with. In both cases, if we want to attenuate disturbances at one frequency, we must amplify some disturbance at another frequency. THEOREM 5.9 [108] Bode’s Integral Theorem for Discrete-time Systems Given a stable closedloop discrete-time feedback system, its sensitivity function has to satisfy the following integral constraint: 1 π

Z

π 0

log|S(ejφ )|dφ =

K X

k=1

ln|βk |,

(5.138)

where βk are the open-loop unstable poles of the system, K is the total number of unstable poles, and φ = Ts ω with the sampling time Ts and the frequency ω in radians/sec. Note that Ts is the sampling period, and the upper limit of the frequency spectrum is π/Ts , the Nyquist frequency. Typical digital control systems assume P C is small and |S| ≈ 1 at or above the Nyquist frequency, which is in general not practical for a physical system. Further research beyond Nyquist frequency is needed to address vibrations at frequencies above the Nyquist frequency, which strives to overcome the limitation due to the integral theorem. Theorem 5.9 implies that if for some frequency |S| < 1, then at some other frequency |S| > 1. Unlike the continuous time result, there is no infinite bandwidth to spread this over. Thus all |S| > 1 happen below the Nyquist frequency, and therefore in a finite frequency range. Since the theorem is limited by frequencies up to the Nyquist frequency, if the closed-loop bandwidth is pushed up, better performance at low frequency may result in worse performance at high frequency. In a word, with the linear feedback control whenever we improve the disturbance rejection at one frequency we pay for it at another. Nevertheless, if we have sufficient knowledge of system disturbance and place the disturbance amplification at places where the disturbance is negligible, then we succeed. Otherwise, most of the disturbances may be amplified.

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Area of vibration amplification

10

Magnitude in dB (20log |S|)

0

Area of vibration rejection

Frequency(Hz)

FIGURE 5.6 Sensitivity function for continuous-time system.

Area of vibration amplification

10

Magnitude in dB (20log |S|)

0

Nyquist frequency

Area of vibration rejection

Frequency(Hz)

FIGURE 5.7 Sensitivity function for discrete-time system.

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5.7.2

Relationship between system gain and phase

In the classical feedback theory, the Bode’s gain-phase integral relation has been used as an important tool to express design constraints in feedback systems. Let L = P C denote the open-loop system. It is noted that ∠L(jω0 ) will be large if the gain L attenuates slowly near ω0 and small if it attenuates rapidly near ω0 . The behavior of ∠L(jω) is particularly important near the crossover frequency ωc , where |L(jωc )| = 1, and π + ∠L(jωc ) is the phase margin of the feedback system. Further |1 + L(jωc )| = |1 + L−1 (jωc )| = 2|sin

π + ∠L(jωc ) | 2

(5.139)

must not be too small for good stability robustness. If π + ∠L(jωc ) is forced to be very small by rapid gain attenuation, the feedback system will amplify disturbances and exhibit little uncertainty tolerance at and near ωc . A non-minimum phase zero contributes an additional phase lag and imposes limitations upon the roll off rate of the open-loop gain. Thus the conflict between attenuation rate and loop quality near crossover is clearly evident. In the classical feedback control theory, it has been common to express design goals in terms of the shape of the open-loop transfer function. A typical design requires that the open-loop transfer function has a high gain at low frequencies and a low gain at high frequencies while the transition should be well behaved [70].

5.7.3

Sampling

Mathematical relations and operations can be handled by digital microprocessor only when they are expressed as a finite set of numbers rather than as functions having an infinite number of possible values. Thus any measured continuous signal must be converted to a set of pulses by sampling, which is the process used to measure a continuous-time variable at separated instants of time. The infinite set of numbers represented by the smooth curve is replaced by a finite set of numbers. Each pulse amplitude is then rounded off to one of a finite number of levels depending on the characteristics of the converter. The process is called quantization. Thus a digital device is one in which signals are quantized in both time and amplitude. In an analog device, signals are analog; that is, they are continuous in time and are not quantized in amplitude. The device that performs the sampling, quantization, and converting to binary form is an analog to digital (A/D) converter. The number of binary digits or bits generated by the device is its word length, which is an important characteristic related to the resolution of the converter. The resolution measures the smallest change in the input signal that will produce a change in the output signal. An example is that if an A/D converter has a word length of 10 bits or more, an input signal can be resolved to 1 in 210 or 1024. If the input signal has a range of 10 V, the resolution is 10/1024, or approximately 0.01 V. Thus in order to produce a change in the output the input must change by at least 0.01 V. A discrete-time signal is extracted by sampling from a continuous-time signal. If the sampling frequency is not selected properly, the resulting sampled sequence will

112

Modeling and Control of Vibration in Mechanical Systems

not accurately represent the original continuous signal. A proper sampling frequency is readily determined in many cases by means of the following sampling theorem. THEOREM 5.10 Sampling Theorem A continuous-time signal y(t) can be reconstructed from its uniformly sampled values y(KTs ) if the sampling period Ts satisfies Ts ≤

π ω ¯

(5.140)

where ω ¯ is the highest frequency contained in the signal, that is, |Y (ω)| = 0 for ω > ω ¯. If a system involves a sampling operation of continuous-time signals to generate discrete-time signals, a time delay may be induced. Any time delay added into a closed-loop control system will decrease the stability of the system and in some cases may even cause system instability. The Nyquist rate shown in Figure 5.7 signifies that the freedom to spread the amplification area around is limited by the Nyquist frequency, which is half of the sampling rate. Figure 5.7 implies that a certain rejection amount |S| < 1 must be accompanied by a certain area of |S| > 1, which has to occur before the Nyquist frequency. With a higher sampling rate and closed-loop system bandwidth being kept constant, the amplification |S| > 1 will essentially spread over a broader frequency band and the height of amplification hump shrinks, as shown in Figure 5.8. If we push the closed-loop system bandwidth from f1 to f2 , as seen in Figure 5.8, better performance in low frequency range may result in worse performance in high frequency range.

5.8

Conclusion

Before presentating a series of advanced vibration control methodologies, this chapter has been used to recall some standard advanced control techniques, which helps understand problems to be addressed in the later chapters and possible solutions. It has reviewed H2 and H∞ performances, H2 and H∞ controls, robust control, controller parametrization, as well as performance limitation of linear feedback control systems.

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Introduction to Optimal and Robust Control

2f f

1

0 f

2

Magnitude in dB (20log |S|)

f

10

N1

Frequency(Hz)

FIGURE 5.8 Sensitivity function in discrete-time domain.

N1

6 Mixed H2/H∞ Control Design for Vibration Rejection

6.1

Introduction

The mixed H2 /H∞ control problem is concerned with the design of a controller that minimizes the H2 norm of a certain closed-loop transfer function while satisfying a H∞ norm constraint on the same or another closed-loop transfer function. One of the important applications of this problem is to address the optimal nominal performance subject to a robust stability constraint. In this chapter, we employ two methods: the one described in Chapter 5 and an improved method. The two methods are applied to the control design in disk drives. In order to meet the robustness requirement against unmodeled high frequency dynamics of the VCM actuator in disk drives, an H∞ constraint is to be satisfied when minimizing the H2 performance of the nominal system. Hence, a mixed H2 /H∞ control can be formulated for disk drive control. Both the simulation and experiment results demonstrate that the mixed H2 /H∞ control design gives a significant performance improvement of TMR over the conventional method of PID control combined with notch filters.

6.2

Mixed H2 /H∞ control problem

This section aims to derive an improved design method for the mixed H2 /H∞ control which will give rise to an equal or better performance than the method in Chapter 5. We consider the state-space representation for linear time-invariant systems: x(k + 1) = Ax(k) + B1 w(k) + B2 u(k), y(k) = C2 x(k) + D21 w(k), z1 (k) = Cz1 x(k) + Dz11 w(k) + Dz12 u(k),

(6.1) (6.2) (6.3)

z2 (k) = Cz2 x(k) + Dz21 w(k) + Dz22 u(k),

(6.4)

where x(k) ∈ Rnx is the state, y(k) ∈ Rny is the measurement output, zi (k) ∈ 115

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Rnz (i = 1, 2) are the controlled outputs, w(k) ∈ Rnw is the disturbance input, u(k) ∈ Rnu is the control input, and A, B1 , B2 , C2 , D21 , Cz1 , Cz2 , Dz11 , Dz12 , Dz21 , Dz22 are of appropriate dimensions. Let a controller of the same dimension as that of the system (6.1)−(6.4) be of the form: xc (k + 1) = Ac xc(k) + Bc y(k), u(k) = Ccxc (k) + Dc y(k),

(6.5) (6.6)

where the matrices (Ac , Bc , Cc , Dc ) are to be determined. First, denote ξ = [xT xTc ]T . It follows from (6.1)−(6.4) and (6.5)−(6.6) that ξ(k + 1) = z1 (k) = z2 (k) =

¯ ¯ Aξ(k) + Bw(k), ¯ 1 w(k), C¯1 ξ(k) + D ¯ 2 w(k), C¯2 ξ(k) + D

(6.7) (6.8) (6.9)

where     A + B2 Dc C2 B2 Cc ¯ = B2 Dc D21 + B1 , A¯ = , B Bc C2 Ac Bc D21 ¯ ¯ C1 = [Cz1 + Dz12 Dc C2 Dz12 Cc ] , D1 = Dz12 Dc D21 + Dz11 , ¯ 2 = Dz22 Dc D21 + Dz21 . C¯2 = [Cz2 + Dz22 Dc C2 H22Cc ] , D

(6.10) (6.11) (6.12)

Denote Tz1 w the transfer function matrix from w to z1 , and Tz2 w the transfer function matrix from w to z2 . The mixed H2 /H∞ control problem is then stated as: Given a positive scalar γ, design a controller of the form (6.5)−(6.6) such that the H2 norm, kTz1 w k2 is minimized subject to the constraint kTz2 w k∞ < γ.

6.3

Method 1: slack variable approach

Given the solutions of H2 control and H∞ control in Chapter 5, the following solution to the mixed H2 /H∞ control follows directly. THEOREM 6.1 Consider the system (6.1)−(6.4). Given scalars µ > 0 and γ > 0, a controller of the form (6.5)−(6.6) that solves the mixed H2 /H∞ control problem exists if the following conditions are satisfied. T race(Π) < µ,  Π Cz1 X + Dz12 E  ∗ X + X T − P2 ∗ ∗

(6.13) 

Cz1 + Dz12 Dc C2  > 0, I + ZT − J Y +YT −H

(6.14)

Mixed H2 /H∞ Control Design for Vibration Rejection 117   P2 J AX + B2 E A + B2 Dc C2 B1 + B2 Dc D21  ∗ H U Y A + W C2 Y B1 + W D21     ∗ ∗ X + X T − P2 I + Z T − J  > 0, (6.15) 0   T  ∗ ∗  ∗ Y +Y −H 0 ∗ ∗ ∗ ∗ I Dz11 + Dz12 Dc D21 = 0,

and  P∞  ∗   ∗   ∗   ∗ ∗

(6.16)

 J˜ AX + B2 E A + B2 Dc C2 B1 + B2 Dc D21 0 ˜  H U Y A + W C2 Y B1 + W D21 0  T T T T T T ˜ ∗ X + X − P∞ I + Z − J 0 X Cz2 + E Dz22    T T ˜ ∗ ∗ Y +YT −H 0 Cz2 + C2T DcT Dz22  T T T T  ∗ ∗ ∗ I Dz21 + D21 Dc Dz22 ∗ ∗ ∗ ∗ γI > 0, (6.17)

where the matrices X, E, Y , W , U , Dc , Z, J, J˜ and the symmetric matrices ˜ and Π are the variables. A feasible mixed H2 /H∞ controller P2 , P∞, H, H is obtained by choosing N1 and M1 nonsingular such that N1 M1 = Z − Y X and calculating Cc = (E − Dc C2 X)M1−1 , Bc = N1−1 (W − Y B2 Dc ),

Dc = Dc ,

(6.18) (6.19)

Ac = N1−1 (U − Y (A + B2 Dc C2 )X − N1 Bc C2 X − Y B2 CcM1 ]M1−1 . (6.20)

In the early development of the mixed H2 /H∞ control, to solve the problem in terms of LMIs, a single Lyapunov matrix is adopted for both the H2 and H∞ performances, which is very conservative in general. A significant improvement was made in [90] where a slack variable technique is introduced which separates the Lyapunov matrices from the controller parameters and hence allows them to be different for the H2 and H∞ performances. This is observed in the above theorem where the Lyapunov matrices P2 , J and H are used for H2 performance and different matrices ˜ are used for H∞ performance. P∞ , J˜ and H

6.4

Method 2: an improved slack variable approach

Recall from Chapter 5 that for a given controller that stabilizes the system (6.7)−(6.9), the H2 norm square, kTz1 w k22 , can be computed by the following minimization [95]: min

(Q=QT ,Π=ΠT )

T race(Π),

(6.21)

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subject to ¯ − Q + C¯ T C¯1 < 0, A¯T QA 1 T ¯ QB ¯ +D ¯ 1T D ¯ 1 < Π. B

(6.22) (6.23)

The following lemma leads to an alternative approach for computing the H2 norm of the system (6.7)−(6.8). LEMMA 6.1 The minimization of the H2 norm square in (6.21) is equivalent to the following minimization: min T race(Π), (6.24) (Q=QT ,Π=ΠT ,Σ)

subject to ˜− A˜T diag{Q, I}A where A˜ =



A¯ C¯1



 Q Σ < 0, ΣT Π

(6.25)

 ¯ B ¯1 . D

Proof First of all, (6.25) can be rewritten as 

 ¯1 − Σ ¯ + C¯ T D A¯T QA¯ − Q + C¯1T C¯1 A¯T QB 1 ¯ 1 − Π < 0. ¯ T QB ¯ +D ¯ 1T D ¯ T QA ¯+D ¯ 1T C¯1 − ΣT B B

(6.26)

It is then clear that if there exists a solution (Q, Π, Σ) to (6.25), the same Q and Π also satisfy (6.22) and (6.23), respectively. On the other hand, if there exist Q and Π satisfying (6.22) and (6.23), then (6.25) is also met by the same Q and Π and ¯ + C¯ T D ¯ 1. Σ = A¯T QB 1 REMARK 6.1 The characterization of the H2 norm in the above lemma has the advantage that it provides a unified treatment of H2 and H∞ designs via an LMI approach, as seen later. Furthermore, the additional parameter Σ in (6.25) will offer an additional freedom in optimization of performance when a mixed H2 and H∞ control design is concerned. Without loss of generality, we shall assume nw = nz , i.e. the disturbance input and the signal to be controlled have the same dimension. Note that, if this is not the case, some simple modification can render the requirement satisfied. For example, ˆ1 = if nw < nz , the matrices B1 , D21 , Dz11 and Dz21 can be augmented as B ˆ ˆ [B1 0nx ×(nz −nw ) ], D21 = [D21 0nx ×(nz −nw ) ], Dz11 = [Dz11 0nz ×(nz −nw ) ] and ˆ z21 = [Dz21 0n ×(n −n ) ]. D z z w

Mixed H2 /H∞ Control Design for Vibration Rejection

119

LEMMA 6.2 There exists a solution (Σ, Q, Π) with Q = QT to (6.25) if and only if there exist matrices (Σ, Π, Q, F, G) with Q = QT and Π = ΠT such that     Q Σ T T ˜ T T ˜ ˜ −F + A G  − ΣT Π + A F + F A  < 0. T ˜ −F + G A diag{Q, I} − (G + GT ) (6.27) Proof First, if (6.25) holds for some Q > 0, by applying the Schur complement, it is easy to know that (6.27) is satisfied with F = 0 and GT = G = diag{Q, I}. On the other hand, if (6.27) holds for some (Σ, Q, F, G), multiplying (6.27) from the left and from the right by ΓT and Γ, respectively, where   I Γ= ˜ , A (6.25) follows. REMARK 6.2 It should be pointed out that (6.21)−(6.23) and Lemma 6.2 give equivalent computations of the H2 norm of the system. For systems without uncertainty, it is well known that (6.21)−(6.23) can be applied to derive the optimal H2 controller [95]. Hence, there is no advantage of using Lemma 6.2. However, as will be seen later, when additional performances such as the H∞ performance are to be met in addition to the H2 performance, the latter will result in a less or equally conservative design due to the additional variables F and G. We observe that when F = 0 and Σ = 0, Lemma 6.2 reduces to the result in Theorem 6.1. While Lemma 6.2 can be applied to compute the H2 norm of the system (6.7)−(6.8) when a controller (6.5)−(6.6) is given, it may not be directly applicable to the H2 control design problem due to the presence of the products of F with A˜ and G with ˜ To overcome this difficulty, we specialize the matrices F and G as follows. A.     λ1 Φ 0 Φ 0 F = , G= , (6.28) 0 λ2 I 0 λ3 I where Φ ∈ R2n×2n and λi , i = 1, 2, 3 are real scaling parameters. While this specialization of F and G generally introduces some conservatism, it contains three additional variables λi , i = 1, 2, 3 as compared to the result of Theorem 6.1 which help reduce the design conservatism in Theorem 6.1. Substituting (6.28) into (6.27) leads to  T ¯−Σ −Q + λ1 A¯T Φ + λ1 ΦT A¯ λ2 C¯1 + λ1 ΦT B
 λ C¯ + λ B T T T ¯ ¯ ¯1 Φ−Σ − Π + λ2 D1 + D 1  2 1  ¯ −λ1 Φ + ΦT A¯ ΦT B ¯1 λ3 C¯1 −λ2 I + λ3 D

120

Modeling and Control of Vibration in Mechanical Systems  −λ1 ΦT + A¯T Φ λ3 C¯1T ¯T Φ ¯T  B − λ2 I + λ3 D 1  (6.29) T  < 0. Q − (Φ + Φ ) 0 0 (1 − 2λ3 )I

A similar characterization for the H∞ performance can be derived. Indeed, recall that when the system (6.7) and (6.9) is known, it is stable with its H∞ norm less than γ if and only if there exists a matrix P = P T > 0 such that A˜T diag{P∞, I}A˜ − diag{P∞ , γ 2 I} < 0, where A˜ =



(6.30)

 ¯ A¯ B ¯2 . C¯2 D

Observe that (6.30) turns out to be a special case of (6.25) with Π = γ 2 I and Σ = 0. Thus, following a similar procedure for deriving (6.29), it can be shown that the system (6.7) and (6.9) has an H∞ performance γ if and only if for some real scalars T εi , i = 1, 2, 3, there exist matrices P∞ = P∞ > 0 and Φ such that T ¯ ¯ −P∞ + ε1 (A¯T Φ + ΦT A) ε2 C¯2 + ε1 ΦT B
T 2 T  ¯ ¯ ¯ ¯2 ε2 C2 + ε1 B Φ −γ I + ε2 D2 + D   ¯ ΦT B −ε1 Φ + ΦT A¯ ¯ ¯2 ε3 C2 −ε2 I + ε3 D  −ε1 ΦT + A¯T Φ ε3 C¯2T ¯T Φ ¯ 2T  B −ε2 I + ε3 D  < 0. T  P − (Φ + Φ ) 0 0 (1 − 2ε3 )I



(6.31)

REMARK 6.3 When setting ε1 = ε2 = 0 (i.e. setting F = 0) and ε3 = 1 and by some row-column exchanges, the above inequality reduces to   ¯ 0 P∞ − Φ − ΦT ΦT A¯ ΦT B  A¯T Φ −P∞ 0 C¯2T    ¯T Φ ¯ 2T  < 0  B 0 −γ 2 I D ¯2 0 C¯2 D −I

which is the result in Chapter 5. Therefore, εi , i = 1, 2, 3 are additional variables which can be exploited to alleviate the conservatism in the mixed H2 /H∞ design of Theorem 6.1. Denote ¯ 11 = Q ¯T , Q ¯ 12 , Q ¯ 22 = Q ¯T , Σ ¯ 1, Ω (Π, X, Y, E, U, W, Z, Dc, Q 11 22 ¯ 2 , λi , i = 1, 2, 3) = Σ

Mixed H2 /H∞ Control Design for Vibration Rejection

121

¯ 11 + λ1 (AY + Y T AT + E T B2T + B2 E) −Q ¯ T12 + λ1 (AT + C2T DcT B2T ) + λ1 U T  − Q  T  λ2 (Cz1 Y + Dz21 E) + λ1 (B1T + D21 ¯ T1 Dc T B2T ) − Σ  T  −λ1 Y + AY + B2 E   −λ1 I + U λ3 (Cz1 Y + Dz21 E) 

∗ ¯ 22 + λ1 (AT X + X T A) + λ1 (C2T W T + W C2 ) −Q T ¯ T2 λ2 (Cz1 + Dz21 DcC2 ) + λ1 B T X + λ1 D21 WT − Σ −λ1 Z + A + B2 Dc C2 −λ1 X + X T A + W C2 λ3 (Cz1 + Dz21 Dc C2 )

∗ ∗ T T T −Π + λ2 (D21 Dc T Dz21 + Dz21 Dc D21 + Dz11 + Dz11 ) B1 + B2 Dc D21 X T B1 + W D21 −λ2 I + λ3 (Dz11 + Dz21 Dc D21 )

∗ ∗ ∗ ¯ 11 − (Y + Y T ) Q ¯ T − (I + Z T ) Q 12 0

 ∗ ∗  ∗ ∗   ∗ ∗ .  ∗ ∗  ¯ 22 − (X + X T )  Q ∗ 0 (1 − 2λ3 )I

(6.32)

We have the following solution to the mixed H2 /H∞ control. THEOREM 6.2 Consider the system (6.1)−(6.4). A controller of the form (6.5)−(6.6) that solves the mixed H2 /H∞ control problem exists if for some λi , εi , i = 1, 2, 3, ¯ 11, Q ¯ 12 , Q ¯ 22, Σ ¯ 1, Σ ¯ 2 , X, Y, Π, U, E, Z, W , there exists a solution (P¯11 , P¯12 , P¯22, Q ¯ = [Q ¯ lj ] = Q ¯ T > 0 and P¯ = [P¯lj ] = P¯ T > 0 to the following Ψ, Dc ) with Q optimization: min T race(Π) subject to ¯ 11 , Q ¯ 12, Q ¯ 22 , Σ ¯ 1, Σ ¯ 2 , λi , Ω (Π, X, Y, E, U, W, Z, Dc, Q i = 1, 2, 3) < 0,

(6.33)

¯ 1, Σ ¯ 2 , εi , Ω (γ 2 I, X, Y, E, U, W, Z, Dc, P¯11 , P¯12, P¯22, Σ i = 1, 2, 3)|Σ¯ 1=Σ¯ 2 =0 < 0.

(6.34)

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Modeling and Control of Vibration in Mechanical Systems

In this situation, a mixed H2 /H∞ controller is given by Ac = M1−T (U − X T (A + B2 Dc C2 )Y − X T B2 Cc N1 −M1T Bc C2 Y )N1−1 , Bc =

M1−T (W

T

− X B2 Dc ),

Cc = (E − Dc C2 Y )N1−1 , Dc = Dc ,

(6.35) (6.36) (6.37)

where M1 , N1 satisfy N1T M1 = Z − Y T X.

