Vibration Control of Active Structures
SOLID MECHANICS AND ITS APPLICATIONS Volume 179
Series Editor:
G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI
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 authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.
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A. Preumont
Vibration Control of Active Structures An Introduction Third Edition
ABC
Author Prof. A. Preumont Universite Libre De Bruxelles Active Structures Laboratory Bruxelles Belgium E-mail:
[email protected] Telephone: +32 2 650 2689 Fax: +32 2 650 2710
ISBN 978-94-007-2032-9
e-ISBN 978-94-007-2033-6
DOI 10.1007/978-94-007-2033-6 Library of Congress Control Number: 2011930521 c 2011 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com
“. . . le travail ´eloigne de nous trois grands maux: l’ennui, le vice et le besoin. ” Voltaire, Candide (XXX)
Preface to the Third Edition
From the outset, this book was intended to be a bridge between the domains of structures and control. This means that both control and structural engineers should feel at home when dealing with their own field (including familiar notations), while having a chance to become acquainted with the other’s discipline and its own specialized vocabulary. That ambition could be summarized by paraphrasing Woody Allen’s movie: Everything You Always Wanted to Know About Control-Structure Interaction (But Were Afraid to Ask). Vocabulary and notations are often major obstacles in communication between different communities, and this is even more so when one deals with smart materials which are multiphysics by nature, forcing us to give up sacrosanct notations. In the nine years that separate this third edition from the previous one, I have enjoyed a considerable “return on experience” from users of this book, in academia as well as in industry, and this has guided me in preparing the present text. Another important lesson has become clear: The success of a structural control project relies more on a sound understanding of the system than on a sophisticated control algorithm. This third edition is about 100 pages longer than the second one. Half of these additional pages constitutes three totally new chapters: Chapter 3 is dedicated to electromagnetic and piezoelectric transducers; the detailed analysis of energy conversion mechanisms is motivated by the increasing importance of energy harvesting devices and passive damping mechanisms. Chapter 5 is devoted to the passive damping of structures with piezoelectric transducers, including the basic principle of the switched inductive shunt. Chapter 16 deals with what will become one of the most challenging structural control problems of the coming years: the active control of extremely large segmented telescopes, with a primary mirror of diameter D = 30m and more. This problem is interesting in many respects: Above all the surface accuracy, because the RMS wavefront error ε cannot exceed a fraction of the wavelength, making the ratio ε/D ∼ 10−9 particularly small. The size of the multivariable control system is also quite unusual: it will involve several thousand sensors
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and actuators. Finally, control-structure interaction is likely to be critical in the design; this offers a wonderful example of the application of multivariable robustness tests. Several other chapters have been reorganized to provide the reader with a deeper physical insight, and better tools for design and robustness assessment. In chapter 7 on active damping, the duality between the Direct Velocity Feedback and the Integral Force Feedback has been stressed. Chapter 8 on isolation has been expanded to include the relaxation isolator which has outstanding performance and uses only passive components. I take this opportunity to thank my co-workers and former students who have helped me in producing this book. I am particularly indebted to the following for their work and contributions as listed below: Ahmed Abu Hanieh and Bruno de Marneffe for damping and isolation; Abhijit Ganguli for machine tool chatter alleviation; Pierre De Man for vibroacoustics; More Thomas Avraam for MR fluids; Renaud Bastaits and Gon¸calo Rodrigues for active control of telescopes and adaptive optics; and Christophe Collette for semiactive suspension and many other things. Bilal Mokrani also contributed to several aspects. The quality of the hardware involved in the various experimental set-ups is due to the care of Mih˘ ai¸ta˘ Horodinc˘ a, Iulian Romanescu and Ioan Burda. Special thanks to Renaud who helped me with the figures. The list of colleagues who have inspired me during my career would be too long to do them justice. Brussels January 2011
Andr´e Preumont
Preface to the Second Edition
My objective in writing this book was to cross the bridge between the structural dynamics and control communities, while providing an overview of the potential of SMART materials for sensing and actuating purposes in active vibration control. I wanted to keep it relatively simple and focused on systems which worked. This resulted in the following: (i) I restricted the text to fundamental concepts and left aside most advanced ones (i.e. robust control) whose usefulness had not yet clearly been established for the application at hand. (ii) I promoted the use of collocated actuator/sensor pairs whose potential, I thought, was strongly underestimated by the control community. (iii) I emphasized control laws with guaranteed stability for active damping (the wide-ranging applications of the IFF are particularly impressive). (iv) I tried to explain why an accurate prediction of the transmission zeros (usually called anti-resonances by the structural dynamicists) is so important in evaluating the performance of a control system. (v) I emphasized the fact that the open-loop zeros are more difficult to predict than the poles, and that they could be strongly influenced by the model truncation (high frequency dynamics) or by local effects (such as membrane strains in piezoelectric shells), especially for nearly collocated distributed actuator/sensor pairs; this effect alone explains many disappointments in active control systems. The success of the first edition confirmed that this approach was useful and it is with pleasure that I accepted to prepare this second edition in the same spirit as the first one. The present edition contains three additional chapters: chapter 6 on active isolation where the celebrated “sky-hook” damper is revisited, chapter 12 on semi-active control, including some material on magneto-rheological fluids whose potential seems enormous, and chapter 14 on the control of cablestructures. It is somewhat surprising that this last subject is finding applications for vibration amplitudes which are nine orders of magnitude apart (respectively meters for large cable-stayed bridges and nanometers for precision space structures). Some material has also been added on the modelling
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of piezoelectric structures (chapter 3) and on the application of distributed sensors in vibroacoustics (chapter 13). I am deeply indebted to my coworkers, particularly Younes Achkire and Fr´ed´eric Bossens for the cable-structures, Vincent Piefort for the modelling of piezoelectric structures, Pierre De Man and Arnaud Fran¸cois in vibroacoustics, Ahmed Abu Hanieh and Mih˘ ai¸ta˘ Horodinc˘ a in active isolation and, last but not least, Nicolas Loix and Jean-Philippe Verschueren who run with enthusiasm and competence our spin-off company, Micromega Dynamics. I greatly enjoyed working with them, exploring not only the concepts and the modelling techniques, but also the technology to make these control systems work. I also express my thanks 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
Andr´e Preumont
Preface to the First Edition
I was introduced to structural control by Rapha¨el Haftka and Bill Hallauer during a one year stay at the Aerospace and Ocean Engineering department of Virginia Tech., during the academic year 1985-1986. At that time, there was a tremendous interest in large space structures in the USA, mainly because of the Strategic Defense Initiative and the space station program. Most of the work was theoretical or numerical, but Bill Hallauer was one of the few experimentalists trying to implement control systems which worked on actual structures. When I returned to Belgium, I was appointed at the chair of Mechanical Engineering and Robotics at ULB, and I decided to start some basic vibration control experiments on my own. A little later, SMART materials became widely available and offered completely new possibilities, particularly for precision structures, but also brought new difficulties due to the strong coupling in their constitutive equations, which requires a complete reformulation of the classical modelling techniques such as finite elements. We started in this new field with the support of the national and regional governments, the European Space Agency, and some bilateral collaborations with European aerospace companies. Our Active Structures Laboratory was inaugurated in October 1995. In recent years, with the downsizing of the space programs, active structures seem to have lost some momentum for space applications, but they gave birth to interesting spin-offs in various fields of engineering, including the car industry, machine tools, consumer products, and even civil engineering. I believe that the field of SMART materials is still in its infancy; significant improvements can be expected in the next few years, that will dramatically improve their recoverable strain and their load carrying capability. This book is the outgrowth of research work carried out at ULB and lecture notes for courses given at the Universities of Brussels and Li`ege. I take this opportunity to thank all my coworkers who took part in this research, particularly Jean-Paul Dufour, Christian Malekian, Nicolas Loix, Younes Achkire,
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Paul Alexandre and Pierre De Man; I greatly enjoyed working with them along the years, and their enthusiasm and creativity have been a constant stimulus in my work. I particularly thank Pierre who made almost all the figures. Finally, I want to thank the Series Editor, Prof. Graham Gladwell who, as he did for my previous book, read the manuscript and corrected many mistakes in my English. His comments have helped to improve the text. Bruxelles July 1996
Andr´e Preumont
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Active versus Passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Vibration Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Smart Materials and Structures . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Various Steps of the Design . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Plant Description, Error and Control Budget . . . . . . . . . . . . . 1.7 Readership and Organization of the Book . . . . . . . . . . . . . . . . 1.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 4 5 7 8 9 10 13 15 16
2
Some Concepts in Structural Dynamics . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equation of Motion of a Discrete System . . . . . . . . . . . . . . . . . 2.3 Vibration Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Structure without Rigid Body Modes . . . . . . . . . . . . . . 2.4.2 Dynamic Flexibility Matrix . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Structure with Rigid Body Modes . . . . . . . . . . . . . . . . . 2.4.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Collocated Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Transmission Zeros and Constrained System . . . . . . . . 2.6 Continuous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Guyan Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Craig-Bampton Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 17 17 18 20 20 21 23 26 27 30 32 33 35 37
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3
Electromagnetic and Piezoelectric Transducers . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Voice Coil Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Proof-Mass Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Geophone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 General Electromechanical Transducer . . . . . . . . . . . . . . . . . . . 3.3.1 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Self-sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Reaction Wheels and Gyrostabilizers . . . . . . . . . . . . . . . . . . . . . 3.5 Smart Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Piezoelectric Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Constitutive Relations of a Discrete Transducer . . . . . 3.6.2 Interpretation of k 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Admittance of the Piezoelectric Transducer . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41 41 41 43 45 46 46 47 48 50 50 51 55 56 58
4
Piezoelectric Beam, Plate and Truss . . . . . . . . . . . . . . . . . . . . . 4.1 Piezoelectric Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Coenergy Density Function . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Piezoelectric Beam Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Piezoelectric Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Laminar Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Current and Charge Amplifiers . . . . . . . . . . . . . . . . . . . . 4.4.2 Distributed Sensor Output . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Charge Amplifier Dynamics . . . . . . . . . . . . . . . . . . . . . . . 4.5 Spatial Modal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Modal Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Modal Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Active Beam with Collocated Actuator-Sensor . . . . . . . . . . . . 4.6.1 Frequency Response Function . . . . . . . . . . . . . . . . . . . . . 4.6.2 Pole-Zero Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Modal Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Admittance of a Beam with a Piezoelectric Patch . . . . . . . . . . 4.8 Piezoelectric Laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Two Dimensional Constitutive Equations . . . . . . . . . . . 4.8.2 Kirchhoff Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Stiffness Matrix of a Multi-layer Elastic Laminate . . . 4.8.4 Multi-layer Laminate with a Piezoelectric Layer . . . . . 4.8.5 Equivalent Piezoelectric Loads . . . . . . . . . . . . . . . . . . . . 4.8.6 Sensor Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.7 Beam Model vs. Plate Model . . . . . . . . . . . . . . . . . . . . . 4.8.8 Additional Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61 61 61 64 66 66 66 68 71 71 71 73 73 73 75 76 77 78 79 80 83 83 83 85 86 87 87 89 92
Contents
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4.9 Active Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Open-Loop Transfer Function . . . . . . . . . . . . . . . . . . . . . 4.9.2 Admittance Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92 96 97 97 98
5
Passive Damping with Piezoelectric Transducers . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Resistive Shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Inductive Shunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Switched Shunt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Equivalent Damping Ratio . . . . . . . . . . . . . . . . . . . . . . . 5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103 103 104 106 109 112 114
6
Collocated versus Non-collocated Control . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Pole-Zero Flipping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Two-Mass Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Collocated Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Non-collocated Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Notch Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Effect of Pole-Zero Flipping on the Bode Plots . . . . . . . . . . . . 6.6 Nearly Collocated Control System . . . . . . . . . . . . . . . . . . . . . . . 6.7 Non-collocated Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 The Role of Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117 117 118 119 120 121 123 124 124 126 129 129
7
Active Damping with Collocated System . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Lead Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Direct Velocity Feedback (DVF) . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Positive Position Feedback (PPF) . . . . . . . . . . . . . . . . . . . . . . . 7.5 Integral Force Feedback(IFF) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Duality between the Lead and the IFF Controllers . . . . . . . . . 7.6.1 Root-Locus of a Single Mode . . . . . . . . . . . . . . . . . . . . . 7.6.2 Open-Loop Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . 7.7 Actuator and Sensor Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Decentralized Control with Collocated Pairs . . . . . . . . . . . . . . 7.8.1 Cross Talk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Force Actuator and Displacement Sensor . . . . . . . . . . . 7.8.3 Displacement Actuator and Force Sensor . . . . . . . . . . . 7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131 131 133 135 137 140 145 145 147 147 149 149 149 150 150
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8
Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Relaxation Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Electromagnetic Realization . . . . . . . . . . . . . . . . . . . . . . 8.3 Active Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Sky-Hook Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Integral Force Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Flexible Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Free-Free Beam with Isolator . . . . . . . . . . . . . . . . . . . . . 8.5 Payload Isolation in Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Interaction Isolator/Attitude Control . . . . . . . . . . . . . . 8.5.2 Gough-Stewart Platform . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Six-Axis Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Relaxation Isolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Integral Force Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Spherical Joints, Modal Spread . . . . . . . . . . . . . . . . . . . . 8.7 Active vs. Passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Car Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155 155 157 160 161 162 163 165 167 170 170 171 172 173 174 175 177 179 184
9
State Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 State Space Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Single Degree of Freedom Oscillator . . . . . . . . . . . . . . . 9.2.2 Flexible Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 System Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Pole Placement by State Feedback . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Example: Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Symmetric Root Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Reduced Order Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.2 Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Transfer Function of the Compensator . . . . . . . . . . . . . . . . . . . 9.10.1 The Two-Mass Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187 187 189 189 190 191 193 194 195 197 198 199 199 201 203 204 206 206 207 208 208 209 213
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10 Analysis and Synthesis in the Frequency Domain . . . . . . . . 10.1 Gain and Phase Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Cauchy’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Nyquist Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . 10.3 Nichols Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Feedback Specification for SISO Systems . . . . . . . . . . . . . . . . . 10.4.1 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Performance Specification . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 Unstructured Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 10.4.5 Robust Performance and Robust Stability . . . . . . . . . . 10.5 Bode Gain-Phase Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 The Bode Ideal Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Non-minimum Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Usual Compensators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 System Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.2 Lead Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.3 PI Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.4 Lag Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.5 PID Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.1 Performance Specification . . . . . . . . . . . . . . . . . . . . . . . . 10.9.2 Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.3 Stability Robustness Tests . . . . . . . . . . . . . . . . . . . . . . . . 10.9.4 Residual Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215 215 216 216 217 221 222 222 223 223 224 225 228 231 233 235 235 237 238 239 239 239 240 241 241 243 244
11 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quadratic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Deterministic LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Stochastic Response to a White Noise . . . . . . . . . . . . . . . . . . . 11.4.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Stochastic LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Asymptotic Behavior of the Closed-Loop . . . . . . . . . . . . . . . . . 11.7 Prescribed Degree of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Gain and Phase Margins of the LQR . . . . . . . . . . . . . . . . . . . . . 11.9 Full State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9.1 Covariance of the Reconstruction Error . . . . . . . . . . . . 11.10 Kalman-Bucy Filter (KBF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.11 Linear Quadratic Gaussian (LQG) . . . . . . . . . . . . . . . . . . . . . . 11.12 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13 Spillover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13.1 Spillover Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
247 247 247 248 250 251 251 252 254 255 257 258 258 259 260 261 263
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11.14 Loop Transfer Recovery (LTR) . . . . . . . . . . . . . . . . . . . . . . . . . 11.15 Integral Control with State Feedback . . . . . . . . . . . . . . . . . . . . 11.16 Frequency Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.16.1 Frequency-Shaped Cost Functionals . . . . . . . . . . . . . . . 11.16.2 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.17 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
264 265 266 266 270 271
12 Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Controllability and Observability Matrices . . . . . . . . . . . . . . . . 12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Cart with Two Inverted Pendulums . . . . . . . . . . . . . . . . 12.3.2 Double Inverted Pendulum . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Two d.o.f. Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 State Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Control Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Left and Right Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Diagonal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 PBH Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Controllability and Observability Gramians . . . . . . . . . . . . . . . 12.10 Internally Balanced Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11.1 Transfer Equivalent Realization . . . . . . . . . . . . . . . . . . . 12.11.2 Internally Balanced Realization . . . . . . . . . . . . . . . . . . . 12.11.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
275 275 275 276 278 278 279 280 281 282 283 284 284 285 286 287 288 289 291 291 292 293 294
13 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Routh-Hurwitz Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Lyapunov’s Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Asymptotic Stability Theorem . . . . . . . . . . . . . . . . . . . . 13.3.4 Lasalle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.5 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.6 Instability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Lyapunov Functions for Linear Systems . . . . . . . . . . . . . . . . . . 13.5 Lyapunov’s Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
299 299 300 301 302 303 303 303 305 305 306 307 308 309
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13.6 An Application to Controller Design . . . . . . . . . . . . . . . . . . . . . 310 13.7 Energy Absorbing Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 13.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 14 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Digital Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Sampling, Aliasing and Prefiltering . . . . . . . . . . . . . . . . 14.1.2 Zero-Order Hold, Computational Delay . . . . . . . . . . . . 14.1.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.4 Discretization of a Continuous Controller . . . . . . . . . . . 14.2 Active Damping of a Truss Structure . . . . . . . . . . . . . . . . . . . . 14.2.1 Actuator Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Implementation, Experimental Results . . . . . . . . . . . . . 14.3 Active Damping Generic Interface . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Active Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Pointing and Position Control . . . . . . . . . . . . . . . . . . . . . 14.4 Active Damping of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Active Damping of a Stiff Beam . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 The HAC/LAC Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Wide-Band Position Control . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Compensator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Vibroacoustics: Volume Displacement Sensors . . . . . . . . . . . . . 14.7.1 QWSIS Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.2 Discrete Array Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.3 Spatial Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.4 Distributed Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
317 317 318 319 320 321 322 322 324 326 326 329 330 330 332 334 334 335 337 339 339 342 343 346 349 351 357
15 Tendon Control of Cable Structures . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Tendon Control of Strings and Cables . . . . . . . . . . . . . . . . . . . . 15.3 Active Damping Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Basic Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Linear Theory of Decentralized Active Damping . . . . . . . . . . . 15.6 Guyed Truss Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Micro Precision Interferometer Testbed . . . . . . . . . . . . . . . . . . . 15.8 Free Floating Truss Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 Application to Cable-Stayed Bridges . . . . . . . . . . . . . . . . . . . . . 15.10 Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.11 Control of Parametric Resonance . . . . . . . . . . . . . . . . . . . . . . . 15.12 Large Scale Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
359 359 360 361 363 364 368 370 372 375 375 375 377
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16 Active Control of Large Telescopes . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Adaptive Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 Active Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Monolithic Primary Mirror . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Segmented Primary Mirror . . . . . . . . . . . . . . . . . . . . . . . 16.4 SVD Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Loop Shaping of the SVD Controller . . . . . . . . . . . . . . . 16.5 Dynamics of a Segmented Mirror . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Control-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.1 Multiplicative Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 16.6.2 Additive Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
385 385 387 390 391 392 394 394 395 397 397 399 400
17 Semi-active Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Magneto-Rheological Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 MR Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Semi-active Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Semi-active Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Narrow-Band Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5.1 Quarter-Car Semi-active Suspension . . . . . . . . . . . . . . . 17.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
403 403 404 406 406 406 410 411 416
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
1
Introduction
1.1
Active versus Passive
Consider a precision structure subjected to varying thermal conditions; unless carefully designed, it will distort as a result of the thermal gradients. One way to prevent this is to build the structure from a thermally stable composite material; this is the passive approach. An alternative way is to use a set of actuators and sensors connected by a feedback loop; such a structure is active. In this case, we exploit the main virtue of feedback, which is to reduce the sensitivity of the output to parameter variations and to attenuate the effect of disturbances within the bandwidth of the control system. Depending on the circumstances, active structures may be cheaper or lighter than passive structures of comparable performances; or they may offer performances that no passive structure could offer, as in the following example. Until a few years ago, the general belief was that atmospheric turbulence would constitute an important limitation to the resolution of earth based telescopes; this was one of the main reasons for developing the Hubble Space Telescope. Nowadays, it is possible to correct in real time the disturbances produced by atmospheric turbulence on the optical wave front coming from celestial objects; this allows us to improve the ultimate resolution of the telescope by one order of magnitude, to the limit imposed by diffraction. The correction is achieved by a deformable mirror coupled to a set of actuators (Fig.1.1). A wave front sensor detects the phase difference in the turbulent wave front and the control computer supplies the shape of the deformable mirror which is required to correct this error. Adaptive optics has become a standard feature in ground-based astronomy. The foregoing example is not the only one where active structures have proved beneficial to astronomy; another example is the primary mirror of large telescopes, which can have a diameter of 8 m or more. Large primary mirrors are very difficult to manufacture and assemble. A passive mirror must be thermally stable and very stiff, in order to keep the right shape in spite of the varying gravity loads during the tracking of a star, and the dynamic A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 1–16. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
2
1 Introduction
Atmospheric turbulence
Degraded image
Focal plane Wavefront sensor
Deformable mirror Imaging camera Corrected image
Control computer
Fig. 1.1 Principle of adaptive optics for the compensation of atmospheric turbulence (by courtesy of G.Rousset-ONERA).
loads from the wind. There are two alternatives to that, both active. The first one, adopted on the Very Large Telescope (VLT) at ESO in Paranal, Chile, consists of having a relatively flexible primary mirror connected at the back to a set of a hundred or so actuators. As in the previous example, the control system uses an image analyzer to evaluate the amplitude of the perturbation of the optical modes; next, the correction is computed to minimize the effect of the perturbation and is applied to the actuators. The influence matrix J between the actuator forces f and the optical mode amplitudes w of the wave front errors can be determined experimentally with the image analyzer: w = Jf
(1.1)
J is a rectangular matrix, because the number of actuators is larger than the number of optical modes of interest. Once the modal errors w∗ have been evaluated, the correcting forces can be calculated from f ∗ = J T (JJ T )−1 w∗
(1.2)
where J T (JJ T )−1 is the pseudo-inverse of the rectangular matrix J. This is the minimum norm solution to Equ.(1.1) (Problem 1.1). The second alternative, adopted on the Keck observatory at Mauna Kea, Hawaii, consists of using a segmented primary mirror. The potential advantages of such a design are lower weight, lower cost, ease of fabrication and assembly. Each segment has a hexagonal shape and is equipped with three computer controlled degrees of freedom (tilt and piston) and six edge sensors measuring the relative displacements with respect to the neighboring segments; the control system is used to achieve the optical quality of a
1.1 Active versus Passive
3
monolithic mirror (by cophasing the segments), to compensate for gravity and wind disturbances, and minimize the impact of the telescope dynamics on the optical performance (Aubrun et al.). Active and adaptive optics will be discussed more deeply in chapter 16. As a third example, also related to astronomy, consider the future interferometric missions. The aim is to use a number of smaller telescopes as an interferometer to achieve a resolution which could only be achieved with a much larger monolithic telescope. One possible spacecraft architecture for such an interferometric mission is represented in Fig.1.2; it consists of a main truss supporting a set of independently pointing telescopes. The relative positions of the telescopes are monitored by a sophisticated metrology and the optical paths between the individual telescopes and the beam combiner are accurately controlled with optical delay lines, based on the information coming from a wave front sensor. Typically, the distance between the telescopes could be 50 m or more, and the order of magnitude of the error allowed on the optical path length is a few nanometers; the pointing error of the individual telescopes is as low as a few nanoradians (i.e. one order of magnitude better than the Hubble Space Telescope). Clearly, such stringent geometrical requirements in the harsh space environment cannot be achieved with a precision monolithic structure, but rather by active means as suggested in Fig.1.2. The main requirement on the supporting truss is not precision but stability, the accuracy of the optical path being taken care of by the wide-band vibration isolation/steering control system of individual telescopes and the optical delay lines (described below). Geometric stability includes thermal stability, vibration damping and prestressing the gaps in deployable structures (this is a critical issue for deployable trusses). In addition to these geometric requirements, this spacecraft would be sent in deep space (e.g. at the Lagrange point L2 ) rather than in low earth orbit, to ensure maximum sensitivity; this makes the weight issue particularly important.
Independent pointing telescopes Vibration isolator
delay line
Large truss
Beam combiner
Vibration isolator
Attitude Control Fig. 1.2 Schematic view of a future interferometric mission.
4
1 Introduction
Another interesting subsystem necessary to achieve the stringent specifications is the six d.o.f. vibration isolator at the interface between the attitude control module and the supporting truss; this isolator allows the low frequency attitude control torque to be transmitted, while filtering out the high frequency disturbances generated by the unbalanced centrifugal forces in the reaction wheels. Another vibration isolator may be used at the interface between the truss and the independent telescopes, possibly combined with the steering of the telescopes. The third component relevant to active control is the optical delay line; it consists of a high precision single degree of freedom translational mechanism supporting a mirror, whose function is to control the optical path length between every telescope and the beam combiner, so that these distances are kept identical to a fraction of the wavelength (e.g. λ/20). These examples were concerned mainly with performance. However, as technology develops and with the availability of low cost electronic components, it is likely that there will be a growing number of applications where active solutions will become cheaper than passive ones, for the same level of performance. The reader should not conclude that active will always be better and that a control system can compensate for a bad design. In most cases, a bad design will remain bad, active or not, and an active solution should normally be considered only after all other passive means have been exhausted. One should always bear in mind that feedback control can compensate for external disturbances only in a limited frequency band that is called the bandwidth of the control system. One should never forget that outside the bandwidth, the disturbance is actually amplified by the control system.
1.2
Vibration Suppression
Mechanical vibrations span amplitudes from meters (civil engineering) to nanometers (precision engineering). Their detrimental effect on systems may be of various natures: Failure: vibration-induced structural failure may occur by excessive strain during transient events (e.g. building response to earthquake), by instability due to particular operating conditions (flutter of bridges under wind excitation), or simply by fatigue (mechanical parts in machines). Comfort: examples where vibrations are detrimental to comfort are numerous: noise and vibration in helicopters, car suspensions, wind-induced sway of buildings. Operation of precision devices: numerous systems in precision engineering, especially optical systems, put severe restrictions on mechanical
1.3 Smart Materials and Structures
5
vibrations. Precision machine tools, wafer steppers1 and telescopes are typical examples. The performances of large interferometers such as the VLTI are limited by microvibrations affecting the various parts of the optical path. Lightweight segmented telescopes (space as well as earth-based) will be impossible to build in their final shape with an accuracy of a fraction of the wavelength, because of the various disturbance sources such as deployment errors and thermal gradients (which dominate the space environment). Such systems will not exist without the capability to control actively the reflector shape. Vibration reduction can be achieved in many different ways, depending on the problem; the most common are stiffening, damping and isolation. Stiffening consists of shifting the resonance frequency of the structure beyond the frequency band of excitation. Damping consists of reducing the resonance peaks by dissipating the vibration energy. Isolation consists of preventing the propagation of disturbances to sensitive parts of the systems. Damping may be achieved passively, with fluid dampers, eddy currents, elastomers or hysteretic elements, or by transferring kinetic energy to Dynamic Vibration Absorbers (DVA). One can also use transducers as energy converters, to transform vibration energy into electrical energy that is dissipated in electrical networks, or stored (energy harvesting). Recently, semiactive devices (also called semi-passive) have become available; they consist of passive devices with controllable properties. The Magneto-Rheological (MR) fluid damper is a famous example; piezoelectric transducers with switched electrical networks is another one. Since they behave in a strongly nonlinear way, semi-active devices can transfer energy from one frequency to another, but they are inherently passive and, unlike active devices, cannot destabilize the system; they are also less vulnerable to power failure. When high performance is needed, active control can be used; this involves a set of sensors (strain, acceleration, velocity, force,. . .), a set of actuators (force, inertial, strain,...) and a control algorithm (feedback or feedforward). Active damping is one of the main focuses of this book. The design of an active control system involves many issues such as how to configurate the sensors and actuators, how to secure stability and robustness (e.g. collocated actuator/sensor pairs); the power requirements will often determine the size of the actuators, and the cost of the project.
