solid mechanics and its applications978-3-319-72296-2/1.pdf · multi-input multi-output (mimo)...

Solid Mechanics and Its Applications

Volume 246

Series editors

J.R. Barber, Ann Arbor, USAAnders Klarbring, Linköping, Sweden

Founding editor

G.M.L. Gladwell, Waterloo, ON, Canada

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and How much?The aim of this series is to provide lucid accounts written by authoritativeresearchers giving vision and insight in answering these questions on the subject ofmechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus itincludes the foundation of mechanics; variational formulations; computationalmechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrationsof solids and structures; dynamical systems and chaos; the theories of elasticity,plasticity and viscoelasticity; composite materials; rods, beams, shells andmembranes; structural control and stability; soils, rocks and geomechanics;fracture; tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is to the first year graduate student. Some textsare monographs defining the current state of the field; others are accessible to finalyear undergraduates; but essentially the emphasis is on readability and clarity.

More information about this series at http://www.springer.com/series/6557

http://www.springer.com/series/6557

André Preumont

Vibration Control of ActiveStructuresAn Introduction

Fourth Edition

123

André PreumontActive Structures LaboratoryUniversité Libre de BruxellesBrusselsBelgium

ISSN 0925-0042 ISSN 2214-7764 (electronic)Solid Mechanics and Its ApplicationsISBN 978-3-319-72295-5 ISBN 978-3-319-72296-2 (eBook)https://doi.org/10.1007/978-3-319-72296-2

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1st edition: © Springer Science+Business Media Dordrecht 19972nd edition: © Kluwer Academic Publishers 20043rd edition: © Springer-Verlag Berlin Heidelberg 20114th edition: © Springer International Publishing AG 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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… le travail éloigne de noustrois grands mauxl’ennui, le vice et le besoin.

Voltaire, Candide (XXX)

Preface to the Fourth Edition

With respect to the previous edition, published 7 years ago, little changes have beenbrought to the first 14 chapters, except some minor alterations. In Chap. 5, thediscussion of the tuning of the inductive shunt has been deepened, and in Chap. 7, asimple demonstration of the important formula ni ¼ ð!i � ziÞ=2zi is (at last)presented.

In the subsequent part of the book, Chap. 15 on cable structures has beenconsiderably enlarged to include our recent work on suspension bridges. The partdevoted to optical telescopes has been split in several chapters: Chap. 16 is devotedto Adaptive Optics, a beautiful application of shape control of flat deformablemirrors. Chapter 17 focused on Active Optics, that is the control of the entiretelescope. These two chapters have a direct relevance to the recently startedextremely large telescope E-ELT (for which the scaling rules lead to surprisingobservations). They can also be viewed as interesting examples of control of largemulti-input multi-output (MIMO) systems. Chapter 18 is more prospective; itaddresses what could be the future of large (D[ 10 m) space reflectors at thehorizon 2030: foldable polymer shells with thin layers of electroactive material; aninteresting aspect of the discussion is the huge difference between the shape controlof a flat plate and that of a shell with double curvature.

The first edition of this book being more than 20 years old, some of its contenthas inevitably less relevance, compared to earlier times or may even be outdated.If I were to rewrite this book, I would probably remove some of the material, inparticular what is now part of most control textbooks (e.g., Chaps. 9, 11–13); thereader familiar with these subjects will forgive me and skip them, to focus on whatis the heart of the matter: the control of large, lightly damped structures.

I wish to thank the group of PhD students who worked with me on this subjectduring the past few years, in particular (in alphabetic order) David Alaluf, RenaudBastaits, Bilal Mokrani, and Kainan Wang. As usual, the quality of the hardwareinvolved in the various experimental set-ups owes a lot to the care of my Romaniancolleagues Mihăiţă Horodincă, Iulian Romanescu, and Ioan Burda.

vii

I am deeply indebted to the numerous funding organizations that supported theActive Structures Laboratory during my years at ULB. A particular thanks goes tothe European Space Agency (ESA) which was always receptive to new researchproposals.