Proof First, observe from (6.29) that Φ is invertible since Φ + ΦT > Q > 0. Denote     X M Y H Φ= , Φ−1 = , M1 U N1 E and



 Y In J= , J1 = diag{J, I, J, I}. N1 0

Further, denote E = Dc C2 Y + Cc N1 , U = X T (A + B2 Dc C2 )Y + X T B2 Cc N1 + M1T Bc C2 Y + M1T Ac N1 , W = Z=

T

X B2 Dc + M1T Bc , Y T X + N1T M1 .

(6.38) (6.39) (6.40) (6.41)

Multiplying from the left and the right of (6.29) by J1T and J1 respectively and ¯ T ] = ΣT J. ¯ = [Q ¯ lj ] = J T QJ and [Σ ¯T Σ applying (6.10), we obtain (6.33) where Q 2 1 By applying a similar procedure to (6.31), (6.34) can be obtained. If there exists a solution for the LMIs (6.33) and (6.34), it is easy to see that   Y +YT I +Z ¯ > 0. >Q I + ZT X + XT Multiplying the above from the left by [Y −T −I] and from the right by [Y −T −I]T , we obtain that (X − Y −T Z) + (X − Y −T Z)T > 0.

It is then clear that Z − Y T X is invertible. Hence, there exist invertible matrices M1 and N1 such that Z − Y T X = N1T M1 . Thus, it follows from (6.38)−(6.41) that the controller parameters of (6.5)−(6.6) can be obtained as in (6.35)−(6.37). REMARK 6.4 It is worth noting that when setting λ1 = λ2 = ε1 = ¯1 = Σ ¯ 2 = 0, Theorem 6.2 reduces to Theorem ε2 = 0, λ3 = ε3 = 1 and Σ 6.1. Hence, by exploring the freedoms offered by these parameters, a less or equally conservative mixed H2 /H∞ design can be achieved. Certainly, the

Mixed H2 /H∞ Control Design for Vibration Rejection

123

controller from Theorem 6.1 is derived by a convex optimization and can be further refined by an iterative procedure. In the disk drive application to be presented later, we demonstrate that Method 2 can give a much better performance than Method 1 with Theorem 6.1 together with an iterative refinement in the latter design. REMARK 6.5 Observe that for given λi , εi , i = 1, 2, 3, (6.33) and (6.34) ¯ 11, Q ¯ 12 , Q ¯ 22, Σ ¯ 1, Σ ¯ 2 , X, Y, Π, U, E, Z, W, Dc), and are linear in (P¯11, P¯12 , P¯22, Q hence can be solved by employing the LMI Tool [69]. The problem of searching for the optimal scaling parameters λi , εi , i = 1, 2, 3 in general may be numerically costly although the function fminsearch in MATLAB Optimization Toolbox may be applied.

6.5 6.5.1

Application in servo loop design for hard disk drives Problem formulation

A block-diagram representation of a typical HDD servo loop is shown in Figure 6.1 with disturbances injected. P (z) and C(z) represent transfer functions of the plant and controller, respectively. v represents all torque disturbances. d represents disturbances that are due to non-repeatable disk and suspension/slider motions. n denotes the PES demodulation and measurement noise. z1 is the true position error, and e is the measured position error or measurement output y in (6.2). D1 , D2 , and N are the disturbance and noise models, and w1 , w2 and w3 are white noises of zero mean and unit variance.

FIGURE 6.1 Mixed H2 /H∞ control scheme for HDD servo loop with disturbance models.

124

Modeling and Control of Vibration in Mechanical Systems

Through experiments, the frequency responses of the actual VCM is obtained and is shown in Figure 6.2. A 5th order model is used to approximate the actual frequency responses of the VCM actuator and is given by P (s) =

5.172 × 1012 s2 + 1.82 × 1017 s + 3.267 × 1021 . (6.42) s5 + 2.117 × 104 s4 + 1.032 × 109 s3 + 1.906 × 1013 s2 +8.587 × 1015 s + 7.345 × 1018

Figure 6.2 shows the comparison between the frequency responses of the actual data and those of P (s). It is clear that their difference is more significant for the frequency range of over 4 kHz. To capture the unmodeled dynamics in high frequencies, dozens of frequency response measurements are carried out and Figure 6.3 shows the multiplicative uncertainty of the VCM actuator defined by   Nmea  Pi (jω) − P (jω)  , (6.43) ∆(ω) = max   i=1  P (jω)

where Nmea is the number of measurements and Pi (jω) is the actual frequency response of the plant in the i-th measurement, and P (jω) is the frequency response of the model in (6.42). An approximate bounding function Wu (s), i.e., the smooth line in Figure 6.3, is obtained as Wu (s) =

3s2 + 2.903 × 104 s + 1.433 × 108 . s2 + 3.016 × 104 s + 1.421 × 109

(6.44)

From Figure 6.3, it is clear that the uncertainty at frequency over 5 kHz is the major concern. We observe that the actual uncertainties at some frequencies below 5 kHz exceed the bounding function, which, however, will not cause any major problem. In fact, we verify that the robust stability of our designed system is guaranteed even with the worst case of uncertainty. By discretization using the zero-order hold, the corresponding z-domain models of the VCM and the bounding function, i.e., P (z) and Wu (z) can be obtained. The disturbance and noise models D1 (s), D2 (s) and N (s) are given by (2.42)−(2.44), and D1 (z), D2 (z) and N (z) are their discrete-time forms. As mentioned, one of the most important performance measures for HDDs is the track misregistration or TMR, the total amount of random fluctuation about the desired track location. TMR is used to judge the required accuracy of positioning and thus to scale the disk capacity. To achieve a high capacity disk drive, one way in servo control is to minimize TMR, which is given in terms of the standard deviation of the true PES, i.e., v u q u 1 X 3σz1 = 3t z1 (i)2 , (6.45) q − 1 i=1 where q is the number of true PES samples.

Mixed H2 /H∞ Control Design for Vibration Rejection

125

60 measured modeled

Magnitude(dB)

40 20 0 −20 −40 2

3

10

10

4

10

0

Phase(deg)

−200 −400 −600 −800 −1000 −1200 2 10

3

10 Frequency(Hz)

FIGURE 6.2 Frequency responses of the VCM actuator.

4

10

126

Modeling and Control of Vibration in Mechanical Systems

20 VCM uncertainty bounding curve 10

0

Magnitude(dB)

−10

−20

−30

−40

−50

−60

2

10

3

10

Frequency(Hz)

FIGURE 6.3 Multiplicative uncertainty of the VCM actuator.

4

10

Mixed H2 /H∞ Control Design for Vibration Rejection

127

Let w ˜ = [w1 w2 w3 ]T and Tz1 w ˜ to z1 . ˜ denote the transfer function matrix from w When q is large enough, the H2 norm of Tz1 w ˜ is given by [88] v u q u 1 X t kTz1 w z1 (i)2 . ˜ k2 ≈ q − 1 i=1

(6.46)

Thus, the control design problem to minimize TMR can be treated as an H2 optimal control problem. On the other hand, we need to ensure the system stability against the unmodeled high frequency dynamics of the VCM actuator, i.e., the constraint kT Wu k∞ < 1 is to be met, where T is the closed-loop transfer function and Wu is the bounding function of the unmodeled dynamics which was derived earlier. Therefore, we have the mixed H2 /H∞ control scheme as shown in Figure 6.1, where w4 ∈ ℓ2 [0, ∞), a disturbance input or a reference. Clearly, the transfer function from w4 to z2 is T Wu . We now derive the state-space representation (6.1)−(6.4) for the system in Figure 6.1 with  T x = xTp xTd1 xTd2 xTn xTu ,  T w = w1 w2 w3 w4 ,   Ap Bp Cd1 0 0 0  0 Ad1 0 0 0    0 0 A 0 0  A= d2  ,  0 0 0 An 0  Bu Cp 0 Bu Cd2 0 Au     Bp Dd1 0 0 0 Bp  Bd1  0  0 0 0       0 , B1 =  0 B 0 0 , B = d2 2      0  0  0 Bn 0  0 Bu Dd2 0 0 0   C2 = −Cp 0 −Cd2 Cn 0 ,     D21 = 0 −Dd2 Dn 1 , Cz1 = −Cp 0 −Cd2 0 0 ,   Dz11 = 0 −Dd2 0 0 , Dz12 = 0,   Cz2 = Du Cp 0 Du Cd2 0 Cu ,   Dz21 = 0 Du Dd2 0 0 , Dz22 = 0,

(6.47) (6.48)

(6.49)

(6.50)

(6.51) (6.52) (6.53) (6.54)

and xp, xd1 , xd2 , xn , and xu are respectively the state variables of the VCM actuator P (z), the input disturbance model D1 (z), the output disturbance model D2 (z), the measurement noise model N (z), and the uncertainty Wu (z). (Ap , Bp , Cp, Dp ), (Ad1 , Bd1 , Cd1 , Dd1 ), (Ad2 , Bd2 , Cd2 , Dd2 ), (An , Bn , Cn , Dn ) and (Au , Bu , Cu , Du ) are respectively the state-space models of P (z), D1 (z), D2 (z), N (z), and Wu (z). Note that for the servo control shown in Figure 6.1, nz = ny = 1.

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Design results

In this section, we will apply the mixed H2 /H∞ control to hard disk drive servo formulated previously. The sampling frequency being used is 20 kHz. By applying Theorem 6.2 and searching for the optimal scaling parameters, we obtain λ1 = 0.3, λ2 = 0.31, λ3 = 0.9, ε1 = 0.3, ε2 = 0.28, ε3 = 1.1 and the minimum H2 norm, i.e., the σ value of the true PES z1 , of 0.00748 µm. For the purpose of comparison, we also design a mixed H2 /H∞ controller for the disk drive using the approach in Theorem 6.1. The minimum H2 norm of 0.01013 µm is obtained. Starting from this controller, we carried out a further iterative procedure between controller variables and Lyapunov parameters: ¯ B, ¯ C, ¯ D) ¯ with the controller parameters Step 1: obtain the closed-loop system (A, ¯ ¯ ¯ ¯ (Ac , Bc , Cc , Dc ), and let A = A, B = B, C = C, and D = D. Step 2: solve LMIs (5.14)−(5.15), (5.20) for P and X, and minimize T race(Π). If T race(Π) does not differ from the previous value, stop. Otherwise, go to step 3. Step 3: With the obtained P and X, solve LMIs (5.14)−(5.15), and (5.20) for (Ac , Bc , Cc , Dc ). Step 4: go to Step 1. The iterative procedure gives a controller that produces a slight improvement of the H2 norm, i.e., 0.01002 µm. Hence, the improved approach represents about 25.3% more improvement on TMR than the design method in Theorem 6.1 together with an iterative refinement. Figure 6.4 shows the comparison of sensitivity functions, where we can see that the sensitivity function designed based on the improved design method is better than that from Theorem 6.1, except that its hump is slightly higher. The comparison of control performances obtained by the improved method and that of Theorem 6.1 is given as in Table 6.1, where the bandwidth from the improved method is much higher. Although the H∞ norm of T Wu of the improved design is slightly higher than that of the design using Method 1, it is below one as required, implying the designed controller makes the closed-loop system robustly stable in the presence of uncertainty bounded by Wu . Further, from Figure 2.23, the disturbance mainly concentrates on frequencies below 1 kHz, hence, the slight higher peak does not degrade the disturbance rejection performance much. Figure 6.5 shows the testing result of sensitivity functions, which is consistent with the simulation results in Figure 6.4. REMARK 6.6 From Table 6.1, one may argue that a reduced H2 norm for the method of Theorem 6.1 may be obtained by a γ value of greater than one as the actual H∞ norm is 0.78, lower than that of the improved method. However, based on our simulations, no obvious improvement on the H2 performance has been observed. For example, with γ = 1.5, a slightly reduced H2 norm of 10.11 nm is obtained whereas the actual H∞ norm is 0.79. With γ = 5, the H2 norm is reduced to 10.08nm with an unchanged H∞ norm of 0.78. This means that with a larger γ, the improvement on the H2 norm for the method of Theorem 6.1 is negligible.

Mixed H2 /H∞ Control Design for Vibration Rejection

129

The improved method is also compared with conventional PID design. As shown in Figure 6.4 and Figure 6.5, the sensitivity function by PID control has a lower bandwidth and almost the same peak value, which gives a higher TMR as listed in Table 6.1. ¯ 1 and Σ ¯ 2 as solutions to the LMI The previous simulation is carried out with Σ (6.32). When (3,1) and (3,2) blocks in (6.32) are set to zeros, it is found that the designed controller gives a 9% lower bandwidth for the closed-loop system, leading to a worse TMR. This demonstrates that the parameter Σ in Lemma 6.1 is useful in achieving a better performance.

10 Method 1 Method 2 PID

5

0

Magnitude(dB)

−5

−10

−15

−20

−25

−30

3

10

4

10

Frequency(Hz)

FIGURE 6.4 Frequency response of sensitivity functions.

Next, we shall calculate the H2 norm using the measured plant frequency response as in Figure 6.3 and sensitivity function as in Figure 6.5. The spectrum of z1 is given by |z1 (fk )|2 = |P (fk )S(fk )|2 |D1 (fk )|2 + |S(fk )|2 |D2 (fk )|2 +|1 − S(fk )|2 |N (fk )|2

(6.55)

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Modeling and Control of Vibration in Mechanical Systems

10 Method 1 Method 2 PID

5 0 −5

Magnitude(dB)

−10 −15 −20 −25 −30 −35 −40 −45

3

10

Frequency(Hz)

FIGURE 6.5 Frequency response of sensitivity functions.

4

10

Mixed H2 /H∞ Control Design for Vibration Rejection

131

TABLE 6.1

Control performance comparison Method Open-loop crossover frequency (Hz) H2 norm (σ (nm)) kT Wu k∞

Method 2 Method 1 PID 1.4k 969 976 7.48 10.13 12.11 0.89 0.78 0.27

where fk (k = 1, 2, 3..., K) are frequency points, P (fk ) represents the measured frequency response of the plant, and S(fk ) is the frequency response of the sensitivity function. The resultant σ value of z1 is 0.01155, better than 0.02714 with the controller designed by the method in Theorem 6.1.

6.6

Conclusion

This chapter has presented two design methods for the mixed H2 and H∞ control. One is a slack variable approach, and another one is a less or equally conservative design in terms of LMIs that contain more free variables than the conventional approaches. Those variables offer additional freedoms in optimization, resulting in a less or equally conservative control design. The improved H2 /H∞ control design has been applied to hard disk drives to minimize the track misregistration while guaranteeing the system robustness in the presence of actuator uncertainties. Compared with the slack variable method, the improved design result for hard disk drives has indicated a marked improvement of 25.3% in the H2 norm or the TMR performance while guaranteeing the system robustness by satisfying the required H∞ constraint. The experimental result validates the advantage of the improved design.

7 Low-Hump Sensitivity Control Design for Hard Disk Drive Systems

7.1

Introduction

In feedback control systems, sensitivity functions are critical to the determination of their ability in disturbance and noise rejections. However, Bode has shown the limitation of using a feedback structure in terms of an integral constraint on the sensitivity function, as discussed in Chapter 5. Briefly speaking, the Bode integral theorem implies that we cannot have a sensitivity function less than unity at all frequencies using output feedback with a finite-bandwidth controller. Such a sensitivity function must amplify the disturbances existing in frequencies higher than the system bandwidth. In view of this, we shall employ the special structure of a secondary actuator system and design appropriate controllers for primary and secondary actuators such that the hump of the sensitivity function comes as low as possible without the cost of lowfrequency performance. This optimized sensitivity function is expected to minimize the amplification of high-frequency disturbances while attenuating low-frequency and mid-frequency disturbances. With this low-hump sensitivity, the dual-stage control system is able to reduce the contribution from all existing disturbances to the error. Two types of microactuator models are considered in this chapter: a MEMS actuator [102] and a PZT actuator [103]. The purpose is to design controls for the primary and the secondary actuators such that a low hump of the sensitivity function can be achieved with the help of secondary actuators. A comparison will be made to evaluate the effectiveness of the proposed method for the two microactuators. Besides simulations, an implementation with a PZT microactuator verifies that the HDD servo loop design method leads to a low-hump sensitivity function.

7.2

Problem statement

Figure 7.1 shows a dual-stage actuation system with one primary actuator Pv (s) and one secondary actuator Pm (s), and two parallel controllers Cv (s) and Cm (s). With disturbances and noise injected, the error is contributed by the disturbances and noise

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Modeling and Control of Vibration in Mechanical Systems

in terms of Se = |Pv (fk )S(fk )|2 |d1 (fk )|2 + |S(fk )|2 |d2 (fk )|2 + |S(fk )|2 |n(fk )|2 , (7.1) where Se is the power spectrum of the error e, S is the sensitivity function, and fk (k = 1, 2, · · · , N ) are frequency points. Figure 7.2 shows an example of main disturbances existing in disk drive systems, where the low frequency components are represented by d1 , the higher frequency portions such as disk vibration and windage are lumped as d2 , and the base line of the spectrum stands for the noise n.

FIGURE 7.1 Parallel structure of a dual-stage actuation system with disturbances and noise injected.

Equation (7.1) implies that the sensitivity function S is important in determining the disturbance rejection of the dual-stage closed loop control system. In a conventional single-stage system using one primary actuator it is difficult to have a lowhump sensitivity function. Thus the dual-stage structure is applied to increase the system bandwidth and lower the sensitivity function peak such that sup |S(jω)| ≤ 1 + τ, ω = 0, · · · , ∞

(7.2)

where 0 < τ 0, then Z π ln |S(ejω )|dω < 0. −π

c). When the orders of the denominator and numerator of G(z) are the same, −2 ≤ kg < 0 and kg 6= −1, then Z π ln |S(ejω )|dω ≥ 0. −π

Thus, we can conclude that only if G(z) is non-strictly proper and kg < −2 or kg > 0, then |S(ejω )| ≤ 1, ω ∈ [0, π], i.e., the sensitivity function is bounded by 0 dB.

Low-Hump Sensitivity Control Design for Hard Disk Drive Systems

153

The open-loop transfer function of the dual-stage parallel system is the sum of each path, i.e., G(z) = Gv (z) + Gm (z), Gv (z) = Pv (z)Cv (z), Gm (z) = Pm (z)Cm (z).

(7.18) (7.19) (7.20)

Note that a VCM actuator can be approximately represented by a double integrator (i.e., k/s2 ) combined with some resonance modes and is often described by a strictly proper model. When Gm (z) is strictly proper, G(z) is also strictly proper and when Gm (z) is non-strictly proper, G(z) is non-strictly proper. It can be concluded that the Bode’s integral of the overall dual-stage loop is determined by the microactuator loop. Only a non-strictly proper model of microactuator could possibly produce a “flat” sensitivity function for dual-stage servo systems. An additional condition for the “flat” sensitivity is kg < −2 or kg > 0. These are necessary conditions and could be used as a criterion to examine the closed-loop design. In what follows, the discrete H∞ loop shaping method will be applied to the control designs of the VCM loop and the microactuator loop such that the sensitivity functions of the two loops are coupled and achieve a low-hump overall sensitivity. The structure of the H∞ loop shaping method is the same as in Figure 7.4. In the discrete time case, Figure 7.4 is formulated as follows. x(k + 1) = Ax(k) + B1 w(k) + B2 u(k), z(k) = C1 x(k) + D11 w(k) + D12 u(k), y(k) = C2 x(k) + D21 w(k) + D22 u(k),

(7.21) (7.22) (7.23)

where 

     Ap 0 0 Bp A= , B1 = , B2 = Bw Cp Aw Bw Bw Dp   C1 = Dw Cp Cw , D11 = Dw , D12 = Dw Dp ,   C2 = Cp 0 , D21 = 1, D22 = Dp ,

(Ap , Bp , Cp , Dp ) and (Aw , Bw , Cw , Dw ) are respectively the state-space realizations of plant P (z) and weighting function W (z). Let (Ac , Bc , Cc , Dc ) be a state space description of C(z). An LMI approach stated in Chapter 5 will be used to design the controller C(z) : (Ac , Bc , Cc , Dc ).

7.4.2

An application example

The VCM actuator and the microactuator are the same as those in Case 2 in Section 7.3.2. Notice that the PZT microactuator is represented using a Pade-delay with two 2nd order resonance terms. The form (7.14) can be regarded as a general model of PZT actuated suspensions. It is strictly proper and thus according to the analysis in

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Section 7.4.1, a “flat” sensitivity function is impossible. The used sampling rate for the controller design is 40 kHz. The discrete-time actuator models are obtained using “zero-order-hold” method to ensure that the designed controllers are implementable. With ω = 2π350, ε = 10−3.2, ζ = 0.4 in (7.6), a VCM controller Cv (z) is designed as in Figure 7.22 using the H∞ loop shaping method in the previous section. Also applying the LMI approach, the controller Cm (z) for the microactuator is designed as in Figure 7.23 with ω = 2π3700, ε = 10−1.38, ζ = 1, and M = 0.071/2 in (7.6). The sensitivity function of the dual-stage system is shown in Figure 7.24, where we can observe that the hump is below 3 dB. The open-loop system has the gain margin of 8.4 dB, the phase margin of 64◦ and the bandwidth of 2.49 kHz. REMARK 7.2 If manufacturing processes could produce an ideal actuator that can be modeled by a Pade-delay only, which is non-strictly proper, it is possible to have a “flat” sensitivity like that in Figure 7.25 with kg > 0.

An actual microactuator, which resembles the ideal case, is the PZT microactuator in [107]. It can be described by a 1-pole roll-off model Pm (s) =

1257 , s + 1.257 × 105

(7.24)

which is strictly proper and does not satisfy the necessary conditions in Section 7.3.1. The sensitivity function with the hump lower than 2 dB, as shown as in Figure 7.26, can be obtained with the same weighting function as that for (7.14).

Low-Hump Sensitivity Control Design for Hard Disk Drive Systems

FIGURE 7.22 VCM controller Cv (z).

FIGURE 7.23 Microactuator controller Cm (z).

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 7.24 Sensitivity function of the dual-stage system.

FIGURE 7.25 Sensitivity function.

Low-Hump Sensitivity Control Design for Hard Disk Drive Systems

157

FIGURE 7.26 Sensitivity function of the dual-stage system.

FIGURE 7.27 Frequency responses of Pv (z)Cv (z) (solid curve) and Pm (z)Cm (z) (dashed curve).