1.3
Smart Materials and Structures
An active structure consists of a structure provided with a set of actuators and sensors coupled by a controller; if the bandwidth of the controller 1
Moore’s law on the number of transistors on an integrated circuit could not hold without a constant improvement of the accuracy of wafer steppers and other precision machines (Taniguchi).
6
1 Introduction
high degree of integration
Sensors
PZT PVDF Fiber optics ...
Structure
Control system
Actuators
SMA PZT Magnetostrictive ...
Fig. 1.3 Smart structure.
includes some vibration modes of the structure, its dynamic response must be considered. If the set of actuators and sensors are located at discrete points of the structure, they can be treated separately. The distinctive feature of smart structures is that the actuators and sensors are often distributed, and have a high degree of integration inside the structure, which makes a separate modelling impossible (Fig.1.3). Moreover, in some applications like vibroacoustics, the behaviour of the structure itself is highly coupled with the surrounding medium; this also requires a coupled modelling. From a mechanical point of view, classical structural materials are entirely described by their elastic constants relating stress and strain, and their thermal expansion coefficient relating the strain to the temperature. Smart materials are materials where strain can also be generated by different mechanisms involving temperature, electric field or magnetic field, etc... as a result of some coupling in their constitutive equations. The most celebrated smart materials are briefly described below: • Shape Memory Alloys (SMA) allow one to recover up to 5 % strain from the phase change induced by temperature. Although two-way applications are possible after education, SMA are best suited to one-way tasks such as deployment. In any case, they can be used only at low frequency and for low precision applications, mainly because of the difficulty of cooling. Fatigue under thermal cycling is also a problem. The best known SMA is called NITINOL; SMA are little used in active vibration control, and will not be discussed in this book.2 • Piezoelectric materials have a recoverable strain of 0.1 % under electric field; they can be used as actuators as well as sensors. There are two broad classes of piezoelectric materials used in vibration control: ceramics 2
The superelastic behavior of SMA may be exploited to achieve damping, for low frequency and low cycle applications, such as earthquake protection.
1.4 Control Strategies
7
and polymers. The piezopolymers are used mostly as sensors, because they require extremely high voltages and they have a limited control authority; the best known is the polyvinylidene fluoride (P V DF or P V F2 ). Piezoceramics are used extensively as actuators and sensors, for a wide range of frequency including ultrasonic applications; they are well suited for high precision in the nanometer range (1nm = 10−9m). The best known piezoceramic is the Lead Zirconate Titanate (PZT); PZT patches can be glued or co-fired on the supporting structure. • Magnetostrictive materials have a recoverable strain of 0.15 % under magnetic field; the maximum response is obtained when the material is subjected to compressive loads. Magnetostrictive actuators can be used as load carrying elements (in compression alone) and they have a long lifetime. They can also be used in high precision applications. The best known is the TERFENOL-D; it can be an alternative to PZT in some applications (sonar). • Magneto-rheological (MR) fluids consist of viscous fluids containing micronsized particles of magnetic material. When the fluid is subjected to a magnetic field, the particles create columnar structures requiring a minimum shear stress to initiate the flow. This effect is reversible and very fast (response time of the order of millisecond). Some fluids exhibit the same behavior under electrical field; they are called electro-rheological (ER) fluids; however, their performances (limited by the electric field breakdown) are currently inferior to MR fluids. MR and ER fluids are used in semi-active devices. This brief list of commercially available smart materials is just a flavor of what is to come: phase change materials are currently under development and are likely to become available in a few years time; they will offer a recoverable strain of the order of 1 % under an electric or magnetic field, one order of magnitude more than the piezoceramics. Electroactive polymers are also slowly emerging for large strain low stiffness applications. The range of available devices to measure position, velocity, acceleration and strain is extremely wide, and there are more to come, particularly in optomechanics. Displacements can be measured with inductive, capacitive and optical means (laser interferometer); the latter two have a resolution in the nanometer range. Piezoelectric accelerometers are very popular but they cannot measure a d.c. component. Strain can be measured with strain gages, piezoceramics, piezopolymers and fiber optics. The latter can be embedded in a structure and give a global average measure of the deformation; they offer a great potential for health monitoring as well. Piezopolymers can be shaped to react only to a limited set of vibration modes (modal filters).
1.4
Control Strategies
There are two radically different approaches to disturbance rejection: feedback and feedforward. Although this text is entirely devoted to feedback
8
1 Introduction
control, it is important to point out the salient features of both approaches, in order to enable the user to select the most appropriate one for a given application.
1.4.1
Feedback
The principle of feedback is represented in Fig.1.4; the output y of the system is compared to the reference input r, and the error signal, e = r − y, is passed into a compensator H(s) and applied to the system G(s). The design problem consists of finding the appropriate compensator H(s) such that the closedloop system is stable and behaves in the appropriate manner.
r
e -
d H(s)
G(s)
y
Fig. 1.4 Principle of feedback control.
In the control of lightly damped structures, feedback control is used for two distinct and somewhat complementary purposes: active damping and model based feedback. The objective of active damping is to reduce the effect of the resonant peaks on the response of the structure. From y(s) 1 = d(s) 1 + GH
(1.3)
(Problem 1.2), this requires GH 1 near the resonances. Active damping can generally be achieved with moderate gains; another nice property is that it can be achieved without a model of the structure, and with guaranteed stability, provided that the actuator and sensor are collocated and have perfect dynamics. Of course actuators and sensors always have finite dynamics and any active damping system has a finite bandwidth. The control objectives can be more ambitious, and we may wish to keep a control variable y (a position, or the pointing of an antenna) to a desired value r in spite of external disturbances d in some frequency range. From the previous formula and y(s) GH F (s) = = (1.4) r(s) 1 + GH we readily see that this requires large values of GH in the frequency range where y r is sought. GH 1 implies that the closed-loop transfer function
1.4 Control Strategies
9
ξi
Modal damping of residual modes
Structural damping
k i
Stability limit Bandwidth
ωc Fig. 1.5 Effect of the control bandwidth on the net damping of the residual modes.
F (s) is close to 1, which means that the output y tracks the input r accurately. From Equ.(1.3), this also ensures disturbance rejection within the bandwidth of the control system. In general, to achieve this, we need a more elaborate strategy involving a mathematical model of the system which, at best, can only be a low-dimensional approximation of the actual system G(s). There are many techniques available to find the appropriate compensator, and only the simplest and the best established will be reviewed in this text. They all have a number of common features: • The bandwidth ωc of the control system is limited by the accuracy of the model; there is always some destabilization of the flexible modes outside ωc (residual modes). The phenomenon whereby the net damping of the residual modes actually decreases when the bandwidth increases is known as spillover (Fig.1.5). • The disturbance rejection within the bandwidth of the control system is always compensated by an amplification of the disturbances outside the bandwidth. • When implemented digitally, the sampling frequency ωs must always be two orders of magnitude larger than ωc to preserve reasonably the behaviour of the continuous system. This puts some hardware restrictions on the bandwidth of the control system.
1.4.2
Feedforward
When a signal correlated to the disturbance is available, feedforward adaptive filtering constitutes an attractive alternative to feedback for disturbance rejection; it was originally developed for noise control (Nelson & Elliott), but it is very efficient for vibration control too (Fuller et al.). Its principle is explained in Fig.1.6. The method relies on the availability of a reference signal correlated to the primary disturbance; this signal is passed through
10
1 Introduction
Primary disturbance source Secondary source
System
Error signal
Adaptive Filter Reference Fig. 1.6 Principle of feedforward control.
an adaptive filter, the output of which is applied to the system by secondary sources. The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized. The idea is to produce a secondary disturbance such that it cancels the effect of the primary disturbance at the location of the error sensor. Of course, there is no guarantee that the global response is also reduced at other locations and, unless the response is dominated by a single mode, there are places where the response can be amplified; the method can therefore be considered as a local one, in contrast to feedback which is global. Unlike active damping which can only attenuate the disturbances near the resonances, feedforward works for any frequency and attempts to cancel the disturbance completely by generating a secondary signal of opposite phase. The method does not need a model of the system, but the adaption procedure relies on the measured impulse response. The approach works better for narrow-band disturbances, but wide-band applications have also been reported. Because it is less sensitive to phase lag than feedback, feedforward control can be used at higher frequency (a good rule of thumb is ωc ωs /10); this is why it has been so successful in acoustics. The main limitation of feedforward adaptive filtering is the availability of a reference signal correlated to the disturbance. There are many applications where such a signal can be readily available from a sensor located on the propagation path of the perturbation. For disturbances induced by rotating machinery, an impulse train generated by the rotation of the main shaft can be used as reference. Table 1.1 summarizes the main features of the two approaches.
1.5
The Various Steps of the Design
The various steps of the design of a controlled structure are shown in Fig.1.7. The starting point is a mechanical system, some performance objectives (e.g.
1.5 The Various Steps of the Design
11
Table 1.1 Comparison of feedback and feedforward control strategies.
Type of control
Advantages
Disadvantages
Feedback Active damping
• no model needed • guaranteed stability when collocated
• effective only near resonances
Model based (LQG,H∞ ...)
• global method • attenuates all disturbances within ωc
• limited bandwidth (ωc ωs ) • disturbances outside ωc are amplified • spillover
• no model necessary • wider bandwidth (ωc ωs /10)
• reference needed • local method (response may be amplified in some part of the system) • large amount of real time computations
Feedforward Adaptive filtering of reference (x-filtered LMS)
• works better for narrow-band disturb.
position accuracy) and a specification of the disturbances applied to it; the controller cannot be designed without some knowledge of the disturbance applied to the system. If the frequency distribution of the energy of the disturbance (i.e. the power spectral density) is known, the open-loop performances can be evaluated and the need for an active control system can be assessed (see next section). If an active system is required, its bandwidth can be roughly specified from Equ.(1.3). The next step consists of selecting the proper type and location for a set of sensors to monitor the behavior of the system, and actuators to control it. The concept of controllability measures the capability of an actuator to interfere with the states of the system. Once the actuators and sensors have been selected, a model of the structure is developed, usually with finite elements; it can be improved by identification if experimental transfer functions are available. Such models generally involve too many degrees of freedom to be directly useful for design purposes; they must be reduced to produce a control design model involving only a few degrees of freedom, usually the vibration modes of the system, which carry the most important information about the system behavior. At this point, if the
12
1 Introduction
Disturbance specification
System
Performance objectives
Sensor / Actuator placement Identification
Model
Controllability Observability
Model reduction Actuator Sensor dynamics
Controller continuous design Digital implementation Closed loop system
iterate until performance objectives are met
Evaluation Fig. 1.7 The various steps of the design.
actuators and sensors can be considered as perfect (in the frequency band of interest), they can be ignored in the model; their effect on the control system performance will be tested after the design has been completed. If, on the contrary, the dynamics of the actuators and sensors may significantly affect the behavior of the system, they must be included in the model before the controller design. Even though most controllers are implemented in a digital manner, nowadays, there are good reasons to carry out a continuous design and transform the continuous controller into a digital one with an appropriate technique. This approach works well when the sampling frequency is two orders of magnitude faster than the bandwidth of the control system, as is generally the case in structural control.
1.6 Plant Description, Error and Control Budget
1.6
13
Plant Description, Error and Control Budget
Consider the block diagram of (Fig.1.8), in which the plant consists of the structure and its actuator and sensor. w is the disturbance applied to the structure, z is the controlled variable or performance metrics (that one wants to keep as close as possible to 0), u is the control input and y is the sensor output (they are all assumed scalar for simplicity). H(s) is the feedback control law, expressed in the Laplace domain (s is the Laplace variable). We define the open-loop transfer functions: Gzw (s): between w and z Gzu (s): between u and z Gyw (s): between w and y Gyu (s): between u and y From the definition of the open-loop transfer functions, y = Gyw w + Gyu Hy
(1.5)
y = (I − Gyu H)−1 Gyw w
(1.6)
u = Hy = H(I − Gyu H)−1 Gyw w = Tuw w
(1.7)
or It follows that
On the other hand z = Gzw w + Gzu u
(1.8)
Combining the two foregoing equations, one finds the closed-loop transmissibility between the disturbance w and the control metrics z: z = Tzw w = [Gzw + Gzu H(I − Gyu H)−1 Gyw ]w Disturbance Control input
w
z Plant
u
y
Performance metrics Output measurement
H(s) Fig. 1.8 Block diagram of the control system.
(1.9)
14
1 Introduction
The frequency content of the disturbance w is usually described by its Power Spectral Density (PSD), Φw (ω) which describes the frequency distribution of the mean-square (MS) value ∞ 2 σw = Φw (ω)dω (1.10) 0
[the unit of Φw is readily obtained from this equation; it is expressed in units of w squared per (rad/s)]. From(1.9), the PSD of the control metrics z is given by: Φz (ω) = |Tzw |2 Φw (ω)
(1.11)
Φz (ω) gives the frequency distribution of the mean-square value of the performance metrics. Even more interesting for design is the cumulative MS response, defined by the integral of the PSD in the frequency range [ω, ∞[ ∞ ∞ 2 σz (ω) = Φz (ν)dν = |Tzw |2 Φw (ν)dν (1.12) ω
ω
It is a monotonously decreasing function of frequency and describes the contribution of all the frequencies above ω to the mean-square value of z. σz (ω) is expressed in the same units as the performance metrics z and σz (0) is the global RMS response; a typical plot is shown in Fig.1.9 for an hypothetical system with 4 modes. For lightly damped structures, the diagram exhibits steps at the natural frequencies of the modes and the magnitude of the steps gives the contribution of each mode to the error budget, in the same units as the performance metrics; it is very helpful to identify the critical modes in a design, at which the effort should be targeted. This diagram can be used to assess the control laws and compare different actuator and sensor configurations. In a similar way, the control budget can be assessed from
sz (w)
RMS error
open-loop
closed-loop H1 (g1) H2 (g2 > g1)
w 0
w1
w2
w3 w4
Fig. 1.9 Error budget distribution in open-loop and in closed-loop for increasing gains.
1.7 Readership and Organization of the Book
σu2 (ω)
∞
=
∞
Φu (ν)dν = ω
|Tuw |2 Φw (ν)dν
15
(1.13)
ω
σu (ω) describes how the RMS control input is distributed over the various modes of the structure and plays a critical role in the actuator design. Clearly, the frequency content of the disturbance w, described by Φw (ω), is essential in the evaluation of the error and control budgets and it is very difficult, even risky, to attempt to design a control system without prior information on the disturbance.
1.7
Readership and Organization of the Book
Structural control and smart structures belong to the general field of Mechatronics; they consist of a mixture of mechanical and electrical engineering, structural mechanics, control engineering, material science and computer science. This book has been written primarily for structural engineers willing to acquire some background in structural control, but it will also interest control engineers involved in flexible structures. It has been assumed that the reader is familiar with structural dynamics and has some basic knowledge of linear system theory, including Laplace transform, root locus, Bode plots, Nyquist plots, etc... Readers who are not familiar with these concepts are advised to read a basic text on linear system theory (e.g. Cannon, Franklin et al.). Some elementary background in signal processing is also assumed. Chapter 2 recalls briefly some concepts of structural dynamics; chapter 3 to 5 consider the transduction mechanisms, the piezoelectric materials and structures and the damping via passive networks. Chapter 6 and 7 consider collocated (and dual) control systems and their use in active damping. Chapter 8 is devoted to vibration isolation. Chapter 9 to 13 cover classical topics in control: state space modelling, frequency domain, optimal control, controllability and observability, and stability. Various structural control applications (active damping, position control of a flexible structure, vibroacoustics) are covered in chapter 14; chapter 15 is devoted to cable-structures and chapter 16 to the wavefront control of large optical telescopes. Finally, chapter 17 is devoted to semi-active control. Each chapter is supplemented by a set of problems; it is assumed that the reader is familiar with MATLAB-SIMULINK or some equivalent computer aided control engineering software. Chapters 1 to 9 plus part of Chapter 10 and some applications of chapter 14 can constitute a one semester graduate course in structural control.
16
1 Introduction
1.8
Problems
P.1.1 Consider the underdeterminate system of equations Jx = w Show that the minimum norm solution, i.e. the solution of the minimization problem min (xT x) such that Jx = w x
is
x = J + w = J T (JJ T )−1 w
J + is called the pseudo-inverse of J. [hint: Use Lagrange multipliers to remove the equality constraint.] P.1.2 Consider the feedback control system of Fig.1.4. Show that the transfer functions from the input r and the disturbance d to the output y are respectively y(s) GH y(s) 1 = = r(s) 1 + GH d(s) 1 + GH P.1.3 Based on your own experience, describe one application in which you feel an active structure may outclass a passive one; outline the system and suggest a configuration for the actuators and sensors.
2
Some Concepts in Structural Dynamics
2.1
Introduction
This chapter is not intended to be a substitute for a course in structural dynamics, which is part of the prerequisites to read this book. The goal of this chapter is twofold: (i) recalling some of the notations which will be used throughout this book, and (ii) insisting on some aspects which are particularly important when dealing with controlled structures and which may otherwise be overlooked. As an example, the structural dynamic analysts are seldom interested in antiresonance frequencies which play a capital role in structural control.
2.2
Equation of Motion of a Discrete System
Consider the system with three point masses represented in Fig.2.1. The equations of motion can be established by considering the free body diagrams of the three masses and applying Newton’s law; one easily gets: Mx ¨1 + k(x1 − x2 ) + c(x˙ 1 − x˙ 2 ) = f m¨ x2 + k(2x2 − x1 − x3 ) + c(2x˙ 2 − x˙ 1 − x˙ 3 ) = 0 m¨ x3 + k(x3 − x2 ) + c(x˙ 3 − x˙ 2 ) = 0 or, in matrix form, ⎛
⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ M 0 0 x ¨1 c −c 0 x˙ 1 k −k 0 x1 f ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ¨2 ⎠ + ⎝ −c 2c −c ⎠ ⎝ x˙ 2 ⎠ + ⎝ −k 2k −k ⎠ ⎝ x2 ⎠ = ⎝ 0 ⎠ (2.1) ⎝ 0 m 0 ⎠⎝x 0 0 m x ¨3 0 −c c x˙ 3 0 −k k x3 0
The general form of the equation of motion governing the dynamic equilibrium between the external, elastic, inertia and damping forces acting on a A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 17–39. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
18
2 Some Concepts in Structural Dynamics
x1 k f
k m
M c
f
k(x2 à x1)
x3
x2
m c
k(x 2 à x1)
k(x3 à x2)
k(x3 à x2)
m
M c(xç 2 à xç 1)
c(xç 2 à xç 1)
m
c(xç 3 à xç 2)
c(xç 3 à xç 2)
Fig. 2.1 Three mass system and its free body diagram.
non-gyroscopic, discrete, flexible structure with a finite number n of degrees of freedom (d.o.f.) is Mx ¨ + C x˙ + Kx = f (2.2) where x and f are the vectors of generalized displacements (translations and rotations) and forces (point forces and torques) and M , K and C are respectively the mass, stiffness and damping matrices; they are symmetric and semi positive definite. M and K arise from the discretization of the structure, usually with finite elements. A lumped mass system such as that of Fig.2.1 has a diagonal mass matrix. The finite element method usually leads to non-diagonal (consistent) mass matrices, but a diagonal mass matrix often provides an acceptable representation of the inertia in the structure (Problem 2.2). The damping matrix C represents the various dissipation mechanisms in the structure, which are usually poorly known. To compensate for this lack of knowledge, it is customary to make assumptions on its form. One of the most popular hypotheses is the Rayleigh damping: C = αM + βK
(2.3)
The coefficients α and β are selected to fit the structure under consideration.
2.3
Vibration Modes
Consider the free response of an undamped (conservative) system of order n. It is governed by Mx ¨ + Kx = 0 (2.4) If one tries a solution of the form x = φi ejωi t , φi and ωi must satisfy the eigenvalue problem (K − ωi2 M )φi = 0 (2.5) Because M and K are symmetric, K is positive semi definite and M is positive definite, the eigenvalue ωi2 must be real and non negative. ωi is the natural
2.3 Vibration Modes
19
frequency and φi is the corresponding mode shape; the number of modes is equal to the number of degrees of freedom, n. Note that Equ.(2.5) defines only the shape, but not the amplitude of the mode which can be scaled arbitrarily. The modes are usually ordered by increasing frequencies (ω1 ≤ ω2 ≤ ω3 ≤ ...). From Equ.(2.5), one sees that if the structure is released from initial conditions x(0) = φi and x(0) ˙ = 0, it oscillates at the frequency ωi according to x(t) = φi cos ωi t, always keeping the shape of mode i. Left multiplying Equ.(2.5) by φTj , one gets the scalar equation φTj Kφi = ωi2 φTj M φi and, upon permuting i and j, one gets similarly, φTi Kφj = ωj2 φTi M φj Substracting these equations, taking into account that a scalar is equal to its transpose, and that K and M are symmetric, one gets 0 = (ωi2 − ωj2 )φTj M φi which shows that the mode shapes corresponding to distinct natural frequencies are orthogonal with respect to the mass matrix. φTj M φi = 0 (ωi = ωj ) It follows from the foregoing equations that the mode shapes are also orthogonal with respect to the stiffness matrix. The orthogonality conditions are often written as (2.6) φTi M φj = μi δij φTi Kφj = μi ωi2 δij
(2.7)
where δij is the Kronecker delta (δij = 1 if i = j, δij = 0 if i = j), μi is the modal mass (also called generalized mass) of mode i. Since the mode shapes can be scaled arbitrarily, it is usual to normalize them in such a way that μi = 1. If one defines the matrix of the mode shapes Φ = (φ1 , φ2 , ..., φn ), the orthogonality relationships read ΦT M Φ = diag(μi )
(2.8)
ΦT KΦ = diag(μi ωi2 )
(2.9)
To demonstrate the orthogonality conditions, we have used the fact that the natural frequencies were distinct. If several modes have the same natural frequency (as often occurs in practice because of symmetry), they form a subspace of dimension equal to the multiplicity of the eigenvalue. Any vector in this subspace is a solution of the eigenvalue problem, and it is always possible to find a set of vectors such that the orthogonality conditions are
20
2 Some Concepts in Structural Dynamics
satisfied. A rigid body mode is such that there is no strain energy associated with it (φTi Kφi = 0). It can be demonstrated that this implies that Kφi = 0; the rigid body modes can therefore be regarded as solutions of the eigenvalue problem (2.5) with ωi = 0.
2.4
Modal Decomposition
2.4.1
Structure without Rigid Body Modes
Let us perform a change of variables from physical coordinates x to modal coordinates according to x = Φz (2.10) where z is the vector of modal amplitudes. Substituting into Equ.(2.2), we get M Φ¨ z + CΦz˙ + KΦz = f Left multiplying by ΦT and using the orthogonality relationships (2.8) and (2.9), we obtain diag(μi )¨ z + ΦT CΦz˙ + diag(μi ωi2 )z = ΦT f
(2.11)
If the matrix ΦT CΦ is diagonal, the damping is said classical or normal. In this case, the modal fraction of critical damping ξi (in short modal damping) is defined by ΦT CΦ = diag(2ξi μi ωi ) (2.12) One can readily check that the Rayleigh damping (2.3) complies with this condition and that the corresponding modal damping ratios are ξi =
1 α ( + βωi ) 2 ωi
(2.13)
The two free parameters α and β can be selected in order to match the modal damping of two modes. Note that the Rayleigh damping tends to overestimate the damping of the high frequency modes. Under condition (2.12), the modal equations are decoupled and Equ.(2.11) can be rewritten z¨ + 2ξ Ω z˙ + Ω 2 z = μ−1 ΦT f (2.14) with the notations ξ = diag(ξi ) Ω = diag(ωi )
(2.15)
μ = diag(μi ) The following values of the modal damping ratio can be regarded as typical: satellites and space structures are generally very lightly damped (ξ
2.4 Modal Decomposition
21
0.001 − 0.005), because of the extensive use of fiber reinforced composites, the absence of aerodynamic damping, and the low strain level. Mechanical engineering applications (steel structures, piping,...) are in the range of ξ 0.01 − 0.02; most dissipation takes place in the joints, and the damping increases with the strain level. For civil engineering applications, ξ 0.05 is typical and, when radiation damping through the ground is involved, it may reach ξ 0.20, depending on the local soil conditions. The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large, as in problems involving soil-structure interaction. Lightly damped structures are usually easier to model, but more difficult to control, because their poles are located very near the imaginary axis and they can be destabilized very easily. If one accepts the assumption of classical damping, the only difference between Equ.(2.2) and (2.14) lies in the change of coordinates (2.10). However, in physical coordinates, the number of degrees of freedom of a discretized model of the form (2.2) is usually large, especially if the geometry is complicated, because of the difficulty of accurately representing the stiffness of the structure. This number of degrees of freedom is unnecessarily large to represent the structural response in a limited bandwidth. If a structure is excited by a band-limited excitation, its response is dominated by the modes whose natural frequencies belong to the bandwidth of the excitation, and the integration of Equ.(2.14) can often be restricted to these modes. The number of degrees of freedom contributing effectively to the response is therefore reduced drastically in modal coordinates.
2.4.2
Dynamic Flexibility Matrix
Consider the steady state harmonic response of Equ.(2.2) to a vector excitation f = F ejωt . The response is also harmonic, x = Xejωt , and the amplitude of F and X are related by X = [−ω 2 M + jωC + K]−1 F = G(ω)F
(2.16)
Where the matrix G(ω) is called the dynamic flexibility matrix ; it is a dynamic generalization of the static flexibility matrix, G(0) = K −1 . The modal expansion of G(ω) can be obtained by transforming (2.16) into modal coordinates x = Φz as we did earlier. The modal response is also harmonic, z = Zejωt and one finds easily that Z = diag{
1 } ΦT F μi(ωi2 + 2jξi ωi ω − ω 2 )
leading to X = ΦZ = Φ diag{
1 } ΦT F μi (ωi2 + 2jξi ωi ω − ω 2 )
22
2 Some Concepts in Structural Dynamics
Comparing with (2.16), one finds the modal expansion of the dynamic flexibility matrix: G(ω) = [−ω 2 M + jωC + K]−1 =
n
μi (ωi2 i=1
φi φTi + 2jξi ωi ω − ω 2 )
(2.17)
where the sum extends to all the modes. Glk (ω) expresses the complex amplitude of the structural response of d.o.f. l when a unit harmonic force ejωt is applied at d.o.f. k. G(ω) can be rewritten G(ω) =
n φi φT i
i=1
μi ωi2
Di (ω)
(2.18)
where Di (ω) =
1−
ω 2 /ωi2
1 + 2jξi ω/ωi
(2.19)
is the dynamic amplification factor of mode i. Di (ω) is equal to 1 at ω = 0, it exhibits large values in the vicinity of ωi , |Di (ωi )| = (2ξi )−1 , and then decreases beyond ωi (Fig.2.2).1 According to the definition of G(ω) the Fourier transform of the response X(ω) is related to the Fourier transform of the excitation F (ω) by X(ω) = G(ω)F (ω) This equation means that all the frequency components work independently, and if the excitation has no energy at one frequency, there is no energy in the response at that frequency. From Fig.2.2, one sees that when the excitation has a limited bandwidth, ω < ωb , the contribution of all the high frequency modes (i.e. such that ωk ωb ) to G(ω) can be evaluated by assuming Dk (ω) 1. As a result, if ωm > ωb , G(ω)
m φi φT
i 2 μ i ωi i=1
Di (ω) +
n φi φTi μi ωi2 i=m+1
(2.20)
This approximation is valid for ω < ωm . The first term in the right hand side is the contribution of all the modes which respond dynamically and the second term is a quasi-static correction for the high frequency modes. Taking into account that n φi φTi −1 G(0) = K = (2.21) μi ωi2 i=1 1
Qi = 1/2ξi is often called the quality factor of mode i.
2.4 Modal Decomposition
23
F
Excitation bandwidth
!b
!
Di
1 2ø i
Mode outside the bandwidth
1
0
!i
!b
!k
!
Fig. 2.2 Fourier spectrum of the excitation F with a limited frequency content ω < ωb and dynamic amplification Di of mode i such that ωi < ωb and ωk ωb .