Brussels, Belgium André PreumontNovember 2017

viii Preface to the Fourth Edition

Preface to the Third Edition

From the outset, this book was intended to be a bridge between the domains ofstructures and control. This means that both control and structural engineers shouldfeel at home when dealing with their own field (including familiar notations), whilehaving a chance to become acquainted with the other’s discipline and its ownspecialized vocabulary. That ambition could be summarized by paraphrasingWoody Allen’s movie: Everything You Always Wanted to Know AboutControl-Structure Interaction (But Were Afraid to Ask). Vocabulary and notationsare often major obstacles in communication between different communities, andthis is even more so when one deals with smart materials which are multiphysics bynature, forcing us to give up sacrosanct notations.

In the nine years that separate this third edition from the previous one, I haveenjoyed a considerable “return on experience” from users of this book, in academiaas well as in industry, and this has guided me in preparing the present text. Anotherimportant lesson has become clear: The success of a structural control project reliesmore on a sound understanding of the system than on a sophisticated controlalgorithm.

This third edition is about 100 pages longer than the second one. Half of theseadditional pages constitutes three totally new chapters: Chapter 3 is dedicated toelectromagnetic and piezoelectric transducers; the detailed analysis of energyconversion mechanisms is motivated by the increasing importance of energy har-vesting devices and passive damping mechanisms. Chapter 5 is devoted to thepassive damping of structures with piezoelectric transducers, including the basicprinciple of the switched inductive shunt. Chapter 16 deals with what will becomeone of the most challenging structural control problems of the coming years: theactive control of extremely large segmented telescopes, with a primary mirror ofdiameter D ¼ 30m and more. This problem is interesting in many respects: Aboveall the surface accuracy, because the RMS wavefront error e cannot exceed afraction of the wavelength, making the ratio e=D� 10�9 particularly small. The sizeof the multivariable control system is also quite unusual: it will involve severalthousand sensors and actuators. Finally, control-structure interaction is likely to be

ix

critical in the design; this offers a wonderful example of the application of multi-variable robustness tests. Several other chapters have been reorganized to providethe reader with a deeper physical insight, and better tools for design and robustnessassessment. In Chapter 7 on active damping, the duality between the DirectVelocity Feedback and the Integral Force Feedback has been stressed. Chapter 8 onisolation has been expanded to include the relaxation isolator which has out-standing performance and uses only passive components.

I take this opportunity to thank my co-workers and former students who havehelped me in producing this book. I am particularly indebted to the following fortheir work and contributions as listed below: Ahmed Abu Hanieh and Bruno deMarneffe for damping and isolation; Abhijit Ganguli for machine tool chatteralleviation; Pierre De Man for vibroacoustics; More Thomas Avraam for MR fluids;Renaud Bastaits and Gonçalo Rodrigues for active control of telescopes andadaptive optics; and Christophe Collette for semi-active suspension and many otherthings. Bilal Mokrani also contributed to several aspects. The quality of the hard-ware involved in the various experimental set-ups is due to the care of MihăițăHorodincă, Iulian Romanescu and Ioan Burda. Special thanks to Renaud whohelped me with the figures. The list of colleagues who have inspired me during mycareer would be too long to do them justice.