158

7.4.3

Modeling and Control of Vibration in Mechanical Systems

Implementation on a hard disk drive

The experimetal setup is the same as in Figure 7.17. An LDV with a range of 2 µm/V is used to measure the position of the dual-stage actuator. Controllers are implemented with dSpace 1103 on a TMS320C240 DSP board. When the dualstage loop is closed and stabilized with the designed controllers, a swept sine signal is injected at point A. A DSA is then used to measure the frequency response of points B over A and obtain the sensitivity function. The resultant sensitivity function is shown in Figure 7.28, where the rough line is the testing result and the smooth line is the simulation result. We can observe that the hump of the sensitivity function is lower than 3 dB, which is better than that by a PID design as shown by the dotted line in Figure 7.28. The testing and simulation results of the dual-stage open loop system are shown in Figure 7.29. The step response in Figure 7.30 shows that the system is stabilized and working in real time. Channel 2 in Figure 7.30 is the control signal of the VCM actuator and Channel 3 is the control signal of the PZT microactuator. The sensitivity function as in Figure 7.28 will amplify the corresponding highfrequency disturbances shown in Figure 7.2 due to the hump above 0 dB. The position error is evaluated from (7.1) with the designed sensitivity functions. The 3σ value of the position error versus frequencies is shown in Figure 7.31, where we can see that the low-hump design outperforms the PID design for disturbance rejection after 2.4 kHz, which is consistent with Figure 7.28. The proposed control design is based on the sensitivity weighting function only and does not consider robustness to plant parameter variations. This, however, will not hamper the practical application of the resulting controllers due to the large gain and phase margins. The system is verified to maintain stability in spite of the variation of resonance frequency, e.g., ±5% shift of PZT resonance frequency around 13.5 kHz. The performance change with the presence of the system parameter uncertainty is illustrated in Figure 7.31, where we can see that the performance varies slightly.

7.5

Conclusion

An H∞ method has been proposed in both continuous and discrete time domains to achieve a low-hump sensitivity function for dual-stage HDD systems using an LMI approach. Two different microactuator models have been studied, which are represented by a MEMS actuator and a PZT actuated suspension. With the proposed selection of sensitivity weighting functions, the sensitivity function with a hump below 3 dB has been achieved in both simulations and experiments. Such a design process can generate a robust servo controller with high disturbance rejection in a low frequency range, and less vibration amplification in a high frequency range, and thus is effective in achieving higher positioning accuracy.

Low-Hump Sensitivity Control Design for Hard Disk Drive Systems

FIGURE 7.28 Sensitivity function of the dual-stage system.

FIGURE 7.29 Open loop frequency responses of the dual-stage system.

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 7.30 Step response of the dual-stage system.

FIGURE 7.31 3σ value of PES NRRO versus frequency.

8 Generalized KYP Lemma-Based Loop Shaping Control Design

8.1

Introduction

To shape frequency responses of closed-loop transfer functions such as sensitivity/complementary sensitivity functions, H∞ optimization together with frequency weighting is a commonly used method. The additional weight functions, however, increase the system and controller complexity since the weighting functions usually have to be of high orders in order to capture the desired specifications accurately. This is especially so when a controller is to be designed, aiming at achieving a wider bandwidth while simultaneously suppressing disturbances of particular frequencies within or beyond the servo bandwidth. Further, the process of choosing appropriate weights is tedious and time-consuming. The KYP Lemma [61], being one of the most fundamental results in systems theory and control, establishes the equivalence between a frequency domain inequality (FDI) for a transfer function and a linear matrix inequality associated with its state space realization. It allows us to characterize various properties of dynamic systems in the frequency domain in terms of linear matrix inequalities. The standard KYP Lemma is only applicable for the infinite frequency range, while the generalized KYP Lemma [62] establishes the equivalence between a frequency domain property and a linear matrix inequality over a finite frequency range, allowing designers to impose performance requirements over chosen finite or infinite frequency ranges. Hence, it is very suitable for analysis and synthesis problems in practical applications where different specifications over different frequency ranges are usually required. In this chapter, the generalized KYP Lemma is applied to design a feedback control such that the specifications of the sensitivity function, required to suppress some specific frequency disturbances, are satisfied. Unlike the standard KYP Lemma, the matrix inequality in the generalized KYP Lemma involves a matrix variable which is not necessarily positive definite and thus the Schur complement cannot be applied to convexify the controller design. To overcome this difficulty, the Youla parametrization approach is used to parameterize the closed-loop transfer function. The search for the coefficients of the parameter Q(z) is then converted to a linear matrix inequality problem within the generalized KYP Lemma framework. An application of the method in the rejection of narrowband high-frequency and mid-frequency distur-

161

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Modeling and Control of Vibration in Mechanical Systems

bances is presented to demonstrate the simplicity of the design and the improvement of the positioning accuracy by the resultant controller.

8.2

Problem description

It is known that the power spectrum of the error e in Figure 2.20 is given by Se = |P (z)S(z)|2 |d1 |2 + |S(z)|2 |d2|2 + |S(z)|2 |n|2

(8.1)

which implies that the sensitivity function S(z) is important in determining the disturbance rejection of a control loop. Thus, the design of a controller that gives rise to a sensitivity function which can reject specific disturbances with known frequencies becomes rather significant. The purpose here is then stated as: to design a dynamic feedback controller C(z) for plant P (z) such that the closed-loop system is stable and for some prescribed positive scalars ri and frequency ranges (fi1 , fi2 ), i = 1, ..., N , |S(f)| < ri , fi1 ≤ f ≤ fi2

(8.2)

where S(f) = 1/(1 + C(f)P (f)). Smaller ri means the less contribution of the disturbance in frequency range (fi1 , fi2 ) to the error. In view of the constraint stated in the Bode integral theorem, it is impossible to achieve disturbance rejection at all frequencies higher than the bandwidth of the control loop for actuators or microactuators used in mechanical motion systems such as hard disk drives. The specification (8.2) is considered for a specific frequency range, which is however possible to be achieved through shaping the sensitivity function. The above design problem may be approached by selecting a proper frequency weighting function and carrying out an H∞ optimization. However, the problem of how to select a proper frequency weighting function that can give an accurate shaping of the sensitivity function is generally difficult and time-consuming. Further, the resultant controller order will depend on the order of the weighting function and the plant. Note that a more accurate frequency shaping usually requires a higher order weighting function. The generalized KYP Lemma [62] gives a necessary and sufficient condition for a given transfer function to satisfy a required frequency domain property over a finite frequency range in terms of a matrix inequality condition. Thus, it may be applied to address the above design problem. In what follows, we employ the generalized KYP Lemma to design a feedback control such that the sensitivity function satisfies the required specifications so as to reject disturbances at specific frequencies. In order to convexify the matrix inequality, the Youla parametrization approach as shown in Figure 8.1 is used to design a controller with the generalized KYP Lemma.

Generalized KYP Lemma-Based Loop Shaping Control Design

163

FIGURE 8.1 Q parameterization for control design.

8.3

Generalized KYP lemma-based control design method

The previous analysis indicates that the sensitivity function plays a key role in disturbance rejection. To reject disturbance of frequency within a certain frequency range, a proper shaping of the sensitivity function can be carried out. In this section, we shall present a frequency shaping method based on the generalized KYP Lemma. First, it is easy to see that the sensitivity function S(z) is equal to the transfer function from w to z in Figure 8.1. The state-space model of the plant under consideration is denoted as (Ap , Bp , Cp , Dp ). Then, a state-space representation of the system in Figure 8.1 is given by x(k + 1) = Ap x(k) + Bp u(k), z(k) = −Cp x(k) + w(k) − Dp u(k),

(8.3) (8.4)

where x ∈ Rnx is the state. Let a state-space representation of the controller C(z) be given by (Ac , Bc , Cc , Dc ). Assuming that D = 1 + Dc Dp is invertible, then a state-space representation ˜ B, ˜ C, ˜ D), ˜ where of the sensitivity function can be given by (A,   Ap − Bp D−1 Dc Cp Bp D−1 Cc , (8.5) A˜ = −Bc Cp + Bc Dp D−1 Dc Dp Ac − Bc Dp D−1 Cc   Bp D−1 Dc ˜= B , Bc − Bc Dp D−1 Dc   ˜ = 1 − Dp D−1 Dc . (8.6) C˜ = −Cp + Dp D−1 Dc Cp − Dp D−1 Cc , D A special case of the generalized KYP Lemma that relates the bounded realness of the sensitivity function over finite frequency ranges to its state space representation is given below.

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LEMMA 8.1 ˜ ˜ −1 B ˜ +D ˜ with A˜ being [62] Consider the sensitivity function S(z) = C(zI − A) jθ stable. Then, for a given scalar r > 0, |S(e )| ≤ r over a finite frequency range if and only if there exist Hermitian matrices U and V ≥ 0 such that ˜ A˜ B  I 0 

∗

Σ

   ˜ 0 0 A˜ B + 0 −r 2 I 0  ˜ ˜ CD





˜ C˜ D −I

∗



 ≤ 0,

(8.7)

where (i) for low frequency range |θ| ≤ θl , 

−U Σ= V

V U − (2cosθl )V



;

(8.8)

(ii) for middle frequency range θ1 ≤ θ ≤ θ2 , Σ=



−U

e−jθc V

ejθc V U − (2cosθd )V



,

θc = (θ1 + θ2 )/2, θd = (θ2 − θ1 )/2;

(8.9)

(iii) for high frequency range |θ| ≥ θh , 

−U Σ= −V

−V U + (2cosθh )V



.

(8.10)

For a given controller C(z), the above gives a necessary and sufficient condition for evaluating if |S(z)| ≤ r over some given frequency range in terms of LMI. However, (8.7) is no longer an LMI when the controller C(z) = (Ac , Bc , Cc, Dc ) ˜ involve the unknown parameters Ac , Bc , Cc and is to be designed since A˜ and B Dc . Further, it is not possible to be converted to an LMI by Schur complement since the matrix Σ is not definite. To overcome this difficulty, in the following we apply a Youla parametrization approach where the controller C(z) is now of the structure as shown in Figure 8.1. Let K(z) be a given observer based controller that can be designed using approaches such as the LQG control: xˆ(k + 1) = Ap xˆ(k) + Bp u(k) + L(z(k) + Cpxˆ(k)),

(8.11)

u(k) = −M xˆ(k).

(8.12)

Then, as stated in Chapter 5, the set of sensitivity functions can be parameterized as S(z) = T11 (z) + T12 (z)Q(z)T21 (z)

(8.13)

Generalized KYP Lemma-Based Loop Shaping Control Design where Q(z) is a stable transfer function to be designed and   T11 (z) T12 (z) = CT (zI − AT )−1 BT + DT , T21 (z) 0   Ap −Bp M 0 Bp    −LCp Ap − Bp M + LCp L Bp  AT BT . =  −Cp CT DT Dp M 1 −Dp  −Cp Cp 1 0

165

(8.14)

If Q(z) has the state-space realization (Aq , Bq , Cq , Dq ), then the designed controller C(z) is given by   Ap − Bp M + LCp + Bp Dq Cp Bp Cq Ac = , (8.15) Bq Cp Aq     L + Bp Dq Bc = , Cc = −M + Dq Cp Cq , Bq Dc = Dq .

(8.16)

Denote the state space representations of T11 (z) and T12 (z)T21 (z) by (At11 , Bt11 , Ct11 , Dt11) and (At , Bt , Ct , Dt ), respectively, then from (8.13) a state space model of S(z) can be written as     At11 0 0 Bt11 ˜ =  Bt  , At 0  , B A˜ =  0 (8.17) 0 Bq Ct Aq Bq Dt   ˜ = Dt11 + Dq Dt . C˜ = Ct11 Dq Ct Cq , D (8.18) Let Q(z) be an FIR filter:

Q(z) = q0 + q1 z −1 + q2 z −2 + ... + qτ z −τ , q = [q0 q1 q2 , ... , qτ ]

(8.19) (8.20)

which is to be designed so that the required bounded realness of the sensitivity function is satisfied. It is known that a state space realization for Q(z) can be given by     0 Iτ−1 0(τ−1)×1 Aq = , Bq = , 0 0 1 Cq = [qτ qτ−1 · · · q1 ] , Dq = q0 ,

where Iτ−1 is the identity matrix of dimension (τ − 1) × (τ − 1) and 0(τ−1)×1 is the zero matrix of dimension (τ − 1) × 1. Note that the filter parameter q to be designed only appears in Cq and Dq . Therefore, from (8.17)−(8.18), we know that q exists ˜ only. In this case, (8.7) defines an LMI in terms of the variables U , in C˜ and D V , and q. Hence, U , V , and the design parameter q can be computed via a convex optimization.

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The KYP Lemma-based control design can be carried out following the steps: Procedure 8.1 Step 1. Compute M and L using MATLAB commands M = dlqr(Ap , Bp , CpT Cp , R) and L = Ap · dlqe(Ap , Bp , − Cp , Wd , Wv ), where R is the weighting for the control input in the cost function X J= (ˆ xT CpT Cp xˆ + uT Ru) for linear quadratic regulator design, and Wd and Wv are the variance matrices of process noise and measurement noise for the Kalman estimator design. Here in the KYP Lemma-based control design, R, Wd , and Wv can be chosen as identity matrices. Step 2. Compute T11 (z), T12 (z) and T21 (z) from (8.14). ˜ B, ˜ C, ˜ D) ˜ in (8.17)−(8.18). Step 3. Obtain the state space model (A, Step 4. Based on disturbance spectrum, specify the positive scalars ri and frequency points fi (i = 1, ..., N ) for the sensitivity function |S(fi )| < ri , fi1 ≤ fi ≤ fi2

(8.21)

where fi1 and fi2 define the frequency range. For each specification on the resultant sensitivity function in the frequency range, construct the LMI (8.7) in terms of the variables U , V , and q with r = ri . Step 5. Obtain q, U and V by solving these LMIs using MATLAB LMI toolbox. If the LMIs are not solvable, the specifications given in Step 4 are to be adjusted. Step 6. Obtain the controller parameters from (8.15)−(8.16).

8.4

Peak filter

Many other control methods are available to reject narrowband disturbance, such as linear time invariant feedback methods using classical design and modern frequency shaping and filter shaping, and adaptive feedforward methods using higher harmonic control and LMS algorithm [57]. It has been shown that most of these methods would result in a compensator with lightly damped complex poles at the center frequency of the disturbance. Such a compensator has high gains at specific frequency. It is named as peak filter according to its shape of frequency response. The peak filter F (s) works in the control loop as in Figure 8.2 with its discretized form F (z).

8.4.1

Conventional peak filter

Conventionally, the peak filter is described as the following model F (s) =

s2 + 2ξ1 ωp s + ωp2 , s2 + 2ξ2 ωp s + ωp2

(8.22)

Generalized KYP Lemma-Based Loop Shaping Control Design

FIGURE 8.2 Peak filter F in the nominal feedback loop.

FIGURE 8.3 Peak filter in the frequency domain.

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Modeling and Control of Vibration in Mechanical Systems

where ωp = 2πfp is the center frequency in radian/sec and ξ1 and ξ2 are the damping ratios with ξ1 > ξ2 . The peak filter F (s) can be designed according to its Bode plot, as illustrated in Figure 8.3. M and N denote the peak height, ∆ stands for the peak width corresponding to N . Let m = 10M/20 , n = 10N/20 .

(8.23)

When M ≫ N , ξ1 and ξ2 are determined approximately by ∆2 + 2∆ √ n − 1, 2∆ + 2 ξ1 ξ2 = . m ξ1 =

(8.24) (8.25)

The maximum phase loss θ caused by the pair of the lightly damped complex poles is given by m−1 θ = arctan √ . 2 m

(8.26)

It is noticed that the conventional version of the peak filter induces additional phase loss. The phase loss negatively impacts the phase margin and distorts the gain of the sensitivity function around the disturbance frequency, particularly for the disturbance near the 0 dB crossover frequency. To overcome the drawback, a phase-lead peak filter [118] was proposed.

8.4.2

Phase lead peak filter

The phase filter is improved by adding a differentiator to provide additional phase lead such that the phase margin is preserved and the sensitivity function curve is smoothly shaped. Let T0 (s) =

P (s)C(s) . 1 + P (s)C(s)

(8.27)

The peak filter adopts the following form F (s) = K

−sin(φ)s2 + ω0 cos(φ)s , s2 + 2ξω0 s + ω02

(8.28)

where ω0 is the disturbance frequency at which high disturbance rejection is required, ξ ∈ (0, 1) is the damping ratio, φ is the phase angle determined by φ = arg[T0 (jω0 )] ∈ [−π, π],

(8.29)

and 0 < K < γ is the positive filter gain. Let G(ω, K) = 1 + T0 (jω)F (jω),

(8.30)

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169

and γ is the minimal positive real solution of the following two equations: Re[G(ω, γ)] = 0, Im[G(ω, γ)] = 0.

(8.31) (8.32)

An estimate of ξ and K in the filter (8.28) is given by ∆d (ω0 + 0.5∆d) , 4ω02 2ξ , K = (10M/20 − 1) T0 (jω0 )

ξ=

(8.33) (8.34)

where the disturbance bandwidth ∆d is defined as the frequency difference between √ the two points whose magnitudes are 1/ 2 times of the peak value, and M in unit dB is the desired reduction ratio of the narrowband disturbance. The disturbance filter in (8.28) is a general high-gain controller to reject narrowband disturbances in a specific frequency range because the filter zero location can be automatically shifted according to the disturbance frequency associated with the baseline servo system with C(s) and P (s).

8.4.3

Group peak filter

The group filtering scheme is used to compensate multiple frequency narrowband disturbance. A parallel structure of the group peak filter is described as F (s) =

n X

F i (s),

(8.35)

i=1

where the sub-filter F i (s) is given by (8.28) with the corresponding filter parameters (ωi , φi , ξi , Ki ). An illustration example is shown in Figure 8.4 with the sensitivity functions plotted before and after activating the group filter with two sub-filters at ω1 = 2π700 and ω2 = 2π2000. The filters associated with solid and dashed curves have different parameters.

8.5

Application in high frequency vibration rejection

In this section, we shall provide an example to demonstrate the design and effectiveness of the proposed KYP based control to reject narrowband disturbances at high frequencies. The previously introduced peak filtering method finds it difficult to deal with disturbances in high frequency range, especially near the actuator resonance modes.

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10 5 0

Magnitude (dB)

−5 −10 −15 −20 −25 with filter with filter without filter

−30 −35

2

10

3

10 Frequency (Hz)

4

10

FIGURE 8.4 Sensitivity functions before and after group peak filtering activated.

FIGURE 8.5 PZT microactuator attached to VCM actuator arm.

Generalized KYP Lemma-Based Loop Shaping Control Design

171

−3

3

x 10

PES NRRO power sepctrum(µm)

2.5

2

1.5

1 8 kHz

10 kHz

0.5

0

0

2000

4000

6000

8000

10000

12000

Frequency(Hz)

FIGURE 8.6 Power spectrum of the position error before servo control.

A. System models Figure 8.5 shows one kind of microactuators for HDDs, which is a PZT actuated suspension attached to a VCM actuator arm. Figure 8.6 shows an example of the main disturbances that exist in the hard disk drive servo system. Besides the disturbances of low frequencies, the system suffers from the high frequency disturbances around 8 kHz and 10 kHz, which are induced by air turbulence to suspensions or sliders in HDDs. For higher rotational speed HDDs, these disturbances appear more prominent and create a significant impact on the positioning accuracy of the R/W head. Here we employ the generalized KYP Lemma to design a feedback control C(z) for the microactuator such that the sensitivity function S(z) satisfies the required specifications so as to reject disturbances around 8 kHz and 10 kHz. The desired specifications for the sensitivity function S(z) are set as Spec.(a) |S(f )| < 0 dB, f ≤ 2 kHz, Spec.(b) |S(f )| < −0.35 dB, 9950 Hz ≤ f ≤ 10050 Hz, Spec.(c) |S(f )| < −1.41 dB, 7950 Hz ≤ f ≤ 8050 Hz.

Spec. (a) means to guarantee a 2 kHz bandwidth at least. The transfer function of the PZT microactuator is given by P zt(s) = 180828605599509(s2 + 3079s + 1.934 × 109 ) , (s + 1.257 × 105 )(s2 + 791.7s + 1.567 × 109 )(s2 + 5089s + 7.195 × 109 ) (8.36)

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which is obtained by curve-fitting to its frequency response measured via DSA. A comparison between the measured and the modeled frequency responses is shown in Figure 8.7, where it is noticed that there are two dominant resonance modes at 6.3 kHz and 13.5 kHz. P zt(s) is discretized using the zero-order-hold method to obtain P zt(z). The sampling frequency being used is 40 kHz.

20

Magnitude(dB)

10

measured modeled

0 −10 −20 −30 −40 3 10

4

10

200

Phase(deg)

100 0 −100 −200 −300 3 10

4

10 Frequency(Hz)

FIGURE 8.7 PZT micro actuator frequency response.

B. Controller design and LDV based experiment results The plant P zt(s) is pre-compensated by the integrator: Int(z) =

6.3z , z−1

(8.37)

and two notch filters for suppressing the two main resonances at 6.3 kHz and 13.5 kHz before the KYP Lemma approach is applied to design a controller so that the specifications (a), (b) and (c) can be satisfied. The notch filter is designed as the following form: N otch =

ω22 s2 + 2ξ1 ω1 s + ω12 · , ω12 s2 + 2ξ2 ω2 s + ω22

(8.38)

where ω1 is the frequency of the resonance to be suppressed , ξ1 < ξ2 , and ω2 and ω1 should be chosen to be close to each other so that the resultant notch filter will

Generalized KYP Lemma-Based Loop Shaping Control Design

173

not influence the system stability. Here for the model (8.36) the notch filters after discretization are as follows. 1.177z 2 − 1.279z + 1.154 , z 2 − 0.8751z + 0.9259 0.4011z 2 + 0.3972z + 0.3532 N otch2(z) = . z 2 + 0.1391z + 0.007922

N otch1(z) =

(Ap , Bp , Cp , Dp ) in (8.3)−(8.4) is a state space description of the combined system P (z) = Int(z) · N otch1(z) · N otch2(z) · P zt(z). Q(z) is chosen as the 1st -order FIR filter (8.19), and q0 and q1 are to be solved via the KYP Lemma. As mentioned previously in Step 1 of the control design procedure, M and L are obtained using MATLAB commands, i.e. M = dlqr(A, B2 , C1T C1 , R) and L = A · dlqe(A, B2 , C1 , Wd , Wv ). Next we use the KYP Lemma to search for the coefficients of Q(z). Three LMIs of the form (8.7) with Σ in (8.8) and (8.9) need to be solved in order to achieve Spec. (a), (b) and (c), respectively. The obtained q0 and q1 are q0 = −0.4117, q1 = 0.6371. As such, C(z) can be obtained via (8.15)−(8.16). The resulting controller for the PZT microactuator is C(z) · Int(z) · N otch1(z) · N otch2(z), and the sensitivity function is shown in Figure 8.8. It can be observed in Figure 8.8 that with Q(z), the sensitivity function is less than −4 dB and −2 dB at 8 kHz and 10 kHz respectively, which means that specifications (a)−(c) have been satisfied by the searched Q(z). The sensitivity function before the KYP Lemma based design is also shown in Figure 8.8 where it can be seen that the required specifications are not met. Moreover, as seen from Figure 8.9, the gain margin and the phase margin are 12 dB and 67 deg, higher than 7.7 dB and 63 deg before the KYP design. Observed from Figures 8.8 and 8.9, the KYP design increases the loop gain around 9 kHz, which results in the reduction of the sensitivity function from 8 to 10 kHz. The price for this compensation is a bit lower loop gain at lower frequencies, which is consistent with the Bode Integral constraint for sensitivity function. In the experiment, the dSpace 1103 on TMS320C240 DSP board was used to implement the controller, and an LDV was used to measure the actuator displacement, as shown in Figure 8.10. Channel 2 over Channel 1 of DSA is the measured frequency response of the sensitivity function with a swept sine signal as the reference. Figure 8.11 shows the experimental sensitivity functions, which agree with the simulation results in Figure 8.8. From (8.1), the σ values of the position errors before and after the KYP Lemma-based control versus frequencies are obtained and shown in Figure 8.12. It can be seen that the performance with the KYP Lemma-based control is much better from 5 kHz onwards, and slightly worse before 5 kHz, which is consistent with the sensitivity functions in Figure 8.8.