G(ω) can be rewritten in terms of the low frequency modes only: G(ω)
m φi φT
i μi ωi2 i=1
Di (ω) + K −1 −
m φi φT i
i=1
μi ωi2
(2.22)
The quasi-static correction of the high frequency modes is often called the residual mode, denoted by R. Unlike all the terms involving Di (ω) which reduce to 0 as ω → ∞, R is independent of the frequency and introduces a feedthrough (constant) component in the transfer matrix. We will shortly see that R has a strong influence on the location of the transmission zeros and that neglecting it may lead to substantial errors in the prediction of the performance of the control system.
2.4.3
Structure with Rigid Body Modes
The approximation (2.22) applies only at low frequency, ω < ωm . If the structure has r rigid body modes, the first sum can be split into rigid and flexible modes; however, the residual mode cannot be used any more, because K −1 no longer exists. This problem can be solved in the following way.
24
2 Some Concepts in Structural Dynamics
The displacements are partitioned into their rigid and flexible contributions according to x = xr + xe = Φr zr + Φe ze (2.23) where Φr and Φe are the matrices whose columns are the rigid body modes and the flexible modes, respectively. Assuming no damping, to make things formally simpler, and taking into account that the rigid body modes satisfy KΦr = 0, we obtain the equation of motion
System loaded with f f − M && xr
Self-equilibrated load P T f = f − M x&&r
f System with dummy constraints, loaded with P T f f Fig. 2.3 Structure with rigid body modes.
M Φr z¨r + M Φe z¨e + KΦe ze = f
(2.24)
ΦTr
Left multiplying by and using the orthogonality relations (2.6) and (2.7), we see that the rigid body modes are governed by ΦTr M Φr z¨r = ΦTr f or
T z¨r = μ−1 r Φr f
(2.25)
Substituting this result into Equ.(2.24), we get M Φe z¨e + KΦe ze = f − M Φr z¨r T −1 T = f − M Φr μ−1 r Φr f = (I − M Φr μr Φr )f
or M Φe z¨e + KΦe ze = P T f
(2.26)
where we have defined the projection matrix T P = I − Φr μ−1 r Φr M
(2.27)
2.4 Modal Decomposition
25
such that P T f is orthogonal to the rigid body modes. In fact, we can easily check that P Φr = 0 (2.28) P Φe = Φe
(2.29)
P can therefore be regarded as a filter which leaves unchanged the flexible modes and eliminates the rigid body modes. If we follow the same procedure as in the foregoing section, we need to evaluate the elastic contribution of the static deflection, which is the solution of Kxe = P T f (2.30) Since KΦr = 0, the solution may contain an arbitrary contribution from the rigid body modes. On the other hand, P T f = f −M x ¨r is the superposition of the external forces and the inertia forces associated with the motion as a rigid body; it is self-equilibrated, because it is orthogonal to the rigid body modes. Since the system is in equilibrium as a rigid body, a particular solution of Equ.(2.30) can be obtained by adding dummy constraints to remove the rigid body modes (Fig.2.3). The modified system is statically determinate and its stiffness matrix can be inverted. If we denote by Giso the flexibility matrix of the modified system, the general solution of (2.30) is xe = Giso P T f + Φr γ where γ is a vector of arbitrary constants. The contribution of the rigid body modes can be eliminated with the projection matrix P , leading to xe = P Giso P T f
(2.31)
P Giso P T is the pseudo-static flexibility matrix of the flexible modes. On the other hand, left multiplying Equ.(2.24) by ΦTe , we get ΦTe M Φe z¨e + ΦTe KΦe ze = ΦTe f where the diagonal matrix ΦTe KΦe is regular. It follows that the pseudo-static deflection can be written alternatively xe = Φe ze = Φe (ΦTe KΦe )−1 ΦTe f
(2.32)
Comparing with Equ.(2.31), we get P Giso P T = Φe (ΦTe KΦe )−1 ΦTe =
n φi φT i
r+1
μi ωi2
(2.33)
This equation is identical to Equ.(2.20) when there are no rigid body modes. From this result, we can extend Equ.(2.22) to systems with rigid body modes:
26
2 Some Concepts in Structural Dynamics r m φi φTi φi φTi G(ω) + +R 2 −μi ω 2 i=r+1 μi (ωi − ω 2 + 2jξi ωi ω) i=1
(2.34)
where the contribution from the residual mode is R=
n m φi φTi φi φTi T = P G P − iso 2 μω μ ω2 m+1 i i r+1 i i
(2.35)
Note that Giso is the flexibility matrix of the system obtained by adding dummy constraints to remove the rigid body modes. Obviously, this can be achieved in many different ways and it may look surprising that they all lead to the same result (2.35). In fact, different boundary conditions lead to different displacements under the self-equilibrated load P T f , but they differ only by a contribution of the rigid body modes, which is destroyed by the projection matrix P , leading to the same P Giso P T . Let us illustrate the procedure with an example.
2.4.4
Example
Consider the system of three identical masses of Fig.2.4. There is one rigid body mode and two flexible ones: ⎛ ⎞ 1 1 1 Φ = (Φr , Φe ) = ⎝ 1 0 −2 ⎠ 1 −1 1 and ΦT M Φ = diag(3, 2, 6)
ΦT KΦ = k.diag(0, 2, 18)
Fig. 2.4 Three mass system: (a) self-equilibrated forces associated with a force f applied to mass 1; (b) dummy constraints.
2.5 Collocated Control System
27
From Equ.(2.27), the projection matrix is ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 1 1 0 0 1 1 1 1 1 P = ⎝ 0 1 0 ⎠ − ⎝ 1 ⎠ . .(1, 1, 1) = ⎝ 0 1 0 ⎠ − ⎝ 1 1 1 ⎠ 3 3 0 0 1 1 0 0 1 1 1 1 ⎛
⎞ 2 −1 −1 1 P = ⎝ −1 2 −1 ⎠ 3 −1 −1 2
or
We can readily check that P Φ = P (Φr , Φe ) = (0, Φe ) and the self-equilibrated loads associated with a force f applied to mass 1 is, Fig.2.4.a ⎛ ⎞⎛ ⎞ ⎛ ⎞ 2 −1 −1 f 2/3 1 P T f = ⎝ −1 2 −1 ⎠ ⎝ 0 ⎠ = ⎝ −1/3 ⎠ f 3 −1 −1 2 0 −1/3 If we impose the statically determinate constraint on mass 1, Fig.2.4.b, the resulting flexibility matrix is ⎛ ⎞ 0 0 0 1 Giso = ⎝ 0 1 1 ⎠ k 0 1 2 leading to
⎛
P Giso P T
⎞ 5 −1 −4 1 ⎝ = −1 2 −1 ⎠ 9k −4 −1 5
The reader can easily check that other dummy constraints would lead to the same pseudo-static flexibility matrix (Problem 2.3).
2.5
Collocated Control System
A collocated control system is a control system where the actuator and the sensor are attached to the same degree of freedom. It is not sufficient to be attached to the same location, but they must also be dual, that is a force actuator must be associated with a translation sensor (measuring displacement, velocity or acceleration), and a torque actuator with a rotation sensor (measuring an angle or an angular velocity), in such a way that the product of the actuator signal and the sensor signal represents the energy (power) exchange between the structure and the control system. Such systems enjoy very interesting properties. The open-loop Frequency Response Function (FRF) of a collocated control system corresponds to a diagonal component
28
2 Some Concepts in Structural Dynamics
of the dynamic flexibility matrix. If the actuator and sensor are attached to d.o.f. k, the open-loop FRF reads Gkk (ω) =
m φ2 (k) i
i=1
μi ωi2
Di (ω) + Rkk
(2.36)
If one assumes that the system is undamped, the FRF is purely real Gkk (ω) =
m i=1
φ2i (k) + Rkk μi (ωi2 − ω 2 )
(2.37)
All the residues are positive (square of the modal amplitude) and, as a result, Gkk (ω) is a monotonously increasing function of ω, which behaves as illustrated in Fig.2.5. The amplitude of the FRF goes from −∞ at the resonance frequencies ωi (corresponding to a pair of imaginary poles at s = ±jωi in the open-loop transfer function) to +∞ at the next resonance frequency ωi+1 . Since the function is continuous, in every interval, there is a frequency zi such that ωi < zi < ωi+1 where the amplitude of the FRF vanishes. In structural dynamics, such frequencies are called anti-resonances; they correspond to purely imaginary zeros at ±jzi , in the open-loop transfer function. Thus, undamped collocated control systems have alternating poles and zeros on the imaginary axis. The pole / zero pattern is that of Fig.2.6.a. For a lightly damped structure, the poles and zeros are just moved a little in the left-half plane, but they are still interlacing, Fig.2.6.b. If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero, zi , the amplitude of the response of the collocated sensor vanishes. This means that the structure oscillates at
Gkk(!)
resonance
Gkk(0) = Kà1 kk static response
Rkk
!i
zi
! i+1
residual
! mode
antiresonance
Fig. 2.5 Open-loop FRF of an undamped structure with a collocated actuator/sensor pair (no rigid body modes).
2.5 Collocated Control System
Im(s)
(a)
Im(s)
x
x
x
x
x
Re(s)
x
29
(b)
Re(s)
Fig. 2.6 Pole/Zero pattern of a structure with collocated (dual) actuator and sensor; (a) undamped; (b) lightly damped (only the upper half of the complex plane is shown, the diagram is symmetrical with respect to the real axis).
the frequency zi according to the shape shown in dotted line on Fig.2.7.b. We will establish in the next section that this shape, and the frequency zi , are actually a mode shape and a natural frequency of the system obtained by constraining the d.o.f. on which the control system acts. We know from control theory that the open-loop zeros are asymptotic values of the closed-loop poles, when the feedback gain goes to infinity. The natural frequencies of the constrained system depend on the d.o.f. where the constraint has been added (this is indeed well known in control theory that the open-loop poles are independent of the actuator and sensor configuration while the open-loop zeros do depend on it). However, from the foregoing discussion, for every actuator/sensor configuration, there will be one and only one zero between two consecutive poles, and the interlacing property applies for any location of the collocated pair. Referring once again to Fig.2.5, one easily sees that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically in such a way that its high frequency asymptote becomes tangent to the frequency axis. This produces a shift in the location of the transmission zeros to the right, and the last one even moves to infinity as the feedthrough (constant) component Rkk disappears from the FRF. Thus, neglecting the residual modes tends to overestimate the frequency of the transmission zeros. As we shall see shortly, the closed-loop poles which remain at finite distance move on loops joining the open-loop poles to the open-loop zeros; therefore, altering the open-loop pole/zero pattern has a direct impact on the closedloop poles. The open-loop transfer function of a undamped structure with a collocated actuator/sensor pair can be written
2 2 (s /z + 1) G(s) = G0 i 2 i2 (ωi < zi < ωi+1 ) (2.38) (s /ωj + 1) j
30
2 Some Concepts in Structural Dynamics
u (a)
y (b)
(c)
g
Fig. 2.7 (a) Structure with collocated actuator and sensor; (b) structure with additional constraint; (c) structure with additional stiffness along the controlled d.o.f.
For a lightly damped structure, it reads
2 2 (s /z + 2ξi s/zi + 1) G(s) = G0 i 2 i2 j (s /ωj + 2ξj s/ωj + 1)
(2.39)
The corresponding Bode and Nyquist plots are represented in Fig 2.8. Every imaginary pole at ±jωi introduces a 1800 phase lag and every imaginary zero at ±jzi a 1800 phase lead. In this way, the phase diagram is always contained between 0 and −1800, as a consequence of the interlacing property. For the same reason, the Nyquist diagram consists of a set of nearly circles (one per mode), all contained in the third and fourth quadrants. Thus, the entire curve G(ω) is below the real axis (the diameter of every circle is proportional to ξi−1 ).
2.5.1
Transmission Zeros and Constrained System
We now establish that the transmission zeros of the undamped system are the poles (natural frequencies) of the constrained system. Consider the undamped structure of Fig.2.7.a (a displacement sensor is assumed for simplicity). The governing equations are Structure: Mx ¨ + Kx = b u (2.40)
2.5 Collocated Control System
Im(G)
w =0
Re(G)
31
G dB
! = zi
wi
zi
w
f 0°
w
-90°
! = !i
-180°
Fig. 2.8 Nyquist diagram and Bode plots of a lightly damped structure with collocated actuator and sensor.
Output sensor : y = bT x
(2.41)
u is the actuator input (scalar) and y is the sensor output (also scalar). The fact that the same vector b appears in the two equations is due to collocation. For a stationary harmonic input at the actuator, u = u0 ejω0 t ; the response is harmonic, x = x0 ejω0 t , and the amplitude vector x0 is solution of (K − ω02 M )x0 = b u0
(2.42)
The sensor output is also harmonic, y = y0 ejω0 t and the output amplitude is given by y0 = bT x0 = bT (K − ω02 M )−1 b u0
(2.43)
Thus, the transmission zeros (antiresonance frequencies) ω0 are solutions of bT (K − ω02 M )−1 b = 0
(2.44)
Now, consider the system with the additional stiffness g along the same d.o.f. as the actuator/sensor, Fig 2.7.c. The stiffness matrix of the modified system is K + gbbT . The natural frequencies of the modified system are solutions of the eigenvalue problem [K + gbbT − ω 2 M ]φ = 0
(2.45)
For all g the solution (ω, φ) of the eigenvalue problem is such that (K − ω 2 M )φ + gbbT φ = 0
(2.46)
32
or
2 Some Concepts in Structural Dynamics
bT φ = −bT (K − ω 2 M )−1 gbbT φ
(2.47)
Since bT φ is a scalar, this implies that bT (K − ω 2 M )−1 b = −
1 g
(2.48)
Taking the limit for g → ∞, one sees that the eigenvalues ω satisfy bT (K − ω 2 M )−1 b = 0
(2.49)
which is identical to (2.44). Thus, ω = ω0 ; the imaginary zeros of the undamped collocated system, solutions of (2.44), are the poles of the constrained system (2.45) at the limit, when the stiffness g added along the actuation d.o.f. increases to ∞: lim [(K + gbbT ) − ω02 M ]x0 = 0
g→∞
(2.50)
This is equivalent to placing a kinematic constraint along the control d.o.f.
2.6
Continuous Structures
Continuous structures are distributed parameter systems which are governed by partial differential equations. Various discretization techniques, such as the Rayleigh-Ritz method, or finite elements, allow us to approximate the partial differential equation by a finite set of ordinary differential equations. In this section, we illustrate some of the features of distributed parameter systems with continuous beams. This example will be frequently used in the subsequent chapters. The plane transverse vibration of a beam is governed by the following partial differential equation (EIw ) + mw ¨=p
(2.51)
This equation is based on the Euler-Bernoulli assumptions that the neutral axis undergoes no extension and that the cross section remains perpendicular to the neutral axis (no shear deformation). EI is the bending stiffness, m is the mass per unit length and p the distributed external load per unit length. If the beam is uniform, the free vibration is governed by wIV +
m w ¨=0 EI
(2.52)
The boundary conditions depend on the support configuration: a simple support implies w = 0 and w = 0 (no displacement, no bending moment); for a clamped end, we have w = 0 and w = 0 (no displacement, no rotation); a
2.7 Guyan Reduction
33
free end corresponds to w = 0 and w = 0 (no bending moment, no shear), etc... A harmonic solution of the form w(x, t) = φ(x) ejωt can be obtained if φ(x) and ω satisfy d4 φ m 2 − ω φ=0 (2.53) dx4 EI with the appropriate boundary conditions. This equation defines a eigenvalue problem; the solution consists of the natural frequencies ωi (infinite in number) and the corresponding mode shapes φi (x). The eigenvalues are tabulated for various boundary conditions in textbooks on mechanical vibrations (e.g. Geradin & Rixen, 1993, p.187). For the pinned-pinned case, the natural frequencies and mode shapes are ωn2 = (nπ)4
EI ml4
(2.54)
nπx (2.55) l Just as for discrete systems, the mode shapes are orthogonal with respect to the mass and stiffness distribution: l m φi (x)φj (x) dx = μi δij (2.56) φn (x) = sin
0
0
l
EI φi (x)φj (x) dx = μi ωi2 δij
(2.57)
The generalized mass corresponding to Equ.(2.55) is μn = ml/2. As with discrete structures, the frequency response function between a point force actuator at xa and a displacement sensor at xs is G(ω) =
∞ i=1
φi (xa )φi (xs ) μi (ωi2 − ω 2 + 2jξi ωi ω)
(2.58)
where the sum extends to infinity. Exactly as for discrete systems, the expansion can be limited to a finite set of modes, the high frequency modes being included in a quasi-static correction as in Equ.(2.34) (Problem 2.5).
2.7
Guyan Reduction
As already mentioned, the size of a discretized model obtained by finite elements is essentially governed by the representation of the stiffness of the structure. For complicated geometries, it may become very large, especially with automated mesh generators. Before solving the eigenvalue problem (2.5), it may be advisable to reduce the size of the model by condensing the degrees of freedom with little or no inertia and which are not excited by external
34
2 Some Concepts in Structural Dynamics
forces, nor involved in the control. The degrees of freedom to be condensed, denoted x2 in what follows, are often referred to as slaves; those kept in the reduced model are called masters and are denoted x1 . To begin with, consider the undamped forced vibration of a structure where the slaves x2 are not excited and have no inertia; the governing equation is M11 0 x ¨1 K11 K12 x1 f1 + = (2.59) 0 0 x ¨2 K21 K22 x2 0 or M11 x ¨1 + K11 x1 + K12 x2 = f1
(2.60)
K21 x1 + K22 x2 = 0
(2.61)
According to the second equation, the slaves x2 are completely determined by the masters x1 : −1 x2 = −K22 K21 x1
(2.62)
Substituting into Equ.(2.60), we find the reduced equation −1 M11 x ¨1 + (K11 − K12 K22 K21 )x1 = f1
(2.63)
which involves only x1 . Note that in this case, the reduced equation has been obtained without approximation. The idea in the so-called Guyan reduction is to assume that the masterslave relationship (2.62) applies even if the degrees of freedom x2 have some inertia (i.e. when the sub-matrix M22 = 0) or applied forces. Thus, one assumes the following transformation I x1 x= = x1 = Lx1 (2.64) −1 x2 −K22 K21 The reduced mass and stiffness matrices are obtained by substituting the above transformation into the kinetic and strain energy: T =
1 T 1 1 ˆ x˙ 1 x˙ M x˙ = x˙ T1 LT M Lx˙ 1 = x˙ T1 M 2 2 2
U=
1 T 1 1 ˆ 1 x Kx = xT1 LT KLx1 = xT1 Kx 2 2 2
with ˆ = LT M L M
ˆ = LT KL K
(2.65)
ˆ = K11 − K12 K −1 K21 as in Equ.(2.63). If The second equation produces K 22 external loads are applied to x2 , the reduced loads are obtained by equating the virtual work δxT f = δxT1 LT f = δxT1 fˆ1
2.8 Craig-Bampton Reduction
or
−1 fˆ1 = LT f = f1 − K12 K22 f2
35
(2.66)
Finally, the reduced equation of motion reads ˆx ˆ 1 = fˆ1 M ¨1 + Kx
(2.67)
Usually, it is not necessary to consider the damping matrix in the reduction, because it is rarely known explicitly at this stage. The Guyan reduction can be performed automatically in commercial finite element packages, the selection of masters and slaves being made by the user. In the selection process the following should be kept in mind: • The degrees of freedom without inertia or applied load can be condensed without affecting the accuracy. • Translational degrees of freedom carry more information than rotational ones. In selecting the masters, preference should be given to translations, especially if large modal amplitudes are expected (Problem 2.7). • It can be demonstrated that the error in the mode shape φi associated with the Guyan reduction is an increasing function of the ratio ωi2 ν12 where ωi is the natural frequency of the mode and ν1 is the first natural frequency of the constrained system, where all the degrees of freedom x1 (masters) have been blocked [ν1 is the smallest solution of det(K22 − ν 2 M22 ) = 0]. Therefore, the quality of a Guyan reduction is strongly related to the natural frequencies of the constrained system and ν1 should be kept far above the frequency band ωb where the model is expected to be accurate. If this is not the case, the model reduction can be improved as follows.
2.8
Craig-Bampton Reduction
Consider the finite element model M11 M12 x ¨1 K11 + M21 M22 x ¨2 K21
K12 K22
x1 x2
=
f1 0
(2.68)
where the degrees of freedom have been partitioned into the masters x1 and the slaves x2 . The masters include all the d.o.f. with a specific interest in the problem: those where disturbance and control loads are applied, where sensors are located and where the performance is evaluated (controlled d.o.f.). The slaves include all the other d.o.f. which have no particular interest in the control problem and are ready for elimination.
36
2 Some Concepts in Structural Dynamics
The Craig-Bampton reduction is conducted in two steps. First, a Guyan reduction is performed according to the static relationship (2.62). In a second step, the constrained system is considered: M22 x¨2 + K22 x2 = 0
(2.69)
(obtained by setting x1 = 0 in the foregoing equation). Let us assume that the eigen modes of this system constitute the column of the matrix Ψ2 , and that they are normalized according to Ψ2T M22 Ψ2 = I. We then perform the change of coordinates I 0 x1 x1 x1 =T (2.70) = −1 x2 α α K21 Ψ2 −K22 Comparing with (2.64), one sees that the solution has been enriched with a set of fixed boundary modes of modal amplitude α. Using the transformation matrix T , the mass and stiffness matrices are obtained as in the previous section: ˆ = T T KT ˆ = TT MT K (2.71) M leading to ˆ 11 M ˆ 12 M
ˆ 12 M I
x ¨1 α ¨
+
ˆ 11 K 0
0 Ω2
x1 α
=
f1 0
(2.72)
ˆ 11 = K11 − In this equation, the stiffness matrix is block diagonal, with K −1 2 T K12 K22 K21 being the Guyan stiffness matrix and Ω = Ψ2 K22 Ψ2 being a diagonal matrix with entries equal to the square of the natural frequenˆ 11 = M11 − M12 K −1 K21 − cies of the fixed boundary modes. Similarly, M 22 −1 −1 −1 K12 K22 M21 + K12 K22 M22 K22 K21 is the Guyan mass matrix [the same as ˆ 11 and M ˆ 11 are fully populated but do not depend that given by (2.65)]. K on the set of constrained modes Ψ2 . The off-diagonal term of the mass maˆ 12 = (M12 − K12 K −1 M22 )Ψ2 . Since all the external loads trix is given by M 22 are applied to the master d.o.f., the right hand side of this equation is unchanged by the transformation. The foregoing equation may be used with an increasing number of constrained modes (increasing the size of α), until the model provides an appropriate representation of the system in the requested frequency band.
2.9 Problems
2.9
37
Problems
P.2.1 Using a finite element program, discretize a simply supported uniform beam with an increasing number of elements (4,8,etc...). Compare the natural frequencies with those obtained with the continuous beam theory. Observe that the finite elements tend to overestimate the natural frequencies. Why is that so? P.2.2 Using the same stiffness matrix as in the previous example and a diagonal mass matrix obtained by lumping the mass of every element at the nodes (the entries of the mass matrix for all translational degrees of freedom are ml/nE , where nE is the number of elements; no inertia is attributed to the rotations), compute the natural frequencies. Compare the results with those obtained with a consistent mass matrix in Problem 2.1. Notice that using a diagonal mass matrix usually tends to underestimate the natural frequencies. P.2.3 Consider the three mass system of section 2.4.4. Show that changing the dummy constraint to mass 2 does not change the pseudo-static flexibility matrix P Giso P T . P.2.4 Consider a simply supported beam with the following properties: l = 1m, m = 1kg/m, EI = 10.266 10−3 N m2 . It is excited by a point force at xa = l/4. (a) Assuming that a displacement sensor is located at xs = l/4 (collocated) and that the system is undamped, plot the transfer function for an increasing number of modes, with and without quasi-static correction for the highfrequency modes. Comment on the variation of the zeros with the number of modes and on the absence of mode 4. Note: To evaluate the quasi-static contribution of the high-frequency modes, it is useful to recall that the static displacement at x = ξ created by a unit force applied at x = a on a simply supported beam is δ(ξ, a) =
(l − a)ξ [a(2l − a) − ξ 2 ] (ξ ≤ a) 6lEI
a(l − ξ) [ξ(2l − ξ) − a2 ] (ξ > a) 6lEI The symmetric operator δ(ξ, a) is often called “flexibility kernel” or Green’s function. δ(ξ, a) =
(b) Including three modes and the quasi-static correction, draw the Nyquist and Bode plots and locate the poles and zeros in the complex plane for a uniform modal damping of ξi = 0.01 and ξi = 0.03.
38
2 Some Concepts in Structural Dynamics
(c) Do the same as (b) when the sensor location is xs = 3l/4. Notice that the interlacing property of the poles and zeros no longer holds. P.2.5 Consider the modal expansion of the transfer function (2.58) and assume that the low frequency amplitude G(0) is available, either from static calculations or from experiments at low frequency. Show that G(ω) can be approximated by the truncated expansion G(ω) = G(0) +
m φi (xa )φi (xs ) i=1
μi ωi2
(ω 2 − 2jξi ωi ω) − ω 2 + 2jξi ωi ω)
(ωi2
P.2.6 Show that the impulse response matrix of a damped structure with rigid body modes reads g(τ ) =
r i=1
where ωdi = ωi
φi φT φi φTi i −ξi ωi τ τ+ e sin ωdi τ μi μ ω r+1 i di n
1(τ )
1 − ξi2 and 1(τ ) is the Heaviside step function.
P.2.7 Consider a uniform beam clamped at one end and free at the other end; it is discretized with six finite elements of equal size. The twelve degrees of freedom are numbered w1 , θ1 to w6 , θ6 starting from the clamped end. We perform various Guyan reductions in which we select x1 according to: (a) all wi , θi (12 degrees of freedom, no reduction); (b) all wi (6 d.o.f.); (c) all θi (6 d.o.f.); (d) w2 , θ2 , w4 , θ4 , w6 , θ6 (6 d.o.f.); (e) w2 , w4 , w6 (3 d.o.f.); (f) θ2 , θ4 , θ6 (3 d.o.f.); For each case, compute the natural frequency ωi of the first three modes and the first natural frequency ν1 of the constrained system. Compare the roles of the translations and rotations. P.2.8 Consider a spacecraft consisting of a rigid main body to which one or several flexible appendages are attached. Assume that there is at least one axis about which the attitude motion is uncoupled from the other axes. Let θ be the (small) angle of rotation about this axis and J be the moment of inertia (of the main body plus the appendages). Show that the equations of motion read m J θ¨ − Γi z¨i = T0 i=1
μi z¨i +
μi Ωi2 zi
− Γi θ¨ = 0
i = 1, ..., m
2.9 Problems
39
where T0 is the torque applied to the main body, μi and Ωi are the modal masses and the natural frequencies of the constrained modes of the flexible appendages and Γi are the modal participation factors of the flexible modes [i.e. Γi is the work done on mode i of the flexible appendages by the inertia forces associated with a unit angular acceleration of the main body] (Hughes, 1974). [Hint: Decompose the motion into the rigid body mode and the components of the constrained flexible modes, express the kinetic energy and the strain energy, write the Lagrangian in the form L=T −V =
1 ˙2 1 1 Jθ − Γi z˙i θ˙ + μi z˙i2 − μi Ωi2 zi 2 2 2 i i i
and write the Lagrange equations.]
3
Electromagnetic and Piezoelectric Transducers
3.1
Introduction
Transducers are critical in active structures technology; they can play the role of actuator, sensor, or simply energy converter, depending on the applications. In many applications, the actuators are the most critical part of the system; however, the sensors become very important in precision engineering where sub-micron amplitudes must be detected. Two broad categories of actuators can be distinguished: “grounded” and “structure borne” actuators. The former react on a fixed support; they include torque motors, force motors (electrodynamic shakers) or tendons. The second category, also called “space realizable”, include jets, reaction wheels, control moment gyros, proof-mass actuators, active members (capable of both structural functions and generating active control forces), piezo strips, etc... Active members and all actuating devices involving only internal, self-equilibrating forces, cannot influence the rigid body motion of a structure. This chapter begins with a description of the voice-coil transducer and its application to the proof-mass actuator and the geophone (absolute velocity sensor). Follows a brief discussion of the single axis gyrostabilizer. The remaining of the chapter is devoted to the piezoelectric materials and the constitutive equations of a discrete piezoelectric transducer. Integrating piezoelectric elements in beams, plates and trusses will be considered in the following chapter.