Brussels, January 2011 André Preumont

x Preface to the Third Edition

Preface to the Second Edition

My objective in writing this book was to cross the bridge between the structuraldynamics and control communities, while providing an overview of the potential ofSMART materials for sensing and actuating purposes in active vibration control.I wanted to keep it relatively simple and focused on systems which worked. Thisresulted in the following: (i) I restricted the text to fundamental concepts and leftaside most advanced ones (i.e. robust control) whose usefulness had not yet clearlybeen established for the application at hand. (ii) I promoted the use of collocatedactuator/sensor pairs whose potential, I thought, was strongly underestimated by thecontrol community. (iii) I emphasized control laws with guaranteed stability foractive damping (the wide-ranging applications of the IFF are particularly impres-sive). (iv) I tried to explain why an accurate prediction of the transmission zeros(usually called anti-resonances by the structural dynamicists) is so important inevaluating the performance of a control system. (v) I emphasized the fact that theopen-loop zeros are more difficult to predict than the poles, and that they could bestrongly influenced by the model truncation (high frequency dynamics) or by localeffects (such as membrane strains in piezoelectric shells), especially for nearlycollocated distributed actuator/sensor pairs; this effect alone explains many disap-pointments in active control systems. The success of the first edition confirmed thatthis approach was useful and it is with pleasure that I accepted to prepare thissecond edition in the same spirit as the first one.

The present edition contains three additional chapters: chapter 6 on active iso-lation where the celebrated “sky-hook” damper is revisited, Chap. 12 onsemi-active control, including some material on magneto-rheological fluids whosepotential seems enormous, and chapter 14 on the control of cable structures. It issomewhat surprising that this last subject is finding applications for vibrationamplitudes which are nine orders of magnitude apart (respectively meters for largecable-stayed bridges and nanometers for precision space structures). Some materialhas also been added on the modelling of piezoelectric structures (chapter 3) and onthe application of distributed sensors in vibroacoustics (chapter 13).

I am deeply indebted to my coworkers, particularly Younes Achkire andFrédéric Bossens for the cable structures, Vincent Piefort for the modelling of

xi

piezoelectric structures, Pierre De Man and Arnaud François in vibroacoustics,Ahmed Abu Hanieh and Mihăiță Horodincă in active isolation and, last but notleast, Nicolas Loix and Jean-Philippe Verschueren who run with enthusiasm andcompetence our spin-off company, Micromega Dynamics. I greatly enjoyedworking with them, exploring not only the concepts and the modelling techniques,but also the technology to make these control systems work. I also express mythanks to David de Salle who did all the editing, and to the Series Editor, Prof.Graham Gladwell who, once again, improved my English.

Brussels, November 2001 André Preumont

xii Preface to the Second Edition

Preface to the First Edition

I was introduced to structural control by Raphaël Haftka and Bill Hallauer during aone year stay at the Aerospace and Ocean Engineering department of VirginiaTech., during the academic year 1985-1986. At that time, there was a tremendousinterest in large space structures in the USA, mainly because of the StrategicDefense Initiative and the space station program. Most of the work was theoreticalor numerical, but Bill Hallauer was one of the few experimentalists trying toimplement control systems which worked on actual structures. When I returned toBelgium, I was appointed at the chair of Mechanical Engineering and Robotics atULB, and I decided to start some basic vibration control experiments on my own.A little later, SMART materials became widely available and offered completelynew possibilities, particularly for precision structures, but also brought new diffi-culties due to the strong coupling in their constitutive equations, which requires acomplete reformulation of the classical modelling techniques such as finite ele-ments. We started in this new field with the support of the national and regionalgovernments, the European Space Agency, and some bilateral collaborations withEuropean aerospace companies. Our Active Structures Laboratory was inauguratedin October 1995.

In recent years, with the downsizing of the space programs, active structuresseem to have lost some momentum for space applications, but they gave birth tointeresting spin-offs in various fields of engineering, including the car industry,machine tools, consumer products, and even civil engineering. I believe that thefield of SMART materials is still in its infancy; significant improvements can beexpected in the next few years, that will dramatically improve their recoverablestrain and their load carrying capability.