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10 Before KYP design After KYP design 0

Magnitude(dB)

−10

−20

−30

−40

−50 1 10

2

10

3

10 Frequency(Hz)

4

10

FIGURE 8.8 Sensitivity functions before and after the KYP lemma-based design: simulation result.

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Generalized KYP Lemma-Based Loop Shaping Control Design

50

Magnitude(dB)

40 30 20 10 0 −10 −20 1 10

After KYP optimization Before KYP optimization 2

3

10

10

4

10

0

Phase(deg)

−200

−400

−600

−800 1 10

2

3

10

10

4

10

Frequency(Hz)

FIGURE 8.9 Open-loop Bode plot before and after the KYP lemma-based design.

FIGURE 8.10 Structure of experimental setup.

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Modeling and Control of Vibration in Mechanical Systems

10 Before KYP design After KYP design 5

Magnitude(dB)

0

−5 < 0dB −10

−15

−20

−25 2 10

3

10 Frequency(Hz)

10

4

FIGURE 8.11 Sensitivity functions before and after the KYP lemma-based design: experimental results. 0.012 Before KYP design After KYP design

0.01

PES σ(µm)

0.008

0.006

0.004

0.002

0

0

2000

4000

6000 Frequency(Hz)

FIGURE 8.12 σ value of PES versus frequency.

8000

10000

12000

Generalized KYP Lemma-Based Loop Shaping Control Design

8.6

177

Application in mid-frequency vibration rejection

The frequency responses of the microactuator are shown in Figure 8.13. Six resonance modes at 3.7, 4.9, 6.9, 9, 12.7 and 15 kHz are included in the model. The disturbance distribution is reflected in the non-repeatable runout power spectrum of the measured PES in Figure 8.14. It is noticed that there is a vibration mode at 650 Hz due to disk vibration. The objective here is to use the above KYP method to design a linear dynamic output feedback controller C(z) for the microactuator in Figure 8.13 such that its closed-loop system is stable and the disturbance centering at 650 Hz is suppressed sufficiently. 45 kHz sampling rate is used in the servo control design. The control algorithm is implemented with the digital position error signal generated from DSP TMS320C6711. Currently, due to the limitation by the DSP speed, with this sampling rate the platform can support up to 10th order controller. Because 650 Hz is at a relative low frequency range, we just involve the static part of the microactuator represented by a pade delay in the control design with the KYP Lemma. After that, notch filters for the resonance modes at 3.7, 9 and 15 kHz will be used to compensate the dynamic part, which will not change a lot the obtained performance of the low frequency part. The 4.9 and 6.9 kHz resonance modes, seen in Figure 8.13, have relatively small magnitudes and can be ignored as long as they are not excited in the control loop. The resonance mode at 12.7 kHz is not considered in the control design as it is not excited easily and does not affect the whole loop stability when the 15 kHz mode is compensated. The pade delay model is given by Ppade−delay = −5.6234

s − 2 · π · 17000 , s + 2 · π · 17000

(8.39)

which is pre-compensated by the proportional-integral (PI) controller: Int(z) = 0.027(−

z + 0.5). z − 0.999

(8.40)

Due to the first order pade-delay model used in the computation of LMIs, the computation of controller can be very efficient. The desired specifications for the sensitivity function S(z) are set as: Spec.(a) |S(f )| < 0 dB, f ≤ 500 Hz, Spec.(b) |S(f )| < −10 dB, 610 Hz ≤ f ≤ 670 Hz, Spec.(c) |S(f )| < 9.54 dB, f ≥ 19 kHz.

Spec. (b) means to attenuate the disturbances centering at 650 Hz by 10 dB at least. The parameters of Q(z) in (8.19) with τ = 1 are attained by solving three LMIs of the form (8.7) corresponding to Spec. (a), (b) and (c). The resultant C(z) is a 10th order controller. For the sake of comparison, the phase-lead peak filter (PLPF) of the form in (8.28) with values K = 0.4, φ = −0.584, ω0 = 2π650, and ξ = 0.0632, is also applied to

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suppress the low frequency disturbances around 650 Hz, and the sensitivity function comparison is shown in Figure 8.15. It can be seen that the KYP method achieves better disturbance rejection from 60 Hz to 1 kHz, although they have almost the same rejection capability in the very narrow band around 650 Hz. However, the KYP method gives a poorer disturbance rejection performance for frequency below 60 Hz than the PLPF method. In the open loop comparison in Figure 8.16, the phase margin (PM) with the PLPF method is much higher, while the bandwidth is lower and the gain margin (GM) is comparable with the KYP Lemma method. Consistent with the sensitivity functions in Figure 8.15, the PES NRRO power spectrum comparison is shown in Figure 8.17 which clearly shows that the KYP based design gives a better disturbance rejection around 650 Hz than the PLPF although at 650 Hz they offer a similar performance. From Figure 2.20, it is known that the spectrum of the true PES y is given by Sy = |P (z)S(z)|2 × |d1 |2 + |S(z)|2 |d2 |2 + |T (z)|2 × |n|2 = Se − |S(z)|2 × |n|2 + |T (z)|2 × |n|2 ,

(8.41) (8.42)

where Se is in (8.1), and T (z) = 1 − S(z) is the closed loop transfer function. Thus the 3σ value of the true PES can be assessed from the power spectrum Se in Figure 8.17 with the known level of noise n. As a result it is improved from 6.4 nm with the PLPF method to 6 nm with the KYP Lemma method. In the above application, only the first order Q(z) is used. A higher order Q(z) offers more design freedom and has the potential of achieving better results. However, whatever Q(z) is used, the resultant sensitivity function has to comply with the Bode integral theorem, meaning that it is not possible to achieve disturbance rejection across the entire frequency range. To further improve the disturbance rejection at low frequency for the KYP Lemma-based design, we shall incorporate a nonlinear compensation in Chapter 13.

8.7

Conclusion

This chapter has applied the generalized KYP Lemma in the microactuator closedloop design to suppress the narrow band disturbances. The system design problem with multiple specifications on the gain properties of the sensitivity function over several frequency ranges has been solved by the LMI optimization based on the KYP Lemma. The Youla parametrization approach has been used in the feedback controller design. Practical applications have been demonstrated for narrowband high frequency and mid-frequency disturbance rejection. The resultant controller verifies that the desired specifications to reject the disturbances have been satisfied via the search for the coefficients of Q(z) in the Youla parametrization approach.

Generalized KYP Lemma-Based Loop Shaping Control Design

40 Measured Modeled

Magnitude(dB)

30 20 10 0 2

10

3

10

4

10

100

Phase(deg)

0 −100 −200 −300 −400 −500 −600 2 10

3

10

Frequency(Hz)

FIGURE 8.13 Frequency response of the PZT microactuator.

4

10

179

180

Modeling and Control of Vibration in Mechanical Systems

−3

x 10

4

NRRO magnitude(µm)

3.5 3 2.5 2 1.5 1 0.5 0

0

1000

2000

3000 4000 Frequency(Hz)

5000

6000

FIGURE 8.14 PES NRRO power spectrum calculated from measured PES signal without servo control, reflecting the vibration distribution of the system (3σ = 21 nm including the noise 3σ = 15.2 nm).

181

Generalized KYP Lemma-Based Loop Shaping Control Design 10 KYP PLPF 0

Magnitude(dB)

−10

−20

−30

−40

−50

−60 1 10

2

10

3

10 Frequency(Hz)

4

10

FIGURE 8.15 Comparison of sensitivity functions.

FIGURE 8.16 Open loop frequency responses (PLPF (GM: 6 dB, PM: 50 deg., Bandwidth 1.4kHz)); KYP(GM: 6 dB, PM: 34 deg., Bandwidth: 1.7 kHz))).

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Modeling and Control of Vibration in Mechanical Systems

−3

x 10

KYP PLPF

1

NRRO magnitude(µm)

0.8

0.6 650Hz

0.4

0.2

0

0

1000

2000

3000

4000

5000

6000

Frequency(Hz)

FIGURE 8.17 NRRO power spectrum with PLPF and KYP (50% reduction before 1 kHz).

9 Combined H2 and KYP Lemma-Based Control Design

9.1

Introduction

As a closed-loop shaping method, the KYP Lemma-based approach allows designers to impose performance requirements over selected finite frequency ranges so as to have the desired sensitivity function that is able to reject the disturbances in these specific frequency ranges. Subsequently, the positioning accuracy can be improved to some extent. However, the KYP Lemma-based loop shaping method does not count for overall positioning error minimization which can be translated into the H2 optimal control problem by taking into consideration the disturbance and noise models. On the other hand, the H2 control design which incorporates all disturbance and noise models can result in an average performance across the entire frequency range and a high order controller. Thus it usually does not have the flexibility to specifically reject disturbances at certain frequency ranges, which however may be dominant factors that influence the overall performance. Therefore there is a need to suppress disturbances of specific frequencies when minimizing the positioning error. This motivates us to incorporate the KYP Lemma-based method with the H2 control method in this chapter. With the selected specific disturbances handled by the KYP Lemma-based design, the H2 control is formulated with a lower order disturbance model, excluding the disturbances covered in the above design. This will not only release the computation burden in the H2 control design but also result in a lower order controller. In this chapter, we will apply the combined control design method to a PZT microactuator such that a disturbance at 650 Hz is rejected with the KYP Lemma-based design and at the same time overall positioning error is minimized via the H2 control design. Then one more disturbance at 2 kHz near the servo bandwidth 1kHz is also considered as a specific disturbance to be rejected via the the KYP Lemma-based design. The design procedure will be illustrated and the resultant controller will be verified via an experiment. A series of simulation and experimental results will show the effectiveness of the control design method in terms of enhancing positioning accuracy.

183

184

9.2

Modeling and Control of Vibration in Mechanical Systems

Problem formulation

FIGURE 9.1 H2 control scheme with Q parametrization for controller design.

In the previous chapter, specifications on sensitivity function S(z) are described as |S(fi )| < ri ,

fi1 < fi < fi2 , i = 1, 2, · · · , m

(9.1)

where ri < 1 is a positive scalar, and fi1 and fi2 define the frequency range. Such an upper-bound specification as in (9.1) will lead to a problem when the frequency fi is larger than and especially near the desired bandwidth or 0-dB crossover frequency of S(z). The 0-dB crossover frequency of S(z) will be pushed away towards a higher frequency, as seen in Figure 9.9, which tends to damage the system stability and deteriorate the system high-frequency performance. In view of this, a lower-bound specification, i.e., |S(fi )| ≥ 1,

fi1 ≤ fi ≤ fi2

(9.2)

is required. This specification helps to fix the bandwidth or 0-dB crossover frequency of S(z), which will be seen later in the application results. The problem of the specific disturbance rejection can be solved by imposing such performance specifications in (9.1) and then using the KYP Lemma-based control design method in Chapter 8. However, as shown in Figure 9.1 associated with Figure 8.14, the servo mechanical system suffers from various kinds of disturbances and sensing noise. The KYP Lemma-based control design cannot include all disturbances and noises which contribute to the position error. In view of this, we also

Combined H2 and KYP Lemma-Based Control Design

185

take into account the overall performance of the servo control system, which is repre T sented as the so-called track misregistration (TMR) induced by w = w1 w2 w3 passing through D1 (s), D2 (s), and N (s). It is expressed by the standard deviation σz of z, and kTzw k2 = σz ,

(9.3)

when w is a white noise with zero mean and identity covariance matrix, where Tzw is the transfer function from w to z. In the next section, we proceed to the controller design to achieve the specifications in (9.1) and (9.2), and meanwhile to optimize (9.3).

9.3

Controller design for specific disturbance rejection and overall error minimization

The generalized KYP Lemma-based design method in Chapter 8 is used to design a controller for specific disturbance attenuation. Let (Ap , Bp , Cp , Dp ) and (Ac , Bc , Cc, Dc ) respectively be the state-space model of plant P (z) and controller C(z). In order to convexify matrix inequalities, the Youla parametrization approach with the Q(z) in a FIR filter form is applied and the controller structure is shown in Figure 9.1. K(z) is an observer based controller that can be designed using the LQG method as in (8.11)−(8.12). For the presentation of the KYP Lemma, we denote  ∗   S S σ(S, Π) := Π (9.4) I I where S(z) = S(ejθ ), I stands for an identity matrix and Π a Hermitian matrix of the form   Π11 Π12 Π= , (9.5) Π∗12 Π22 which specifies the frequency domain property to be investigated.

9.3.1

Q parametrization to meet specific specifications

A. Specification (9.1) ˜ B, ˜ C, ˜ D) ˜ Recall from (8.13)−(8.14) that a set of sensitivity functions S(z): (A, can be Q-parameterized. According to the denotation (9.4)−(9.5), the specification |S(z)| ≤ r is written as σ(S, Π) ≤ 0 with     Π11 Π12 1 0 Π= = . (9.6) Π∗12 Π22 0 − r2

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Modeling and Control of Vibration in Mechanical Systems


Thus based on the KYP Lemma in Chapter 8, achieving S ejθ ≤ r for the frequency range θ1 ≤ θ ≤ θ2 can be obtained by solving the following matrix inequality         ˜ ∗ ˜ ˜ ∗ ˜ A˜ B A˜ B C˜ D C˜ D Σ + Π ≤ 0, (9.7) I 0 I 0 0 I 0 I which is, since Π11 > 0, equivalent to       ˜ ∗ ˜ 0 0 A˜ B A˜ B  I 0 Σ I 0 + 0 −r 2   ˜ Π11 C˜ D



˜ C˜ D

∗

−Π11

where

Σ=



−U

ejθc V U − (2 cos θd ) V





Π11 

≤ 0,

(9.8)

,

(9.9)

θc = (θ1 + θ2 )/2, θd = (θ2 − θ1 )/2,

(9.10)

e−jθc V

U and V are Hermitian matrices and V ≥ 0. To convexify the matrix inequality (9.8), we shall give a state space realization of S(z) = T11 (z) + T12 (z) Q (z) T21 (z). Denote the state-space representation of T11 (z) and T12 (z)T21 (z) by (At11 , Bt11 , Ct11 , Dt11 ) and (At , Bt , Ct , Dt ), respectively. Then a state-space model of S(z) can be written as (8.17)−(8.18). B. Specification (9.2) Again, according to the denotation (9.4)−(9.5) the specification |S(z)| ≥ r is equivalent to σ(S, Π) ≤ 0 with     Π11 Π12 −1 0 Π= = . (9.11) Π∗12 Π22 0 r2 However, because Π11 < 0, (9.7) can not be converted equivalently to (9.8), which means (9.7) is not possibly convexified according to the method in Section 9.3.1. Hence we resort to the following specification 

0 a + jb σ (S, Π) = aR (S) + bI (S) + c, Π := a − jb 2c



(9.12)

where R and I denote the real and the imaginary parts of S(ejθ ). When a, b and c are properly selected, |S(z)| ≥ r can be achieved. A simple selection is a = 0, b = −1, c = r, and σ (S, Π) = −I (S) + r.

(9.13)

Thus σ(S, Π) ≤ 0 means I (S) ≥ r, and subsequently |S(z)| ≥ r. In this situation,     Π11 Π12 0 −j Π= = , (9.14) Π∗12 Π22 j 2r

Combined H2 and KYP Lemma-Based Control Design

187

where Π11 = 0 and (9.7) is equivalent to 

˜ A˜ B I 0

∗

   ˜B ˜ 0 C˜ ∗Π12 A Σ + ˜ ∗ Π12 + Π∗ D ˜ + Π22 ≤ 0, I 0 Π∗12 C˜ D 12 

(9.15)

˜ only, and can which is a linear matrix inequality with unknown variables in C˜ and D be solved using the same method as in Section 9.3.1A. It should be mentioned that R (S) ≥ r can also be used to achieve |S(z)| ≥ r, if it is suitable for a specific application. In this case,     Π11 Π12 0 −1 Π= = , (9.16) Π∗12 Π22 −1 2r and the linear matrix inequality (9.15) remains applicable.

9.3.2

Q parametrization to minimize H2 performance

Next we focus on the design of Q(z) to minimize the H2 norm kTzw k2 . From Figure 9.1 we have −z = N (z) w3 + S (z) [P (z) D1 (z) w1 + D2 (z) w2 − N (z) w3 ] . (9.17) Denote a state-space realization of P (z)D1 (z), D2 (z) and N (z) by (A1 , B1 , C1 , D1 ), (A2 , B2 , C2 , D2 ), and (A3 , B3 , C3 , D3 ), respectively. It follows from (8.13) and (8.17)−(8.18) that ¯ (k) + Bw(k), ¯ x (k + 1) = Ax ¯ (k) + Dw(k), ¯ −z (k) = Cx where, 

A1  0 A¯ =   0 ˜ 1 BC  ˜ 1 C¯ = DC

(9.18) (9.19)

   B1 0 0 0 0 0  A2 0 0 B2 0  , , B ¯= 0 (9.20)  0 0 B3  0 A3 0  ˜ 2 −BC ˜ 3 A˜ ˜ 1 BD ˜ 2 −BD ˜ 3 BD BC    ¯ = DD ˜ 2 −DC ˜ 3 + C3 C˜ , D ˜ 1 DD ˜ 2 −DD ˜ 3 + D3 . DC

The H2 norm kTzw k2 can be minimized as follows: min

(Ξ=ΞT >0, Ω=ΩT >0)

T race (Ω)

(9.21)

subject to A¯T ΞA¯ − Ξ + C¯ T C¯ < 0 ¯ 2 kHz) frequency disturbances, it works like phase-stabilized control [86]. This feature is beneficial to the stability of the closed-loop system.

213

Blending Control for Multi-Frequency Disturbance Rejection 10

0

Magnitude(dB)

−10

−20

−30

−40

−50

−60

−70 1

10

10

2

3

10

4

10

Frequency(Hz)

FIGURE 10.11 Sensitivity function with disturbance rejections at 0.65 and 2 kHz.

10.5

Conclusion

The rejection problem of several disturbances around different frequencies has been formulated as a control blending problem. A controller for each disturbance rejection has been designed individually by using the H2 optimal control method. The ultimate controller has been obtained by blending all these H2 controllers so that all these disturbances can be rejected simultaneously. The control blending method has been respectively applied to design a controller for a 1.8-inch HDD VCM actuator in three cases, (1) rejecting two disturbances of frequencies higher than bandwidth and close to actuator resonance frequencies, (2) rejecting three disturbances with frequencies higher than bandwidth and different from resonance frequencies, and, (3) rejecting one disturbance with frequency lower than bandwidth and another higher than bandwidth. Simulation and experimental results have shown that the control blending technique results in a simultaneous attenuation for these disturbances. In addition, it is worth noting that the method is able to prevent phase loss when it is used to deal with disturbances near the bandwidth.

11 H∞ -Based Design for Disturbance Observer

11.1

Introduction

The idea of observing disturbance to improve the performance of a servomechanism was first introduced in [123]. It was suggested that if the disturbances were supposedly generated by a linear dynamic system and the model of the system was known, they could be estimated from the system output measurement by an asymptotic estimator (Luenberger observer) and the effect of the disturbances could be neutralized by feeding the disturbance estimates back into the system [124]. Over the years, the method has been modified and applied. However, it is not always easy to identify the disturbance model. Further, it is not always true that the disturbance model is linear time-invariant. Subsequently, a new type of disturbance observer (DOB) has been introduced [128]. This new method does not require control designers to have the full information of the disturbance model and does not need the assumption that the disturbance model is linear time-invariant. However, it requires the model of the controlled plant to be accurately known and invertible, at least within the bandwidth of interest [129]. Recently, it has been proven that under certain assumptions imposed on the plant and disturbance models, the two different methods are equivalent in that the original disturbance observer introduced in [123] is actually a generalization of the latter method [126]. In this chapter, we study the latter method where a general form of disturbance observers, which does not need to solve the plant model inverse, will be introduced. If majority of the disturbances are of relatively lower frequency, when a DOB is added on to attenuate the effect of disturbances, the standard and conventional way of designing a Q-filter is to design it to be a low-pass filter with unity DC gain [128]. In this chapter, we introduce the conventional disturbance observer first, and then present a general form of disturbance observer. An H∞ control based method is applied to the design of a Q-filter. The designed disturbance observer is applied to an HDD servo system and its effectiveness in disturbance attenuation is demonstrated by simulations and experiments.

215

216

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Modeling and Control of Vibration in Mechanical Systems

Conventional disturbance observer

Figure 11.1 shows the block diagram of the conventional disturbance observer (DOB) structure, where C is the feedback controller, Pn is the nominal model of the plant P , d1 is the input disturbance, n is the noise, dˆ1 is an estimate of d1 , and Pn−1 is the inverse of Pn . τ represents a delay in the plant P . Theoretically, P (z) = z −τ Pn (z). Ignoring the nominal feedback loop with C, the transfer functions from the disturbance d1 and noise n to the output y are given by

P Pn (1 − Qz −τ ) , Pn + Q(P − Pn z −τ ) −P Q = . Pn + Q(P − Pn z −τ )

Tyd1 =

(11.1)

Tyn

(11.2)

To reject the disturbance d1 , Q can be set as unity because Tyd1 ≈ 0 when the delay is negligible. However, when Q = 1, Tyn ≈ −1 which means the measurement noise n is not attenuated. Thus, to eliminate the noise effect, it is known from (11.2) that an ideal solution of Q is zero, but this will mean Tyd1 ≈ P and hence the disturbance will be amplified.

FIGURE 11.1 Block diagram of the control loop with a conventional disturbance observer.