3.2
Voice Coil Transducer
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa. The system consists of a permanent magnet (Fig.3.1) which produces a uniform magnetic flux density B normal to the gap, and a coil which is free to move axially within the gap. Let v be the velocity of the coil, f the external force acting to maintain the coil in A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 41–59. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
42
3 Electromagnetic and Piezoelectric Transducers (b)
Fig. 3.1 Voice-coil transducer: (a) Physical principle. (b) Symbolic representation.
equilibrium against the electromagnetic forces, e the voltage difference across the coil and i the current into the coil. In this ideal transducer, we neglect the electrical resistance and the self inductance of the coil, as well as its mass and damping (if necessary, these can be handled by adding R and L to the electrical circuit of the coil, or a mass and damper to its mechanical model). The voice coil actuator is one of the most popular actuators in mechatronics (e.g. it is used in electromagnetic loudspeakers), but it is also used as sensor in geophones. The first constitutive equation of the voice coil transducer follows from Faraday’s law: e = 2πnrBv = T v (3.1) where T = 2πnrB
(3.2)
is the transducer constant, equal to the product of the length of the coil exposed to the magnetic flux, 2πnr, and the magnetic flux density B. The second equation follows from the Lorentz force law: The external force f required to balance the total force of the magnetic field on n turns of the conductor is f = −i 2πnrB = −T i
(3.3)
where T is again the transducer constant (3.2). Equation (3.1) and (3.3) are the constitutive equations of the voice coil transducer. Notice that the transducer constant T appearing in Faraday’s law (3.1), expressed in volt.sec/m, is the same as that appearing in the Lorentz force (3.3), expressed in N/Amp.
3.2 Voice Coil Transducer
43
The total power delivered to the moving-coil transducer is equal to the sum of the electric power, ei, and the mechanical power, f v. Combining with (3.1) and (3.3), one gets ei + f v = T vi − T iv = 0
(3.4)
Thus, at any time, there is an equilibrium between the electrical power absorbed by the device and the mechanical power delivered (and vice versa). The moving-coil transducer cannot store energy, and behaves as a perfect electromechanical converter. In practice, however, the transfer is never perfect due to eddy currents, flux leakage and magnetic hysteresis, leading to slightly different values of T in (3.1) and (3.3).
3.2.1
Proof-Mass Actuator
A proof-mass actuator (Fig.3.2) is an inertial actuator which is used in various applications of vibration control. A reaction mass m is connected to the support structure by a spring k, a damper c and a force actuator f which can be either magnetic or hydraulic. In the electromagnetic actuator discussed here, the force actuator consists of a voice coil transducer of constant T excited by a current generator i; the spring is achieved with membranes which also guide the linear motion of the moving mass. The system is readily modelled as in Fig.3.2.a. Combining the equation of a single d.o.f. oscillator with the Lorentz force law (3.3), one finds
(a)
(b) Moving mass
x
Membranes
Coil
m
c f = à Ti
k
N S
Support
F
Magnetic circuit
i=q Permanent magnet
Fig. 3.2 Proof-mass actuator (a) model assuming a current generator; (b) conceptual design of an electrodynamic actuator based on a voice coil transducer.
44
3 Electromagnetic and Piezoelectric Transducers
m¨ x + cx˙ + kx = T i
(3.5)
or, in the Laplace domain, x=
ms2
Ti + cs + k
(3.6)
(s is the Laplace variable). The total force applied to the support is equal and opposite to that applied to the mass: F = −ms2 x =
−ms2 T i ms2 + cs + k
(3.7)
It follows that the transfer function between the total force F and the current i applied to the coil is F −s2 T = 2 (3.8) i s + 2ξp ωp s + ωp2 where T is the transducer constant (in N/Amp), ωp = (k/m)1/2 is the natural frequency of the spring-mass system and ξp is the damping ratio, which in practice is fairly high, typically 20 % or more.1 The Bode plots of (3.8) are shown in Fig.3.3; one sees that the system behaves like a high-pass filter with a high frequency asymptote equal to the transducer constant T ; above some critical frequency ωc 2ωp , the proof-mass actuator can be regarded as an ideal force generator. It has no authority over the rigid body modes and
F i
T
!p
!c
!
180
Phase
! Fig. 3.3 Bode plot F/i of an electrodynamic proof-mass actuator. 1
The negative sign in (3.8) is irrelevant.
3.2 Voice Coil Transducer
45
the operation at low frequency requires a large stroke, which is technically difficult. Medium to high frequency actuators (40 Hz and more) are relatively easy to obtain with low cost components (loudspeaker technology). If the current source is replaced by a voltage source (Fig.3.4), the modelling is slightly more complicated and combines the mechanical equation (3.5) and an electrical equation which is readily derived from Faraday’s law: T x˙ + L
di + Ri = E(t) dt
(3.9)
where L is the inductance and R is the resistance of the electrical circuit.
Fig. 3.4 Model of a proof-mass actuator with a voltage source.
3.2.2
Geophone
The geophone is a transducer which behaves like an absolute velocity sensor above some cut-off frequency which depends on its mechanical construction. The system of Fig.3.2.a is readily transformed into a geophone by using the voltage e as the sensor output (Fig.3.5). If x0 is the displacement of the support and if the voice coil is open (i = 0), the governing equations are m¨ x + c(x˙ − x˙ 0 ) + k(x − x0 ) = 0 T (x˙ − x˙ 0 ) = e combining these equations, one readily finds that x − x0 = e = T s(x − x0 ) =
−ms2 x0 ms2 + cs + k −s2 T sx0 s2 + (c/m)s + k/m
46
3 Electromagnetic and Piezoelectric Transducers
x0 e Fig. 3.5 Model of a geophone based on a voice coil transducer.
e −s2 T = 2 x˙ 0 s + 2ξp ωp s + ωp2
(3.10)
Thus, there is a perfect duality between a proof-mass actuator used with a current source and a geophone (connected to an infinite resistor); above the corner frequency, the gain of the geophone is equal to the transducer constant T . Designing geophones with very low corner frequency is in general difficult, especially if their orientation with respect to the gravity vector is variable; active geophones where the corner frequency is lowered electronically may constitute a good alternative option.
3.3
General Electromechanical Transducer
i
e
Ze
T me T em
Zm
v = xç f
Fig. 3.6 Electrical analog representation of an electromechanical transducer.
3.3.1
Constitutive Equations
The constitutive behavior of a wide class of electromechanical transducers can be modelled as in Fig.3.6, where the central box represents the conversion mechanism between electrical energy and mechanical energy, and vice versa. In Laplace form, the constitutive equations read
3.3 General Electromechanical Transducer
47
e = Ze i + Tem v
(3.11)
f = Tme i + Zm v
(3.12)
where e is the Laplace transform of the input voltage across the electrical terminals, i the input current, f the force applied to the mechanical terminals, and v the velocity of the mechanical part. Ze is the blocked electrical impedance, measured for v = 0; Tem is the transduction coefficient representing the electromotive force (voltage) appearing in the electrical circuit per unit velocity in the mechanical part (in volt.sec/m). Tme is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input on the electrical side (in N/Amp), and Zm is the mechanical impedance, measured when the electrical side is open (i = 0). As an example, it is easy to check that the proof-mass with voltage source (Fig.3.4) can be written in this form with Ze = Ls + R, Zm = ms + c + k/s, Tem = T and Tme = −T . The same representation applies also to the piezoelectric transducer analyzed below. In absence of external force (f = 0), v can be eliminated between the two foregoing equations, leading to e = (Ze −
Tem Tme )i Zm
−Tem Tme /Zm is called the motional impedance. The total driving point electrical impedance is the sum of the blocked and the motional impedances.
3.3.2
Self-sensing
Equation (3.11) shows that the voltage drop across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals. Thus, if Ze i can be measured and subtracted from e, a signal proportional to the velocity is obtained. This suggests the bridge structure of Fig.3.7. The bridge equations are as follows: for the branch containing the transducer, e = Ze I + Tem v + Zb I 1 I= (e − Tem v) Ze + Zb V4 = Zb I =
Zb (e − Tem v) Ze + Zb
For the other branch, e = kZe i + kZb i V2 = kZb i =
Zb e Ze + Zb
48
3 Electromagnetic and Piezoelectric Transducers
Transducer
Fig. 3.7 Bridge circuit for self-sensing actuation.
and the bridge output V4 − V2 = (
−Zb Tem )v Ze + Zb
(3.13)
is indeed a linear function of the velocity v of the mechanical terminals. Note, however, that −Zb Tem /(Ze + Zb ) acts as a filter; the bridge impedance Zb must be adapted to the transducer impedance Ze to avoid amplitude distortion and phase shift between the output voltage V4 − V2 and the transducer velocity in the frequency band of interest.
3.4
Reaction Wheels and Gyrostabilizers
These devices are torque actuators normally used in attitude control of satellites. They have authority over the rigid body modes as well as the flexible modes. A reaction wheel consists of a rotating wheel whose axis is fixed with respect to the spacecraft; a torque is generated by increasing or decreasing the angular velocity. If the angular velocity exceeds the specification, the wheel must be unloaded, using another type of actuator (jets or magnetic). In control moment gyros (CMG), the rotating wheel is mounted on gimbals, and the gimbal torques are used as control inputs. The principle of a one-axis gyrostabilizer is described in Fig.3.8. Rotating the gimbal about the x axis with an angular velocity θ˙x produces torques: Ty = Jz Ω θ˙x cos θx
(3.14)
3.4 Reaction Wheels and Gyrostabilizers
z
Ω
Rotor
49
Servo Motor
y
.
Gimbal
x
θx
Fig. 3.8 One-axis gyrostabilizer.
Tz = Jz Ω θ˙x sin θx
(3.15)
where Jz Ω is the angular momentum along the z axis, and θx is the deviation of the rotor axis with respect to the vertical. The servo motor on the gimbal axis is velocity controlled. The angle θx is measured also, and a small gain feedback maintains the axis of the rotor in the vertical position (for a deeper discussion of the use of CMG in attitude control, see Jacot & Liska). Output Strain
Electric charge
Magnetic flux
Stress
Elasticity
Piezoelectricity
Magnetostriction
Electric field
PiezoPermittivity electricity
Input
Magnetic Magnetostriction field Heat
Light
Magnetoelectric effect
Thermal expansion
Pyroelectricity
Photostriction
Photovoltaic effect
Temperature
Light Photoelasticity Electro -optic effect
Magneto -optic
Permeability Specific heat
Refractive index
Fig. 3.9 Stimulus-response relations indicating various effects in materials. The smart materials correspond to the non-diagonal cells.
50
3 Electromagnetic and Piezoelectric Transducers
3.5
Smart Materials
Piezoelectric materials belong to the so-called smart materials, or multifunctional materials, which have the ability to respond significantly to stimuli of different physical natures. Figure 3.9 lists various effects that are observed in materials in response to various inputs: mechanical, electrical, magnetic, thermal, light. The coupling between the physical fields of different types is expressed by the non-diagonal cells in the figure; if its magnitude is sufficient, the coupling can be used to build discrete or distributed transducers of various types, which can be used as sensors, actuators, or even integrated in structures with various degrees of tailoring and complexity (e.g. as fibers), to make them controllable or responsive to their environment (e.g. for shape morphing, precision shape control, damage detection, dynamic response alleviation,...).
3.6
Piezoelectric Transducer
The piezoelectric effect was discovered by Pierre and Jacques Curie in 1880. The direct piezoelectric effect consists in the ability of certain crystalline materials to generate an electrical charge in proportion to an externally applied force; the direct effect is used in force transducers. According to the inverse piezoelectric effect, an electric field parallel to the direction of polarization induces an expansion of the material. The piezoelectric effect is anisotropic; it can be exhibited only by materials whose crystal structure has no center of symmetry; this is the case for some ceramics below a certain temperature called the Curie temperature; in this phase, the crystal has built-in electric dipoles, but the dipoles are randomly orientated and the net electric dipole on a macroscopic scale is zero. During the poling process, when the crystal is cooled in the presence of a high electric field, the dipoles tend to align, leading to an electric dipole on a macroscopic scale. After cooling and removing of the poling field, the dipoles cannot return to their original position; they remain aligned along the poling direction and the material body becomes permanently piezoelectric, with the ability to convert mechanical energy to electrical energy and vice versa; this property will be lost if the temperature exceeds the Curie temperature or if the transducer is subjected to an excessive electric field in the direction opposed to the poling field. The most popular piezoelectric materials are Lead-Zirconate-Titanate (PZT) which is a ceramic, and Polyvinylidene fluoride (PVDF) which is a polymer. In addition to the piezoelectric effect, piezoelectric materials exhibit a pyroelectric effect, according to which electric charges are generated when the material is subjected to temperature; this effect is used to produce heat detectors; it will not be discussed here. In this section, we consider a transducer made of a one-dimensional piezoelectric material of constitutive equations (we use the notations of the IEEE Standard on Piezoelectricity)
3.6 Piezoelectric Transducer
D = εT E + d33 T S = d33 E + sE T
51
(3.16) (3.17)
where D is the electric displacement (charge per unit area, expressed in Coulomb/m2 ), E the electric field (V /m), T the stress (N/m2 ) and S the strain. εT is the dielectric constant (permittivity) under constant stress, sE is the compliance when the electric field is constant (inverse of the Young’s modulus) and d33 is the piezoelectric constant, expressed in m/V or Coulomb/N ewton; the reason for the subscript 33 is that, by convention, index 3 is always aligned to the poling direction of the material, and we assume that the electric field is parallel to the poling direction. More complicated situations will be considered later. Note that the same constant d33 appears in (3.16) and (3.17). In the absence of an external force, a transducer subjected to a voltage with the same polarity as that during poling produces an elongation, and a voltage opposed to that during poling makes it shrink (inverse piezoelectric effect). In (3.17), this amounts to a positive d33 . Conversely (direct piezoelectric effect), if we consider a transducer with open electrodes (D = 0), according to (3.16), E = −(d33 /εT )T , which means that a traction stress will produce a voltage with polarity opposed to that during poling, and a compressive stress will produce a voltage with the same polarity as that during poling.
3.6.1
Constitutive Relations of a Discrete Transducer
Equations (3.16) and (3.17) can be written in a matrix form T D ε d33 E = S d33 sE T
(3.18)
where (E, T ) are the independent variables and (D, S) are the dependent variables. If (E, S) are taken as the independent variables, they can be rewritten d33 d33 2 D = E S + εT 1 − E T E s s ε T =
1 d33 S− EE E s s
or
D T
=
εT (1 − k 2 ) −e33
e33 cE
E S
(3.19)
where cE = 1/sE is the Young’s modulus under E = 0 (short circuited electrodes), in N/m2 (P a); e33 = d33 /sE , the product of d33 by the Young modulus, is the constant relating the electric displacement to the strain for
52
3 Electromagnetic and Piezoelectric Transducers
Electrode
Cross section: A Thickness: t # of disks in the stack: n l = nt
E = V=t
_
+
t
Electric charge: Q = nAD Capacitance: C = n2"A=l
Free piezoelectric expansion:
Voltage driven: î = d 33nV Charge driven:
î = d 33nQ C
Fig. 3.10 Piezoelectric linear transducer.
short-circuited electrodes (in Coulomb/m2), and also that relating the compressive stress to the electric field when the transducer is blocked (S = 0). d33 2 e33 2 = (3.20) sE εT cE εT k is called the Electromechanical coupling factor of the material; it measures the efficiency of the conversion of mechanical energy into electrical energy, and vice versa, as discussed below. From (3.19), we note that εT (1 − k 2 ) is the dielectric constant under zero strain. If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a stack of n disks of thickness t and cross section A (Fig.3.10), the global constitutive equations of the transducer are obtained by integrating Equ.(3.18) or (3.19) over the volume of the transducer; one finds (Problem 3.1) Q C nd33 V = (3.21) Δ nd33 1/Ka f or Q C(1 − k 2 ) nd33 Ka V = (3.22) f −nd33 Ka Ka Δ k2 =
where Q = nAD is the total electric charge on the electrodes of the transducer, Δ = Sl is the total extension (l = nt is the length of the transducer), f = AT is the total force and V the voltage applied between the electrodes of the transducer, resulting in an electric field E = V /t = nV /l. C = εT An2 /l is the capacitance of the transducer with no external load (f = 0), Ka = A/sE l
3.6 Piezoelectric Transducer
53
is the stiffness with short-circuited electrodes (V = 0). Note that the electromechanical coupling factor can be written alternatively k2 =
d33 2 n2 d33 2 Ka = sE ε T C
Equation (3.21) can be inverted
Ka V 1/Ka = f C(1 − k 2 ) −nd33
−nd33 C
(3.23)
Q Δ
(3.24)
from which we can see that the stiffness with open electrodes (Q = 0) is Ka /(1 − k 2 ) and the capacitance for a fixed geometry (Δ = 0) is C(1 − k 2 ). Note that typical values of k are in the range 0.3 − 0.7; for large k, the stiffness changes significantly with the electrical boundary conditions, and similarly the capacitance depends on the mechanical boundary conditions. Next, let us write the total stored electromechanical energy and coenergy functions.2 Consider the discrete piezoelectric transducer of Fig.3.11; the total
Fig. 3.11 Discrete Piezoelectric transducer.
power delivered to the transducer is the sum of the electric power, V i and ˙ The net work on the transducer is the mechanical power, f Δ. ˙ = V dQ + f dΔ dW = V idt + f Δdt
(3.25)
For a conservative element, this work is converted into stored energy, dWe , and the total stored energy, We (Δ, Q) can be obtained by integrating (3.25) from the reference state to the state (Δ, Q).3 Upon differentiating We (Δ, Q), 2 3
Energy and coenergy functions are needed in connection with energy formulations such as Hamilton principle, Lagrange equations or finite elements. Since the system is conservative, the integration can be done along any path leading from (0, 0) to (Δ, Q).
54
3 Electromagnetic and Piezoelectric Transducers
dWe (Δ, Q) =
∂We ∂We dΔ + dQ ∂Δ ∂Q
(3.26)
and, comparing with (3.25), we recover the constitutive equations f=
∂We ∂Δ
V =
∂We ∂Q
(3.27)
Substituting f and V from (3.24) into (3.25), one gets dWe = V dQ + f dΔ =
Q dQ nd33 Ka Ka − (Δ dQ + Q dΔ) + Δ dΔ C(1 − k 2 ) C(1 − k 2 ) 1 − k2
which is the total differential of We (Δ, Q) =
Q2 nd33 Ka K a Δ2 − QΔ + 2C(1 − k 2 ) C(1 − k 2 ) 1 − k2 2
(3.28)
This is the analytical expression of the stored electromechanical energy for the discrete piezoelectric transducer. Equations (3.27) recover the constitutive equations (3.24). The first term on the right hand side of (3.28) is the electrical energy stored in the capacitance C(1 − k 2 ) (corresponding to a fixed geometry, = 0); the third term is the elastic strain energy stored in a spring of stiffness Ka/(1 − k 2 ) (corresponding to open electrodes, Q = 0); the second term is the piezoelectric energy. The electromechanical energy function uses Δ and Q as independent state variables. A coenergy function using Δ and V as independent variables can be defined by the Legendre transformation We∗ (Δ, V ) = V Q − We (Δ, Q)
(3.29)
The total differential of the coenergy is dWe∗ = Q dV + V dQ −
∂We ∂We dΔ − dQ ∂Δ ∂Q
dWe∗ = Q dV − f dΔ
(3.30)
where Equ.(3.27) have been used. It follows that Q=
∂We∗ ∂V
and
f =−
∂We∗ ∂Δ
Introducing the constitutive equations (3.22) into (3.30), dWe∗ = C(1 − k 2 )V + nd33 KaΔ dV + (nd33 Ka V − Ka Δ) dΔ = C(1 − k 2 )V dV + nd33 Ka (ΔdV + V dΔ) − Ka Δ dΔ
(3.31)
3.6 Piezoelectric Transducer
55
which is the total differential of We∗ (Δ, V ) = C(1 − k 2 )
V2 Δ2 + nd33 Ka V Δ − Ka 2 2
(3.32)
This is the analytical form of the coenergy function for the discrete piezoelectric transducer. The first term on the right hand side of (3.32) is recognized as the electrical coenergy in the capacitance C(1 − k 2 ) (corresponding to a fixed geometry, Δ = 0); the third is the strain energy stored in a spring of stiffness Ka (corresponding to short-circuited electrodes, V = 0). The second term of (3.32) is the piezoelectric coenergy; using the fact that the uniform electric field is E = nV /l and the uniform strain is S = Δ/l, it can be rewritten Se33 E dΩ (3.33) Ω
where the integral extends to the volume Ω of the transducer. The analytical form (3.28) of the electromechanical energy, together with the constitutive equations (3.27) can be regarded as an alternative definition of a discrete piezoelectric transducer, and similarly for the analytical expression of the coenergy (3.32) and the constitutive equations (3.31).
3.6.2
Interpretation of k2
Consider a piezoelectric transducer subjected to the following mechanical cycle: first, it is loaded with a force F with short-circuited electrodes; the resulting extension is F Δ1 = Ka where Ka = A/(sE l) is the stiffness with short-circuited electrodes. The energy stored in the system is W1 =
Δ1
f dx = 0
F Δ1 F2 = 2 2Ka
At this point, the electrodes are open and the transducer is unloaded according to a path of slope Ka /(1 − k 2 ), corresponding to the new electrical boundary conditions, F (1 − k 2 ) Δ2 = Ka The energy recovered in this way is W2 =
Δ2
f dx = 0
F Δ2 F 2 (1 − k 2 ) = 2 2Ka
56
3 Electromagnetic and Piezoelectric Transducers
leaving W1 − W2 stored in the transducer. The ratio between the remaining stored energy and the initial stored energy is W1 − W2 = k2 W1 Similarly, consider the following electrical cycle: first, a voltage V is applied to the transducer which is mechanically unconstrained (f = 0). The electric charges appearing on the electrodes are Q1 = CV where C = εT An2 /l is the unconstrained capacitance, and the energy stored in the transducer is Q1 V Q1 CV 2 W1 = v dq = = 2 2 0 At this point, the transducer is blocked mechanically and electrically unloaded from V to 0. The electrical charges are removed according to Q2 = C(1 − k 2 )V where the capacitance for fixed geometry has been used. The energy recovered in this way is Q2 C(1 − k 2 )V 2 W2 = v dq = 2 0 leaving W1 − W2 stored in the transducer. Here again, the ratio between the remaining stored energy and the initial stored energy is W1 − W2 = k2 W1 Although the foregoing relationships provide a clear physical interpretation of the electromechanical coupling factor, they do not bring a practical way of measuring k 2 ; the experimental determination of k 2 is often based on impedance (or admittance) measurements.
3.6.3
Admittance of the Piezoelectric Transducer
Consider the system of Fig.3.12, where the piezoelectric transducer is assumed massless and is connected to a mass M . The force acting on the mass is the negative of that acting on the transducer, f = −M x ¨; using (3.22), Q C(1 − k 2 ) nd33 Ka V = (3.34) −M x ¨ −nd33 Ka Ka x
3.6 Piezoelectric Transducer (a)
57
(b)
dB
Transducer
Fig. 3.12 (a) Elementary dynamical model of the piezoelectric transducer. (b) Typical admittance FRF of the transducer, in the vicinity of its natural frequency.
From the second equation, one gets (in Laplace form) x=
nd33 Ka M s2 + K a
and, substituting in the first one and using (3.23), one finds
Q M s2 + Ka /(1 − k 2 ) 2 = C(1 − k ) V M s2 + K a
(3.35)
It follows that the admittance reads: I sQ s2 + z 2 = = sC(1 − k 2 ) 2 V V s + p2
(3.36)
where the poles and zeros are respectively p2 =
Ka M
and
z2 =
Ka /(1 − k 2 ) M
(3.37)
p is the natural frequency with short-circuited electrodes (V = 0) and z is the natural frequency with open electrodes (I = 0). From the previous equation one sees that z 2 − p2 = k2 (3.38) z2 which constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of admittance (or impedance) FRF measurements (Fig.3.12.b).
58
3 Electromagnetic and Piezoelectric Transducers
3.7
Problems
P.3.1 Consider the piezoelectric linear transducer of Fig.3.10; assuming that all the electrical and mechanical quantities are uniformly distributed, show that the constitutive equations of the transducer, Equ.(3.22) can be derived from those of the material, Equ.(3.19). P.3.2 A piezoelectric transducer supporting an inertial mass M can be used as an accelerometer or as a proof-mass actuator.
(a)
(b)
voltage amplifier
Q charge amplifier
V
V =0
piezoelectric transducer
0
f Fig. 3.13 (a) Piezoelectric accelerometer: the input is the support acceleration x ¨0 and the output is the electric charge Q measured by the charge amplifier. (b) Proof-mass actuator: the input is the voltage V applied to the piezo stack and the output is the reaction force f applied to the support.
(a) Accelerometer (Fig.3.13.a): the transducer is placed on a surface subjected to an acceleration x ¨0 and it is connected to a charge amplifier enforcing the electrical boundary conditions V 0 (see section 4.4.1).4 Show that the transfer function between the support acceleration x ¨0 and the electric charge Q is given by Q −nd33 M = (3.39) x ¨0 1 + 2ξs/ωn + s2 /ωn2 where ωn2 = Ka /M and c/M = 2ξωn . Comment on what would be the requirements for a good accelerometer. 4
Some damping is introduced in the system by assuming a mechanical stiffness Ka + cs instead of Ka in the mechanical part of the transducer constitutive equations (3.22) Q C(1 − k2 ) nd33 Ka V = f −nd33 Ka Ka + cs Δ
3.7 Problems
59
(b) Proof-mass actuator (Fig.3.13.b): The input is the voltage V applied to the piezo stack and the output is the reaction force f applied to the support. Show that the transfer function between the voltage V and the reaction force f is given by f s2 nd33 Ka = 2 (3.40) V s + 2ξsωn + ωn2 Compare with the solution based on a voice-coil transducer, Equ.(3.8). P.3.3 Draw the cycle diagrams (f, Δ) and (V, Q) of the physical interpretations of the electromechanical coupling factor k 2 , in section 3.6.2. P.3.4 Represent the discrete piezoelectric transducer (3.24) in the electrical analog form of Fig.3.6. P.3.5 Electromagnetic damper: Consider a beam with modal properties μi , ωi , φi (x) attached to a voice coil transducer of constant T at x = a. Evaluate the modal damping ξi when the coil is shunted on a resistor R. How can this damper be optimized?