This book is the outgrowth of research work carried out at ULB and lecture notesfor courses given at the Universities of Brussels and Liège. I take this opportunity tothank all my coworkers who took part in this research, particularly Jean-PaulDufour, Christian Malekian, Nicolas Loix, Younes Achkire, Paul Alexandre andPierre De Man; I greatly enjoyed working with them along the years, and theirenthusiasm and creativity have been a constant stimulus in my work. I particularlythank Pierre who made almost all the figures.

xiii

Finally, I want to thank the Series Editor, Prof. Graham Gladwell who, as he didfor my previous book, read the manuscript and corrected many mistakes in myEnglish. His comments have helped to improve the text.

Bruxelles, July 1996 André Preumont

xiv Preface to the First Edition

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Active Versus Passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Smart Materials and Structures . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 The Various Steps of the Design . . . . . . . . . . . . . . . . . . . . . . 111.6 Plant Description, Error and Control Budget . . . . . . . . . . . . . . 121.7 Pseudo-inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7.1 Under-actuated System . . . . . . . . . . . . . . . . . . . . . . 141.7.2 Over-actuated System . . . . . . . . . . . . . . . . . . . . . . . 151.7.3 Singular Value Decomposition . . . . . . . . . . . . . . . . 151.7.4 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . 16

1.8 Readership and Organization of the Book . . . . . . . . . . . . . . . . 171.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Some Concepts in Structural Dynamics . . . . . . . . . . . . . . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Equation of Motion of a Discrete System . . . . . . . . . . . . . . . . 222.3 Vibration Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 Structure Without Rigid Body Modes . . . . . . . . . . . 242.4.2 Dynamic Flexibility Matrix . . . . . . . . . . . . . . . . . . . 262.4.3 Structure with Rigid Body Modes . . . . . . . . . . . . . . 282.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Collocated Control System . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Transmission Zeros and Constrained System . . . . . . 35

xv

2.6 Continuous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.7 Guyan Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8 Craig–Bampton Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Electromagnetic and Piezoelectric Transducers . . . . . . . . . . . . . . . . 473.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Voice Coil Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 Proof-Mass Actuator . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Geophone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 General Electromechanical Transducer . . . . . . . . . . . . . . . . . . 523.3.1 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . 523.3.2 Self-sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 Reaction Wheels and Gyrostabilizers . . . . . . . . . . . . . . . . . . . 543.5 Smart Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.6 Piezoelectric Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6.1 Constitutive Relations of a Discrete Transducer . . . . 573.6.2 Interpretation of k2 . . . . . . . . . . . . . . . . . . . . . . . . . 613.6.3 Admittance of the Piezoelectric Transducer . . . . . . . 63

3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Piezoelectric Beam, Plate and Truss . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Piezoelectric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1.1 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . 674.1.2 Coenergy Density Function . . . . . . . . . . . . . . . . . . . 69

4.2 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Piezoelectric Beam Actuator . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 734.3.2 Piezoelectric Loads . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 Laminar Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.1 Current and Charge Amplifiers . . . . . . . . . . . . . . . . 774.4.2 Distributed Sensor Output . . . . . . . . . . . . . . . . . . . . 774.4.3 Charge Amplifier Dynamics . . . . . . . . . . . . . . . . . . 79

4.5 Spatial Modal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5.1 Modal Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5.2 Modal Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6 Active Beam with Collocated Actuator/Sensor . . . . . . . . . . . . 824.6.1 Frequency Response Function . . . . . . . . . . . . . . . . . 834.6.2 Pole-Zero Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 844.6.3 Modal Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Admittance of a Beam with a Piezoelectric Patch . . . . . . . . . . 86

xvi Contents

4.8 Piezoelectric Laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.8.1 Two-Dimensional Constitutive Equations . . . . . . . . . 894.8.2 Kirchhoff Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 894.8.3 Stiffness Matrix of a Multilayer Elastic Laminate . . . 914.8.4 Multilayer Laminate with a Piezoelectric Layer . . . . 924.8.5 Equivalent Piezoelectric Loads . . . . . . . . . . . . . . . . 924.8.6 Sensor Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.8.7 Beam Model Versus Plate Model . . . . . . . . . . . . . . 944.8.8 Additional Remarks . . . . . . . . . . . . . . . . . . . . . . . . 97