H∞ -Based Design for Disturbance Observer

217

Considering the overall system in Figure 11.1, the sensitivity function is given by S(z) =

1 − Qz −τ . 1 − Qz −τ + P C + P QPn−1

(11.3)

Theoretically, the plant model is P = Pn z −τ , thus (11.3) becomes S(z) =

1 − Qz −τ , 1 + PC

(11.4)

which is stable as long as Q is stable since the loop is stable before the disturbance observer is added on. Moreover, it is deduced from (11.3) that when Qz −τ = 1 with zero phase around the disturbance frequency, the disturbance can be rejected because S(z) → 0. Hence Q is designed such that the phase of Qz −τ is almost zero degree and the magnitude is close to one in the frequency range where the disturbance d1 dominates. Note that in Figure 11.1, it is needed to solve the inverse Pn−1 of the nominal plant model. In what follows, a general form of the disturbance observer is proposed and the plant model inverse is not needed.

11.3

A general form of disturbance observer

Let Pn (z) = B(z)/A(z), B(z) = bm z m + bm−1 z m−1 + · · · + b0 , A(z) = z n + an−1 z n−1 + · · · + a0

(11.5)

be the nominal model of the plant P . With the same notations as in Figure 11.1, Figure 11.2 displays a general form of disturbance observer, where M (z) =

A(z) B(z) , N (z) = d , z dm z n

(11.6)

and d2 is the output disturbance. In the conventional disturbance observer in Figure 11.1, M (z) = z −τ , N (z) = Pn−1 (z).

(11.7)

Denote the state-space descriptions P (z) : (Ap , Bp , Cp , Dp ); C(z) : (Ac , Bc , Cc , Dc ); M (z) : (AM , BM , CM , DM ); N (z) : (AN , BN , CN , DN ). From Figure 11.2, x(k + 1) = Ax(k) + B1 w(k) + B2 uq (k), yq (k) = Cy x(k) + Dyw w(k) + Dyu uq (k), y(k) = Cz x(k),

(11.8) (11.9) (11.10)

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 11.2 Block diagram of the control loop with a general disturbance observer. where 

 Ap − Bp Dc Cp Bp Cc 0 0  −Bc Cp Ac 0 0   A=  −BM Dc Cp BM Cc AM 0  , −BN Cp 0 0 AN     −Bp Bp −Bp Dc Bp Dc  0   0 −Bc Bc     B1 =   0 −BM Dc BM Dc  , B2 =  −BM  , 0 −BN BN 0   Cy = −DN Cp − DM Dc Cp DM Cc CM CN ,   Dyw = 0 −(DN + Dm Dc ) DN + DM Dc , Dyu = −DM ,   Cz = −Cp 0 0 0 , Dzw = [0 − 1 0], Dzu = 0,

T

[xTp (k)

xTc (k) T

xTM (k)

(11.11)

(11.12) (11.13)

xTN (k)]

and x (k) = is the augmented state of P (z), C(z), M (z), and N (z), and w (k) = [d1 (k) d2 (k) n(k)]. Denote the transfer function from w to y as Tyw = [Tyd1 Tyd2 Tyn ]. The H∞ optimization method will be applied to design Q(z) to minimize the H∞ norm

Wd1 Tyd1

Wd2 Tyd2 . (11.14)

Wn Tyn ∞

Wd1 , Wd2 and Wn are weightings. Here the models for disturbances d1 , d2 and noise n are not needed. However, in order to have a desired suppression of the

H∞ -Based Design for Disturbance Observer

219

disturbances d1 , d2 and the noise n, appropriately chosen weightings Wd1 , Wd2 and Wn are required which relies on our knowledge of the disturbances. Different from the conventional disturbance observer, the general disturbance observer does not need the inverse of the nominal plant model. As shown later, it is able to suppress the disturbances in a low frequency range without much performance degradation to higher frequency disturbances and noise. The objective of the general disturbance observer design is then stated as: Given a positive scalar γ and appropriate weightings Wd1 , Wd2 and Wn , design a stable Q(z) : (AQ , BQ , CQ , DQ ) such that k[Wd1 Tyd1 Wd2 Tyd2 Wn Tyn ]T k∞ < γ.

(11.15)

The H∞ control design problem can be solved via Theorem 5.4. REMARK 11.1 The sensitivity function of the control system in Figure 11.2 with the general disturbance observer is given by S(z) =

1 + QM . 1 + P C + QM − P QN

(11.16)

With the conventional disturbance observer in (11.7), (11.16) is equal to S(z) =

1 + Qz −d . 1 + P C + Qz −d − P QPn−1

(11.17)

Assume that in (11.16) Q(z) = Qn (z) ×

z dn , P = Pn , B(z)

(11.18)

then S(z) =

1 + Qn z dn /z dm , 1 + P C + Qn z dn /z dm − Qn

(11.19)

which recovers the form in (11.17) when dm ≥ dn , meaning that the conventional disturbance observer is a special case of the general disturbance observer. REMARK 11.2 In Figure 11.2, if the loop is cut at u, the transfer function from u to e˜ or y is considered as a new plant, and derived as Pequ =

P y = . u 1 − (M − P N )Q

(11.20)

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An equivalent open loop is denoted by TEQ−OL (z), and TEQ−OL (z) = Pequ C =

PC . 1 − (M − P N )Q

(11.21)

Assume ideally that Pn = P and dm = dn . Equation (11.20) becomes Pequ = P , which implies that the proposed general disturbance observer does not influence the characteristics of open-loop P (z)C(z) greatly and thus the stability and performance achieved by the nominal control loop with controller C(z) are maintained. This is similar to the conventional disturbance observer.

11.4

Application results

It has been shown that a disturbance observer is capable of estimating disturbances and modeling error [128]. Hence, disturbance observers can be used to increase the R/W head-positioning accuracy in hard disk drives (HDDs) by using its estimation results to cancel the effect of the disturbances and modeling error. Further, due to its cost effectiveness and easy “add-on” implementation with minimal change required to the existing feedback controller, the disturbance observer without using additional sensors is frequently used to enhance the tracking performance of a hard disk drive servo system, such as the attenuation of disturbances [127], and the compensation of VCM pivot friction [130]. Consider the VCM plant with model P (s) given in Chapter 10. The discretized model with the sampling time Ts = 1/30000 sec is given by (11.5) with B(z) = 0.02399z 5 + 0.1868z 4 − 0.03483z 3 − 0.02165z 2 + 0.1728z + 0.02025, (11.22) A(z) = z 6 − 3.052z 5 + 4.657z 4 − 4.979z 3 + 3.997z 2 − 2.399z + 0.7765. (11.23) The controller C(z) is the combination of a PID controller and two notch filters, and given by 1.474 × 10−5 z 6 − 4.416 × 10−5 z 5 + 6.517 × 10−5 z 4 − 6.621 × 10−5 z 3 +5.015 × 10−5 z 2 − 2.872 × 10−5 z + 9.053 × 10−6 C(z) = . 3.333 × 10−5 z 6 − 7.59 × 10−5 z 5 + 8.081 × 10−5 z 4 − 4.846 × 10−5 z 3 +1.545 × 10−5 z 2 − 5.234 × 10−6 z

(11.24)

Let M (z) =

A(z) B(z) , N (z) = 6 , z6 z

(11.25)

H∞ -Based Design for Disturbance Observer

221

Wd1 = 0.3, Wd2 = 1 and Wn = 1.5. A stable Q(z) is then obtained via the H∞ optimization in (11.14) and its frequency response is shown in Figure 11.3. Note that the plant model inverse is not required in the general disturbance observer. This benefit is of great significance, especially for nonminimum phase plant. The sensitivity function |S(z)| is plotted in Figure 11.4, from which it can be seen that the designed disturbance observer is able to suppress disturbance with frequency lower than 1 kHz without causing much degradation for rejection of higher frequency disturbance. The servo performance, such as bandwidth, will change with different weightings Wd1 , Wd2 and Wn . Thus by adjusting the weightings according to the weights of d1 , d2 , and noise n in the position error signal, the designed disturbance observer will result in a desired reduction rate of the error. To demonstrate the effectiveness of the disturbance estimation in the time domain, we assume that the disturbances d1 and d2 and the noise n are generated by d1 = D1 (s)w 1 , d2 = D2 (s)w2 and n = Nn (s)w3 , where D1 (s) =

0.0004(s2 − 83.39s + 9.741 × 105 )(s2 + 1616s + 9.626 × 106 ) , (s2 + 125.7s + 3.948 × 105 )(s2 + 10.05s + 1.011 × 106 )

D2 (s) and Nn (s) are in (2.60) and (2.61), and wi (i = 1, 2, 3) are independent white noises with unity variance. With the designed general disturbance observer, the estimate dˆ1 of d1 is shown in Figure 11.6. It follows the original d1 approximately. As a result, the error signal is shown in Figure 11.7. 50% reduction is achieved. The general disturbance observer is more effective to compensate for the input disturbance d1 than d2 and n. Wd2 and Wn are selected as 1 and 1.5 is to keep the attenuation to d2 and n achieved by the nominal feedback controller C(z). With lower Wd2 and Wn and higher Wd1 , the attenuation to d2 and n will be degraded, although more suppression to d1 will be attained by using the disturbance observer. Moreover, the conventional disturbance observer is designed for comparison. M (z) and N (z) are given by M (z) = z −1 ,

(11.26)

N (z) = Pn−1 (z) 5.2494(z 2 − 1.983z + 0.9834)(z 2 − 1.253z + 0.9654)(z 2 + 0.1836z + 0.8179) . = z(z + 0.9501)(z + 0.1259)(z + 0.116)(z 2 − 1.22z + 0.9646) (11.27) A stable Q(z) for the conventional disturbance observer is designed with the H∞ control method. Figure 11.5 shows the resultant sensitivity function, which is similar to the one from the general disturbance observer. However, the plant model inverse needs to be calculated. Experiment has been done with a LDV and a dSpace 1103. The measured sensitivity functions are shown in Figure 11.8, which agree with the simulation results in Figure 11.4. To evaluate the effect of the disturbance observer on the stability and performance achieved by the nominal controller C(z), TEQ−OL is measured with the

222

Modeling and Control of Vibration in Mechanical Systems

general disturbance observer and the conventional disturbance observer, and shown in Figure 11.9 and Figure 11.10. As expected, performance measures such as gain margin and phase margin are not affected.

Bode Diagram 40 35 30

Magnitude (dB)

25 20 15 10 5 0 −5 −10 315

Phase (deg)

270 225 180 135 90 45 1

2

10

3

10

10

4

10

Frequency (Hz)

FIGURE 11.3 Frequency response of the designed Q(z).

11.5

Conclusion

A general form of disturbance observer has been presented and designed based on the H∞ control method to achieve desired disturbance/noise rejection. The disturbance observer does not need to solve the plant model inverse, and thus its design is simplified and has great advantages over the conventional disturbance observer, especially for nonminimum phase plant. The simulation and implementation results show that the general disturbance observer designed using the method employed in this chapter is able to effectively improve the attenuation of disturbance in low frequency, and will not sacrifice the stability and performance of the nominal feedback control loop.

H∞ -Based Design for Disturbance Observer

223

10

0

Magnitude(dB)

−10

−20

−30

−40

No DOB With DOB −50

2

3

10

4

10 Frequency(Hz)

10

FIGURE 11.4 The sensitivity functions without and with the general disturbance observer.

10

0

Magnitude(dB)

−10

−20

−30

−40 Conventional DOB General DOB −50 1 10

2

3

10

10

4

10

Frequency(Hz)

FIGURE 11.5 The sensitivity function comparison with the general and the conventional disturbance observers.

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Modeling and Control of Vibration in Mechanical Systems

−3

Amplitude of disturbance d1 and its estimate (µm)

8

x 10

d1 estimate of d1

6 4 2 0 −2 −4 −6 −8

0

0.05

0.1

0.15 Time(sec)

0.2

0.25

0.3

FIGURE 11.6 Disturbance d1 .

0.03 No DOB With DOB

Amplitude of error e (µm)

0.02

0.01

0

−0.01

−0.02

−0.03

0

FIGURE 11.7 Error signal e.

0.05

0.1

0.15 Time(sec)

0.2

0.25

0.3

H∞ -Based Design for Disturbance Observer

225

10

0

Magnitude(dB)

−10

−20

−30

−40 No DOB With general DOB −50

2

3

10

4

10 Frequency(Hz)

10

FIGURE 11.8 Measured sensitivity functions without and with the general disturbance observer.

Magnitude(dB)

60 Nominal with the general DOB

40 20 0 −20 −40 1

10

2

3

10

10

4

10

0 Phase(deg)

−100 −200 −300 −400 −500 1 10

2

3

10

10

4

10

Freuqency(Hz)

FIGURE 11.9 Comparison of TEQ−OL about the general disturbance observer.

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Modeling and Control of Vibration in Mechanical Systems

Magnitude(dB)

60 Nominal with the conventional DOB

40 20 0 −20 −40 1

10

2

3

10

10

4

10

0 Phase(deg)

−100 −200 −300 −400 −500 1 10

2

3

10

10

4

10

Freuqency(Hz)

FIGURE 11.10 Comparison of TEQ−OL about conventional disturbance observer.

12 Two-Dimensional H2 Control for Error Minimization

12.1

Introduction

The H2 optimal control for 1-D systems is a classical problem in linear systems theory. Its objective is to minimize the error energy of the system when the system is subject to a unit impulse input or, equivalently, a white noise input of unit variance. Because of this analytically and practically meaningful specification, the H2 problem and solution has been well studied and applied for several decades. Recently, the H2 control problem has been studied for 2-D systems and a sufficient condition for the evaluation of 2-D system H2 performance in terms of LMIs is derived [139]. Using the condition, a systematic method for the design of the H2 controller for 2D systems in terms of LMIs has been developed. The developed 2-D H2 control design method is of great importance to those systems that have 2-D behavior and can be modeled using 2-D linear system models. Self-servo track writer (SSTW) for data storage devices is one of these systems [17]. In the self-servo track writing, due to vibrations and noise the servo controller causes the actuator to follow the resulting non-circular trajectory in the next burst writing step, so that the new bursts are written at locations reflecting the errors present in the preceding track via the closed-loop response of the servo loop, as well as in the present track. Consequently, each step in the process carries a “memory” of all preceding track shape errors. This “memory” depends on the particular closed-loop response of the servo loop. Because of the interdependency of propagation tracks, track shape errors may be amplified from one track to the next through the closed-loop response when writing the propagation tracks. Thus self-servo writing systems must provide a means of accurately writing servo-patterns while controlling the propagation of track shape errors. Therefore, error propagation containment is critically important. The target of preventing error propagation is to reject the track shape error due to track noncircularity recorded in propagated tracks so that the circular concentric tracks are achieved in every propagation trace. In this chapter we describe the SSTW process with a two-dimensional (2-D) model. Then the error propagation containment problem of the SSTW process is formulated as a 2-D stabilization problem. Instead of the conventional feedforward control, the 2-D stabilizing control is able to prevent the error propagation. Furthermore, the

227

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Modeling and Control of Vibration in Mechanical Systems

TMR minimization problem of the SSTW process is formulated as a 2-D H2 control problem. A 2-D H2 controller is designed which is able to prevent the error propagation and minimize the TMR.

12.2

2-D stabilization control

We consider the following 2-D system model [138]:  h   h  x (i + 1, j) x (i, j) =A v + B1 w(i, j) + B2 u(i, j), xv (i, j + 1) x (i, j)  h  x (i, j) + D11 w(i, j) + D12 u(i, j), y(i, j) = C1 v x (i, j)  h  x (i, j) e(i, j) = C2 v + D21 w(i, j) + D22 u(i, j), x (i, j)

(12.1) (12.2) (12.3)

where xh ∈ Rn1 , xv ∈ Rn2 , w(i, j) ∈ Rq , u(i, j) ∈ Rm , y(i, j) ∈ Rp and e(i, j) ∈ Rl are, respectively, the horizontal state, the vertical state, the disturbance input, the control input, the controlled output, and the measurement of the plant. Let  h  x (i, j) x(i, j) = , (12.4) xv (i, j) the above system is equivalent to x(i + 1, j + 1) = A1 x(i, j + 1) + A2 x(i + 1, j) +B11 w(i, j + 1) + B12 w(i + 1, j) +B21 u(i, j + 1) + B22 u(i + 1, j), y(i, j) = C1 x(i, j) + D11 w(i, j) + D12 u(i, j), e(i, j) = C2 x(i, j) + D22 u(i, j)

(12.5) (12.6) (12.7)

where 

   In1 0 0 0 A1 = A, A2 = A, 0 0 0 In2     In1 0 0 0 Bk1 = Bk , Bk2 = Bk , k = 1, 2. 0 0 0 In2

(12.8)

Introduce the following 2-D output feedback controller C(z1 , z2 ): xc(i + 1, j + 1) = Ac1 xc(i, j + 1) + Ac2 xc (i + 1, j) +Bc1 e(i, j + 1) + Bc2 e(i + 1, j), u(i, j) = Ccxc (i, j) + Dc e(i, j).

(12.9) (12.10)

Two-Dimensional H2 Control for Error Minimization

229

Associated with the 2-D controller (12.9)−(12.10), the 2-D stabilization problem is stated as follows: for the 2-D system (12.1)−(12.3) or (12.5)−(12.7) with w(i, j) = 0, design a dynamic output feedback controller of the form in (12.9)−(12.10) such that the resulting closed-loop SSTW servo system is asymptotically stable. Define Z = Dc CR + Cc ΞT , Vk = SB2k Dc + ΛBck , (12.11) T T Uk = S(Ak + B2k Dc C)R + SB2k Cc Ξ + ΛBck CR + ΛAck Ξ , k = 1, 2, (12.12) where R > 0, S > 0 and Ξ and Λ are invertible matrices satisfying ΞΛT = I − RS. THEOREM 12.1 [136] Consider the 2-D system (12.5)−(12.7). Then, there exists a full order output feedback controller of the form in (12.9)−(12.10) that asymptotically stabilizes the system (12.5)−(12.7) if there exist matrices R > 0, S > 0, ˜ X > 0, Dc , Uk , Vk (k = 1, 2) and Z such that the following LMI holds: Ω   ˜X ˜T −Ω 0 Ω A1 ˜F − Ω ˜ X) Ω ˜T  < 0  0 −(Ω (12.13) A2 ˜ ˜ ˜ ΩA1 ΩA2 −ΩF where

    SA1 + V1 C U1 ˜F = S I , Ω ˜ A1 = Ω , I R A1 + B21 Dc C A1 R + B21 Z   SA2 + V2 C U2 ˜ A2 = Ω . A2 + B22 Dc C A2 R + B22 Z

(12.14)

In this situation, the controller parameters of (12.9)−(12.10) can be given by Cc = (Z − Dc CR)Ξ−T , Bck = Λ−1 (Vk − SB2k Dc ),

Ack = Λ

12.3

−1

T

[Uk − S(Ak + B2k Dc C2 )R − SB2k Cc Ξ − ΛBck C2 R]Ξ

(12.15) −T

, k = 1, 2. (12.16)

2-D H2 control

Let Tyw : w → y denote the closed-loop system subject to the white noise w. The H2 norm of Tyw is approximately given by v u X u 1 1 L,K−1 kTyw k2 = t y(i, j)2 (12.17) LK−1 i, j=1

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where L and K are large enough. The control design problem to minimize the 2D H2 norm is stated as follows: find a 2-D output feedback controller of the form in (12.9)−(12.10) for the 2-D system (12.1)−(12.3) or (12.5)−(12.7) such that the closed-loop system is stable and the H2 performance kTyw k2 is minimized. An LMI approach will be given as follows to design a 2-D H2 controller for the 2-D system (12.1)−(12.3) such that the closed-loop system is stable and the error is minimized. THEOREM 12.2 [139] The 2-D H2 control problem for the plant (12.1)−(12.3) is solvable if there exist matrices S > 0, Θ, Λ, Γ, Dc and block-diagonal matrices X = h diag{X h , X v }, Y = diag{Y h , Y v }, M = diag{M h , M v }, H11 = diag{H11 , v h v h v H11 } > 0, H22 = diag{H22 , H22} > 0, and H12 = diag{H12, H12}, of appropriate dimensions, such that   H11 ∗ ∗ ∗ ∗ T  H12 H22 ∗ ∗ ∗   T  XA + ΓC2  > 0, Θ X + X − H ∗ ∗ 11   T  A + B2 Dc C2 AY T + B2 Λ I + M T − H12 Y + Y T − H22 ∗  C1 + D12 Dc C2 C1 Y T + D12 Λ 0 0 I (12.18)   S ∗ ∗ ∗  XB1 + ΓD21 X + X T − H11 ∗ ∗   > 0, (12.19) T T  B1 + B2 Dc D21 I + M T − H12 Y + Y − H22 ∗  D11 + D12 Dc D21 0 0 I T race(S) < λ2 ,

(12.20)

are satisfied. If the above stated conditions are satisfied, a feasible H2 controller is given by  h   h  xc (i + 1, j) xc (i, j) = Ac v + Bc e(i, j), xhc ∈ Rnh , xvc ∈ Rnv , (12.21) xvc (i, j + 1) xc (i, j)  h  xc (i, j) u(i, j) = Cc v + Dc e(i, j), (12.22) xc (i, j) with Ac = U −1 [Θ − X(A + B2 Dc C2 )Y T − XB2 CcV T − U Bc C2 Y T ]V −T ,(12.23) Cc = (Λ − Dc C2 Y T )V −T , Bc = U −1 (Γ − XB2 Dc ), (12.24) where U = diag{U h , U v } and V = diag{V h , V v } satisfy XY T + U V T = M . REMARK 12.1 The solution of the 2-D controller in the above theorem is in terms of LMIs which can be efficiently solved by convex optimization just

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231

like other 1-D control problems involving LMIs. In [139], it is proved that if there exist solutions X, Y , M , Dc , S > 0, Θ, Λ, Γ, H11, H22 , and H12 for the LMIs (12.18)-(12.20), M − XY T is invertible. From U V T = M − XY T , nonsingular matrices U and V can be computed. Then, the computation of the controller parameters Cc , Ac , Bc can be carried out by solving (12.23) and (12.24).

12.4

SSTW process and modeling

FIGURE 12.1 SSTW process.

The process of self-servo writing is shown in Figure 12.1 and is generally known to involve the following distinct steps [135]: writing some tracks or at least one track called seed tracks; reader reads back the seed track and writer writes actual product servopattern for the next track based on the readback signal; writing servopattern for

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the next track based on the readback signal from the previous written track till the whole process is completed. During the process, SSTW generates radial information progressively to deploy servopattern. The radial error in track N is inevitably compounded to the following tracks: tracks N + 1, N + 2, · · ·. This leads to error propagation in SSTW. In self-servo track writing, track shape errors such as non-circularity are introduced by mechanical disturbances, spindle motor vibration and other factors when writing the propagation tracks. The servo controller causes the actuator to follow the resulting non-circular trajectory in the next burst writing step, so that the new bursts are written at locations reflecting the errors present in the preceding step via the closed-loop response of the servo loop, as well as in the present step. Consequently, each step in the process carries a “memory” of all preceding track shape errors. This “memory” depends on the particular closed-loop response of the servo loop. Self-servo writing systems must provide a means of accurately writing servopatterns while controlling the propagation of track shape errors.