4
Piezoelectric Beam, Plate and Truss
4.1
Piezoelectric Material
4.1.1
Constitutive Relations
The constitutive equations of a general piezoelectric material are Tij = cE ijkl Skl − ekij Ek
(4.1)
Di = eikl Skl + εSik Ek
(4.2)
where Tij and Skl are the components of the stress and strain tensors, respectively, cE ijkl are the elastic constants under constant electric field (Hooke’s tensor), eikl the piezoelectric constants (in Coulomb/m2 ) and εSij the dielectric constant under constant strain. These formulae use classical tensor notations, where all indices i, j, k, l = 1, 2, 3, and there is a summation on all repeated indices. The above equations are a generalization of (3.19), with Skl and Ej as independent variables; they can be written alternatively with Tkl and Ej as independent variables: Sij = sE ijkl Tkl + dkij Ek
(4.3)
Di = dikl Tkl + εTik Ek
(4.4)
where sE ijkl is the tensor of compliance under constant electric field, dikl the piezoelectric constants (in Coulomb/N ewton) and εTik the dielectric constant under constant stress. The difference between the properties under constant stress and under constant strain has been stressed earlier. As an alternative to the above tensor notations, it is customary to use the engineering vector notations
A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 61–101. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
62
4 Piezoelectric Beam, Plate and Truss
⎫ ⎧ T11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ T33 T = ⎪ ⎪ T23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T31 ⎪ ⎪ ⎪ ⎭ ⎩ T12
⎫ ⎧ S11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ S33 S= ⎪ 2S23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2S31 ⎪ ⎪ ⎪ ⎭ ⎩ 2S12
(4.5)
With these notations, Equ.(4.1) (4.2) can be written in matrix form {T } = [c]{S} − [e]{E} {D} = [e]T {S} + [ε]{E}
(4.6)
and (4.3), (4.4), {S} = [s]{T } + [d]{E} {D} = [d]T {T } + [ε]{E}
(4.7)
where the superscript T stands for the transposed; the other superscripts have been omitted, but can be guessed from the equation itself. Assuming that the coordinate system coincides with the orthotropy axes of the material and that the direction of polarization coincides with direction 3, the explicit form of (4.7) is: Actuation: ⎫ ⎡ ⎧ s11 ⎪ ⎪ ⎪ S11 ⎪ ⎪ ⎪ ⎪ ⎢ s12 ⎪ S ⎪ ⎪ 22 ⎪ ⎪ ⎬ ⎢ ⎨ ⎢ s13 S33 =⎢ ⎢ ⎪ ⎪ ⎢ 0 ⎪ 2S23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ 0 ⎪ ⎪ ⎪ 2S31 ⎪ ⎭ ⎩ 0 2S12
s12 s22 s23 0 0 0
s13 s23 s33 0 0 0
0 0 0 s44 0 0
0 0 0 0 s55 0
⎫ ⎡ ⎤⎧ 0 ⎪ 0 T11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ 0 ⎪ 0 ⎥ T ⎪ ⎪ 22 ⎪ ⎥⎪ ⎬ ⎢ ⎨ ⎢ 0 ⎥ ⎥ T33 + ⎢ 0 ⎢ 0 ⎥ 0 ⎥⎪ T23 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ d15 ⎦ 0 ⎪ T31 ⎪ ⎪ ⎪ ⎭ ⎩ s66 0 T12
0 0 0 d24 0 0
⎤ d31 ⎧ ⎫ d32 ⎥ ⎥ ⎨ E1 ⎬ ⎥ d33 ⎥ E2 0 ⎥ ⎥ ⎩ E3 ⎭ 0 ⎦ 0 (4.8)
Sensing: ⎫ ⎧ T11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎡ ⎪ ⎧ ⎫ ⎡ ⎫ ⎤⎪ ⎤⎧ T22 ⎪ ⎪ ⎪ ⎪ ⎪ D ε11 0 0 ⎨ E1 ⎬ 0 0 0 0 0 d ⎬ ⎨ 1⎬ ⎨ 15 T33 D2 = ⎣ 0 0 0 d24 0 0 ⎦ + ⎣ 0 ε22 0 ⎦ E2 (4.9) T23 ⎪ ⎩ ⎭ ⎭ ⎪ ⎩ ⎪ d31 d32 d33 0 0 0 ⎪ 0 0 ε D3 E ⎪ ⎪ 33 3 ⎪ T31 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ T12 Typical values of the piezoelectric constants for piezoceramics (PZT) and piezopolymers (PVDF) are given in Table 4.1. Examining the actuator equation (4.8), we note that when an electric field E3 is applied parallel to the
4.1 Piezoelectric Material
63
direction of polarization, an extension is observed along the same direction; its amplitude is governed by the piezoelectric coefficient d33 . Similarly, a shrinkage is observed along the directions 1 and 2 perpendicular to the electric field, the amplitude of which is controlled by d31 and d32 , respectively (shrinkage, because d31 and d32 are negative). Piezoceramics have an isotropic behaviour in the plane, d31 = d32 ; on the contrary, when PVDF is polarized under stress, its piezoelectric properties are highly anisotropic, with d31 ∼ 5d32 . Equation (4.8) also indicates that an electric field E1 normal to the direction of polarization 3 produces a shear deformation S13 , controlled by the piezoelectric constant d15 (similarly, a shear deformation S23 occurs if an electric field E2 is applied; it is controlled by d24 ). An interesting feature of this type of actuation is that d15 is the largest of all piezoelectric coefficients (500 10−12 C/N for PZT). The various modes of operation associated with the piezoelectric coefficients d33 , d31 and d15 are illustrated in Fig.4.1.
ÉL = nd 33V P P +
d33 V
_
d31
L E P t
V
ÉL
ÉL = Ed 31L
E = V=t Supporting structure
d15
ÉL
í = d15E1
1
í
E1
P
L0
V
ÉL = íL0
3
Fig. 4.1 Actuation modes of piezoelectric actuators. P indicates the direction of polarization.
64
4 Piezoelectric Beam, Plate and Truss Table 4.1 Typical properties of piezoelectric materials. Material properties
PZT
PVDF
Piezoelectric constants d33 (10−12 C/N or m/V ) d31 (10−12 C/N or m/V )
300 -150
-25 uni-axial: d31 = 15 d32 = 3 bi-axial: d31 = d32 = 3 0 0.025
d15 (10−12 C/N or m/V ) e31 = d31 /sE (C/m2 ) Electromechanical coupling factor k33 k31 k15
500 -7.5 0.7 0.3 0.7
∼ 0.1
Dielectric constant εT /ε0 (ε0 = 8.85 10−12 F/m) Max. Electric field (V /mm)
1800
10
2000
5 105
Max. operating (Curie) T ◦ (◦ C) 80◦ − 150◦
4.1.2
90◦
Density (Kg/m3 ) Young’s modulus 1/sE (GPa)
7600 50
1800 2.5
Maximum stress (MPa) Traction Compression Maximum strain
80 600 Brittle
200 200 50 %
Coenergy Density Function
With an approach parallel to that of the discrete transducer, the total stored energy density in a unit volume of material is the sum of the mechanical work and the electrical work, dWe (S, D) = {dS}T {T } + {dD}T {E}
(4.10)
[compare with (3.25)]. For a conservative system, We (S, D) can be obtained by integrating (4.10) from the reference state to the state (S, D); since the system is conservative, the integration can be done along any path from
4.1 Piezoelectric Material
65
(0, 0) to (S, D). Upon differentiating We (S, D) and comparing with (4.10) we recover the constitutive equations ∂We ∂We {T } = and {E} = (4.11) ∂S ∂D which are the distributed counterparts of (3.27). The coenergy density function is defined by the Legendre transformation We∗ (S, E) = {E}T {D} − We (S, D)
(4.12)
[compare with (3.29)]. The total differential is dWe∗ = {dE}T {D} + {E}T {dD} − {dS}T
∂We ∂S
− {dD}T
= {dE}T {D} − {dS}T {T } where (4.11) have been used. It follows that ∂We∗ ∂We∗ {D} = and {T } = − ∂E ∂S
∂We ∂D
(4.13)
(4.14)
Substituting (4.6) into (4.13), dWe∗ = {dE}T [e]T {S} + {dE}T [ε]{E} − {dS}T [c]{S} + {dS}T [e]{E} (4.15) which is the total differential of We∗ (S, E) =
1 1 {E}T [ε]{E} + {S}T [e]{E} − {S}T [c]{S} 2 2
(4.16)
[compare with (3.32)]. The first term in the right hand side is the electrical coenergy stored in the dielectric material (ε is the matrix of permittivity under constant strain); the third term is the strain energy stored in the elastic material (c is the matrix of elastic constants under constant electric field); the second term is the piezoelectric coenergy, which generalizes (3.33) in three dimensions. Taking the partial derivatives (4.14), one recovers the constitutive equations (4.6). In that sense, the analytical form of the coenergy density function, (4.16) together with (4.14), can be seen as an alternative definition of the linear piezoelectricity. In the literature, H(S, E) = −We∗ (S, E) is known as the electric enthalpy density.
(4.17)
66
4 Piezoelectric Beam, Plate and Truss
4.2
Hamilton’s Principle
According to Hamilton’s principle, the variational indicator t2 V.I. = [δ(T ∗ + We∗ ) + δWnc ]dt = 0
(4.18)
t1
is stationary for all admissible (virtual) variations δui and δEi of the path between the two fixed configurations at t1 and t2 . 1 ∗ T = {u} ˙ T {u}dΩ ˙ (4.19) 2 Ω is the kinetic (co)energy ( is the density) and ! " 1 We∗ = {E}T [ε]{E} + 2{S}T [e]{E} − {S}T [c]{S} dΩ 2 Ω
(4.20)
has been defined in the previous section. T ∗ + We∗ is the Lagrangian and δWnc is the virtual work of nonconservative external forces and applied currents.
4.3
Piezoelectric Beam Actuator
Consider the piezoelectric beam of Fig.4.2; it is covered with a single piezoelectric layer of uniform thickness hp , polarized along the z axis; the supporting structure is acting as electrode on one side and there is an electrode of variable width bp (x) on the other side. The voltage difference between the electrodes is controlled, so that the part of the piezoelectric material located between the electrodes is subjected to an electric field E3 parallel to the polarization (note that the piezoelectric material which is not covered by the electrode on both sides is useless as active material). We denote by w(x, t) the transverse displacements of the beam; according to the Euler-Bernoulli assumption, the stress and strain fields are uniaxial, along Ox; the axial strain S1 is related to the curvature w by S1 = −zw
(4.21)
where z is the distance to the neutral axis. We also assume that the piezoelectric layer is thin enough, so that E3 is constant across the thickness.
4.3.1
Hamilton’s Principle
The kinetic coenergy reads 1 T = 2 ∗
l 0
Aw˙ 2 dx
(4.22)
4.3 Piezoelectric Beam Actuator Piezoelectric material
hp
z
67
bp(x)
w(x; t)
Electrode
p(x; t)
zm
x
h1
h2
V ral Neut Axis
Fig. 4.2 Piezoelectric beam covered by a single piezoelectric layer with an electrode profile of width bp (x).
where A is the cross-section of the beam. Both the electric field and the strain vectors have a single non-zero component, respectively E3 and S1 ; the coenergy function (4.20) is therefore We∗ =
1 2
" ε33 E32 + 2S1 e31 E3 − c11 S12 dA
(4.23)
# $ 2 ε33 E32 − 2w ze31 E3 − c11 w z 2 dA
(4.24)
l
!
dx 0
A
and, combining with (4.21), We∗ =
1 2
l
dx 0
A
The first contribution to We∗ is restricted to the piezoelectric part of the beam under the electrode area; the integral over the cross section can be written ε33 E32 bp hp . The second contribution is also restricted to the piezoelectric layer; taking into account that
h2
zdA = A
bp z dz = bp hp zm h1
where zm is the distance between the mid-plane of the piezoelectric layer and the neutral axis (Fig.4.2), it can be written −2w e31 E3 bp hp zm . The ∗ third term in We can be rewritten by introducing the bending stiffness (we give up the classical notation EI familiar to structural engineers to avoid confusion) c11 z 2 dA
D= A
(4.25)
68
4 Piezoelectric Beam, Plate and Truss
Thus, We∗ reads We∗ =
1 2
l# $ 2 ε33 E32 bp hp − 2w e31 E3 bp hp zm − Dw dx 0
Next, we can apply Hamilton’s principle, recalling that only the vertical displacement is subject to virtual changes, δw, since the electric potential is fixed (voltage control). Integrating by part the kinetic energy with respect to time and taking into account that δw(x, t1 ) = δw(x, t2 ) = 0,
t2
δT ∗ dt =
t1
t2
0
t1
Similarly, δWe∗ =
l
0
l
dt
Aw˙ δ w˙ dx = −
t2
l
Aw ¨ δw dx
dt 0
t1
[−δw (e31 E3 bp hp zm ) − Dw δw ]dx
and, integrating by part twice with respect to x, δWe∗ = − (e31 E3 bp hp zm ) δw
l
l
0
−Dw δw
0
l
l
+ (Dw ) δw 0
0
−
−
+ (e31 E3 bp hp zm ) δw l
l 0
(e31 E3 bp hp zm ) δw dx
(Dw ) δw dx 0
The virtual work of nonconservative forces is l δWnc = p(x, t)δw dx 0
where p(x, t) is the distributed transverse load applied to the beam. Introducing in Hamilton’s principle (4.18), one gets that
l
t2
V.I. =
dt t1
0
# $ −Aw ¨ − (e31 E3 bp hp zm ) − Dw + p δw dx
l %# $ &l # $ − e31 E3 bp hp zm + Dw δw + {(e31 E3 bp hp zm ) + Dw }δw = 0 0
0
for all admissible variations δw compatible with the kinematics of the system (i.e. boundary conditions); let us discuss this equation.
4.3.2
Piezoelectric Loads
It follows from the previous equation that the differential equation governing the problem is
4.3 Piezoelectric Beam Actuator
$ # Aw ¨ + Dw = p − (e31 E3 bp hp zm )
69
(4.26)
If one takes into account that only bp depends on the spatial variable x and that E3 hp = V , the voltage applied between the electrodes of the piezoelectric layer, it becomes $ # Aw ¨ + Dw = p − e31 V zm bp (x)
(4.27)
This equation indicates that the effect of the piezoelectric layer is equivalent to a distributed load proportional to the second derivative of the width of the electrode. Examining the remaining terms, one must also have $ # e31 E3 bp hp zm + Dw δw = 0
$ # (e31 E3 bp hp zm ) + Dw δw = 0 at x = 0 and x = l (4.28) The first condition states that at an end where the rotation is free (where a virtual rotation is allowed, δw = 0), one must have
e31 V bp zm + Dw = 0
(4.29)
This means that the effect of the piezoelectric layer is that of a bending moment proportional to the width of the electrode. Similarly, the second condition states that at an end where the displacement is free (where a virtual displacement is allowed, δw = 0), one must have $ # e31 V bp zm + Dw =0
(4.30)
$ Dw represents the transverse shear force along the beam in classical beam theory, and step changes of the shear distribution occur where point loads are applied. This means that the effect of a change of slope bp in the width of the electrode is equivalent to a point force proportional to change of the first derivative of the electrode width. One should always keep in mind that the piezoelectric loading consists of internal forces which are always self-equilibrated. Figure 4.3 shows a few examples of electrode shapes and the corresponding piezoelectric loading. A rectangular electrode (Fig.4.3.a) is equivalent to a pair of bending moments Mp applied at the ends of the electrode. A triangular electrode (Fig.4.3.b) is equivalent to a pair of point forces P and a bending moment Mp ; note that if the beam is clamped on the left side, the corresponding loads will be taken by the support, and the only remaining force is the point load at the right end. A parabolic electrode (Fig.4.3.c) is #
70
4 Piezoelectric Beam, Plate and Truss
(a) Mp = à e31V b z m
Mp
b
zm
Mp
(b) Mp
V
P
b
Mp = à e31V b z m P = e 31V b z m l
l
P
(c)
bp = 4b2 x(l à x)
p
P
l P = à e 31V z m 4b l p = e 31V z m 8b2 l
b
l x P Fig. 4.3 Examples of electrode shapes and corresponding piezoelectric loading: (a) rectangular electrode, (b) triangular electrode, (c) parabolic electrode. The piezoelectric loading is always self-equilibrated.
equivalent to a uniform distributed load p and a pair of point forces P at the ends. As another example, consider the electrode shape of Fig.4.4. It consists of a rectangular part of length l1 , followed by a part with constant slope, of length l2 . According to the foregoing discussion, this is equivalent to bending moments M1 and M2 at the extremities of the electrodes, and point forces P at the location where there is a sudden change in the first derivative b (x). Once again, the piezoelectric loading is self-equilibrated.
4.4 Laminar Sensor
71
M1 = à e 31V b1 z m M1
b1 P
M 2 = à e 31V b2 z m P = à e 31V (b2àb1)z m l2
l1
l2
P
b2
M2
Fig. 4.4 Self-equilibrated equivalent piezoelectric loading for an electrode with a sudden change in bp (x).
4.4
Laminar Sensor
4.4.1
Current and Charge Amplifiers
When used in sensing mode, a piezoelectric transducer is coupled to an operational amplifier (Fig.4.5.a) to form either a current amplifier (Fig.4.5.b), or a charge amplifier (Fig.4.5.c). An operational amplifier is an active electrical circuit working as a high gain linear voltage amplifier with infinite input resistance (so that the input currents i− and i+ are essentially zero), and zero output resistance, so that the output voltage e0 is essentially proportional to the voltage difference e+ − e− ; the open loop gain A is usually very high, which means that the allowable input voltage is very small (millivolt). As a result, when the electrodes of a piezoelectric transducer are connected to an operational amplifier, they can be regarded as short-circuited and the electric field through the piezo can be considered as E3 = 0. Then, it follows from the constitutive equation (4.2) that the electric displacement is proportional to the strain D3 = e31 S1 (4.31)
4.4.2
Distributed Sensor Output
If one assumes that the piezoelectric sensor is thin with respect to the beam, the strain can be regarded as uniform over its thickness, S1 = −zm w , and E3 = 0 is enforced by the charge amplifier; integrating over the electrode area (Fig.4.2), one gets Q=
D3 dA = −
b
b
bp (x)zm e31 w dx = −zm e31 a
bp (x)w dx a
(4.32)
72
4 Piezoelectric Beam, Plate and Truss
i
e
i ,i ' 0
io
(a)
e
i
e o = A( e A > 105
e
e
Very small
e )
R
i i
(b)
e Piezo
e à = e o + Ri
e
}
àA e0 = 1+A Ri ' à Ri
C
i
(c)
e0 = à Aeà
eo
i
e Piezo
e0 = à Aeà e = eà = i=sC + eo 0
}
eo
e
àA i (1+A) sC
' à i=sC = à Q=C
Fig. 4.5 (a) Operational amplifier, (b) Current amplifier, (c) Charge amplifier.
with a constant polarization profile e31 . It is assumed that the sensor extends from x = a to x = b along the beam. Thus, the amount of electric charge is proportional to the weighted average of the curvature, the weighing function being the width of the electrode. For an electrode with constant width,
Q = −zm e31 bp [w (b) − w (a)]
(4.33)
The sensor output is proportional to the difference of slopes (i.e. rotations) at the extremities of the sensor strip. We note that this result is dual of that of Fig.4.3.a, where the piezoelectric transducer is used in actuation mode. Equation (4.32) can be integrated by parts, twice, leading to
b
b
a
b
− wbp
w bp (x)dx = w bp a
+
a
b
w b dx
(4.34)
a
If, as an example, one considers the case of a cantilever beam clamped at x = 0 and covered with a piezoelectric strip and an electrode of triangular shape extending over the whole length as in Fig.4.3.b (a = 0 and b = l), w(0) = w (0) = 0 (cantilever beam) and bp = 0, bp (l) = 0, bp = −bp (0)/l (triangular electrode). Substituting into the foregoing equations, one gets
4.5 Spatial Modal Filters
73
bp (0) w(l) ∼ w(l) (4.35) l Thus, the output signal is proportional to the tip displacement of the cantilever beam. Once again, this result is dual of that obtained in actuation mode (the piezoelectric loading is a point force at the tip). Similarly, if one considers a parabolic electrode as in Fig.4.3.c and if the beam is such that w(0) = w(l) = 0 (this includes pinned-pinned, pinned-clamped, etc), we have bp (0) = bp (l) = 0 and bp (x) = −8b/l2 and, substituting into (4.34), Q = −zm e31
8b Q = zm e31 2 l
l
0
w(x)dx ∼
l
w(x)dx
(4.36)
0
Thus, the output signal is proportional to the volume displacement, which is, once again, dual of the uniform distributed load in actuation mode. All the above results are based on the beam theory which is essentially onedimensional; their accuracy in practical applications will depend very much on the relevance of these assumptions for the applications concerned. This issue is important in applications, especially in collocated control systems.
4.4.3
Charge Amplifier Dynamics
According to Fig.4.5.c, the output voltage is proportional to the amount of electric charge generated on the electrode; the amplifier gain is fixed by the capacitance C. This relation is correct at frequencies beyond some corner frequency depending on the amplifier construction, but does not apply statically (near ω = 0). If a refined model of the charge amplifier is required, this behavior can be represented by adding a second order high-pass filter F (s) =
s2 s2 + 2ξc ωc s + ωc2
(4.37)
with appropriate parameters ωc and ξc . For frequencies well above the corner frequency ωc , F (s) behaves like a unit gain.
4.5
Spatial Modal Filters
4.5.1
Modal Actuator
According to (4.27), a piezoelectric layer with an electrode of width bp (x) is equivalent to a distributed transverse load proportional to bp (x). Let w(x, t) =
i
zi (t)φi (x)
(4.38)
74
4 Piezoelectric Beam, Plate and Truss
be the modal expansion of the transverse displacements, where zi (t) are the modal amplitudes, and φi (x) the mode shapes, solutions of the eigenvalue problem % & Dφi (x) − ωi2 Aφi = 0 (4.39) They satisfy the orthogonality conditions 0
0
l
l
A φi (x)φj (x)dx = μi δij
(4.40)
D φi (x)φj (x)dx = μi ωi2 δij
(4.41)
where μi is the modal mass, ωi the natural frequency of mode i, and δij is the Kronecker delta (δij = 1 if i = j, δij = 0 if i = j). Substituting (4.38) into (4.27) (assuming p = 0), one gets z¨i φi + zi (Dφi ) = −e31V bp zm A i
i
or using (4.39), A
z¨i φi + A
i
zi ωi2 φi = −e31 V bp zm
i
where the sums extend over all modes. Upon multiplying by φk (x), integrating over the length of the beam, and using the orthogonality condition (4.40), one finds easily the equation governing the modal amplitude zk : μk (z¨k + ωk2 zk ) = −e31 V zm
l 0
bp (x)φk (x)dx
(4.42)
The right hand side is the modal force pk applied by the piezoelectric strip to mode k. From the first orthogonality condition (4.40), it is readily seen that if the electrode profile is chosen in such a way that
bp ∼ Aφl (x)
(4.43)
all the modal forces pk vanish, except pl : pk ∼ −e31 V zm
l 0
Aφl φk dx ∼ −e31 V zm μl δkl
(4.44)
such an electrode profile will excite only mode l; it constitutes a modal actuator (for mode l).
4.5 Spatial Modal Filters
4.5.2
75
Modal Sensor
Similarly, if the piezoelectric layer is used as a sensor, the electric charge appearing on the sensor is given by (4.32). Introducing the modal expansion (4.38), l zi (t) bp (x)φi (x)dx (4.45) Q = −zm e31 0
i
Comparing this equation with the second orthogonality conditions (4.41), one sees that any specific mode can be made unobservable by choosing the electrode profile in such a way that the integral vanishes. If the electrode profile is chosen according to
bp (x) ∼ Dφl (x)
(4.46)
(proportional to the distribution of the bending moment of mode l), the output charge becomes Q ∼ −zm e31 μl ωl2 zl (t)
(4.47)
It contains only a contribution from mode l. This electrode profile constitutes a modal sensor. Note that, for a uniform beam, (4.39) implies that the mode shapes satisfy φIV i (x) ∼ φi (x). It follows that the electrode profile of a modal sensor also satisfies that of a modal actuator: from (4.46), Mode 2
Mode 1
Cantilever
Simply supported Modal filter for mode 1
Modal filter for mode 2
+
-
-
+
Fig. 4.6 Electrode profile of modal filters for the first two modes of a uniform beam for various boundary conditions: left: cantilever, right: simply supported.
76
4 Piezoelectric Beam, Plate and Truss
bp (x) ∼ φIV l (x) ∼ φl (x)
(4.48)
which satisfies (4.43). Figure 4.6 illustrates the electrode profile of modal filters used for a uniform beam with various boundary conditions; the change of sign indicates a change in polarity of the piezoelectric strip, which is equivalent to negative values of bp (x). As an alternative, the part of the sensor with negative polarity can be bonded on the opposite side of the beam, with the same polarity. The reader will notice that the electrode shape of the simply supported beam is the same as the mode shape itself, while for the cantilever beam, the electrode shape is that of the mode shape of a beam clamped at the opposite end. Modal filters constitute an attractive option for spillover alleviation, because they allow one to minimize the controllability and observability of a known set of modes. In practical applications, however, the beam approximation often provides fairly poor modal filters, because the piezoelectric layer reacts as an orthotropic material rather than a unidirectional one (Preumont et al., 2003).
4.6
Active Beam with Collocated Actuator-Sensor
Consider a beam provided with a pair of rectangular piezoelectric actuator and sensor (Fig.4.7). The two patches do not have to be of the same size, nor have the same material properties, but they are collocated in the sense of the Euler-Bernoulli beam theory, which means that they extend over the same length along the beam. The system can, for example, be modelled by finite elements; the mesh is such that there is a node at both ends of the piezo patches (each node has two degrees of freedom, one translation yi and one rotation θi ). We seek the open-loop FRF between the voltage V (t) applied Sensor
Actuator
M
yi
M x1
qi
x2 F.E. model
Fig. 4.7 Active cantilever beam with collocated piezoelectric actuator and sensor. Every node has 2 d.o.f. (yi and θi ).
4.6 Active Beam with Collocated Actuator-Sensor
77
to the actuator, and the output voltage v0 (t) of the sensor (assumed to be connected to a charge amplifier).
4.6.1
Frequency Response Function
According to Fig.4.3.a, the rectangular piezoelectric actuator is equivalent to a pair of torques M with opposite signs and proportional to V : M = −e31 zm bp V = ga V
(4.49)
where ga is the actuator gain which can be computed from the actuator size and the material properties. In the general form of the equation of motion, the external force vector in a FE model is f = bM = bga V
(4.50)
where the influence vector b has the form bT = (.., 0, −1, 0, 1, ...); the only non-zero components correspond to the rotational degrees of freedom of the nodes located at x = x1 and x = x2 in the model. In modal coordinates, the system dynamics is governed by a set of independent second order equations z¨k + 2ξk ωk z˙k + ωk2 zk =
φTk f pk = μk μk
(4.51)
where ωk is the natural frequency of mode k, ξk the modal damping ratio and μk the modal mass. Using the Laplace variable s, we can write it alternatively as pk zk = (4.52) 2 μk (s + 2ξk ωk s + ωk2 ) The modal forces pk represent the work of the external loading on the various mode shapes: pk = φTk f = φTk bga V = ga V Δθka
(4.53)
where Δθka = φTk b is the relative rotation [difference of slope w (x2 ) − w (x1 )] between the extremities of the actuator, for mode k. Similarly, according to (4.33), the sensor output is also proportional to the difference of slopes, that is the relative rotation of the extremities of the sensor, θ s . In modal coordinates, v0 = gs Δθs = gs zi Δθis (4.54) i
where gs is the sensor gain, depending on the sensor size, material properties and on the charge amplifier gain (which converts the electric charge into voltage), and Δθis are the modal components of the relative rotation between the extremities of the sensor. Note that if the sensor and the actuator extend
78
4 Piezoelectric Beam, Plate and Truss
over the same length of the beam, they can be considered as collocated in the sense of the Euler-Bernoulli beam theory, and Δθis = Δθia = Δθi
(4.55)
Combining the foregoing equations, one easily gets the transfer function between the actuator voltage V and the sensor output v0 ; the FRF follows by substituting s = jω. v0 Δθi2 = G(ω) = ga gs 2 V μi (ωi − ω 2 + 2jξi ωi ω) n
(4.56)
i=1
4.6.2
Pole-Zero Pattern
For an undamped system, the FRF is purely real: n v0 Δθi2 = G(ω) = ga gs V μ (ωi2 − ω 2 ) i=1 i
(4.57)
All the residues of the modal expansion are positive and G(ω) is an increasing function of ω similar to that represented in Fig.2.5; the pole-zero pattern is that of Fig.2.6.a. As explained in chapter 2, for a lightly damped structure, the poles and zeros are slightly moved to the left half plane as in Fig.2.6.b. The position of the zeros in the complex plane depends on the position of the
Fig. 4.8 Experimental open-loop FRF G(ω) of a piezoelectric beam similar to that of Fig.4.7.