4.9 Active Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.9.1 Open-Loop Transfer Function . . . . . . . . . . . . . . . . . 1014.9.2 Admittance Function . . . . . . . . . . . . . . . . . . . . . . . . 101

4.10 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Passive Damping with Piezoelectric Transducers . . . . . . . . . . . . . . 1075.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2 Resistive Shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3 Inductive Shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3.1 Equal Peak Design . . . . . . . . . . . . . . . . . . . . . . . . . 1135.3.2 Robustness of the Equal Peak Design . . . . . . . . . . . 116

5.4 Switched Shunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4.1 Equivalent Damping Ratio . . . . . . . . . . . . . . . . . . . 120


6 Collocated Versus Non-collocated Control . . . . . . . . . . . . . . . . . . . 1256.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2 Pole-Zero Flipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3 The Two-Mass Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3.1 Collocated Control . . . . . . . . . . . . . . . . . . . . . . . . . 1296.3.2 Non-collocated Control . . . . . . . . . . . . . . . . . . . . . . 129

6.4 Notch Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5 Effect of Pole-Zero Flipping on the Bode Plots . . . . . . . . . . . . 1326.6 Nearly Collocated Control System . . . . . . . . . . . . . . . . . . . . . 1326.7 Non-collocated Control Systems . . . . . . . . . . . . . . . . . . . . . . 1336.8 The Role of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7 Active Damping with Collocated System . . . . . . . . . . . . . . . . . . . . . 1397.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.2 Lead Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.3 Direct Velocity Feedback (DVF) . . . . . . . . . . . . . . . . . . . . . . 143

Contents xvii

7.4 Positive Position Feedback (PPF) . . . . . . . . . . . . . . . . . . . . . . 1457.5 Integral Force Feedback (IFF) . . . . . . . . . . . . . . . . . . . . . . . . 1487.6 Duality Between the Lead and the IFF Controllers . . . . . . . . . 153

7.6.1 Root Locus of a Single Mode . . . . . . . . . . . . . . . . . 1537.6.2 Open-Loop Poles and Zeros . . . . . . . . . . . . . . . . . . 153

7.7 Actuator and Sensor Dynamics . . . . . . . . . . . . . . . . . . . . . . . 1547.8 Decentralized Control with Collocated Pairs . . . . . . . . . . . . . . 156

7.8.1 Cross talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.8.2 Force Actuator and Displacement Sensor . . . . . . . . . 1567.8.3 Displacement Actuator and Force Sensor . . . . . . . . . 157

7.9 Proof of Equation (7.18)–(7.32) . . . . . . . . . . . . . . . . . . . . . . . 1587.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8 Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.2 Relaxation Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.2.1 Electromagnetic Realization . . . . . . . . . . . . . . . . . . . 1708.3 Active Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

8.3.1 Sky-Hook Damper . . . . . . . . . . . . . . . . . . . . . . . . . 1728.3.2 Integral Force Feedback . . . . . . . . . . . . . . . . . . . . . 173

8.4 Flexible Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768.4.1 Free-Free Beam with Isolator . . . . . . . . . . . . . . . . . 177

8.5 Payload Isolation in Spacecraft . . . . . . . . . . . . . . . . . . . . . . . 1808.5.1 Interaction Isolator/Attitude Control . . . . . . . . . . . . . 1808.5.2 Gough–Stewart Platform . . . . . . . . . . . . . . . . . . . . . 181

8.6 Six-Axis Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.6.1 Relaxation Isolator . . . . . . . . . . . . . . . . . . . . . . . . . 1838.6.2 Integral Force Feedback . . . . . . . . . . . . . . . . . . . . . 1848.6.3 Spherical Joints, Modal Spread . . . . . . . . . . . . . . . . 186