12.4.1

SSTW servo loop

FIGURE 12.2 SSTW servo loop with disturbances and noise models.

Figure 12.2 shows the SSTW servo loop with disturbances and noises. D1 (s), D2 (s), and N (s) are respectively the models of disturbances d1 , d2 and noise n. y(k) is the position of the write head with respect to a perfectly circular track on the disk and P ES(k) is the position error signal. Let Tp be the rotational period of the disk, and Ts be the sampling rate of the position error signal. Then the sector number K = Tp /Ts . y(k − K) represents the track profile of the previous track. Similarly, P ES(k −K) represents the position error when writing the previous track. The read head follows on the track y(k − K) which is the reference input for the SSTW servo system, i.e., one revolution of y(k) becomes the reference of the next written track

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233

or a few subsequent tracks due to the action of self-servo writing. It follows from Figure 12.2 that y(k) = T (z)y(k − K) + P (z)S(z)d1 (k) + S(z)d2 (k) + T (z)n(k), (12.25) where S(z) = 1/(1 + P (z)C(z)) is the sensitivity function, T (z) = 1 − S(z) is the closed-loop transfer function. The typical closed-loop transfer function will amplify the error at the frequency where its magnitude is more than 0 dB. In other words, these frequency components in disturbances or noise will be amplified during propagation, while others will compound to following tracks and decay gradually. With the same disturbance models D1 (s) and D2 (s) and noise model N (s) as in Chapter 11, Figure 12.3 shows PES NRRO in the time domain and its σ value versus track number when a 1-D feedback controller C(z) is used in the closed loop. The error propagation mentioned previously is clearly observed. Hence the error propagation problem must be addressed in the servo control design for SSTW.

0.06 Position error signal Sigma value

0.04

PES and σ of PES

0.02

0

−0.02

−0.04

−0.06

0

1

2

3

4

5 Track number

6

7

8

9

10

FIGURE 12.3 PES NRRO and its σ values versus track number during propagation. (The time sequence and the σ value increase with the track number.)

12.4.2

Two-dimensional model

For the 2-D modeling, the following notations are used:

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FIGURE 12.4 SSTW servo loop modeling in two dimensions. i = 0, 1, 2, · · ·, L: the ith track. j · Ts : time with j = 0, 1, 2, · · · , K − 1. As shown in Figure 12.4, y(i, j) is the position of the write head at track i in the radial dimension and time j · Ts in the axial dimension and P ES(i, j) the position error. y(i − 1, j) (i.e., y(j − K)) represents the track profile of the (i − 1)-th track. Similarly, P ES(i − 1, j) represents the position error of the (i − 1)-th track. The read head follows the track y(i − 1, j) which is the reference input for the SSTW servo system, i.e., one revolution of y(i, j) becomes the reference of the next written track due to the action of self-servo track writing. Based on Figure 12.4, we have xp (i, j + 1) = Ap xp (i, j) + Bp (u(i, j) + d1 (i, j)), y(i, j) = Cp xp (i, j) + Dp (u(i, j) + d1 (i, j)) + d2 (i, j), P ES(i, j) = y(i − 1, j) − y(i, j) + n(i, j).

(12.26) (12.27) (12.28)

Denote xp , xd1 , xd2 and xn the corresponding state vectors of P (z), D1 (z), D2 (z) and N (z), respectively. Let xh (i + 1, j) = y(i, j), xh ∈ Rnh , xv ∈ Rnv , xv (i, j + 1)T = [xp (i, j + 1)T xd1 (i, j + 1)T xd2 (i, j + 1)T xn (i, j + 1)T ], e(i, j) = P ES(i, j), w(i, j)T = [w1 (i, j)T w2 (i, j)T w3 (i, j)T ], (12.29) it follows from (12.26)−(12.28) that 

  h  xh (i + 1, j) x (i, j) =A v + B1 w(i, j) + B2 u(i, j), xv (i, j + 1) x (i, j)  h  x (i, j) y(i, j) = C1 v + D11 w(i, j) + D12 u(i, j), x (i, j)

(12.30) (12.31)

Two-Dimensional H2 Control for Error Minimization  h  x (i, j) e(i, j) = C2 v + D21 w(i, j) + D22 u(i, j), x (i, j)

235 (12.32)

where   Dp Dd1 Dd2 0 Cp Dp Cd1 Cd2 0  Bp Dd1 0  0 Ap Bp Cd1 0 0      0 A= 0 0   , B1 =  Bd1  0 0 Ad1  0 0 0 Bd2 0 Ad2 0  0 0 0 0 0 0 An   Dp  Bp       B2 =   0  , C1 = 0 Cp Dp Cd1 Cd2 0 ,  0  0   D11 = Dp Dd1 Dd2 0 , D12 = Dp ,   C2 = 1 − Cp − Dp Cd1 − Cd2 Cn ,   D21 = −Dp Dd1 − Dd2 Dn , D22 = −Dp . 

 0 0   0  , 0  Bn

(12.33)

(12.34)

(12.35) (12.36) (12.37)

As such, the SSTW servo loop is modeled as the 2-D Roesser model (12.30)−(12.37) [138] where disturbance and noise models are taken into consideration. The SSTW error propagation problem can then be simplified as the stabilization problem of a 2-D system. Unlike 1-D feedback control plus feedforward compensation, it does not need an additional feedforward controller to prevent the error propagation. In the next section, feedforward compensation on the basis of 1-D feedback control will be presented, followed by 2-D control in a later section.

12.5

Feedforward compensation method

To contain the error propagation, it is easy to come up with the idea of injecting a correction signal f(k) to the PES by using the available signal P ES(k −K) through a feedforward compensator F (z). Thus a feedforward (FF) compensation method is used as shown in Figure 12.5. During the servo writing process, by tracking the previously written adjacent tracks, adjusting servo reference and closing the existing VCM loop, the head can be offset to a known radial position with reference to the adjacent tracks and generates radial information progressively to deploy servo pattern. The radial error in one track is inevitably compounded to the following tracks and propagates according to the closed-loop transfer function from y(k − K) to y(k), as seen in (12.25). The target of error propagation containment is thus to reject the written-in error due to track non-circularity recorded in propagated tracks so that the circular concentric

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 12.5 SSTW servo loop.

tracks are achieved in every propagation trace. Obviously, when there is no hump in the closed-loop transfer function T (z), the error propagation will be contained. Next, we use three control schemes for the self-servo track writer: (1). PD control since it can produce a flat closed-loop response such that the error propagation is contained even without a feedforward compensation; (2). PID control plus feedforward compensation; (3). To minimize TMR or equivalently the H2 norm of the transfer function from w = [w1 w2 w3 ]T to track profile y, the H2 control technique is employed to design the feedback controller C(z) and then a feedforward compensator F (z) is designed to contain the error propagation. The frequency response of the VCM plant under consideration is shown in Figure 12.6, and its transfer function P (s) is described by the following zeros, poles and gain: zeros = 104 × [3.4558, − 0.6158 ± 8.7749j, 0.7540 ± 4.9697j, −0.1960 ± 3.2614j, −0.9425 ± 1.6324j], (12.38) 5 poles = 10 × [−0.2513, − 0.3456, − 0.0726 ± 1.0342j, − 0.0090 ± 0.5968j, −0.0377 ± 0.3751j, − 0.0283 ± 0.2813j, −0.0011 ± 0.0036j], gain = 5.8987 × 1012 .

(12.39) (12.40)

(1). PD feedback control A typical form of PID controllers is given by C(z) = Kp + Ki

z z−1 + Kd . z−1 z

(12.41)

Let Ki = 0, a PD controller leading to a flat closed-loop transfer function can be obtained by adjusting Kp and Kd . Denote the closed-loop transfer function as

Two-Dimensional H2 Control for Error Minimization

237

60 Measured Modeled

Magnitude(dB)

40 20 0 −20 −40 −60

2

10

3

10 Frequency(Hz)

4

10

Phase(deg)

0 −200 −400 −600 −800 −1000

2

10

3

10

FIGURE 12.6 Frequency response of a VCM actuator.

4

10

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 12.7 Frequency response of the closed-loop transfer function with the PD controller, PID controller, and H2 controller. TP D (z). Figure 12.7 shows a flat magnitude response of TP D (z) resulting from the PD controller given below. The sampling rate is 12.64 kHz. C(z) =

4.79 × 10−5 − 9.573 × 10−5 + 4.783 × 10−5 . z2 − z

(12.42)

Theoretically the error propagation would disappear as kTP D k∞ ≤ 1. However, the PD control is not able to deal with bias force due to lack of integrator. Simulation shows that the propagation stops after several tracks when a small bias force of 0.001 is added. Therefore, PD control is not practically applicable. (2). PID feedback control plus feedforward compensation The closed-loop transfer function with the PID controller C(z) =

2.435 × 10−5 z 2 − 4.406 × 10−5 z + 1.999 × 10−5 z 2 − 1.209 + 0.24

(12.43)

has a region greater than 0 dB as seen in Figure 12.7. As mentioned earlier, this will cause the error propagation problem. Thus a feedforward compensator F (z) will be designed to contain the error propagation. According to Figure 12.5,

y(k) =



PC F + 1 + PC 1 + PC



y(k − K) +

P 1 d1 (k) + d2 (k) 1 + PC 1 + PC

Two-Dimensional H2 Control for Error Minimization +

239

PC PF F n(k) − d1 (k − K) − d2 (k − K), (12.44) 1 + PC 1 + PC 1 + PC

which means that if the feedforward compensator F (z) is designed such that the magnitude of the transfer function Φ=

PC F + 1 + PC 1 + PC

(12.45)

is less than one, i.e., kΦk∞ < 1, the error propagation can be contained. It is straightforward from (12.45) that F (z) = Φ(z)(1 + P (z)C(z)) − P (z)C(z).

(12.46)

When Φ is selected as the closed-loop transfer function with the previous PD control, a 7th order F (z) is obtained from (12.46) after order reduction. The designed F (z) can prevent the error propagation when the PID feedback control is applied in the servo loop since kΦk∞ < 1 with the designed F (z), as observed in Figure 12.8. Figure 12.9 shows the 3σ value of the PES versus frequency with error propagation containment.

1

0

−1 FF with H2 −2

FF with PID

Magnitude(dB)

−3

−4

−5

−6

−7

−8

−9 1 10

2

3

10

10 Frequency(Hz)

FIGURE 12.8 |Φ| versus frequency.

4

10

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14 PID 12

10 3σ (percent of track)

H2 8

6

4

2

0

1000

2000

3000

4000

5000

6000

7000

Frequency(Hz)

FIGURE 12.9 3σ of PES NRRO. (3). H2 feedback control plus feedforward compensation With the identified disturbance models in Chapter 2, the augmented system for the optimal H2 control design is described as follows. x(k + 1) = Ax(k) + B1 w(k) + B2 u(k), P ES(k) = C1 x(k) + D11 w(k),

(12.47) (12.48)

z(k) = C2 x(k) + D21 w(k) + D22 u(k),

(12.49)

where 

  Bp Dd1 Ap Bp Cd1 0 0  0 Ad1  Bd1 0 0     , B1 =  A= 0 0 Ad2 0  0 0 0 0 An 0     C1 = Cp 0 Cd2 Cn , D11 = 0 Dd2 Dn ,   C2 = Cp 0 Cd2 0 , D21 = 0, D22 = 0,

0 0 Bd2 0

   0 Bp   0   , B2 =  0  ,(12.50)   0 0  Bn 0

(12.51)

(12.52)

x is the combined state variables from the VCM actuator model P (z), the input disturbance model D1 (z), the output disturbance model D2 (z), and the measurement noise model N (z). (Ap , Bp , Cp , Dp ), (Ad1 , Bd1 , Cd1 , Dd1 ), (Ad2 , Bd2 , Cd2 , Dd2 ), and (An , Bn , Cn , Dn ) are respectively the state-space models of P (z), D1 (z), D2 (z) and N (z). P ES is the measured position error signal and z stands for y in Figure 12.5. The H2 control problem can be solved using the method in Chapter 5.

Two-Dimensional H2 Control for Error Minimization

FIGURE 12.10 Frequency response of the H2 controller.

FIGURE 12.11 Frequency response of the open-loop system with the H2 controller.

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 12.12 Comparison of sensitivity functions.

Figures 12.10 and 12.11 show the frequency responses of the designed H2 feedback controller and the compensated open-loop system with bandwidth of 1.07 kHz, gain margin 7.3 dB, and phase margin 45.9 deg. Figure 12.12 shows the comparison of error rejection functions with the PID control and the H2 control, where we can see that the H2 control outperforms the PID control in error rejection. The frequency response of the closed-loop transfer function with the H2 control is shown in Figure 12.7, which implies that a feedforward compensator is needed to contain the error propagation when the servo loop uses the H2 feedback controller. For the design of the feedforward compensator F (z), the selected Φ depends on the feedback controller and it is found that lower Φ may not give a better error propagation containment. When Φ = 0.9TP D , a satisfactory error propagation containment is guaranteed when the H2 feedback control is applied in the SSTW servo loop. A 6th order F (z) is thus obtained after model reduction. The resultant |Φ| with the designed F (z) is shown in Figure 12.8, and it is seen that |Φ| < 1. Figure 12.13 shows the σ value comparison of PES at different tracks with error propagation containment. It can be observed that PES σ is up and down as track propagation is going on. Compared with that of the PID control, the σ of PES versus track number is improved by around 27% when the H2 control is employed in the SSTW servo loop.

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243

FIGURE 12.13 σ value of PES NRRO versus track number.

12.6

2-D control formulation for SSTW

With the 2-D model (12.30)−(12.37), the SSTW error propagation problem is simplified as the stabilization problem of a 2-D system. That is to design a dynamic output feedback controller of the form in (12.9)−(12.10) for the model (12.30)−(12.32) with w(i, j) = 0 such that the resulting closed-loop system is asymptotically stable. Unlike the design of 1-D feedback plus feedforward compensation previously, it does not need an additional feedforward controller to prevent the error propagation. It is not difficult to see from (12.9)−(12.10) that (Ac2 , Bc2 , Cc , Dc ) is acting along the time direction only, and thus actually it works like the 1-D feedback controller that we are concerned with conventionally. The 2-D controller (12.21)−(12.22) can be written equivalently to the form (12.9)(12.10). Let     I 0 0 0 Ac1 = nh Ac , Ac2 = Ac , (12.53) 0 0 0 Inv     I 0 0 0 Bc1 = nh Bc , Bc2 = Bc . (12.54) 0 0 0 Inv The performance of (Ac2 , Bc2 , Cc, Dc ) can thus be evaluated as a normal 1-D

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control in time dimension. As is known that one of the most important performance measures for SSTW is the track misregistration or TMR, the total amount of random fluctuation about the desired track location. TMR is used to judge the required accuracy of positioning. To achieve a high positioning accuracy, one way in servo control is to minimize TMR, which is expressed as the standard deviation of the true PES, i.e. v u X u 1 1 L,K−1 σy(i,j) = t y(i, j)2 . (12.55) L K − 1 i,j=0

Let Tyw : w → y denote the closed-loop system subject to a white noise w. When L and K are large enough, the H2 norm of Tyw can be approximately given by v u X u 1 1 L,K−1 y(i, j)2 . (12.56) kTyw k2 = t LK −1 i, j=1

Thus, the control design problem to minimize TMR can be treated as a 2-D H2 optimal control problem, which is stated as follows: find a 2-D output feedback controller of the form in (12.9)−(12.10) for the SSTW plant P (s) such that the closed-loop system is stable and the H2 performance kTyw k2 is minimized. The problem of minimizing the TMR of the SSTW process is thus formulated as the 2-D H2 control problem. A 2-D H2 controller will subsequently be designed to minimize the track mis-registration. Note that since the 2-D controller stabilizes the system, it also contains the error propagation. Thus, the 2-D H2 control approach simultaneously addresses the error propagation and TMR minimization problems, which is different from the previous 1-D method where the problems are addressed separately by feedback and feedforward controls.

12.7 12.7.1

2-D stabilization control for error propagation containment Simulation results

The simulation block diagram on 2-D control is shown in Figure 12.14. The simulation is carried out in MATLAB/Simulink. In the simulation, the sector number K = 270, the spindle rotational speed is 7200 RPM, and thus the sampling frequency is 270 × (7200/60) = 32400 Hz. To show the capability of the designed 2-D controller to prevent the error propagation, 100 tracks are propagated in the simulation. The results show that the SSTW process is stabilized and the error propagation is contained. The σ value of PES NRRO is plotted versus the track number in Figure 12.15, which implies that the error amplitude is oscillating steadily.

Two-Dimensional H2 Control for Error Minimization

245

FIGURE 12.14 2-D controller for SSTW servo loop.

From (12.9)−(12.10) and Figure 12.14, it is not difficult to see that (Ac2 , Bc2 , Cc , Dc ) is acting along the time direction only, and thus actually it works like the 1-D controller we usually take into account. Subsequently, the open-loop and the sensitivity function frequency responses are obtained and shown in Figures 12.16 and 12.17. The open-loop crossover frequency is 1.6 kHz, the gain margin is 6 dB, and the phase margin is 50 degrees. A bad high frequency part of the sensitivity function has appeared, because here only the stabilization problem is considered and no performance optimization is involved. Thus the 2-D H2 control will be presented which gives a better performance than the stabilizing only controller.

12.8 12.8.1

2-D H2 control for error minimization Simulation results

The σ value of PES NRRO is plotted versus the track number in Figure 12.15, where the error amplitude is oscillating steadily, which implies that error propagation is contained. Additionally, the position error has been minimized in the H2 norm sense. The σ values of PES NRRO with the stabilization control and the H2 control are compared in Figure 12.15 and it is seen that the error is reduced by an average of 60% via the 2-D H2 control. The conventional 1-D H2 feedback control has also been designed to minimize the error signal, and based on the feedback control a feedforward control is further designed to contain the error propagation. This method is also compared with the above two methods in Figure 12.15. It is seen that the proposed 2-D control method is comparable to the previous 1-D control method.

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Modeling and Control of Vibration in Mechanical Systems −3

9

x 10

8

σ of pes NRRO (µm)

7

6

5

4

3

2

1

0

2−D H2 control 2−D Stabiliation 1−D H2 feedback+feedforward control 0

10

20

30

40

50

60

70

80

90

100

Track number

FIGURE 12.15 σ of PES NRRO versus track number. 30

Magnitude(dB)

20 10 0 −10 −20 2 10

3

10

4

10

−50

Phase(deg)

−100

−150

−200

−250 2 10

3

10

Frequency(Hz)

FIGURE 12.16 Open-loop frequency response with stabilization controller.

4

10

Two-Dimensional H2 Control for Error Minimization

247

20

15

10

Magnitude(dB)

5

0

−5

−10

−15

−20

−25

−30 2 10

3

10

4

10

Frequency(Hz)

FIGURE 12.17 Sensitivity function with stabilization controller (Ac2 , Bc2 , Cc, Dc ).

12.8.2

Experimental results

It is noted that in the 2-D controller (12.9)−(12.10), (Ac2 , Bc2 , Cc , Dc ) acts in the time dimension, and runs on the same track as a 1-D controller C(z) we have usually considered. Thus in this section we particularly take into account the 1-D controller C(z) : (Ac2 , Bc2 , Cc , Dc ) for the plant P (s) (12.38)−(12.40) in the time dimension. Straightforwardly from the designed 2-D controller C(z1 , z2 ) : (Ac1 , Ac2 , Bc1 , Bc2 , Cc, Dc ), the 1-D controller C(z) : (Ac2 , Bc2 , Cc , Dc ) can be obtained and its frequency response is shown in Figure 12.18, where the notches around 4, 6 and 9 kHz are to suppress the resonances as observed in Figure 12.6. With the 1-D controller C(z), the simulated frequency responses of the open loop C(z)P (z) and the sensitivity function S(z) = 1/(1 + C(z)P (z)) are drawn in Figures 12.19 and 12.20. To practically verify the 1-D controller C(z), experiment is carried out with dSpace 1103 on TMS320C240 DSP board and the LDV used to measure the displacement of the actuator. The measured open-loop frequency response compared with the simulated one is shown in Figure 12.19, and the crossover frequency is 1.2 kHz, the gain margin is 9 dB, and the phase margin is 54 degrees. Figure 12.20 also shows the measured sensitivity function with comparison to the simulated one. Additionally, the step response and the control signal of the closed loop are taken from the oscillo-

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scope as a bmp picture shown in Figure 12.21, which testifies that the 1-D controller drawn from the designed 2-D controller works well. The results in this section show that the 1-D controller C(z) extracted from the 2D controller can stabilize the plant P (s) in the time dimension. However, as shown in Figure 12.3, one 1-D feedback controller C(z) is not capable enough to contain error propagation and stabilize the SSTW process modeled in (12.30)−(12.37) in two dimensions. On the basis of the 2-D model, the designed 2-D controller can stabilize the SSTW process and minimize the position error simultaneously.

Bode Diagram 30

Magnitude (dB)

20 10 0 −10 −20 −30 135

Phase (deg)

90

45

0

−45

−90 0

10

1

10

2

10

3

10

4

10

Frequency (Hz)

FIGURE 12.18 Frequency response of controller (Ac2 , Bc2 , Cc , Dc ).

12.9

Conclusion

This chapter has employed 2-D controllers in stabilization and error minimization of systems that have 2-D behavior and can be modeled as a 2-D model. The application in a self-servo track writing process described by a 2-D model has been addressed in detail. By applying the two-dimensional model, the error propagation

Two-Dimensional H2 Control for Error Minimization

249

40 Measured Simulated

Magnitude(dB)

20 0 −20 −40 −60

2

3

10

10

4

10

100 0 Phase(deg)

−100 −200 −300 −400 −500 −600

2

3

10

10

4

10

Frequency(Hz)

FIGURE 12.19 Open-loop frequency response with controller (Ac2 , Bc2 , Cc, Dc ). 10 Measured Simulated 5

0

Magnitude(dB)

−5

−10

−15

−20

−25

−30

−35

−40 10

2

3

10 Frequency(Hz)

FIGURE 12.20 Sensitivity function with controller (Ac2 , Bc2 , Cc , Dc ).

10

4

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 12.21 Step response (Channel 1/2/3: Reference/Output/Control signal).

containment problem and the TMR minimization problem of the self-servo track writing process has been formulated as a 2-D H2 control problem. With the stored error information of the preceding track, the adopted 2-D controller is applicable. The 2-D approach provides a systematic way of achieving both the error propagation containment and the TMR minimization simultaneously. The simulation results have demonstrated that the error propagation is prevented by the 2-D control scheme, and good positioning accuracy is achieved with the 2-D H2 control scheme. Also, the time dimensional portion, as a 1-D controller, of the designed 2-D controller has been implemented via LDV and dSpace and the implementation results have been shown to verify that the 1-D controller performs well.