4.6 Active Beam with Collocated Actuator-Sensor
79
actuator/sensor pair along the beam, while the poles do not. The Bode and Nyquist plots of such a system are always similar to those of Fig.2.8. Once again, this interlacing property of the poles and zeros is of fundamental importance in control system design for lightly damped vibrating systems, because it is possible to find a fixed controller with guaranteed stability, irrespective to changes in the mass and stiffness distribution of the system. Figure 4.8 shows typical experimental results obtained with a system similar to that of Fig.4.6. Observe that G(ω) does not exhibit any roll-off (decay) at high frequency; this indicates a feedthrough component in the system, which is not apparent from the modal expansion (4.56) (according to which the high frequency behavior is as ω −2 ). It will become clearer when we consider the modal truncation.1
4.6.3
Modal Truncation
Let us now examine the modal truncation of (4.56) which normally includes all the modes of the system (a finite number n with a discrete model, or infinite if one looks at the system as a distributed one). Obviously, if one wants an accurate model in some frequency band [0, ωc ], all the modes (with significant residues) which belong to this frequency band must be included in the truncated expansion, but the high frequency modes cannot be completely ignored. To analyze this, one rewrites (4.56) n Δθi2 .Di (ω) μi ωi2 i=1
(4.58)
1 1 − ω 2 /ωi2 + 2jξi ω/ωi
(4.59)
G(ω) = ga gs where Di (ω) =
n i=1
is the dynamic amplification of mode i. For any mode with a natural frequency ωi substantially larger than ωc , one sees from Fig.2.2 that Di (ω) 1 within [0, ωc ] and the sum (4.58) may be replaced by m n Δθi2 Δθi2 G(ω) = gags 2 .Di (ω) + ga gs μω μ ω2 i=1 i i i=m+1 i i
(4.60)
where m has been selected in such a way that ωm ωc . This equation recognizes the fact that, at low frequency, the high frequency modes respond in a quasi-static manner. The sum over the high frequency modes can be eliminated by noting that the static gain satisfies 1
Another observation is that a small linear shift appears in the phase diagram, due to the fact that these results have been obtained digitally (the sampling is responsible for a small delay in the system).
80
4 Piezoelectric Beam, Plate and Truss n Δθi2 G(0) = ga gs μi ωi2 i=1
(4.61)
leading to m m Δθi2 Δθi2 G(ω) = ga gs ] 2 .Di (ω) + [G(0) − ga gs μi ωi μi ωi2 i=1 i=1
(4.62)
The term between brackets, independent of ω, which corresponds to the high frequency modes is often called the residual mode. This equation can be written alternatively m Δθi2 G(ω) = G(0) + ga gs .[Di (ω) − 1] μi ωi2 i=1
or G(ω) = G(0) + ga gs
m Δθi2 (ω 2 − 2jξi ωi ω) μi ωi2 (ωi2 − ω 2 + 2jξi ωi ω) i=1
(4.63)
The feedthrough component observed in Fig.4.8 is clearly apparent in (4.62). Note that the above equations require the static gain G(0), but do not require the knowledge of the high frequency modes. It is important to emphasize the fact that the quasi-static correction has a significant impact on the open-loop zeros of G(ω), and consequently on the performance of the control system. Referring to Fig.2.5, it is clear that neglecting the residual mode (quasi-static correction) amounts to shifting the diagram G(ω) along the vertical axis; this operation alters the location of the zeros which are at the crossing of G(ω) with the horizontal axis. Including the quasi-static correction tends to bring the zeros closer to the poles which, in general, tends to reduce the performance of the control system. Thus, it is a fairly general statement to say that neglecting the residual mode (high frequency dynamics) tends to overestimate the performance of the control system. Finally, note that since the piezoelectric loads are self-equilibrated, they would not affect the rigid body modes if there were any.
4.7
Admittance of a Beam with a Piezoelectric Patch
Let us consider a beam provided with a single piezoelectric patch and establish the analytical expression of the admittance FRF, or equivalently of the dynamic capacitance. Assuming a rectangular patch of length l (from x1 to x2 ), width bp , thickness t, and distant zm from the mid-plane of the beam, applying a voltage V generates a pair of self-equilibrated moments M = −e31 bp zm V . As in the previous section, the response of mode i is governed by (assuming no damping)
4.7 Admittance of a Beam with a Piezoelectric Patch
μi z¨i + μi ωi2 zi = M Δθi or zi = −e31 bp zm V
Δθi μi (s2 + ωi2 )
81
(4.64) (4.65)
where Δθi = φi (x2 ) − φi (x1 ) is the difference of slope of mode i at the ends of the patch. The beam deflection is w=
n
zi φi (x) = −e31 bp zm V
i=1
n Δθi φi (x) 2 2 μ i (s + ωi ) i=1
(4.66)
In the previous section, the charge amplifier cancelled the electric field across the sensor. Here, we must use the second constitutive equation of a unidirectional piezoelectric material D = εT (1 − k 2 )E + e31 S
(4.67)
with the electric field E = V /t; it is assumed that t 600 , and a infinite gain margin. The first result comes from the observation that any open-loop transfer function with a phase margin less than 600 would have to cross the unit circle within the forbidden disk. The second one is the consequence of the fact that the number of encirclements of −1 cannot be changed if one increases the gain by any factor larger than 1. Note that, for curve (2) in Fig.11.4.b, point A could cross −1 if the gain were reduced. However, since A is outside the forbidden circle, this cannot occur if the multiplying factor remains larger than 1/2. This result can be extended to multivariable systems: For each control channel, the LQR has guaranteed margins of GM = 1/2 to
∞
and
P M > 600
11.9 Full State Observer
g e − jφ
-
Forbidden circle
G ( jω I − A )−1 B
Unit circle
(2)
A
-1
60°
257
-1
A (a)
(b)
(1)
Fig. 11.4 Possible Nyquist plots of G0 (jω). (a) Stable open-loop transfer function. (b) Open-loop transfer function with two unstable poles.
For the situation depicted in Fig.11.4.a, the phase is close to 900 above crossover; as a result, the gain roll-off rate at high frequency is at most −20 dB/decade.
11.9
Full State Observer
The state feedback assumes that the states are known at all times. In most practical situations, a direct measurement of all the states would not be feasible. As we already saw earlier, if the system is observable, the states can be reconstructed from a model of the system and the output measurement y. One should never forget, however, that good state reconstruction requires a good model of the system. If the state feedback is based on the reconstructed states, the separation principle tells us that the design of the regulator and that of the observer can be done independently. Consider the system x˙ = Ax + Bu + w (11.47) y = Cx + v
(11.48)
where v and w are uncorrelated white noise processes with zero mean and covariance intensity matrices V and W , respectively. From chapter 9, we know that without noise, a full state observer of the form x ˆ˙ = Aˆ x + Bu + K(y − C x ˆ)
(11.49)
258
11 Optimal Control
x ˆ(0) = 0 converges to the actual states, provided that the eigenvalues of (A − KC) are in the left half plane. In fact, the poles of the observer can be assigned arbitrarily if the system is observable. In presence of plant and measurement noise, w and v, the error equation is the following e = x − xˆ e˙ = (A − KC)e + w − Kv
(11.50)
It shows that both the system noise w and the measurement noise v act as excitations on the measurement error. Note that the measurement noise is amplified by the gain matrix of the observer, which suggests that noisy measurements will require moderate gains in the observer.
11.9.1
Covariance of the Reconstruction Error
Comparing Equ.(11.50) with Equ.(11.19) to (11.21), we see that the steady state reconstruction error has zero mean, and a covariance matrix P = E[eeT ] given by the Lyapunov equation: (A − KC)P + P (A − KC)T + W + KV K T = 0
(11.51)
where we have used the fact that v and w are uncorrelated. It can be rewritten AP + P AT + W − (KCP + P C T K T ) + KV K T = 0
(11.52)
This equation expresses the equilibrium between (as they appear in the equation) the dissipation in the system, the covariance of the disturbance acting on the system, the reduction of the error covariance due to the use of the measurement, and the measurement noise. The latter two contributions depend of the gain matrix K of the observer.
11.10
Kalman-Bucy Filter (KBF)
Since the error covariance matrix depends on the gain matrix K of the observer, one may look for the optimal choice of K which minimizes a quadratic objective function J = E[(aT e)2 ] = aT E[eeT ]a = aT P a
(11.53)
where a is a vector of arbitrary coefficients. There is a universal choice of K which makes J minimum for all a: K = P C T V −1
(11.54)
11.11 Linear Quadratic Gaussian (LQG)
259
where P is the covariance matrix of the optimal observer, solution of the Riccati equation AP + P AT + W − P C T V −1 CP = 0
(11.55)
This solution has been obtained as a parametric optimization problem for the assumed structure of the observer given by Equ.(11.49), but it is in fact optimal. The minimum variance observer is also called steady state KalmanBucy filter (KBF).
11.11
Linear Quadratic Gaussian (LQG)
The so-called Linear Quadratic Gaussian problem is formulated as follows: Consider the completely controllable and observable linear time-invariant system x˙ = Ax + Bu + w (11.56) y = Cx + v
(11.57)
where w and v are uncorrelated white noise processes with intensity matrices W ≥ 0 and V > 0. Find the control u such that the cost functional J = E[xT Qx + uT Ru]
Q ≥ 0, R > 0
(11.58)
is minimized. The solution of this problem is a linear, constant gain feedback u = −Gˆ x
(11.59)
where G is the solution of the LQR problem and x ˆ is the reconstructed state obtained from the Kalman-Bucy filter. Combining Equ.(11.56), (11.59) and (11.50), we obtain the closed-loop equation w I 0 x A − BG BG x˙ (11.60) + = v I −K e 0 A − KC e˙ Its block triangular form implies the separation principle: The eigenvalues of the closed-loop system consist of two decoupled sets, corresponding to the regulator and the observer. Note that the separation principle is related to the state feedback and the structure of the observer, rather than to the optimality; it applies to any state feedback and any full state observer of the form (11.49). The compensator equation has exactly the same form as that of Fig.9.9, except that it is no longer restricted to SISO systems. As we already mentioned, the stability of the compensator is not guaranteed, because only the closed-loop poles have been considered in the design.
260
11.12
11 Optimal Control
Duality
Although no obvious relationship exists between the physical problems of optimal state feedback with a quadratic performance index, and the minimum variance state observer, the algebras of the solution of the two problems are closely related, as summarized in Table 11.1. Table 11.1 Duality between LQR and KBF. LQR: Gain: G = R−1 B T P Riccati: P A + AT P − P BR−1 B T P + Q = 0 Closed loop: x˙ = (A − BR−1 B T P )x KBF: Gain: K = P C T V −1 Riccati: AP + P AT − P C T V −1 CP + W = 0 Error equation: e˙ = (A − P C T V −1 C)e
The duality between the design of the regulator and that of the Kalman filter can be expressed as follows. Consider the fictitious dual control problem: Find u that minimizes the performance index J = E[z T W z + uT V u] for the system z˙ = AT z + C T u The solution is u = −Gz with
G = V −1 CP
where P is solution of P AT + AP + W − P C T V −1 CP = 0 It is readily observed that this Riccati equation is that of the Kalman filter of the original problem, and that the gain matrix of the minimum variance observer for the original problem is related to the solution of the fictitious regulator problem by K = GT
11.13 Spillover
11.13
261
Spillover
Flexible structures are distributed parameter systems which, in principle, have an infinite number of degrees of freedom. In practice, they are discretized by a finite number of coordinates (e.g. finite elements) and this is in general quite sufficient to account for the low frequency dynamical behavior in most practical situations. When it comes to control flexible structures with state feedback and full state observer, the designer cannot deal directly with the finite element model, which is by far too big. Instead, a reduced model must be developed, which includes the few dominant low frequency modes. Due to the inherent low damping of flexible structures, particularly in the space environment, there is a danger that a state feedback based on a reduced model destabilizes the residual modes, which are not included in the model of the structure contained in the observer. The aim of this section is to point out the danger of spillover instability. It is assumed that the state variables are the modal amplitudes and the modal velocities (as in section 9.2.2). In what follows, the subscript c refers to the controlled modes, which are included in the control model, and the subscript r refers to the residual modes which are ignored in the control design. Although they are not included in the state feedback, the residual modes are excited by the control input and they also contribute to the output measurement (Fig.11.5); it is this closed-loop interaction, together with the low damping of the residual modes, which is the origin of the problem. With the foregoing notation, the dynamics of the open-loop system is x˙ c = Ac xc + Bc u + w (11.61) x˙ r = Ar xr + Br u
(11.62)
y = Cc xc + Cr xr + v
(11.63)
A perfect knowledge of the controlled modes is assumed. The full state observer is xˆ˙ c = Ac xˆc + Bc u + Kc (y − Cc x ˆc ) (11.64) and the state feedback u = −Gc x ˆc
(11.65)
The interaction between the control system and the residual modes can be analyzed by considering the composite system formed by the state variables (xTc , eTc , xTr )T , where ec = xc − x ˆc . The governing equation is ⎞⎛ ⎞ ⎛ ⎞ ⎛ Bc Gc 0 Ac − Bc Gc xc x˙ c ⎝ e˙ c ⎠ = ⎝ 0 Ac − Kc Cc −Kc Cr ⎠ ⎝ ec ⎠ (11.66) x˙ r −Br Gc Br Gc Ar xr This equation is the starting point for the analysis of the spillover (Fig.11.5). The key terms are Kc Cr and Br Gc . They arise from the sensor output being contaminated by the residual modes via the term Cr xr (observation spillover),
262
11 Optimal Control
Flexible System Dynamics Actuators
Controlled Modes
Ac
Bc
xc
Sensors
Cc
u
+
Residual Modes
Ar
Br
Regulator u
u = − Gc x$ c
x$c
xr
y
Cr
Linear observer x&$ c = Ac x$ c + B c u + K c ( y − C c x$ c )
y
Fig. 11.5 Spillover mechanism.
and the feedback control exciting the residual modes via the term Br u (control spillover). Equ.(11.66) shows that if either Cr = 0 or Br = 0, the eigenvalues of the system remain decoupled, that is identical to those of the regulator (Ac − Bc Gc ), the observer (Ac − Kc Cc ) and the residual modes Ar . They are typically located in the complex plane as indicated in Fig.11.6. The poles of the regulator (controlled modes) have a substantial stability margin, and the poles of the observer are located even farther left. On the contrary, the poles corresponding to the residual modes are barely stable, their only stability margin being provided by the natural damping. When both Cr = 0 and Br = 0, i.e. when there is both control and observation spillover, the eigenvalues of the system shift away from their decoupled locations. The magnitude of the shift depends on the coupling terms Br Gc and Kc Cr . Since the stability margin of the residual modes is small, even a small shift can make them unstable. This is spillover instability. Not all the residual modes are potentially critical from the point of view of spillover, but only those which are observable, controllable, and are close to the bandwidth of the controller Problem 11.6).
11.13 Spillover
263
Im(s) Stability margin Residual modes
Observer poles
Controlled modes
Re(s)
Fig. 11.6 Typical location of the closed-loop poles in the complex plane, showing the small stability margin of the residual modes (only the upper half is shown).
11.13.1
Spillover Reduction
In the previous section, we have seen that the spillover phenomenon arises from the excitation of the residual modes by the control (control spillover, Br u) and the contamination of the sensor output by the residual modes (observation spillover, Cr xr ). For MIMO systems, both terms can be reduced by a judicious design of the regulator and the Kalman filter. Control spillover can be alleviated by minimizing the amount of energy fed into the residual modes. This can be achieved by supplementing the cost functional used in the regulator design by a quadratic term in the control spillover: ∞ ∞ T T (xc Qxc + u Ru)dt + uT BrT W Br u dt (11.67) J= 0
0
where the weighting matrix W allows us to penalize some specific modes. This amounts to using the modified control weighting matrix R + BrT W Br
(11.68)
This control weighting matrix penalizes the excitation u whose shape is favorable to the residual modes; this tends to produce a control which is orthogonal to the residual modes. Of course, this is achieved more effectively when there are many actuators, and it cannot be achieved at all with a single actuator. Similarly, a reduction of the observation spillover can be achieved by designing the observer as a Kalman filter with a measurement noise intensity matrix V + Cr V1 CrT (11.69)
264
11 Optimal Control
The extra contribution to the covariance intensity matrix indicates to the filter that the measurement noise has the spatial shape of the residual modes (Cr V1 CrT is the covariance matrix of the observation spillover Cr xr if E[xr xTr ] = V1 ). This tends to desensitize the reconstructed states to the residual modes. Here again, the procedure works better if many sensors are available. The foregoing methodology for spillover reduction was introduced by Sezak et al. under the name of Model Error Sensitivity Suppression (MESS). It is only one of the many methods for spillover reduction, but it is interesting because it stresses the role of the matrices R and V in the LQG design. Another interesting situation where the plant noise statistics have a direct impact on the stability margins is discussed in the next section.
11.14
Loop Transfer Recovery (LTR)
In section 11.8, we have seen that the LQR has guaranteed stability margins of GM = 1/2 to ∞ and P M > 600 for each control channel. This property is lost when the state feedback is based on an observer or a Kalman filter. In that case, the margins can become substantially smaller. The Loop Transfer Recovery (LTR) is a robustness improvement procedure consisting of using a Kalman filter with fictitious noise parameters: If W0 is the nominal plant noise intensity matrix, the KBF is designed with the following plant noise intensity matrix W (q) = W0 + q2 BW1 B T
(11.70)
where W1 is an arbitrary symmetric semi positive definite matrix and q is a scalar adjustment parameter. From the presence of the input matrix B in the second term of (11.70), we see that the extra plant noise is assumed to enter the system at the input. Of course, for q = 0, the resulting KBF is the nominal one. As q → ∞, it can be proved (Doyle & Stein) that, for square, minimum phase open-loop systems G(s), the loop transfer function H(s)G(s) from the control signal u to the compensator output u (loop broken at the input of the plant, as indicated in Fig.11.7) tends to that of the LQR: lim G(sI − A + BG + KC)−1 K C(sI − A)−1 B = G(sI − A)−1 B (11.71)
q→∞
As a result, the LQG/LTR recovers asymptotically the margins of the LQR as q → ∞. Note that • The loop breaking point at the input of the plant, as indicated by × in Fig.11.7, is a reasonable one, because this is typically one of the locations where the uncertainty enters the system.
11.15 Integral Control with State Feedback
(a)
265
LQG
H (s) G ( sI − A + BG + KC )−1 K
-
(b)
G(s) u
u'
C ( sI − A )−1 B
y
LQR -
G ( sI − A ) −1 B
Fig. 11.7 Loop transfer functions of the LQG and the LQR with loop breaking at the input of the plant.
• The KBF gain matrix, K(q) is a function of the scalar parameter q. For q = 0, K(0) is the optimal filter for the true noise parameters. As q increases, the filter does a poorer job of noise rejection, but the stability margins are improved, with essentially no change in the bandwidth of the closed-loop system. Thus, the designer can select q by trading off noise rejection and stability margins. • The margins of the LQG/LTR are indeed substantial; they provide a good protection against delays and nonlinearities in the actuators. They are not sufficient to guarantee against spillover instability, however, because the phase uncertainty associated with a residual mode often exceeds 600 (it may reach 1800 if the residual mode belongs to the bandwidth). • The LTR procedure is normally applied numerically by solving a set of Riccati equations for increasing values of q 2 , until the right compromise is achieved. For SISO systems, it can also be applied graphically on a symmetric root locus, by assuming that the noise enters the plant at the input [E = E0 + qB in Equ.(9.72)] (Problem 11.7).
11.15
Integral Control with State Feedback
Consider a linear time invariant system subject to a constant disturbance w: x˙ = Ax + Bu + w
(11.72)
y = Cx
(11.73)
If we use a state feedback u = −Gx to stabilize the system, there will always be a non-zero steady state error in the output y. Increasing the gain G would
266
11 Optimal Control
reduce the error at the expense of a wider bandwidth and a larger noise sensitivity. An alternative approach consists of introducing an integral action by supplementing Equ.(11.72) by p˙ = y (11.74) leading to the augmented state vector (xT , pT )T . With the state feedback u = −Gx − Gp p the closed-loop equation reads A − BG −BGp x˙ w x = + p˙ 0 p C 0
(11.75)
(11.76)
If G and Gp are chosen in such a way that they stabilize the system, we have lim p˙ = 0
t→∞
(11.77)
which means that the steady state error will be zero (y∞ = 0), without knowledge of the disturbance w.
11.16
Frequency Shaping
As we saw in earlier chapters, the desirable features of control systems include some integral action at low frequency to compensate for steady state errors and very low frequency disturbances, and enough roll-off at high frequency for noise rejection, and to stabilize the residual dynamics. Moreover, there are special situations where the system is subjected to a narrow-band disturbance at a known frequency. The standard LQG does not give the proper answer to these problems (no integral action, and the roll-off rate of the LQR is only −20 dB/decade). We have seen in the previous section how the state space model can be modified to include some integral action; in this section, we address the more general question of frequency shaping. The weakness of the standard LQG formulation lies in the use of a frequency independent cost functional, and of noise statistics with uniform spectral distribution (white noise). Frequency shaping can be achieved either by considering a frequency dependent cost functional in the LQR formulation, or by using colored (i.e. non-white) noise statistics in the LQG problem.
11.16.1
Frequency-Shaped Cost Functionals
According to Parseval’s theorem, the cost functional (11.10) of the LQR can be written in the frequency domain as
11.16 Frequency Shaping
J=
1 2π
∞
[x∗ (ω)H T Hx(ω) + u∗ (ω)Ru(ω)]dω
267
(11.78)
−∞
where x(ω) and u(ω) are the Fourier transforms of x and u, and ∗ indicate the complex conjugate transposed (Hermitian). Equ.(11.78) shows clearly that the weighing matrices Q = H T H and R = S T S do not depend on ω, meaning that all the frequency components are treated equally. Next, assume that we select frequency dependent weighing matrices Q(ω) = H ∗ (ω)H(ω)
and
R(ω) = S ∗ (ω)S(ω)
(11.79)
Clearly, if the shaping objectives are to produce a P+I type of controller and to increase the roll-off, we must select Q(ω) to put more weight on low frequency, to achieve some integral action, and R(ω) to put more weight on high frequency, to attenuate the high frequency contribution of the control. Examples of such functions in the scalar case are Q(ω) =
ω02 + ω 2 ω2
R(ω) =
ω12n + ω 2n ω12n
(11.80)
where the corner frequencies ω0 and ω1 and the exponent n are selected in the appropriate manner. Typical penalty functions are represented in Fig.11.8. Likewise, a narrow-band disturbance can be handled by including a lightly damped oscillator at the appropriate frequency in Q(ω). Equation (11.78) can be rewritten ∞ 1 J= [x∗ (ω)H ∗ (ω)H(ω)x(ω) + u∗ (ω)S ∗ (ω)S(ω)u(ω)]dω (11.81) 2π −∞
Q( jω )
R( jω )
dB
ω (rad / s) Fig. 11.8 Frequency dependent weighting matrices.
268
11 Optimal Control
We assume that all the input and output channels are filtered in the same way, so that the weighing matrices are restricted to the form H(ω) = h(ω)H and S(ω) = s(ω)S, with h(ω) and s(ω) being scalar functions. If we introduce the modified controlled variable z1 = H(ω)x = h(ω)Hx = h(ω)z
(11.82)
u1 = s(ω)u
(11.83)
and control we get the frequency independent cost functional ∞ 1 J= [z ∗ z1 + u∗1 Ru1 ]dω 2π −∞ 1
(11.84)
or, in the time domain,
∞
J= 0
(z1T z1 + uT1 Ru1 )dt
(11.85)
This cost functional refers to the augmented system of Fig.11.9, including input filters s−1 (ω) on all input channels and output filters h(ω) on all controlled variables. If a state space realization of these filters is available (Problem 11.9), the complete system is governed by the following equations: • Structure x˙ = Ax + Bu
(11.86)
y = Cx + Du
(11.87)
• Output filter [state space realization of h(ω)] x˙ 0 = A0 x0 + B0 z
(11.88)
z1 = C0 x0 + D0 z
(11.89)
s −1 (ω )
u1
Ai Bi Ci Input filter
h(ω )
u
A
B
C
D
Structure
z
AO BO CO
z1
DO
Output filter
Fig. 11.9 State space realization of the augmented system including frequency shaping.
11.16 Frequency Shaping
269
• Input filter [state space realization of s−1 (ω)] x˙ i = Ai xi + Bi u1
(11.90)
u = Ci xi
(11.91)
These equations can be combined together as x˙ = A x + B u1
(11.92)
z1 = C x
(11.93)
with the augmented state vector x = (xT , xTi , xT0 )T and the notations
⎛
⎞ 0 0 ⎠ A0
A A = ⎝ 0 B0 C
BCi Ai B0 DCi ⎛ ⎞ 0 B = ⎝ Bi ⎠ 0
(11.94)
(11.95)
C = (D0 C , D0 DCi , C0 )
(11.96)
The state feedback −Gc x is obtained by solving the LQR problem for the augmented system with the quadratic performance index (11.85). Notice that, since the input and output filter equations are solved in the computer, the
u
y=z
Structure
Observer
x$
xi
Ai Bi Ci Input filter
u1
x0 AO BO
-Gc LQR
CO x*
DO
Output filter
Fig. 11.10 Architecture of the frequency-shaped LQG controller (y = z).
270
11 Optimal Control
states xi and x0 are known; only the states x of the structure must be reconstructed with an observer. The overall architecture of the controller in shown in Fig.11.10. It can be shown that the poles of the output filter (eigenvalues of A0 ) appear unchanged in the compensator (Problem 11.11); this property can be used to introduce a large gain over a narrow frequency range, by introducing a lightly damped pole in A0 (Problem 11.10).
11.16.2
Noise Model
As an alternative to the frequency-shaped cost functionals, loop shaping can be achieved by assuming that the plant noise w has an appropriate power spectral density, instead of being a white noise. Thus, we assume that w is the output of a filter excited by a white noise at the input. If the system is governed by x˙ = Ax + Bu + Ew (11.97) y = Cx + Du + v
(11.98)
and the plant noise is modelled according to z˙ = Aw z + Bw w∗
(11.99)
w = Cw z
(11.100)
where Aw is stable and w∗ is a white noise (Problem 11.12), the two sets of equations can be coupled together to form the augmented system 0 A ECw B x˙ x (11.101) u+ = + w∗ Bw 0 Aw 0 z˙ z !
y= C
x 0 + Du + v z "
(11.102)
or, with x∗ = (xT , z T )T and the appropriate definitions of A∗ , B ∗ , C ∗ and E∗, x˙ ∗ = A∗ x∗ + B ∗ u + E ∗ w∗ (11.103) y = C ∗ x∗ + Du + v ∗
(11.104)
Since w and v are white noise processes, the augmented system fits into the LQG framework and a full state feedback and a full state observer can be constructed by solving the two problems LQR{A∗ , B ∗ , Q∗ , R∗ } KBF{A∗ , C ∗ , W = E ∗ E ∗T , V } with the appropriate matrices Q∗ , R∗ and V .
11.17 Problems
271
In Equ.(11.101), note that the filter dynamics is not controllable from the plant input, but this is not a problem provided that Aw is stable, that is if the augmented system is stabilizable (see next chapter). In principle, a large gain over some frequency range can be obtained by proper selection of the poles of Aw and the input and output matrices Bw and Cw . However, in contrast to the previous section, the poles of Aw do not appear unchanged in the compensator (Problem 11.13) and this technique may lead to difficulties for the rejection of narrow-band perturbations (Problem 11.14).