8.7 Active Versus Passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878.8 Car Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

9 State Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1999.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1999.2 State Space Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

9.2.1 Single Degree of Freedom Oscillator . . . . . . . . . . . . 2019.2.2 Flexible Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 2029.2.3 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 204

9.3 System Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.3.1 Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9.4 Pole Placement by State Feedback . . . . . . . . . . . . . . . . . . . . . 2089.4.1 Example: Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 209

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9.5 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.5.1 Symmetric Root Locus . . . . . . . . . . . . . . . . . . . . . . 2119.5.2 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 212

9.6 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.7 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

9.7.1 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 2169.8 Reduced-Order Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

9.8.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2189.8.2 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 219

9.9 Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2209.10 Transfer Function of the Compensator . . . . . . . . . . . . . . . . . . 221

9.10.1 The Two-Mass Problem . . . . . . . . . . . . . . . . . . . . . 2229.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

10 Analysis and Synthesis in the Frequency Domain . . . . . . . . . . . . . . 22710.1 Gain and Phase Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22710.2 Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

10.2.1 Cauchy’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . 22810.2.2 Nyquist Stability Criterion . . . . . . . . . . . . . . . . . . . . 229

10.3 Nichols Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23310.4 Feedback Specification for SISO Systems . . . . . . . . . . . . . . . . 234

10.4.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23410.4.2 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23510.4.3 Performance Specification . . . . . . . . . . . . . . . . . . . . 23510.4.4 Unstructured Uncertainty . . . . . . . . . . . . . . . . . . . . . 23610.4.5 Robust Performance and Robust Stability . . . . . . . . . 237

10.5 Bode Gain–Phase Relationships . . . . . . . . . . . . . . . . . . . . . . . 24010.6 The Bode Ideal Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24210.7 Non-minimum Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . 24410.8 Usual Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

10.8.1 System Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24710.8.2 Lead Compensator . . . . . . . . . . . . . . . . . . . . . . . . . 24810.8.3 PI Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.8.4 Lag Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . 25010.8.5 PID Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10.9 Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25110.9.1 Performance Specification . . . . . . . . . . . . . . . . . . . . 25110.9.2 Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . 25210.9.3 Stability Robustness Tests . . . . . . . . . . . . . . . . . . . . 25310.9.4 Residual Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 254


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11 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25911.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25911.2 Quadratic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25911.3 Deterministic LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26011.4 Stochastic Response to a White Noise . . . . . . . . . . . . . . . . . . 262

11.4.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26411.5 Stochastic LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26411.6 Asymptotic Behavior of the Closed Loop . . . . . . . . . . . . . . . . 26511.7 Prescribed Degree of Stability . . . . . . . . . . . . . . . . . . . . . . . . 26711.8 Gain and Phase Margins of the LQR . . . . . . . . . . . . . . . . . . . 26811.9 Full State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

11.9.1 Covariance of the Reconstruction Error . . . . . . . . . . 27011.10 Kalman Filter (KF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27111.11 Linear Quadratic Gaussian (LQG) . . . . . . . . . . . . . . . . . . . . . 27111.12 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27211.13 Spillover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

11.13.1 Spillover Reduction . . . . . . . . . . . . . . . . . . . . . . . . 27611.14 Loop Transfer Recovery (LTR) . . . . . . . . . . . . . . . . . . . . . . . 27711.15 Integral Control with State Feedback . . . . . . . . . . . . . . . . . . . 27811.16 Frequency Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

11.16.1 Frequency-Shaped Cost Functionals . . . . . . . . . . . . . 27911.16.2 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282


12 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . . . 28912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29012.2 Controllability and Observability Matrices . . . . . . . . . . . . . . . 29012.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

12.3.1 Cart with Two Inverted Pendulums . . . . . . . . . . . . . 29312.3.2 Double Inverted Pendulum . . . . . . . . . . . . . . . . . . . 29312.3.3 Two d.o.f. Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 295