13 Nonlinearity Compensation and Nonlinear Control

13.1

Introduction

Nonlinearities such as friction in the actuator pivot are known to limit the low frequency gain of a control loop. Translated to the error rejection function or sensitivity function, it lifts the magnitude of the sensitivity function at low frequencies, and thus reduces the ability of the control loop to reject vibrations at low frequencies and affects the system performance. Based on an identified friction model, the friction can be compensated for by injecting an estimated friction force into the actuator. A friction compensation method based on a nonlinear hysteresis model is thus studied. Moreover, on the basis of a linear feedback control, to further improve the rejection of low-frequency disturbances such as nonlinear disturbances arising from friction torque or bias or other unknown disturbances, an adaptive nonlinear compensation scheme will be adopted in this chapter to cancel their effects through a proper estimation of the disturbances.

13.2

Nonlinearity compensation

As stated in Chapter 2, the nonlinear friction model fe = F (x(k)) is identified as in the form of (2.20) using the operator based method. With the model, the friction can be compensated for by injecting the friction force fe into the plant, as seen in Figure 13.1. A sinusoidal signal of 50 Hz and 1 V amplitude is injected as the reference. The input u versus the actuator displacement x is compared for the cases of with and without the nonlinear compensation in Figure 13.2. It is evident that with the nonlinear compensation the relationship between u and x is linearized very well. In the frequency domain, the actuator frequency responses before and after the compensation are measured with a swept sine wave via a DSA and shown in Figure 13.3. The compensated magnitude and phase responses approach those of the pure double integrator much more closely than those before compensation. With the friction compensation, the VCM actuator frequency responses are mea-

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sured with different sinusoidal reference amplitudes and plotted in Figures 13.4 and 13.5, where the straight smooth lines are from the pure double integrator. It is seen that the linearization effect becomes better when the reference amplitude is higher than 1 V (0.5 µm/V), and it is not so satisfactory for 0.25 V. Although the friction model is obtained on the basis of the measurement for 0.5, 1, and 3 V displacement amplitudes, the compensation based on the obtained model is able to achieve good linearization effect for any displacement ranging from 0.5 V and above.

FIGURE 13.1 Friction compensation for the actuation system.

Figure 13.6 shows the corresponding simulated and measured sensitivity functions. With the compensation the magnitude at 10 Hz is reduced by around 20 dB due to the increased open-loop gain at low frequencies as seen in Figures 13.4 and 13.5. The slightly lower magnitude from 60 to 100 Hz before compensation, corresponding to the higher magnitude of the original actuator model from 60 to 100 Hz in Figure 13.3, is caused by the nonlinearity of the original actuator. In the above, a model based nonlinearity compensation has improved the ability of the closed control loop to reject vibrations in low frequency range. In the rest of the chapter, a non-model based compensation, which is an adaptive and a more flexible scheme, will be applied to compensate the nonlinear or unknown vibrations in low frequency range.

13.3

Nonlinear control

While a KYP Lemma-based linear control can achieve disturbance rejection over some chosen frequency ranges, it cannot run away from performance limitation,

253

Nonlinearity Compensation and Nonlinear Control −3

1.5

x 10

No compensation 1

0.5

Input u (V)

With compensation 0

−0.5

−1

−1.5 −1.5

−1

−0.5

0

0.5

1

1.5

Displacement (V, 0.5 µm/V)

FIGURE 13.2 Input u versus displacement x with and without compensation. 100

After comp.(measured) After comp.(simulated) 1/s2 Before comp.

Magnitude(dB)

90 80 70 60 50 40 1 10

2

10

0

Phase (deg)

−100

−200

−300

−400 1 10

2

10 Frequency(Hz)

FIGURE 13.3 Actuator frequency responses with and without friction compensation.

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Magnitude(dB)

100 0.25V 0.5V

80 60 40 1

2

10

10

Phase (deg)

0

−100

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−300 1 10

2

10 Frequency(Hz)

FIGURE 13.4 Actuator frequency responses with friction compensation for different displacements in voltage with 0.5µm/V (Straight smooth lines: the pure double integrator).

Magnitude(dB)

100 1V 3V

80 60 40 1

2

10

10

Phase (deg)

0

−100

−200

−300 1 10

2

10 Frequency(Hz)

FIGURE 13.5 Actuator frequency responses with friction compensation for different displacements in voltage with 0.5 µm/V (Straight smooth lines: the pure double integrator).

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Nonlinearity Compensation and Nonlinear Control 10 After compensation After compensation Before compensation

0

−10

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−20

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−40

−50

−60

−70

−80

−90 1 10

10

2

10

3

Frequency(Hz)

FIGURE 13.6 Sensitivity functions with and without friction compensation.

as observed from the well known Bode integral constraint. For example, the KYP Lemma-based control design in Chapter 8 results in an excellent attenuation of disturbance around 650 Hz. But its ability in rejecting low frequency disturbance is not very desirable. On the other hand, in Chapter 2, the low-frequency disturbance is modeled as the output of an adaptive nonlinear scheme with the error signal as the input. Here it is used to compensate for the low-frequency disturbance, i.e., a nonlinear controller is augmented with the KYP Lemma-based linear feedback controller, which remarkably improves the system disturbance rejection capability in low frequency range without sacrificing performance at other frequencies. Considering Figure 13.7 with plant P (z): (Ap , Bp , Cp , Dp ), we have the following discrete time state-space realization: x(k + 1) = Ap x(k) + Bp u(k) + Bp d1 (k),

(13.1)

z(k) = −Cp x(k) + w(k) − Dp u(k) − Dp d1 (k) − d2 (k), e(k) = −Cp x(k) + w(k) − Dp u(k) − Dp d1 (k) − d2 (k) + n(k).

(13.2) (13.3)

The design of control law u = uL + uN includes two parts: 1. The first is to design a linear dynamic output feedback controller uL = C(z)e for the plant P (z) such that the closed-loop system is stable and satisfies the performance in (8.2).

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FIGURE 13.7 Control structure of a plant P (s) with Youla parametrization approach and adaptive nonlinear compensation.

2. The second is to design a nonlinear control law uN such that the contribution of the low-frequency disturbance in d1 to the error can be compensated.

13.3.1

Design of a composite control law

The linear control law design based on the KYP Lemma in Chapter 8 can give good rejection of disturbance of particular frequencies such as that around 650 Hz. However, the performance at low frequency needs to be improved. Thus we shall design a nonlinear compensation based on the nonlinear modeling discussed in Section 2.4.2. Note that the modeling in Section 2.4.2 is based on the time history of the measurement e. We consider the modeled disturbance dˆ1 given by dˆ1 = ω ˜ T (k)s(Φk )

(13.4)

ω ˜ (k + 1) = (1 − δ)˜ ω(k) − Γs(Φk )e(k),

(13.5)

with the update law

where s(Φk ) is given in (2.56). As demonstrated later, by injecting uN (k) = −dˆ1

(13.6)

to the plant combined with uL as shown in Figure 13.7, we are able to compensate the low-frequency disturbance in d1 , which is modeled as in (2.55).

Nonlinearity Compensation and Nonlinear Control

13.3.2

257

Experimental results in hard disk drives

The plant under consideration and the linear controller are the same as in Section 8.6 of Chapter 8. The nonlinear control signal uN in Figure 13.7 is calculated from (13.5)−(13.6) and (2.56) in Section 2.4.2 of Chapter 2. The center positions cei and c∆ei for the measurement e and velocity e˙ are chosen as zero. The variances are 2 σe2i = σ∆e = 10, i = 1, ..., p. The forgetting factor δ = 0.5. Γ affects the learning i speed and should be selected to be as large as possible. To evaluate the disturbance rejection performance of the combined linear control C(z) designed in Section 8.6 in Chapter 8 and the nonlinear control (13.6), a sinusoidal signal with the logarithmically spaced frequency from 10 Hz to 22.5 kHz is respectively injected as w in Figure 13.7. For each frequency sinusoidal input, the error signal e will involve multiple frequency components due to the nonlinear control. In this situation, it is reasonable that the error rejection capability is directly measured as the amplitude ratio e/w in time domain. At each frequency point, the error rejection e/w is then plotted and shown in Figure 13.8. The error rejection capability is evaluated for each value of p = 1, 5, 9. As in Figure 13.8, the two cases with p = 1 and p = 5 give similar results, and both are better than that given by p = 9. This implies that a higher p may not necessarily lead to a better result. This phenomenon is consistent with the observation in modeling. Overall, from the simulation result, the nonlinear control produces a better rejection of disturbances of low frequencies, while not affecting the high frequency disturbance rejection performance. Figure 13.8 also shows the effect of Γ on the error rejection. 10% of the Γ value in (2.58) is used in the calculation. It is observed that the larger Γ yields a better accuracy. This also agrees with the modeling result in Section 2.4.2 in Chapter 2. Consequently, corresponding to the error rejection in Figure 13.8 with p = 1, the power spectrum of PES NRRO is shown in Fig 13.9. It is observed that the error is much lowered by 80% before 400 Hz, and no cost in higher frequency range is paid. Overall it is evaluated from calculation that the 3σ of the true PES NRRO is improved from 6.0 nm with the KYP Lemma method to 5.5 nm with the KYP Lemma-based linear control augmented with the nonlinear control.

REMARK 13.1 It should be mentioned that the nonlinear compensation scheme can be combined with any linear control to improve low frequency vibration rejection without sacrificing disturbance rejection capability in other frequency ranges.

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10

0

−10

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−20

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−50

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−70 p=1 p=5 p=9 No uN

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Frequency(Hz)

FIGURE 13.8 Comparison of error rejection frequency response without and with uN of different p and Γ.

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Nonlinearity Compensation and Nonlinear Control −3

1

x 10

KYP KYP+nonlinear control 0.9

0.8

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0.4

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0.1

0

0

1000

2000

3000

4000

5000

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FIGURE 13.9 NRRO power spectrum with KYP lemma-based linear control and nonlinear compensation (80% reduction before 400 Hz).

13.4

Conclusion

The nonlinear friction has been compensated for by injecting the modeled friction force in Chapter 2 into the actuator. With the model based compensation, the linearization effect for the VCM actuator has been verified via the measurement of the hysteresis in time domain and the frequency response in frequency domain. The measured error rejection function showed an increased error rejection capability in low frequency range, as a result of the compensation. To further improve the disturbance rejection in low frequency range, the linear control is combined with an adaptive nonlinear compensation. The simulation results have demonstrated that the proposed controller can effectively reject disturbances at low frequencies, resulting in a marked improvement for the 3σ value of the PES NRRO in the data storage system.

14 Quantization Effect on Vibration Rejection and Its Compensation

14.1

Introduction

A/D and D/A converters are inevitable in digital control systems. Chapter 5 mentioned quantization performed by the A/D converter. This chapter investigates the quantization effect on vibration rejection capability of the closed-loop control system. With a proposed quantizer model, the influence of different quantizer bits on the error rejection ability in low frequency range is analyzed and evaluated based on frequency response measurement. On the other hand, it is known that nonlinear behaviors such as actuator pivot friction limit the low frequency gain of the open loop. Translated to the error rejection function or sensitivity function, it lifts the magnitude at low frequencies and thus reduces the ability of the closed-loop system to reject vibrations in the low frequency range. Therefore, quantization and friction induced problems should be treated differently for more effective control. A simple and low cost scaling scheme is developed in this chapter to compensate for the effect of the quantizer. It is demonstrated that the compensation scheme is effective in improving the sensitivity function in the low frequency range without deteriorating performances at other frequencies. With the compensation for the quantization effect, the impact on the rejection ability is mostly due to the friction. As such, the effects from quantization and friction on the error rejection function can be differentiated, and the quantization and friction induced problems can be tackled separately in the control loop. Additionally, through the proposed quantization model and measurement methodology, suitable bit resolution for the quantizer can be identified with and without the compensation.

14.2

Description of control system with quantizer

The system under investigation is the VCM actuator in a commercial 1.8−inch disk drive with a spindle motor rotational speed of 4200 RPM. The experiment setup is shown in Figure 14.1, where the spindle driver is used to spin the spindle motor, the

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LDV is to measure the position of the read/write head, and a VCM driver to drive the VCM actuator. The LDV displacement range is set as 2 µm/V. The frequency response of the VCM actuator is measured via DSA by injecting a swept sine wave of 5 mV amplitude. Due to experimental limitations, measurements in this chapter are taken when the head is positioned in the middle-diameter region of the disk. The measured frequency response of the VCM actuator is shown in Figure 14.2, and its model P (s) is obtained as zeros = 105 × [1.6965; −0.0077 ± 0.2575j]; poles = 105 × [−1.6965; −0.0251 ± 0.5020j; −0.0101 ± 0.2511j; −0.0019 ± 0.0073j];

gain = −5.3167 × 1017.

A PID controller combined with notch filters is designed for the VCM actuator and shown in Figure 14.4. The closed servo loop of the VCM actuator is seen in Figure 14.3, where dSpace DS1103 with TMS320C240 DSP on board are used to implement the controller with the sampling rate of 30 kHz. The quantizer model Q(·) is given by   max(e) 2n Q(e) = × floor e × + 0.5 , (14.1) 2n max(e) where n is the quantizer bit, max(e) means the maximum amplitude of the error signal e, and floor(e) means rounding e to the nearest integer towards minus infinity. The sensitivity function of the control loop without Q(e) is given by S(z) =

1 . 1 + P (z)C(z)

(14.2)

A swept sine wave with 50 mV amplitude is injected as the reference signal to the closed control loop and the sensitivity function S(z) is measured and plotted in Figure 14.5. It is known that as the excitation level to the VCM actuator decreases, the VCM actuator gain in the low frequency range is lowered due to friction nonlinearity effect [40], which correspondingly leads to the increased |S(z)|. This is illustrated also in Figure 14.5, where S(z) changes for different reference amplitudes. In this chapter, we focus on the effect of quantizer (14.1) on the sensitivity function and seek to differentiate the quantization and friction nonlinearity effect. With the quantizer Q(e), the sensitivity function is given by SQ (z) =

1 e = reference 1 + P (z)Qe (z)C(z)

(14.3)

e3 (z) e(z)

(14.4)

where Qe (z) =

stands for an approximation of the quantizer (14.1).

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Quantization Effect on Vibration Rejection and Its Compensation

FIGURE 14.1 Experimental setup. 60 Measured Modeled

Magnitude(dB)

40 20 0 −20 −40 1 10

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4

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0

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FIGURE 14.2 Frequency response of the VCM actuator measured by injecting a swept sine wave of 5mV amplitude.

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 14.3 The servo loop in experiment.

15

Magnitude(dB)

10 5 0 −5 −10 −15 −20 1 10

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3

10

10

4

10

100

Phase(deg)

50

0

−50

−100 1 10

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10 Frequency(Hz)

FIGURE 14.4 Frequency response of the controller C(z).

4

10

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Quantization Effect on Vibration Rejection and Its Compensation

10 0 −10

Magnitude(dB)

−20 −30 −40 −50 −60 0.05v 0.1v 0.5v 1v

−70 −80 1 10

2

3

10

10

4

10

Frequency(Hz)

FIGURE 14.5 Frequency response of the sensitivity function S(z) with different reference levels (i.e., actuator moving ranges are different).

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14.3

Modeling and Control of Vibration in Mechanical Systems

Quantization effect on error rejection

In this section, we investigate how the quantizer (14.1) with different bits affects the error rejection function, and show that the lower low-frequency disturbance rejection reflected in the sensitivity function may also be caused by quantization in addition to friction.

14.3.1

Quantizer frequency response measurement

Before we proceed to investigate the quantization effect on error rejection, we examine the transfer function (14.4) for the quantizer (14.1) with different bits n through the measurement of its frequency response. The frequency response of e3 over e in Figure 14.3 was measured via DSA by injecting the swept sinusoidal signal as the reference signal. The quantizer model of the form in (14.1) with max(e) = 0.2 V and different resolution bits of n = 6, 8, 10 are investigated respectively. The dashed curves in Figures 14.6−14.8 are the corresponding measured frequency responses of Q(e). It is observed that the magnitude difference at frequencies less than 100 Hz between bits 6 and 8 are almost 10 dB and the phase difference is about 250 deg. From 200 Hz upwards, the quantizer gains in the three cases are exactly unity. Moreover, as the bit number increases, the quantizer approaches unity gain. When bit n = 10, it can be approximated as 1. The frequency response when n = 12 is almost the same as that when n = 10, and thus is omitted here.

14.3.2

Quantization effect on error rejection

Figures 14.6−14.8 show that the bit number n affects Q(e) mainly in low frequency range. In addition, we shall see that Q(e) with lower bit number n will deteriorate the error rejection capability of the servo system in low frequency range. The measured sensitivity functions with different quantizer bits are shown in Figure 14.9, where the effect of the bit n on the low-frequency part can be seen. The averaged difference of |S(z)| at the low-frequency range when the bit changes from 10 to 6 is about 10 dB. The trend is that the lower number of bits leads to a lower effective magnitude of the compensated open loop, and thus higher |S(z)|, which means poorer error rejection ability. Note that a lower bit number means that the known part due to the quantization is less. When the error signal is too low for the A/D converter to differentiate, a high level of error rejection can not be reflected in the sensitivity transfer function. In Figure 14.5, the lowest sensitivity function level at low frequencies is below −60 dB. This implies that a gain of at least 1000 requires more than 10 bit resolution. Hence, as shown in Figure 14.9, for the two cases of n = 10 and n = 12, no obvious impact on the sensitivity functions is seen.

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Quantization Effect on Vibration Rejection and Its Compensation 20 10

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0 −10 −20 −30 After compensation Before compensation

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200 100 Phase(deg)

0 −100 −200 −300 −400 1 10

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FIGURE 14.6 Frequency response of the quantizer before and after compensation (bit number n = 6). 20

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After compensation Before compensation 10

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2

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3

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150

Phase(deg)

100 50 0 −50 −100 −150 1 10

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3

10

FIGURE 14.7 Frequency response of the quantizer with compensation (bit number n = 8).

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Modeling and Control of Vibration in Mechanical Systems

10

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After compensation Before compensation 5

0

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10

FIGURE 14.8 Frequency response of the quantizer with compensation (bit number n = 10). 10

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−30

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−50

−60 n=12 n=10 n=8 n=6

−70

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10

FIGURE 14.9 Measured sensitivity function SQ (z) with different bits n.

4

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Quantization Effect on Vibration Rejection and Its Compensation

We take |SQ (f)| at f = 10 Hz as the representative value of |SQ (z)| at low frequencies. Figure 14.10 shows the relation of |SQ (f)| versus bit number n at f = 10 Hz, which is decreasing dramatically until n = 10. This means that bit number n = 10 is necessary for satisfactory performance. Approximately |SQ (z)| = 0.4604n3 −12.1375n2 +99.2833n−303.2000. Similar to Figure 14.5, |SQ (z)| with a fixed bit number n also changes with plant excitation levels. The trend of |SQ (z)| with the bit number n is almost the same for each excitation level. −44

−46

−48

−50

|S(f)| f=10

−52

−54

−56

−58

−60

−62

−64

6

7

8

9 Bit n

10

11

12

FIGURE 14.10 Sensitivity function |SQ (f)| with f = 10 Hz versus bit n.

14.4

Compensation of quantization effect on error rejection

In this section, a scaling method is used to compensate for the effect of the quantizer on the sensitivity function at low frequencies. The compensation scheme is shown in Figure 14.11. When the error signal amplitude is less than a threshold δ, it is scaled up by a factor a > 1, and then scaled down by the factor a−1 after undergoing quantization. The method of choosing the threshold δ and the scaling factor a is illustrated in Figure 14.12. M is the value at the beginning point where the quantization effect

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 14.11 Compensation scheme of quantization effect. can be seen obviously. D is the biggest difference between SQ and S. δ and a can be generally chosen as follows. δ = 10M/20 · reference, a = 10D/20 .

(14.5)

10

0

−10

Magnitude(dB)

−20

−30

−40 M −50 D −60

−70 Target S Measured SQ −80 1 10

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10

4

10

Frequency(Hz)

FIGURE 14.12 Choosing the threshold δ and the scaling factor a.

The compensation is carried out for three cases with n = 6, 8, and 10 respectively, and the improvement of the error rejection after compensation will be discussed. A. n = 6

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Quantization Effect on Vibration Rejection and Its Compensation

In this case, from the sensitivity function in Figure 14.9, δ and a are obtained as δ = 10−40/20 · 50 mV = 0.5 mV, a = 1018/20 = 8. Figure 14.6 shows the improved frequency response of the quantizer after compensation. The sensitivity function improvement after compensation in the low frequency range is shown in Figures 14.9 and 14.13. There is no further improvement when a is increased. When a = 5, the result is as good as when a = 8, which means that the optimal value of a is roughly between 5 and 8.

10

0

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−20

−30

−40

−50

−60

−70

n=10 n=8 n=6

−80 1 10

10

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3

10

4

Frequency(Hz)

FIGURE 14.13 Sensitivity function with quantization compensation.

B. n = 8 When n = 8, δ = 10−48/20 · 50 mv = 0.2 mv, a = 1015/20 = 5.6. Figures 14.7 and 14.13 show the obvious improvement of the quantizer and the sensitivity function after compensation. When a is increased to 10 and above, no further improvement is observed. Thus the best value of a can be chosen around 5.6. C. n = 10 In the case with bit n = 10, δ = 10−48/20 · 50 mV = 0.2 mV, a = 1010/20 = 3.

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Modeling and Control of Vibration in Mechanical Systems TABLE 14.1

Quantization and friction effect Quantizer bit n 6 8 Actually measured |S|f=10Hz (dB) −45 −50 Compensated quantization effect (dB) 13 18 Estimated |S|f=10Hz with friction effect (dB) −58 −68

10 −65 4 −69

No obvious difference can be seen in Figure 14.8. When compared with Figure 14.9, Figure 14.13 shows the obvious improvement of the sensitivity function after compensation. When a = 6, the result is almost the same. As a is increased to 10, the result becomes worse. Thus in this case a = 3 can be chosen as one of the best scaling factors. It can be seen from Figure 14.13 that after compensation, |SQ (z)| is much closer to |S(z)| in the dotted line in Figure 14.5, which is measured without the quantizer and thus considered as subjected to nonlinear friction effect only, e.g., |S(z)|f=10Hz = −69 dB with friction effect alone. From Figure 14.13, we can also see that instead of bit number n = 10 previously shown in Section 14.3.2 without compensation, bit number n = 8 is adequate with the compensation for a similar performance. It has been shown that the low frequency range of the sensitivity function is also affected by quantization in addition to actuator pivot friction. When the quantization effect is compensated for sufficiently, the low frequency portion of the sensitivity function can be regarded as the estimated effect from friction alone. Thus, by comparing the compensated and the uncompensated |SQ (z)| with Figure 14.13, the quantization and the friction effect on SQ (z) can then be differentiated and shown in Table 14.1. We can see that the estimated friction effect for n = 6 is not as accurate as the other two cases. This is because the compensation for n = 6 is insufficient, as seen in Figure 14.13. In this situation, a multi-stage scaling, i.e., a = ai for δi1 < |e| < δi2 , i = 1, 2, ..., may be necessary. To this end, the scaling compensation scheme is conducted for quantization compensation in the disk drive. The series of results demonstrates that the compensation scheme is effective in improving the sensitivity function in the low frequency range without influencing other frequencies. Additionally, it is noted that the implementation of the scheme is simple and therefore incurs low cost.