11.17
Problems
P.11.1 Consider the linear system (11.19) subjected to a white noise excitation with covariance intensity matrix W1 . Show that the quadratic performance index J = E[xT Qx] can be written alternatively J = tr[P DW1 DT ] where P is the solution of the Lyapunov equation (11.3). P.11.2 Consider the inverted pendulum of section 9.2.3. Using the absolute displacement as control variable, design a LQR by solving the Riccati equation, for various values of the control weight . Compare the result to that obtained in section 9.5.2 with the symmetric root locus. P.11.3 Same as Problem 11.2 but with the α − shif t procedure of section 11.7. Check that for all values of , the closed-loop poles lie to the left of the vertical line at −α (select −α to the left of −ω0 ). Compare the state feedback gains to those of the previous problem. P.11.4 For one of the LQR designed at Problem 11.2, draw the Nyquist plot of G0 (ω) = G(jωI − A)−1 B. Evaluate the gain and phase margins. P.11.5 Consider the state space equation (9.14) of a flexible structure in modal coordinates and assume that the mode shapes have been normalized in such a way that μi = 1. Show that the total energy (kinetic + strain) can be written in the form T + U = z T Qz
with
Q=
1 I 2
where z is the state vector defined as z = (η T Ω, η˙ T )T . P.11.6 Consider a simply supported uniform beam with a point force actuator at x = l/6 and a displacement sensor at 5l/6. Assume that the system is undamped and that EI = 1N m2 , m = 1kg/m, and l = 1m. (a) Write the equations in state variable form using the state variable z defined as z = (η T Ω, η˙ T )T .
272
11 Optimal Control
(b) Design a LQR for a model truncated after the first three modes, using Q = I (see Problem 11.5); select the control weight in such a way that the closed-loop poles are (−0.788±j9.87), (−1.37±j39.48), and (−1.58±j88.83). (c) Check that a full state Luenberger observer with poles located at −175.39, −20.92, −24.40 ± j50.87, −7.3 ± j9.34 shifts the residual mode from p4 = (0 ± j157.9) to p∗4 = (+0.177 ± j157.5) (this example was used by Balas to demonstrate the spillover phenomenon). (d) Using a model with 3 modes and assuming that the plant noise intensity matrix has the form W = wI, design a KBF and plot the evolution of the residual modes 4 and 5 (in closed-loop) as the noise intensity ratio q = w/v increases (and the observer becomes faster). (e) For the compensator designed in (d), assuming that all the modes have a structural damping of ξi = 0.001, plot the evolution with the parameter q of the open-loop transfer function G5 H3 corresponding to 5 structural modes (including 2 residual modes). [Hint: Use the result of Problem 2.5 to compute G5 (ω).] P.11.7 Reconsider the inverted pendulum of Problem 11.4. Assume that the output is the absolute position of the pendulum. Design a Kalman filter assuming that the plant noise enters the system at the input (E = B). Apply the LTR procedure and check that, as q 2 increases, the open-loop transfer function GH(ω) tends to that of the LQR (Problem 11.4). Check the effect of the procedure on the bandwidth of the control system. [Note: The assumption that the output of the system is the absolute position x rather than the tilt angle θ may appear as a practical restriction, but it is not, because x can always be obtained indirectly from θ and u by Equ.(9.21). It is necessary to remove the feedthrough component from the output before applying the LTR procedure.] [Hint: The KBF/LTR is the limit as q → ∞ of the symmetric root locus (9.71) based on E = B.] P.11.8 Consider the two-mass problem of section 9.10.1. (a) Design a LQR by solving the Riccati equation for various values of the control weight . Show that for some , we obtain the same gains as those obtained with the symmetric root locus in section 9.10.1. (b) For these gains, draw the Nyquist plot of the LQR, G0 (ω) = G(sI − A)−1 B evaluate the gain and phase margins. (c) Assuming that the plant noise enters at the input, design a KBF by solving the Riccati equation for various values of the noise intensity ratio q = w/v.
11.17 Problems
273
Show that for some q, we obtain the same gains as those obtained with the symmetric root locus. Calculate the gain and phase margins. (d) Apply the LTR technique with increasing q; draw a set of Nyquist plots of GH(ω) showing the evolution of the gain and phase margins. Check that GH(ω) → G0 (ω) as q → ∞. P.11.9 Find a state space realization of the input and output filters h(ω) and s−1 (ω) corresponding to the weighting matrices (11.80): |h(ω)|2 = |s−1 (ω)|2 =
ω02 + ω 2 ω2
ω12n + ω 2n
ω12n
(n = 2)
The latter is known as Butterworth filter of order n; its poles are located on a circle of radius ω1 according to Fig.11.1. P.11.10 Consider the two-mass problem of section 9.10.1. Assume that the system is subjected to a sinuso¨ıdal disturbance at ν0 = 0.5 rad/s acting on the main body. Using a frequency-shaped cost functional, design a LQG controller with good disturbance rejection capability. Compare the performance of the new design to the nominal one (time response, sensitivity function,...). [Hint: use a lightly damped oscillator as output filter h(ω) =
ν02
−
ω2
ν02 + 2jξων0
where ξ is kept as design parameter.] P.11.11 Show that the compensator obtained by the frequency-shaped cost functional has the following state space realization: ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ˆ˙ A − Kf C (B − Kf D)Ci 0 x ˆ Kf x ⎝ x˙ i ⎠ = ⎝ −Bi Gcx Ai − Bi Gci −Bi Gco ⎠ ⎝ xi ⎠ + ⎝ 0 ⎠ y 0 0 Ao xo Bo x˙ 0 u = Ci xi where Kf is the gain of the observer for x ˆ and Gc = (Gcx , Gci , Gco ) is the gain of the state feedback. Note that, as a result of the structure of the system matrix, the poles of the compensator include those of the output filter, Ao . P.11.12 Find a state space realization of the noise model (11.99) (11.100) achieving the following power spectral density: Φw (ω) =
ω02 + ω 2 ω12n . ω2 ω12n + ω 2n
(n = 2)
274
11 Optimal Control
(this filter combines in cascade the two filters used in Problem 11.9). P.11.13 Show that the compensator obtained by using a noise model in the loop shaping has the following state space realization: A − BGcx − Kf x C ECw − BGcw x ˆ xˆ˙ = + Kf y z −Kf w C Aw z˙ ˆ − Gcw z u = −Gcx x where Gc = (Gcx , Gcw ) is the gain matrix of the regulator of the augmented system and KfT = (KfTx , KfTw ) is the corresponding observer gain matrix. Note that the system matrix is no longer block triangular, so that the poles of the compensator differ from those of Aw . P.11.14 Repeat Problem 11.10 using a noise model (w is the output of a second order filter). Compare the results.
12
Controllability and Observability
12.1
Introduction
Controllability measures the ability of a particular actuator configuration to control all the states of the system; conversely, observability measures the ability of the particular sensor configuration to supply all the information necessary to estimate all the states of the system. Classically, control theory offers controllability and observability tests which are based on the rank deficiency of the controllability and observability matrices: The system is controllable if the controllability matrix is full rank, and observable if the observability matrix is full rank. This answer is often not enough for practical engineering problems where we need more quantitative information. Consider for example a simply supported uniform beam; the mode shapes are given by (2.55). If the structure is subject to a point force acting at the center of the beam, it is obvious that the modes of even orders are not controllable because they have a nodal point at the center. Similarly, a displacement sensor will be insensitive to the modes having a nodal point where it is located. According to the rank tests, as soon as the actuator or the sensor are slightly moved away from the nodal point, the rank deficiency disappears, indicating that the corresponding mode becomes controllable or observable. This is too good to be true, and any attempt to control a mode with an actuator located close to a nodal point would inevitably lead to difficulties, because this mode is only weakly controllable or observable. In this chapter, after having discussed the basic concepts, we shall turn our attention to the quantitative measures of controllability and observability, and apply the concept to model reduction.
12.1.1
Definitions
Consider the linear time-invariant system x˙ = Ax + Bu
A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 275–297. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
(12.1)
276
12 Controllability and Observability
y = Cx
(12.2)
• The system is completely controllable if the state of the system can be transferred from zero to any final state x∗ within a finite time. • The system is stabilizable if all the unstable eigenvalues are controllable or, in other words, if the non controllable subspace is stable. • The system is completely observable if the state x can be determined from the knowledge of u and y over a finite time segment. In the specialized literature, observability refers to the determination of the current state from future output, while the determination of the state from past output is called reconstructibility. For linear, time-invariant systems, these concepts are equivalent and do not have to be distinguished. • The system is detectable if all the unstable eigenvalues are observable, or equivalently, if the unobservable subspace is stable.
12.2
Controllability and Observability Matrices
The simplest way to introduce the controllability matrix is to consider the single input n-dimensional discrete-time system governed by the difference equation xk+1 = Axk + buk (12.3) where A is the n × n system matrix and b the n-dimensional input vector. Assuming that the system starts from rest, x0 = 0, the successive values of the state vector resulting from the scalar input uk are x1 = bu0 x2 = Ax1 + bu1 = Abu0 + bu1 ... n−1
xn = A or
bu0 + An−2 bu1 + ... + bun−1 ⎛
⎞ un−1 ⎜ ... ⎟ ⎟ xn = (b, Ab, A2 b, ..., An−1 b) ⎜ ⎝ u1 ⎠ u0
(12.4)
where n is equal to the order of the system. The n × n matrix C = (b, Ab, A2 b, ..., An−1 b)
(12.5)
is called the controllability matrix; its columns span the state space which can be reached after exactly n samples. If C is full rank, the state vector can be transferred to any final value x∗ after only n samples. By solving Equ.(12.4), one finds
12.2 Controllability and Observability Matrices
⎞ un−1 ⎜ ... ⎟ −1 ∗ ⎜ ⎟ ⎝ u1 ⎠ = C x u0
277
⎛
(12.6)
Next, consider the values of xN for N > n. Once again, ⎛ ⎞ uN −1 ⎜ ... ⎟ ⎟ xN = (b, Ab, A2 b, ..., AN −1 b) ⎜ ⎝ u1 ⎠ u0 It turns out that the rank of the rectangular matrix (b, Ab, A2 b, ..., AN −1 b) is the same as that of C, and that the columns of the two matrices span the same space. This is a consequence of the Cayley-Hamilton theorem, which states that every matrix A satisfies its own characteristic equation. Thus, if the characteristic equation of A is α(s) = det(sI − A) = sn + a1 sn−1 + ... + an−1 s + an = 0
(12.7)
A satisfies the matrix equation An + a1 An−1 + ... + an−1 A + an = 0
(12.8)
It follows that for any m > n, Am b is linearly dependent on the columns of the controllability matrix C; as a result, increasing the number of columns Am b does not enlarge the space which is spanned (Problem 12.1). In conclusion, the system (12.3) if completely controllable if and only if (iff) the rank of the controllability matrix C is n. This result has been established for a single-input discrete-time linear system, but it also applies to multi-input discrete as well as continuous time linear systems. The linear time-invariant system (12.1) with r inputs is completely controllable iff the n × (n × r) controllability matrix C = (B, AB, A2 B, ..., An−1 B)
(12.9)
rank(C) = n
(12.10)
is such that We then say that the pair (A, B) is controllable. If C is not full rank, the subspace spanned by its columns defines the controllable subspace of the system. In a similar manner, the system (12.1) (12.2) is observable iff the observability matrix
278
12 Controllability and Observability
⎞ C ⎜ CA ⎟ ⎟ O=⎜ ⎝ ... ⎠ CAn−1 ⎛
(12.11)
is such that rank(O) = n
(12.12)
In this case, we say that the pair (A, C) is observable. From the fact that OT = (C T , AT C T , ..., (AT )n−1 C T ) we conclude that the pair (A, C) is observable iff the dual system (AT , C T ) is controllable. Conversely, the pair (A, B) is controllable iff the dual system (AT , B T ) is observable. The duality between observability and controllability has already been stressed in section 11.12.
12.3
Examples
12.3.1
Cart with Two Inverted Pendulums
Consider two inverted pendulums with the same mass m and lengths l1 and l2 placed on a cart of mass M (Fig.12.1.b). Assume that the input variable u is the force applied to the cart (in contrast to section 9.2.3, where the input was the displacement of the support). Using the state variables x = (θ1 , θ2 , θ˙1 , θ˙2 )T , we can write the linearized equations near θ1 = θ2 = 0 as ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ x˙ 1 0 0 1 0 0 x1 ⎜ x˙ 2 ⎟ ⎜ 0 0 0 1 ⎟ ⎜ x2 ⎟ ⎜ 0 ⎟ ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎜ ⎟ (12.13) ⎝ x˙ 3 ⎠ ⎝ a1 a2 0 0 ⎠ ⎝ x3 ⎠ + ⎝ b1 ⎠ u a3 a4 0 0 x˙ 4 x4 b2 where a1 = (g/l1 )(1 + m/M ), a2 = (g/l1 )(m/M ), a3 = (g/l2 )(m/M ), a4 = (g/l2 )(1 + m/M ), b1 = −1/M l1 and b2 = −1/M l2 (Problem 12.3). The controllability matrix is ⎛ ⎞ 0 b1 0 a1 b1 + a2 b2 ⎜ 0 b2 0 a3 b1 + a4 b2 ⎟ ⎟ C=⎜ (12.14) ⎝ b 1 0 a1 b 1 + a 2 b 2 ⎠ 0 b 2 0 a3 b 1 + a 4 b 2 0 It can be checked easily that this matrix is full rank provided that l1 = l2 . If l1 = l2 , the rank of C is reduced to 2. Thus, when the time constants of the two pendulums are the same, the system is not controllable (in practical applications, it is likely that the difficulties in controlling the system will appear long before reaching l1 = l2 ).
12.3 Examples
y
m
m
(b)
(a)
m
θ 1 l1
θ
θ2
l x
M
279
u (force)
x
θ2
l2
M u
m l
(c)
θ1
m l
x
M
u
Fig. 12.1 Various configurations of inverted pendulum.
Next, consider the observability of the system from the measurement of θ1 . We have C = (1, 0, 0, 0) and the observability matrix is ⎛ ⎞ 1 0 0 0 ⎜ 0 0 1 0 ⎟ ⎟ O=⎜ (12.15) ⎝ a 1 a2 0 0 ⎠ 0 0 a1 a2 Since det(O) = −a22 = 0, we conclude that the system is always observable from a single angle measurement; this result is somewhat surprising, but true.
12.3.2
Double Inverted Pendulum
Next, consider a double inverted pendulum on a cart as in Fig.12.1.c. To simplify the equations without losing any generality in the discussion, we assume that the two arms have the same length, and that the two masses are the same. The equations of motion can be written more conveniently by using the absolute tilt angles of the two arms (Problem 12.4). Using the state
280
12 Controllability and Observability
vector x = (θ1 , θ2 , θ˙1 , θ˙2 )T , we can write the linearized equations about the vertical position as ⎛ ⎞ ⎛ ⎞ 0 0 1 0 0 ⎜ ⎜ 0 ⎟ 0 0 0 1 ⎟ ⎟ ⎜ ⎟ x˙ = ⎜ (12.16) ⎝ 2ω02 (1 + a) −ω02 0 0 ⎠ x + ⎝ −1/M l ⎠ u 2 2 −2ω0 2ω0 0 0 0 where ω02 = g/l and a = m/M . The controllability matrix reads ⎛ ⎞ 0 −1 0 −2ω02 (1 + a) ⎟ 1 ⎜ 0 −2ω02 ⎜ 0 0 ⎟ C= 2 ⎝ ⎠ −1 0 −2ω (1 + a) 0 Ml 0 0 0 −2ω02 0
(12.17)
Since det(C) = −4ω04 /M 4 l 4 = 0, the system is always controllable. Similarly, the observability matrix from θ1 reads ⎛ ⎞ 1 0 0 0 ⎜ 0 0 1 0 ⎟ ⎟ O=⎜ (12.18) ⎝ 2ω02 (1 + a) −ω02 0 0 ⎠ 2 2 0 0 2ω0 (1 + a) −ω0 We have det(O) = −ω04 = 0, which indicates that the system is indeed observable from θ1 alone.
12.3.3
Two d.o.f. Oscillator
Consider the mechanical system of Fig.12.2. It consists of two identical undamped single d.o.f. oscillators connected with a spring of stiffness εk. The input of the system is the point force applied to mass 1. The mass and stiffness matrices are respectively m 0 1 + ε −ε M= K=k (12.19) 0 m −ε 1 + ε
u
k
m
εk
m
x1 Fig. 12.2 Two d.o.f. oscillator.
k
x2
12.4 State Transformation
Defining the state vector x = (x1 , x2 , x˙ 1 , x˙ 2 )T the state space equation ⎛ 0 0 1 0 ⎜ 0 0 0 1 x˙ = ⎜ ⎝ −ωn2 (1 + ε) ωn2 ε 0 0 ωn2 ε −ωn2 (1 + ε) 0 0
281
and using Equ.(9.11), we find ⎞
⎛
⎞ 0 ⎟ ⎜ ⎟ ⎟x + ⎜ 0 ⎟u ⎠ ⎝ 1/m ⎠ 0
where ωn2 = k/m. The controllability matrix reads ⎛ ⎞ 0 1 0 −ωn2 (1 + ε) ⎟ 1 ⎜ 0 ωn2 ε ⎜ 0 0 ⎟ C= 2 ⎝ ⎠ 1 0 −ω (1 + ε) 0 m n 0 0 ωn2 ε 0
(12.20)
(12.21)
det(C) = −ωn4 ε2 /m4 indicates that the system is no longer controllable as ε approaches 0. Indeed, when the stiffness of the coupling spring vanishes, the two masses become uncoupled and mass 2 is uncontrollable from the force acting on mass 1.
12.4
State Transformation
Consider a SISO system x˙ = Ax + bu y = cT x Since A is n×n and b and c are both n×1, the system has n2 +2n parameters. If we consider the non singular transformation of the state, x = T xc
(12.22)
x˙ c = T −1 AT xc + T −1 bu
(12.23)
y = cT T xc
(12.24)
x˙ c = Ac xc + bc u
(12.25)
y = cTc xc
(12.26)
the transformed state equation is
or
with the proper definition of Ac , bc and cc . The non singular transformation matrix T contains n2 free parameters which can be chosen to achieve special properties for the transformed system; we shall discuss an example in detail in the next section. It can be shown (Problem 12.5) that the controllability
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12 Controllability and Observability
matrix of the transformed system, Cc , is related to that of the original system by Cc = T −1 C (12.27) For any non singular transformation T , the rank of Cc is the same as that of C. Thus, the property of controllability is preserved by any non singular transformation.
12.4.1
Control Canonical Form
We have seen in the previous section that the transformation matrix T can be selected in such a way that the transformed system has special properties. A form which is especially attractive from the state feedback point of view is the control canonical form, where the transformed system is expressed in terms of the 2n coefficients ai and bj appearing in the system transfer function G(s) =
y(s) b(s) b1 sn−1 + ... + bn = = n u(s) a(s) s + a1 sn−1 + ... + an
The transformed matrices are ⎛ −a1 −a2 ⎜ 1 ... ⎜ ⎜ 0 1 Ac = ⎜ ⎜ ⎝ 0 0 ...
(Problem 12.6) ⎞ ... −an 0 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ .. . 0 ⎠ 1 0
⎛ ⎞ 1 ⎜0⎟ ⎜ ⎟ ⎜ ⎟ bc = ⎜ ... ⎟ ⎜ ⎟ ⎝0⎠ 0
(12.28)
(12.29)
cTc = (b1 , ..., bn ) Besides the fact that the transformation between the state space model in control canonical form and the input-output model is straightforward, it is easy to compute the state feedback gains to achieve a desired closed-loop characteristic equation. Indeed, if the state feedback u = −gcT xc is applied, the closed-loop system matrix becomes ⎛ ⎞ −a1 − gc1 −a2 − gc2 ... −an − gcn ⎜ ⎟ 1 ... 0 0 ⎜ ⎟ ⎜ ⎟ T 0 1 0 Ac − bc gc = ⎜ (12.30) ⎟ ⎜ ⎟ .. ⎝ ⎠ 0 . 0 0 ... 1 0 The corresponding characteristic equation is αc (s) = sn + (a1 + gc1 )sn−1 + ... + (an + gcn ) = 0
(12.31)
12.4 State Transformation
283
Thus, in control canonical form, the state feedback gains can be obtained directly from the coefficients of the closed-loop characteristic equation, making pole placement very simple. The state feedback gains in the original state space system are slightly more difficult to compute, as we now examine. In principle, the linear transformation matrix leading from the original state space representation to the control canonical form can be obtained from Equ.(12.27): T = CCc−1 (12.32) where C and Cc are the controllability matrices of the original system and of the control canonical form (Problem 12.7), respectively. From u = −gcT xc = −gcT T −1x = −g T x it follows that the state feedback gains g of the original model are related to those in control canonical form, gc , by g T = gcT (Cc C −1 )
(12.33)
This formula is not very practical, because it requires the inverse of the controllability matrix. However, it can be expressed alternatively by Ackermann’s formula (12.34) g T = eTn C −1 αc (A) where eTn = (0, 0, ..., 1) and αc (A) is the closed-loop characteristic polynomial, expressed in terms of the open-loop system matrix A. Equation (12.34) states that the gain vector is in fact the last row of C −1 αc (A). The demonstration uses the Cayley-Hamilton theorem; it is left to the reader (Problem 12.8). Note that C −1 does not have to be calculated explicitly; instead, it is more convenient to proceed in two steps, by first solving the equation bT C = eTn for b, and then computing g T = bT αc (A)
12.4.2
Left and Right Eigenvectors
If the non-symmetric system matrix A has distinct eigenvalues, its eigenvectors will be linearly independent and can be taken as the columns of a regular matrix P : AP = P Λ (12.35) where Λ = diag(λi ) is a diagonal matrix with the eigenvalues of A. It follows that P −1 AP = Λ (12.36)
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12 Controllability and Observability
If we define QT = P −1 and right multiply the foregoing equation by QT , we get QT A = ΛQT (12.37) The columns pi of P and qi of Q (i.e. the rows of QT ) are called the right and left eigenvectors of A, respectively, because Api = λi pi
and
qiT A = λi qiT
(12.38)
From the definition of QT , the left and right eigenvectors are orthogonal qiT pj = δij
(12.39)
From Equ.(12.36), we have QT AP = Λ
12.4.3
and
A = P ΛQT
(12.40)
Diagonal Form
Let us use the right eigenvector matrix P as state transformation matrix x = P xd
(12.41)
Following the procedure described earlier in this section, we can write the transformed state equation as x˙ d = Λxd + QT bu
(12.42)
y = c T P xd
(12.43)
Since Λ is a diagonal matrix with entries equal to the poles of the system, Equ.(12.42) shows that the transformed system behaves like a set of independent first order systems. The diagonal form is also called the modal form, and the states xc are the modes of the system. Note that this concept of mode is related only to the matrix A and is different from the vibration modes as defined in section 2.3 (for an undamped structure, the entries of Λ are identical to the natural frequencies of the structure, as illustrated in the example of section 12.7). For MIMO systems, Equ.(12.42) and (12.43) become
12.5
x˙ d = Λxd + QT Bu
(12.44)
y = CP xd
(12.45)
PBH Test
It is easy to show (Problem 12.10) that the controllability matrix in diagonal form reads
12.6 Residues
⎛
1 λ1 ⎜ 1 λ2 T Cd = diag(qi b) ⎜ ⎝ 1 λn
⎞ ... λn−1 1 ⎟ ... λn−1 2 ⎟ ⎠ ... n−1 ... λn
285
(12.46)
The second matrix in this expression is called a Vandermonde matrix; it is non-singular if the eigenvalues are distinct. In this case, the rank of Cd is the same as that of diag(qiT b). As a result, the system is controllable iff qiT b = 0
for all i
(12.47)
Thus, any left eigenvector orthogonal to the input vector is uncontrollable. From Equ.(12.42), we see that qiT b is in fact a measure of the effective input of the control in mode i and can therefore be regarded as a measure of controllability of mode i. From the duality between controllability and observability, the foregoing results can readily be extended to observability; the observability matrix reads (Problem 12.10) ⎛ ⎞ 1 1 ... 1 ⎜ λ1 λ2 ... λn ⎟ ⎟ diag(cT pi ) (12.48) Od = ⎜ ⎝ ⎠ ... n−1 n−1 λ2 ... λn−1 λ1 n Once again, a system with distinct eigenvalues is controllable iff cT pi = 0
for all i
(12.49)
Any right eigenvector orthogonal to the output vector is unobservable. From Equ.(12.43), we see that cT pi is a measure of the contribution of mode i to the output y. From Equ.(12.46) and (12.48), we conclude that a system with multiple eigenvalues cannot be controlled from a single input, nor observed from a single output. The tests (12.47) and (12.49) are often called the Popov-BelevitchHautus (in short PBH) eigenvector tests of controllability and observability. For a MIMO system, qiT B is a row vector; its entry k measures the controllability of mode i from the input k. Similarly, the component j of Cpi measures the observability of mode i from the component j of the output vector.
12.6
Residues
Next, consider the open-loop transfer function of the system, G(s) = cT (sI − A)−1 b
(12.50)
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12 Controllability and Observability
From Equ.(12.42) and (12.43), it can be written alternatively G(s) = cT P (sI − Λ)−1 QT b
(12.51)
Since sI − Λ is diagonal, we easily obtain the following partial fraction decomposition n n (cT pi )(qiT b) Ri G(s) = = (12.52) s − λi s − λi i=1 i=1 where the residue of mode i, Ri = (cT pi )(qiT b)
(12.53)
is the product of the observability and controllability measures of mode i. For MIMO systems, the partial fraction decomposition becomes G(s) = C
n n pi qiT Ri B= s − λi s − λi i=1
(12.54)
i=1
with the residue matrix Ri = Cpi qiT B
(12.55)
Its entry (k, l) combines the observability of mode i from output k and the controllability from input l.
12.7
Example
In order to dissipate any confusion about the eigenvectors of A and the mode shapes of the structure (section 2.3), let us consider a flexible structure with one input and one output; we assume that the dynamic equations are written in state variable form (9.14) and, to make things even clearer, we further assume that the system is undamped (ξ = 0) and that the mode shapes are normalized according to μ = 1. We use the notation φ(a) = ΦTu and φT (s) = Φy to emphasize the fact that φ(a) and φ(s) contain the amplitude of the mode shapes at the actuator and sensor locations, respectively. With these notations, the state space equation reads 0 Ω 0 z˙ = z+ f (12.56) −Ω 0 φ(a) y = (φ(s)T Ω −1 , 0)z
(12.57)
In this equation, the state vector is z=
Ωη η˙
(12.58)
12.8 Sensitivity
287
where η is the vector of the amplitudes of the structural modes. The non-diagonal system matrix can be brought to diagonal form according to Equ.(12.40); we get 1 1 I I I −jI jΩ 0 T P =√ ,Q = √ ,Λ = I jI 0 −jΩ 2 jI −jI 2 (12.59) We see that the natural frequencies of the system appear with positive and negative signs on the diagonal of Λ, but the eigenvectors of A have nothing to do with the mode shapes of the structure. The PBH eigenvector tests read " 1 1 ! −jφ(a) QT b = √ cT P = √ φT (s)Ω −1 φT (s)Ω −1 (12.60) jφ(a) 2 2 Thus, the controllability and observability measures qiT b and cT pi are proportional to the modal amplitudes φi (a) and φi (s), respectively. Introducing this in Equ.(12.52) and combining the complex conjugate eigenvalues, the partial fraction decomposition can be reduced to G(s) =
m φi (a)φi (s) i=1
s2 + ωi2
(12.61)
where the sum extends to all the structural modes (m = n/2). This result is identical to Equ.(2.58). To conclude this example, we see that when the state equation is written in modal coordinates as in Equ.(12.56), the PBH tests and the associated controllability and observability measures provide no more information than the amplitude of the mode shapes, φ(a) and φ(s).