12.4 State Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29612.4.1 Control Canonical Form . . . . . . . . . . . . . . . . . . . . . 29612.4.2 Left and Right Eigenvectors . . . . . . . . . . . . . . . . . . 29812.4.3 Diagonal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

12.5 PBH Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29912.6 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30012.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30112.8 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30212.9 Controllability and Observability Gramians . . . . . . . . . . . . . . . 30312.10 Internally Balanced Coordinates . . . . . . . . . . . . . . . . . . . . . . . 304

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12.11 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30612.11.1 Transfer Equivalent Realization . . . . . . . . . . . . . . . . 30612.11.2 Internally Balanced Realization . . . . . . . . . . . . . . . . 30712.11.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308


13 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

13.1.1 Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31413.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

13.2.1 Routh–Hurwitz Criterion . . . . . . . . . . . . . . . . . . . . . 31613.3 Lyapunov’s Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . 317

13.3.1 Introductory Example . . . . . . . . . . . . . . . . . . . . . . . 31713.3.2 Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 31813.3.3 Asymptotic Stability Theorem . . . . . . . . . . . . . . . . . 32013.3.4 Lasalle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 32013.3.5 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . 32113.3.6 Instability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 321

13.4 Lyapunov Functions for Linear Systems . . . . . . . . . . . . . . . . . 32313.5 Lyapunov’s Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . . 32413.6 An Application to Controller Design . . . . . . . . . . . . . . . . . . . 32513.7 Energy Absorbing Controls . . . . . . . . . . . . . . . . . . . . . . . . . . 32613.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

14 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33314.1 Digital Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

14.1.1 Sampling, Aliasing, and Prefiltering . . . . . . . . . . . . . 33414.1.2 Zero-Order Hold, Computational Delay . . . . . . . . . . 33514.1.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33614.1.4 Discretization of a Continuous Controller . . . . . . . . . 337

14.2 Active Damping of a Truss Structure . . . . . . . . . . . . . . . . . . . 33814.2.1 Actuator Placement . . . . . . . . . . . . . . . . . . . . . . . . . 33914.2.2 Implementation, Experimental Results . . . . . . . . . . . 339

14.3 Active Damping Generic Interface . . . . . . . . . . . . . . . . . . . . . 34214.3.1 Active Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 34214.3.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34514.3.3 Pointing and Position Control . . . . . . . . . . . . . . . . . 345

14.4 Active Damping of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 34714.4.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

14.5 Active Damping of a Stiff Beam . . . . . . . . . . . . . . . . . . . . . . 35014.5.1 System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

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14.6 The HAC/LAC Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35214.6.1 Wide-Band Position Control . . . . . . . . . . . . . . . . . . 35314.6.2 Compensator Design . . . . . . . . . . . . . . . . . . . . . . . . 35514.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

14.7 Vibroacoustics: Volume Displacement Sensors . . . . . . . . . . . . 35814.7.1 QWSIS Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35914.7.2 Discrete Array Sensor . . . . . . . . . . . . . . . . . . . . . . . 36214.7.3 Spatial Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . 36514.7.4 Distributed Sensor . . . . . . . . . . . . . . . . . . . . . . . . . 368


15 Tendon Control of Cable Structures . . . . . . . . . . . . . . . . . . . . . . . . 37715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37715.2 Tendon Control of Strings and Cables . . . . . . . . . . . . . . . . . . 37915.3 Active Damping Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37915.4 Basic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38115.5 Linear Theory of Decentralized Active Damping. . . . . . . . . . . 38315.6 Guyed Truss Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38815.7 Microprecision Interferometer Testbed . . . . . . . . . . . . . . . . . . 39115.8 Free-Floating Truss Experiment . . . . . . . . . . . . . . . . . . . . . . . 39215.9 Application to Cable-Stayed Bridges . . . . . . . . . . . . . . . . . . . 394