14.5

Conclusion

The quantization effect on the closed-loop system performance has been investigated. The frequency responses of the quantizers with different bits have been measured and analyzed. Its effect on the error rejection function or sensitivity function at low frequencies has shown that the rejection ability of the servo loop for low

Quantization Effect on Vibration Rejection and Its Compensation

273

frequency disturbances is relevant to the quantizer in addition to the pivot friction. Moreover, a simple and low cost scaling scheme has been used to compensate for the effect of the quantizer. With sufficient compensation, the effects due to quantization and friction can be differentiated, and thus the friction and the quantization impact on the system performance such as disturbance rejection capability can be treated separately. Through the proposed quantization model and measurement methodology, a suitable bit resolution for the quantizer can be easily identified while the pivot friction nonlinearity effect can be decoupled and tackled for more effective closed-loop system control.

15 Adaptive Filtering Algorithms for Active Vibration Control

15.1

Introduction

Many advanced control techniques have been studied for active noise and vibration control, including optimal and robust control and adaptive control strategies. The advance of computer technologies has made digital signal processing techniques much more useful in modern control systems. Adaptive filters have been widely used in the implementation of adaptive algorithms for active vibration control (AVC). Sampling time constraint is the major drawback of a digital control system that requires very high processing speed for real time control. Adaptive filtering offers significant advantages over passive silencers at low frequencies where lower sampling rates are adequate. There are two main approaches to adaptive filtering in AVC, feedforward and feedback adaptive filtering. The feedforward algorithm requires the source information in order to attenuate the vibration. In many applications, the source of vibration is impractical or expensive to measure, and a feedforward algorithm becomes impossible or difficult to implement in those applications. A feedback algorithm has an advantage of utilizing only the vibration signal to be controlled.

15.2

Adaptive feedforward algorithm

Figure 15.1 shows the so-called filtered-X LMS (FXLMS) algorithm which was first introduced by Bernard Widrow in 1981 [37]. The filtered-X LMS algorithm is derived as follows with the notation defined in Chapter 3. The error signal e(k) is given by e(k) = d(k) + y(k) = d(k) + P (z)u(k) = d(k) + P (z)(W T (z)x(k)).

(15.1)

275

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Modeling and Control of Vibration in Mechanical Systems

FIGURE 15.1 Block diagram of FXLMS algorithm.

Considering that ∇e(k) =

∂e(k) = p(k) ∗ x(k), ∂W (k)

(15.2)

the gradient estimate of the mean-squared error will be ˆ = ∇e2 (k) = 2[∇e(k)]e(k) ∇ξ(k) = 2[p(k) ∗ x(k)]e(k),

(15.3)

where p(k) is the impulse response of P (z). In a practical situation, an exact model P (z) is not available, and therefore its estimated model Pˆ (z) is used to represent P (z) in the algorithm. Either an FIR or IIR filter can be used to model the AVC system. Then the reference signal for the LMS algorithm will be x ˆ(k) = Pˆ (z)x(k).

(15.4)

xˆ(k) is called the filtered reference signal because the reference input signal is passed through the estimated model of the AVC system. Equation (15.3) becomes ˆ = 2[pˆ(k) ∗ x(k)]e(k) = 2ˆ ∇ξ(k) x(k)e(k).

(15.5)

The coefficients of the weight vector will be updated by the following equation.

277

Adaptive Filtering Algorithms for Active Vibration Control

µ ˆ ∇ξ(k) 2 = W (k) − µˆ x(k)e(k).

W (k + 1) = W (k) −

(15.6)

The resulting adaptive algorithm is known as the Filtered-X LMS algorithm. In the normalized case, the algorithm is known as the Filtered-X NLMS (FXNLMS) algorithm whereby the weight vector will be updated by W (k + 1) = W (k) − µ(k)ˆ x(k)e(k), α µ(k) = . ε+x ˆT (k)ˆ x(k)

(15.7) (15.8)

So far, the feedforward adaptive algorithm has been discussed. The feedforward algorithm requires the reference signal that cannot be available in many applications. To overcome this problem an adaptive feedback algorithm is considered below.

15.3

Adaptive feedback algorithm

There is another algorithm that combines the traditional feedback control and adaptive filtering approach, and is therefore referred to as the Adaptive Feedback Algorithm. The algorithm utilizes only the feedback error signal to cancel the disturbance vibration. Since there is no direct reference information available for the vibration control, the disturbance signal is regenerated (extracted) from the error signal and the approximated input reference signal is then fed back to the adaptive filter, as shown in Figure 15.2. An estimate of the reference signal x(k) is obtained by subtracting the estimated cancellation signal yˆ(k) of y(k) from the error signal e(k). ˆ = e(k) − yˆ(k). xˆ(k) = d(k)

(15.9)

The weight updating algorithm is the same as the feedforward control scheme, FXLMS or FXNLMS. The feedback loop W (z)Pˆ (z) will introduce poles to the system. The characteristic equation of the system will be: α(z) = 1 + W (z)Pˆ (z).

(15.10)

Adaptive feedback control can be seen as adaptive inverse control where an adaptive filter is used to track the inverse model of P (z). The analogy is illustrated in Figure 15.3. In this control system, a compensator C(z) will adapt to track the inverse model P −1 (z) [38]. The system is analyzed as follows. X (z) = R(z) − X(z) = R(z) − E(z) + Pˆ (z)C(z)X (z), ′

′

(15.11)

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i.e., ′

X (z) =

E(z) = D(z) +

R(z) − E(z)

1 − Pˆ (z)C(z)

.

P (z)C(z)R(z) − P (z)C(z)E(z) , 1 − Pˆ (z)C(z)

(15.12)

(15.13)

from which, E(z) =

R(z) D(z)(1 − Pˆ (z)C(z)) + . (15.14) 1 − Pˆ (z)C(z) + P (z)C(z) 1 − Pˆ (z)C(z) + P (z)C(z)

With the reference signal R(z) set to zero, the transfer function from D(z) to E(z) will be: E(z) 1 − Pˆ (z)C(z) . = D(z) 1 − Pˆ (z)C(z) + P (z)C(z)

(15.15)

If the estimated model Pˆ (z) is exactly the same as P (z) , (15.15) will be reduced to: E(z) = 1 − P (z)C(z). D(z)

(15.16)

An adaptive filter will be applied in the place of C(z) and the filter will adapt its weight vector to approach the inverse model of P (z), P −1 (z) [38]. When C(z) becomes the inverse model of P (z), the effect of D(z) over E(z) will be totally eliminated. Therefore correct modeling of P (z) is crucial to the adaptive feedback control system. In a practical situation, it is difficult to obtain the exact model of the system. Modeling error may lead the system to an unstable situation.

Adaptive Filtering Algorithms for Active Vibration Control

FIGURE 15.2 Filtered-X LMS adaptive feedback algorithm.

FIGURE 15.3 Adaptive inverse control scheme.

279

280

15.4

Modeling and Control of Vibration in Mechanical Systems

Comparison between feedforward and feedback controls

In spite of the advantage of not requiring the information over feedforward control, adaptive feedback control can only be applicable for narrow band disturbance rejection. The traditional feedback controller cannot react to the disturbance before the control error has already occurred. But the adaptive filter has the ability to capture the statistics of the disturbance signal. For periodic disturbance signals, it is possible for the adaptive feedback control to track the frequency of the signal and progressively attenuate the disturbance signal. Therefore adaptive feedback control is still efficient for pure sinusoid signals. On the other hand, feedforward control has an ability to reject the wide band disturbances, because the controller receives the disturbance signal before it reaches the point to be controlled and takes a control action in advance to eliminate the disturbance impact. An approximated model of the system appears in the feedback loop of the adaptive feedback system, which introduces poles to the system. Therefore robustness of stability of the adaptive feedback system can be improved by minimizing the modeling error of the system.

15.5

Application in Stewart platform

We have chosen the adaptive feedback algorithm for hexapod smart structure introduced in Chapter 3. Since the structure has six actuators, a multiple-channel adaptive feedback control system is studied.

15.5.1

Multi-channel adaptive feedback AVC system

In a multiple-channel AVC system, let the number of the secondary sources be K and the number of error sensors be M . The reference signal synthesizer uses K secondary signals, M error signals, and K ×M secondary path estimates to generate M reference signals for K × M adaptive filters. The synthesized reference signals are expressed as:

xm (k) = em (k) +

K X

n=1

pˆmn (k) ∗ un (k), m = 1, 2, · · · , M,

(15.17)

Adaptive Filtering Algorithms for Active Vibration Control

281

where pˆmn (k) is the impulse response of the secondary-path estimate Pˆmn (z) and un (k) is the nth secondary signal expressed as un (k) =

M X

m=1

wnm(k) ∗ xn (k), n = 1, 2, · · · , K,

(15.18)

where wnm (k) is the impulse response of the adaptive filter Wnm (z) . A 2 × 2 adaptive feedback AVC system shown in Figure 15.4 is presented as an example for multiple channel AVC systems. Two secondary signals, u1 (k) and u2 (k) are generated as u1 (k) = w11 (k) ∗ x1 (k) + w12(k) ∗ x2 (k),

u2 (k) = w21 (k) ∗ x1 (k) + w22(k) ∗ x2 (k).

(15.19) (15.20)

The reference signals are synthesized as: x1 (k) = e1 (k) + pˆ11 (k) ∗ u1 (k) + pˆ12 (k) ∗ u2 (k),

x2 (k) = e2 (k) + pˆ21 (k) ∗ u1 (k) + pˆ22 (k) ∗ u2 (k).

(15.21) (15.22)

The coefficients of the four adaptive filters are adjusted using the FXLMS algorithm expressed as: W11 (k + 1) = W11 (k) + µ{[pˆ11(k) ∗ X1 (k)]e1 (k) + [pˆ21 (k) ∗ X1 (k)]e2 (k)}, W21 (k + 1) = W21 (k) + µ{[pˆ12(k) ∗ X1 (k)]e1 (k) + [pˆ22 (k) ∗ X1 (k)]e2 (k)},

W12 (k + 1) = W12 (k) + µ{[pˆ11(k) ∗ X2 (k)]e1 (k) + [pˆ21 (k) ∗ X2 (k)]e2 (k)}, W22 (k + 1) = W22 (k) + µ{[pˆ12(k) ∗ X2 (k)]e1 (k) + [pˆ22 (k) ∗ X2 (k)]e2 (k)}.

15.5.2

Multi-channel adaptive feedback algorithm for hexapod platform

The hexapod smart structure has six secondary sources. If we use six error sensors, there will be 72 adaptive filters including 36 filters for reference signal synthesizer and 36 filters for Filtered-X purpose. There will be a very high computational burden in real time implementation. Therefore a (6 × 1) adaptive control scheme, as shown in Figure 15.5, has been chosen for the smart structure with only one sensor placed at the center of the upper plate surface. In this system, there are six secondary path actuators and one error sensor. The error signal will be: e(k) = d(k) +

6 X

n=1

pn (k) ∗ un (k),

(15.23)

where pn (k), n = 1, 2, · · · , 6 is the impulse response of the secondary path Pn (z), and un (k), n = 1, 2, · · · , 6 is the secondary signal of the adaptive filter Wn (z).

282

Modeling and Control of Vibration in Mechanical Systems

FIGURE 15.4 Block diagram of 2 × 2 adaptive feedback algorithm.

Adaptive Filtering Algorithms for Active Vibration Control

283

The reference signal x(k) is synthesized as an estimate of the primary disturbance. ˆ = e(k) − x(k) = d(k)

6 X

n=1

pˆn (k) ∗ un (k),

(15.24)

where pˆn (k) is the impulse response of Pˆn (z). The FXLMS algorithm is used to minimize the error signal e(k) by adjusting the weight vector for each adaptive filter Wn (z) according to: ′

Wn (k + 1) = Wn (k) + µXn (k)e(k), n = 1, 2, · · · , 6, ′

(15.25)

where Xn (k) = pˆn (k) ∗ X(k) is the reference signal vector filtered by the secondary path estimate, Pˆn (z).

FIGURE 15.5 Block diagram of 6 × 1 FXLMS adaptive feedback control system.

284

15.5.3

Modeling and Control of Vibration in Mechanical Systems

Simulation and implementation

Identification of the actuators in the platform can be found in Chapter 3. The 6 × 1 adaptive feedback controller is developed using the MATLAB Simulink platform. Improvement and modification on the controller structure was performed during an experiment. There are six Filtered-X LMS adaptive filters which form the main parts of the controller. The outputs of the adaptive filters are fed into the six DA converters of the dSPACE real time interface board (DS1104). Another major portion is the primary disturbance synthesizer with the six estimated filters Pˆn (z) of the secondary paths Pn (z). The secondary signals from the adaptive filters are also inputed into these filters in order to regenerate the primary disturbance signal. Normally a feedback controller can get into oscillation due to internally generated noise. To prevent this unstable situation, small nonlinear dead zones are placed at the output of the primary signal synthesizer and the receiving point of the error signal. The dead zone levels are small enough to ensure that only the noise signal is prohibited from passing through. An automatic gain control, which will be introduced in Section 15.5.3.2, is placed at the reference signal input to the six adaptive filters to improve the stability of the controller. Sampling frequency is set at 1 kHz. Therefore, the entire control process is carried out within 1 ms for each sampled signal and the secondary signals are sent out with 1 kHz sampling rate. We may obtain better performance if the sampling frequency is increased. But the computing demand of the six adaptive filters limits the sampling frequency. The adaptation step size µ is set between 0.0005 and 0.002 depending on the vibration frequency. During the experiment, the controller is able to achieve 20 dB to 30 dB attenuation for vibration of frequency from 60 Hz to 220 Hz. The result can be further improved by minimizing the modeling error of the secondary path. Figure 15.6 shows the experiment setup and the connections between various devices. The Stewart platform is mounted on a shaker (Labworks ET139) through a custom−made mounting interface. The shaker is powered by a power amplifier, which in turn is driven by a signal generator. A sine wave is used as the reference signal for the shaker. The piezoelectric accelerometer (357B21 from PCB Piezoelectronics) is placed at the center of the top plate to detect the error signal of the system as a whole. A charge amplifier (Sinocera YE5852) will amplify the detected error signal from the accelerometer and filter the high frequency noise before the signal is sent to the controller via the ADC unit. An interface unit of DS1104 DSP board, which includes peripheral outlets for the DS1104 card, serves as a junction point among the DSP board, charge amplifier and DMU (Drive and Monitoring Unit).

Adaptive Filtering Algorithms for Active Vibration Control 15.5.3.1

285

Experimental results

Experiments are conducted for different frequencies of vibration. Depending on the frequency of the disturbance (vibration) signal, an adaptation step size µ is chosen. Recall that the step size is inversely proportional to the reference input power; a smaller step size µ is chosen for higher frequency since the signal power increases as the frequency is higher. The PZT actuator has a maximum stroke length of 50 µm. In order not to exceed the maximum stroke length of the actuator, the vibration signal is chosen with 200 mV peak-to-peak at the ADC input. Error signal at the input of the algorithm will be 20 mV peak-to-peak (−40 dB) due to the scaling factor of the A/D converter. During the experiments, the controller is observed to be able to attenuate disturbances with frequency up to 220 Hz. The actual resonance frequency of the system is approximately at 230 Hz. When the vibration frequency is around 230 Hz, the system runs into an unstable state. This may be due to a large modeling error around the resonance frequency. Theoretically, there is a phase shift of close to 180◦ around the resonance frequency. The phase response of the secondary path also shows that there is a major phase shift around the resonance frequency. But the phase response in the identification result may not have sufficient changes to represent the resonance region. Therefore, the controller is modified to compensate the phase error. Inverted gain (−1) is inserted in the error signal path and the controller is tested with frequency starting from 240 Hz. With this modification, disturbance frequency up to 280 Hz is observed to be attenuated. Figures (15.7)−(15.10) are the experimental results captured by dSPACE control desk with frequency 60 Hz, 210 Hz, 240 Hz, and 270 Hz, respectively. The straight line is the point where AVC is switched on. The error signal is inverted when the disturbance frequency is set at 240 Hz and 270 Hz.

286

Modeling and Control of Vibration in Mechanical Systems

FIGURE 15.6 General layout of the experimental setup.

Adaptive Filtering Algorithms for Active Vibration Control

FIGURE 15.7 60 Hz error signal in dB unit.

FIGURE 15.8 210 Hz error signal in dB unit.

287

288

Modeling and Control of Vibration in Mechanical Systems

FIGURE 15.9 240 Hz error signal in dB unit.

FIGURE 15.10 270 Hz error signal in dB unit.

Adaptive Filtering Algorithms for Active Vibration Control 15.5.3.2

289

Controller modification and discussion

Observe from Figure 15.7−Figure 15.10 that the level of vibration attenuation varies with frequency. It may be because of the error of magnitude in the frequency response of the secondary path. An accurate modeling is required not only at the fundamental frequency but also at the harmonics of the disturbance signal. Because once the controller starts to suppress the disturbance signal, harmonics will be introduced in the reference signal path of the controller. Therefore, modeling is a big challenge. Nonlinearity of the secondary path actuator introduces modeling error and instability. Modeling errors include magnitude error and phase error. Phase error is a crucial factor since the feedback control system must be able to predict the disturbance before the control signal is sent out. Phase error in the secondary path model of the synthesizer section will introduce the wrong phase in the reference signal. Then the phase error in the reference signal makes the adaptive controller output the wrong phase of cancellation signal to the PZT actuator as well as to the synthesizer again. The magnitude response of the secondary path model is altered by ± 6 dB, which means that the FIR models of the secondary path are multiplied by 2 or divided by 2, to introduce the magnitude error. Then the experiments are conducted again. The stability of the controller is not affected and there is only a slight decrease in vibration attenuation level. Furthermore, there is a magnitude peak at 240 Hz in the frequency response of the secondary path model. Based on the experiments, the actual peak response is estimated to be at 230 Hz. But the controller is able to attenuate the disturbance frequency of 240 Hz with the phase compensating gain (−1) in the error signal path. Therefore, we can say that the adaptive feedback controller is able to tolerate some magnitude error in the secondary path model. During the experiment, the controller is observed to have a faster convergence rate for a higher frequency vibration. It verifies that a larger signal power at higher frequency can drive the adaptive filter to converge at a faster rate. But the faster convergence rate due to the higher signal power can lead the system to instability if the adaptation step size “µ” is not sufficiently small. Minimizing the adaptation step size can improve the stability of the controller but a slow convergence rate has to be borne. Another strategy used to compromise between the stability and convergence rate is to use the normalized LMS algorithm. The NFXLMS algorithm is able to work well in the feedforward algorithm. But in the practical implementation of adaptive feedback system, surged convergence of the normalized adaptive filter is again observed to cause some instability to the feedback system. Therefore, instead of applying the normalized algorithm, another strategy is developed and applied to the system to ensure the stability and to improve the convergence of the adaptive algorithm. That is an automatic gain control (AGC), introduced in the reference signal path as shown in Figure 15.11. When the adaptive controllers start to adapt their filter weights, it is observed that the reference signal becomes larger than the normal level before the adaptation is triggered. It may be because of the error of magnitude of the secondary path model.

290

Modeling and Control of Vibration in Mechanical Systems

In order to prevent the actuator from being overdriven and the control system from being driven into an unstable situation, an automatic gain control, AGC is inserted in the reference signal path. AGC is a simple nonlinear attenuator. The reference signal generated by the synthesizer will be transformed into a 16-order vector form by a delay line. “Max” block will filter out the maximum value of the vector signal and this signal will be transformed back to a scalar value by “To Sample” block. This scalar value is amplified with “Slider Gain” which can be adjusted during the experiment to obtain the best performance. Constant value 1 is added to avoid the division by zero. The summed value will be reciprocated by the math function “1/µ”. This results in a variable attenuating factor for the reference signal by inputting it to the “Cross Product” block together with the reference signal. Therefore, when the reference signal becomes larger, AGC will attenuate more and maintain the reference signal within a safe level. The mathematical description of the automatic gain control is Xi (k + 1) Xo (k + 1) = , (15.26) 1 + gXo (k) where g is the AGC gain (slider gain). The different results between with and without automatic gain control are shown Figures 15.12 and 15.13. It can be seen that automatic gain control can greatly improve the performance of the adaptive feedback controller. Another way of minimizing the overgrowing of reference signal is to restrict the adaptive filter weights by putting in a leakage factor. The leakage factor of the adaptive filter prevents the weight vector from growing without bound after the convergence of the error signal. A leakage factor of about 0.9999 is chosen so that 1 − 0.9999 = 0.0001 is still less than the smallest adaptation size 0.0005 in the experiments. But experiments with the leakage factor show that the attenuation level is decreased from 30 dB to 20 dB. Although the leakage factor can improve the stability, it can degrade the performance of the AVC.

15.6

Conclusion

In this chapter, with the adaptive filtering algorithm being conceptualized, a multiplereference adaptive feedback controller has been developed. A 6×1 adaptive feedback system has been implemented using Simulink and Control Desk. Adaptive identification has been used to model the six PZT actuators in the secondary path. Experiments have been performed with different adaptation step µ. Lower step size has to be chosen for a high frequency vibration signal in order to take into account its high signal power. Improperly chosen µ has been observed to affect the stability and performance of the control system. The regenerated reference signal has been observed to become larger than the normal level before the adaptive filter has started to adapt. The leaky FXLMS algorithm

Adaptive Filtering Algorithms for Active Vibration Control

291

has been applied to restrict the overflow of the weight vector of the adaptive filter. But the controller performance is degraded with introducing the leakage factor. Finally the automatic gain controller has been considered and developed in the reference signal path. The experimental results are greatly improved by fine-tuning the slider gain of the automatic gain controller. Actually, the nonlinear property of the automatic gain controller partially compensates for the modeling error so that the performance and stability of the adaptive feedback controller are comparatively better than those without using the automatic gain controller. For vibrations with frequencies from 60 to 220 Hz, an attenuation level of up to 30 dB has been achieved. Therefore, the adaptive feedback controller can provide satisfactory performance among various constraints.

FIGURE 15.11 Simulink diagram of automatic gain control.

292

Modeling and Control of Vibration in Mechanical Systems

FIGURE 15.12 180 Hz error signal without automatic gain control.

FIGURE 15.13 180 Hz error signal with automatic gain control.

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