15.9.1 Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . . 39415.9.2 Control of Parametric Resonance . . . . . . . . . . . . . . . 39415.9.3 Large Scale Experiment . . . . . . . . . . . . . . . . . . . . . 396

15.10 Application to Suspension Bridges . . . . . . . . . . . . . . . . . . . . . 40315.10.1 Footbridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40315.10.2 Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . . 407

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

16 Active Control of Large Telescopes: Adaptive Optics . . . . . . . . . . . 41716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

16.1.1 Wavefront Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . 42016.1.2 Zernike Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42016.1.3 Fried Length, Seeing . . . . . . . . . . . . . . . . . . . . . . . . 42216.1.4 Kolmogorov Turbulence Model . . . . . . . . . . . . . . . . 42216.1.5 Strehl Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42316.1.6 Power Spectral Density of the Zernike Modes . . . . . 424

16.2 Deformable Mirror for Adaptive Optics . . . . . . . . . . . . . . . . . 42516.2.1 Stoney Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 42616.2.2 Stroke Versus Natural Frequency . . . . . . . . . . . . . . . 428

16.3 Feedback Control of an AO Mirror . . . . . . . . . . . . . . . . . . . . 42916.3.1 Quasi-static Control . . . . . . . . . . . . . . . . . . . . . . . . 42916.3.2 Control of the Mirror Based on the Jacobian . . . . . . 43016.3.3 Control of Zernike Modes . . . . . . . . . . . . . . . . . . . . 431

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16.4 Dynamic Response of the AO Mirror . . . . . . . . . . . . . . . . . . . 43516.4.1 Dynamic Model of the Mirror . . . . . . . . . . . . . . . . . 43516.4.2 Control-Structure Interaction . . . . . . . . . . . . . . . . . . 43716.4.3 Passive Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 43916.4.4 Active Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

16.5 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44416.5.1 Segmented AO Mirror . . . . . . . . . . . . . . . . . . . . . . 44416.5.2 Initial Curvature of the AO Mirror . . . . . . . . . . . . . . 447

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

17 Active Control of Large Telescopes: Active Optics . . . . . . . . . . . . . 44917.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44917.2 Monolithic Primary Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . 45017.3 Segmented Primary Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . 45117.4 SVD Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

17.4.1 Loop Shaping of the SVD Controller . . . . . . . . . . . . 45517.5 Dynamics of a Segmented Mirror . . . . . . . . . . . . . . . . . . . . . . 45717.6 Control-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 459

17.6.1 SISO System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45917.6.2 MIMO System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46117.6.3 Spillover Alleviation . . . . . . . . . . . . . . . . . . . . . . . . 463

17.7 Scaling Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46417.7.1 Static Deflection Under Gravity . . . . . . . . . . . . . . . . 46417.7.2 First Resonance Frequency . . . . . . . . . . . . . . . . . . . 46517.7.3 Control Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 466

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

18 Adaptive Thin Shell Space Reflectors . . . . . . . . . . . . . . . . . . . . . . . 46918.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46918.2 Adaptive Plates Versus Adaptive Shells . . . . . . . . . . . . . . . . . 47218.3 Adaptive Spherical Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47318.4 Quasi-static Control: Hierarchical Approach . . . . . . . . . . . . . . 47718.5 Petal Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47818.6 MATS Demonstrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

18.6.1 Manufacturing of the Demonstrator . . . . . . . . . . . . . 484References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

19 Semi-active Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48719.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48719.2 Magneto-Rheological Fluids . . . . . . . . . . . . . . . . . . . . . . . . . 48819.3 MR Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49019.4 Semi-active Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

19.4.1 Semi-active Devices . . . . . . . . . . . . . . . . . . . . . . . . 494

Contents xxiii

19.5 Narrow-Band Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . 49519.5.1 Quarter-Car Semi-active Suspension . . . . . . . . . . . . 496


Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

xxiv Contents

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