12.8
Sensitivity
The ultimate goal of the control system is to relocate the closed-loop poles at desirable locations in the complex plane; this should be done, preferably, with moderate values of the gain, in order to limit the control effort and the detrimental effects of noise and modelling errors. The closed-loop poles sk of a SISO system are solutions of the characteristic equation 1 + gH(s)G(s) = 0; they start from the open-loop poles λk for g = 0 and move gradually away as g increases, in a direction which is dictated by the compensator H(s). The rate of change of the closed-loop pole sk near g = 0 is a direct measure of the authority of the control system on this pole; it can be evaluated as follows: for a small gain g = Δg, sk = λk + Δsk ; if the open-loop poles are distinct, we can approximate sk − λi λk − λi
(k = i)
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12 Controllability and Observability
The partial fraction decomposition (12.52) becomes G(sk )
Ri Rk + Δsk λk − λi
(12.62)
Ri Rk + }=0 Δsk λk − λi
(12.63)
i=k
and the characteristic equation 1 + ΔgH(λk ){
i=k
or
Ri ΔgH(λk )Rk = −1 − ΔgH(λk ) Δsk λk − λi i=k
Upon taking the limit Δg → 0, we get (
∂sk Δsk )g=0 = lim = −H(λk )Rk Δg−→0 Δg ∂g
(12.64)
This result shows that the rate of change of the closed-loop poles near g = 0 is proportional to the corresponding residue Rk and to the magnitude of the transfer function of the compensator H(λk ). The latter observation explains why the poles located in the roll-off region of the compensator move only very slowly for small g.
12.9
Controllability and Observability Gramians
Consider the linear time-invariant system (12.1); the controllability measures the ability of the controller to control all the system states from the particular actuator configuration, or equivalently, the abilty to excite all the states from the input u. Consider the response of the system to a set of independent white noises of unit intensity: E[u(t1 )uT (t2 )] = Iδ(t1 − t2 )
(12.65)
If the system is asymptotically stable (i.e. if all the poles of A have negative real parts), the response of the system is bounded, and the steady state covariance matrix is finite; it reads (Problem 12.12.a) ∞ T Wc = E[xxT ] = eAτ BB T eA τ dτ (12.66) 0
Wc is called the Controllability Gramian. According to section 11.4, it is solution of the Lyapunov equation AWc + Wc AT + BB T = 0
(12.67)
12.10 Internally Balanced Coordinates
289
The system is controllable if all the states of the system can be excited; this condition is fulfilled iff Wc is positive definite. From the duality between the observability and controllability, we know that the pair (A, C) is observable iff the pair (AT , C T ) is controllable. It follows that the system is observable iff the observability Gramian ∞ T Wo = eA τ C T CeAτ dτ (12.68) 0
is positive definite. Substituting (AT , C T ) to (A, B) in Equ.(12.67), we see that, if A is asymptotically stable, Wo is solution of AT Wo + Wo A + C T C = 0
(12.69)
Just as the controllability Gramian reflects the ability of the input u to perturb the states of the system, the observability Gramian reflects the ability of non-zero initial conditions x0 of the state vector to affect the output y of the system. This can be seen from the following result (Problem 12.12.b): ∞ y T y dt = xT0 Wo x0 (12.70) 0
If we perform a coordinate transformation x = Tx ˜
(12.71)
the Gramians are transformed according to ˜c = T −1Wc T −T Wc (T ) = W
(12.72)
Wo (T ) = W˜o = T T Wo T
(12.73)
where the notation Wc (T ) refers to the controllability Gramian after the coordinate transformation (12.71). The proof is left to the reader (Problem 12.13).
12.10
Internally Balanced Coordinates
As we have just seen, the Gramians depend on the choice of state variables. Since, in most cases, the latter are not dimensionally homogeneous, nor normalized in an appropriate manner, the magnitude of the entries of the Gramians are not physically meaningful for identifying the least controllable or least observable part of a system. This information would be especially useful for model reduction. It is possible to perform a coordinate transformation such that the controllability and observability Gramians are diagonal and equal; this unique set of coordinates is called internally balanced (Moore).
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12 Controllability and Observability
Let Wc and Wo be the controllability and observability Gramians of an asymptotically stable time-invariant linear system. We perform a spectral decomposition of Wc according to Wc = Vc Σc2 VcT
(12.74)
where Vc is a unitary matrix (Vc VcT = I) and Σc2 is the diagonal matrix of eigenvalues (all positive if Wc is positive definite). If we define Lc = Vc Σc , we can write equivalently Wc = Lc LTc (12.75) (when Lc is a lower triangular matrix, this decomposition is called a Choleski factorization). From Equ.(12.73) and (12.74), if we perform a change of coordinates (12.76) x = T1 x1 with T1 = Lc , the Gramians become −T Wc (T1 ) = L−1 =I c Wc Lc
(12.77)
Wo (T1 ) = LTc Wo Lc
(12.78)
Next, we perform the spectral decomposition of Wo (T1 ) according to Wo (T1 ) = U Σ 2 U T
(12.79)
(with U U T = I) and use the transformation matrix T2 = U Σ −1/2 to perform another change of coordinates x1 = T2 x2
(12.80)
Equ.(12.72) and (12.73) show that the Gramians in the new coordinate system read (12.81) Wc (T1 T2 ) = Σ 1/2 U T U Σ 1/2 = Σ Wo (T1 T2 ) = Σ −1/2 U T U Σ 2 U T U Σ −1/2 = Σ
(12.82)
Thus, in the new coordinate system, the controllability and observability Gramians are equal and diagonal Wc (T1 T2 ) = Wo (T1 T2 ) = Σ
(12.83)
For this reason, the new coordinate system is called internally balanced; it is denotated xb . The global coordinate transformation is x = T1 T2 xb
(12.84)
and the internally balanced model is readily obtained from Equ.(12.23) and (12.24). From Equ.(12.72) and (12.73), we see that, for any transformation T,
12.11 Model Reduction
It follows that
291
Wc Wo = T [Wc (T )Wo (T )]T −1
(12.85)
Wc Wo = (T1 T2 )Σ 2 (T1 T2 )−1
(12.86)
Thus, the eigenvalues of Wc Wo are the entries of Σ 2 , and the transformation matrix T1 T2 contains the right eigenvectors of Wc W0 . The eigenvalues of the Gramians change with the coordinate transformation, but the eigenvalues of the Gramian product Wc W0 is invariant (Problem 12.13). The square root of the eigenvalues of the Gramian product, σi , are called the Hankel singular values of the system.
12.11
Model Reduction
Consider the partition of a state space model according to A11 A12 x1 B1 x˙ 1 = + u x˙ 2 A21 A22 x2 B2 y = C1 x1 + C2 x2
(12.87)
(12.88)
If, in some coordinate system, the subsystem (A11 , B1 , C1 ) has the same impulse response as the full order system, it constitutes an exact lower order model of the system; the model of minimum order is called the minimum realization. Model reduction is concerned with approximate models, and involves a trade-off between the order of the model and its ability to duplicate the behavior of the full order model within a given frequency range.
12.11.1
Transfer Equivalent Realization
If we consider the partial fraction decomposition (12.54), one reduction strategy consists of truncating all the modes with poles far away from the frequency domain of interest (and possibly including their contribution to the static gains) and also those with small residues Ri , which are only weakly controllable or observable (or both). This procedure produces a realization which approximates the transfer function within the frequency band. However, since the uncontrollable part of the system is deleted, even if it is observable, the reduced model cannot reproduce the response to disturbances that may excite the system. This may lead to problems in the state reconstruction. To understand this, recall that the transfer function Ge (s), which is the relevant one for the observer design, is that between the plant noise and the output (section 9.7). If the plant noise does not enter at the input, Ge (s) does have contributions from all observable modes, including those which are uncontrollable from the input.
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12 Controllability and Observability
The procedure can be improved by including all the modes which have a significant contribution to Ge (s) too.
12.11.2
Internally Balanced Realization
Internally balanced coordinates can be used to extend the concept of minimum realization. The idea consists of using the entries of the joint Gramian Σ to partition the original system into a dominant subsystem, with large entries σi , and a weak one, with small σi . The reduction is achieved by cutting the weak subsystem from the dominant one. The following result guarantees that the reduced system remains asymptotically stable: If the internally balanced system is partitioned according to (12.87) and if the joint Gramian is Σ1 0 (12.89) Wc = Wo = 0 Σ2 the two subsystems (A11 , B1 , C1 ) and (A22 , B2 , C2 ) are asymptotically stable and internally balanced, such that Wc1 = Wo1 = Σ1 = diag(σ1 ...σk )
(12.90)
Wc2 = Wo2 = Σ2 = diag(σk+1 ...σn )
(12.91)
The proof is left to the reader (Problem 12.14). Thus, if we order the internally balanced coordinates by decreasing magnitude of σi and if the subsystems 1 and 2 are selected in such a way that σk+1 σk , the global system is clearly dominated by subsystem 1. The model reduction consists of severing subsystem 2, as indicated in Fig.12.3, which produces the reduced system (A11 , B1 , C1 ). Dominant subsystem (Σ1)
u
y
A11 B1 C1 A12
+
A2 1 A2 2 B2 C 2 Weak subsystem
(Σ2)
Fig. 12.3 Model reduction.
Reduced model obtained by cutting these connections
12.11 Model Reduction
Gain dB
293
Full model
Reduced models 8 states Gain dB
Reduced models 12 states
Full model ω (rad / s)
Fig. 12.4 Input-output frequency response of the full model and the reduced models based on internally balanced coordinates and modal truncation.
12.11.3
Example
Consider a simply supported uniform beam with a point force actuator at xa = 0.331 l and a displacement sensor at xs = 0.85 l. We assume that l = 1 m, EI = 10.266 10−3 N m2 , m = 1kg/m and ξ = 0.005. The natural frequencies and the mode shapes are given by (2.54) and (2.55); we find ω1 = 1 rad/s, ω2 = 4 rad/s, etc... The system can be written in state variable form according to (9.14). In a second step, the system can be transformed into internally balanced coordinates following the procedure of section 12.10.1. Two kinds of reduced models have been obtained as follows: • Transform into internally balanced coordinates and delete the subsystem corresponding to the smallest entries of the joint Gramian. • Delete the modal coordinates corresponding to the smallest static gains φ(a)φ(s) μi ωi2 in the modal expansion of the transfer function G(s) =
m φi (a)φi (s) i=1
μi ωi2
.
ωi2 s2 + 2ξωi s + ωi2
(12.92)
294
12 Controllability and Observability
Gain dB
Full model
Reduced models 8 states
Gain dB
Full model
Reduced models 12 states
ω (rad / s) Fig. 12.5 Disturbance-output frequency response of the full model and the reduced models based on internally balanced coordinates and modal truncation.
Figure 12.4 compares the amplitude plots of the input-output frequency response function G(ω) of the reduced models with 8 and 12 states, with that of the full model; the internally balanced realization and the modal truncation based on the static gains are almost identical (they cannot be distinguished on the plot). Figure 12.5 compares the results obtained with the same reduced models, for the frequency response function between a disturbance applied at xd = 0.55 l and the output sensor. Once again, the results obtained with the internally balanced realization and the modal truncation based on the static gains are nearly the same (we can notice a slight difference near ω = 30 rad/s for the reduced models with 12 states); the reduced models with 8 states are substantially in error in the vicinity of 9 rad/s, because mode 3, which has been eliminated during the reduction process (it is almost not controllable from the input), is excited by the disturbance.
12.12
Problems
P.12.1 Show that for a n-dimensional system, the rank of the matrix (b, Ab, A2 b, ..., AN −1 b) is the same as that of the controllability matrix C, for any N > n.
12.12 Problems
295
P.12.2 Consider the inverted pendulum of Fig.12.1.a, where the input variable u is the force applied to the cart. Show that the equation of motion near θ = 0 is g m u θ¨ − (1 + )θ = − l M Ml ˙ T . Compute the Write the equation in state variable form using x = (θ, θ) controllability matrix. [Hint: use Lagrange’s equations] P.12.3 Consider two inverted pendulums on a cart as in Fig.12.1.b. Show that the equations of motion near θ1 = 0 and θ2 = 0 are g m g m u θ¨1 − (1 + )θ1 − θ2 = − l1 M l1 M M l1 g m g m u θ¨2 − θ1 − (1 + )θ2 = − l2 M l2 M M l2 P.12.4 Consider the double inverted pendulum of Fig.12.1.c. Show that the equations of motion near θ1 = 0 and θ2 = 0 are 1 θ¨1 = 2ω02 (1 + a)θ1 − ω02 θ2 − u Ml θ¨2 = −2ω02 θ1 + 2ω02 θ2 where θ1 and θ2 are the absolute angles of the two arms, ω02 = g/l and a = m/M . P.12.5 Show that for two sets of state variables related by the non singular transformation x = T xc , the controllability matrices are related by Cc = T −1 C P.12.6 Show that the control canonical form (12.29) is a state space realization of the transfer function (12.28). P.12.7 Show that for n = 3, the controllability matrix of the control canonical form reads ⎛ ⎞ 1 −a1 a21 − a2 ⎝ 0 1 −a1 ⎠ 0 0 1 P.12.8 Demonstrate Ackermann’s formula (12.34) for SISO systems.
296
12 Controllability and Observability
[Hint: Proceed according to the following steps: (1) (2) (3) (4)
Show that eTi Ac = eTi−1 Using the Cayley-Hamilton theorem, show that eTn αc (Ac ) = gcT Show that αc (Ac ) = T −1 αc (A) = Cc C −1 αc (A) Using the result of Problem 12.7, show that eTn Cc = eTn .]
P.12.9 Consider the single degree of freedom oscillator of section 9.4.1. Calculate the state feedback gains leading to the characteristic equation (9.49) using Ackermann’s formula. Compare with (9.50) and (9.51). P.12.10 Show that for a system in diagonal form, the controllability and observability matrices are given by Equ.(12.46) and (12.48). P.12.11 The PBH rank tests state that • The pair (A, b) is controllable iff rank[sI − A, b] = n • The pair (cT , A) is observable iff cT =n rank sI − A
for all s
for all s
Show that these tests are equivalent to the eigenvector tests (12.47) and (12.49). P.12.12 Consider an asymptotically stable linear time-invariant system. Show that (a) The steady state covariance matrix due to independent white noise inputs of unit intensity E[u(t1 )uT (t2 )] = Iδ(t1 − t2 ) is equal to the controllability Gramian: ∞ T eAτ BB T eA τ dτ Wc = E[xxT ] = 0
(b) The free response from initial conditions x0 satisfies ∞ y T y dτ = xT0 Wo x0 0
where Wo is the observability Gramian.
12.12 Problems
297
[Hint: the state impulse response is x(τ ) = eAτ B and the free output response from non-zero initial conditions x0 is y(τ ) = CeAτ x0 .] P.12.13 Show that for the coordinate transformation x = T x ˜, the Gramians are transformed according to ˜cT T Wc = T W Wo = T −T W˜o T −1 Show that the eigenvalues of the Gramian product Wc W0 are invariant with respect to a coordinate transformation. P.12.14 Show that if an internally balanced system is partitioned according to (12.87), the two subsystems (A11 , B1 , C1 ) and (A22 , B2 , C2 ) are internally balanced with joint Gramians Σ1 and Σ2 . [Hint: Partition the Lyapunov equations governing Wc and Wo .]
13
Stability
13.1
Introduction
A basic knowledge of stability of linear systems has been assumed throughout the previous chapters. Stability was associated with the location of the poles of the system in the left half plane. In chapter 9, we saw that the poles are the eigenvalues of the system matrix A when the system is written in state variable form. In chapter 10, we examined the Nyquist criterion for closed-loop stability of a SISO system; we concluded on the stability of the closed-loop system G(1 + G)−1 from the number of encirclements of −1 by the open-loop transfer function G(s). In this chapter, we examine the salient results of Lyapunov’s theory of stability; it is attractive for mechanical systems because of its exceptional physical meaning and its wide ranging applicability, especially for the analysis of nonlinear systems, and also in controller design. We will conclude this chapter with a class of collocated controls that are especially useful in practice, because of their guaranteed stability, even for nonlinear systems; we will call them energy absorbing controls. The following discussion will be restricted to time-invariant systems (also called autonomous), but most of the results can be extended to time-varying systems. As in the previous chapters, most of the general results are stated without proof and the discussion is focused on vibrating mechanical systems; a deeper discussion can be found in the references. Consider a time-invariant system, linear or not • The equilibrium state x = 0 is stable in the sense of Lyapunov if, for every ε > 0 there is some δ > 0 (depending on ε) such that, if ||x0 || < δ, then ||x|| < ε for all t > t0 . In this statement, ||.|| stands for a norm, measuring the distance to the equilibrium; the Euclidean norm is defined as ||x|| = (xT x)1/2 . States which are not stable in the sense of Lyapunov are unstable. Stability is a local property; if it is independent of the size of the initial perturbation x0 , it is global. A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 299–316. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
300
13 Stability
• The equilibrium state x = 0 is asymptotically stable if it is stable in the sense of Lyapunov and if, for any x0 close to 0, x(t) → 0 as t → ∞. Thus, for a mechanical system, asymptotic stability implies some damping, unlike Lyapunov stability. For a linear time-invariant system, since x(t) = eAt x0 , asymptotic stability is always global; nonlinear systems exhibit more complicated behaviors and they can have more than one equilibrium point (Problem 13.1). The stability of an equilibrium point is related to the behavior of the free trajectories starting in its neighborhood; if all the trajectories eventually converge towards the equilibrium point, it is asymptotically stable; if the trajectories converge towards a limit cycle, the system is unstable (Problem 13.2). The above definitions of internal stability refer to the free response from non-zero initial conditions. In some cases, we are more interested in the inputoutput response: • A system is externally stable if every bounded input produces a bounded output. For obvious reasons, this is also called BIBO stability. External stability has no relation to internal (zero-input) stability in general, except for linear time-invariant systems, where it is equivalent to asymptotic stability (if the system is both controllable and observable).
13.1.1
Phase Portrait
As we have already mentioned, the stability of an equilibrium point is related to the behavior of the trajectories in its vicinity. If we can always find a small domain containing the equilibrium point, such that all trajectories starting within this domain remain arbitrarily close to the origin, the equilibrium is stable; if all trajectories starting in a small domain eventually converge towards the origin, the equilibrium is asymptotically stable, and if this occurs for any initial condition, we have global asymptotic stability. The complete set of trajectories is called the phase portrait; to visualize it, consider the second order system x ¨ + a1 (x)x˙ + a2 (x) = 0 (13.1) ˙ we can easily represent the Defining the state variables x1 = x and x2 = x, trajectories in the phase plane (x1 , x2 ); various situations are considered in Fig.13.1. Figure 13.1.a corresponds to a linear oscillator with viscous damping; all the trajectories consist of spirals converging towards the origin (the decay rate is governed by the damping); the system is globally asymptotically stable. Figure 13.1.b shows the phase portrait of an unstable linear system (poles at -2 and +1); all the trajectories are unbounded. The situation depicted on Fig.13.1.c is that of a Van der Pol oscillator (Problem 13.2), all the trajectories converge towards a limit cycle; the system is unstable, although all the trajectories are bounded.
13.2 Linear Systems
(a)
x&
301
x&
(b)
x
x
x&
Limit cycle
x
(c) Fig. 13.1 Phase portrait for various second order systems: (a) x ¨ + 2ξ x˙ + x = 0, (b) x ¨ + x˙ − 2x = 0, (c) x ¨ − μ(1 − x2 )x˙ + x = 0.
13.2
Linear Systems
Since the stability of a system is independent of the state space coordinates, it is convenient to consider the diagonal form (12.44), where Λ = diag(λi ) is the diagonal matrix with the eigenvalues of A. The free response from non-zero initial conditions reads xd (t) = eΛt xd (0)
(13.2)
Each state coordinate follows an exponential xi (t) = eλi t xi (0). The system is stable in the sense of Lyapunov if Re(λi ) ≤ 0. If Re(λi ) < 0 (strictly negative), the system is globally asymptotically stable (and also externally stable). If the characteristic equation is available in the form d(s) = a0 sn + a1 sn−1 + ... + an = 0
(13.3)
302
13 Stability
it is not necessary to compute all the eigenvalues to assess the asymptotic stability of the system; this can be done directly from the coefficients ai of the characteristic polynomial by the Routh-Hurwitz criterion.
13.2.1
Routh-Hurwitz Criterion
Assume that the characteristic polynomial is written in the form (13.3) with a0 > 0. 1. If not all the coefficients ai are positive, that is if ak ≤ 0 for some k, the system is not asymptotically stable (it may still be stable in the sense of Lyapunov if some ak = 0). 2. If all the coefficients ai > 0, a necessary and sufficient condition for all the roots λi to have negative real parts is that all the determinants Δ1 , Δ2 , ... Δn defined below must be positive. The determinants are constructed as follows: Step 1. Form the array: a1 a3 a5
a0 a2 a4
a2n−1 a2n−2
0 0 ... a1 a0 . . . a3 a2 ... a2n−3 ...
0 0 0 an
where a1 , . . . , an are the coefficients of the characteristic polynomial, and ai = 0 (i > n). Step 2. Compute the determinants: Δ1 = a *1 * a a0 Δ2 = ** 1 * a3 a2 * a1 a 0 * Δ3 = ** a3 a2 * a5 a4 ... * * a1 * * a3 * Δn = ** a5 * * * a2n−1
* * * * 0 a1 a3
* * * * * *
a0 a2 a4
0 a1 a3 ...
a2n−2
a2n−3
* 0 . . . 0 ** a0 . . . 0 ** a2 0 ** * * . . . an *
All the eigenvalues λi have negative real parts iff Δi > 0 for all i.
(13.4)
13.3 Lyapunov’s Direct Method
13.3
303
Lyapunov’s Direct Method
13.3.1
Introductory Example
Consider the linear oscillator x˙ 1 0 1 x1 = x˙ 2 −k/m −c/m x2
(13.5)
We know that it is asymptotically stable for positive damping (c > 0); its phase portrait is represented in Fig.13.1.a; for any disturbed state x0 , the system returns to the equilibrium x = 0. The total energy of the system is the sum of the kinetic energy of the mass and the strain energy in the spring: E(x) =
m 2 k 2 x + x 2 2 2 1
(13.6)
E(x) is positive definite because it satisfies the two conditions E(0) = 0 E(x) > 0
for all x = 0
(13.7)
The time derivative of the total energy during the free response is E˙ = mx2 x˙ 2 + kx1 x˙ 1 and, upon substituting x˙ 1 and x˙ 2 from Equ.(13.5), E˙ = −cx22
(13.8)
We see that E˙ is always negative for a structure with positive damping. Since E is positive and decreases along all trajectories, it must eventually go to E = 0 which, from (13.6), corresponds to the equilibrium state x = 0. This implies that the system is asymptotically stable. Here, we have proved asymptotic stability by showing that the total energy decreases along all trajectories; Lyapunov’s direct method (also called second method, for chronological reasons), generalizes this concept. Unlike other techniques (Eigenvalues, Routh-Hurwitz, Nyquist,...), the method is also applicable to nonlinear and time-varying systems. In what follows, we shall restrict our attention to time-invariant systems for which the theorems have a simpler form; more general results can be found in the literature (e.g. Vidyasagar).
13.3.2
Stability Theorem
A time-invariant Lyapunov function candidate V (x) is a continuously differentiable, locally positive definite function, i.e. satisfying
304
13 Stability
V (0) = 0 V (x) > 0
for all x = 0
in D
(13.9)
where D is a certain domain containing the origin. Theorem: Consider a system governed by the vector differential equation x˙ = f (x)
(13.10)
such that f (0) = 0. The equilibrium state x = 0 is stable (in the sense of Lyapunov) if one can find a Lyapunov function candidate V (x) such that V˙ (x) ≤ 0
(13.11)
for all trajectories in the neighborhood of the origin. If condition (13.11) is satisfied, V (x) is called a Lyapunov function for the system (13.10). The Lyapunov function is a generalization of the total energy of the linear oscillator considered in the introductory example. The foregoing theorem is only a sufficient condition; the fact that no Lyapunov function can be found does not mean that the system is not stable. There is no general procedure for constructing a Lyapunov function, and this is the main weakness of the method. As an example, consider the simple pendulum, governed by the equation g θ¨ + sin θ = 0 l
(13.12)
where l is the length of the pendulum, θ the angle and g the acceleration of ˙ we rewrite it gravity. Introducing the state variables x1 = θ and x2 = θ, x˙ 1 = x2 x˙ 2 = −g/l sin x1
(13.13)
Let us again use the total energy (kinetic plus potential) as Lyapunov function candidate: ml2 2 V (x) = x + mgl(1 − cos x1 ) (13.14) 2 2 It is indeed positive definite in the vicinity of x = 0. We have V˙ (x) = ml2 x2 x˙ 2 + mgl sin x1 x˙ 1 and, substituting x˙ 1 and x˙ 2 from Equ.(13.13), we obtain the time derivative along the trajectories: V˙ (x) = −mglx2 sin x1 + mglx2 sin x1 = 0
13.3 Lyapunov’s Direct Method
305
which simply expresses the conservation of energy. Thus, V˙ (x) satisfies condition (13.11), V (x) is a Lyapunov function for the pendulum and the equilibrium point x = 0 is stable. We now examine a stronger statement for asymptotic stability.
13.3.3
Asymptotic Stability Theorem
Theorem: The state x is asymptotically stable if one can find a continuously differentiable, positive definite function V (x) such that V˙ (x) < 0
(13.15)
for all trajectories in some neighborhood of the origin. Besides, if V (x) is such that there exists a nondecreasing scalar function α(.) of the distance to the equilibrium (Fig.13.2), such that α(0) = 0 and 0 < α(||x||) ≤ V (x)
for all x = 0
(13.16)
then the system is globally asymptotically stable under condition (13.15).
V(x)
α(
x )
x Fig. 13.2 Definition of V (x) and α(||x||) for global stability.
13.3.4
Lasalle’s Theorem
Going back to the linear oscillator, we see that Equ.(13.8) does not comply with the requirement (13.15) to be strictly negative; indeed, E˙ = 0 whenever x2 = 0, even if x1 = 0 (i.e. whenever the trajectory crosses the x axis in Fig.13.1.a). The answer to that is given by Lasalle’s theorem, which extends the asymptotic stability even if V˙ is not strictly negative. Theorem: The state x = 0 is asymptotically stable if one can find a differentiable positive definite function V (x) such that
306
13 Stability
V˙ (x) ≤ 0
(13.17)
for all trajectories, provided that the set of points where V˙ = 0, S = {x ∈ Rn : V˙ (x) = 0} contains no trajectories other than the trivial one x = 0. As an example, consider the nonlinear spring with friction governed by x˙ 1 = x2 x˙ 2 = −f (x2 ) − g(x1 )
(13.18)
where g(x1 ) is the nonlinear restoring force and f (x2 ) is the friction. We assume that g and f are continuous functions such that σ g(σ) > 0,
σ f (σ) > 0,
σ = 0
(13.19)
[f (σ) and g(σ) are entirely contained in the first and third quadrant]. It is easy to see that the linear oscillator is the particular case with g(x1 ) =
k x1 m
c x2 m
(13.20)
f (x2 ) = 0
(13.21)
f (x2 ) =
and that the simple pendulum corresponds to g(x1 ) =
g sin x1 l
The total energy is taken as Lyapunov function candidate x1 1 V (x1 , x2 ) = x22 + g(u) du 2 0
(13.22)
where the first term is the kinetic energy (per unit of mass), and the second one, the potential energy stored in the spring. The time derivative is V˙ (x1 , x2 ) = x2 x˙ 2 + g(x1 )x˙ 1 = −x2 f (x2 ) ≤ 0
(13.23)
Since the set of points where x2 = 0 does not contain trajectories, the system is globally asymptotically stable.
13.3.5
Geometric Interpretation
To visualize the concept, it is useful to consider, once again, a second order system for which the phase space is a plane. In this case, V (x1 , x2 ) can be visualized by its contours (Fig.13.3). The stability is associated with the behavior of the trajectories with respect to the contours of V . If we can find a locally positive definite function V (x) such that all the trajectories cross the
13.3 Lyapunov’s Direct Method
307
x2 3 2 1 0
x1 c1 c2 > c1
V=c
Fig. 13.3 Contours of V (x1 , x2 ) in the phase plane.
contours downwards (curve 1), the system is asymptotically stable; if some trajectories follow the contours, V˙ = 0, the system is stable in the sense of Lyapunov (curve 2). The trajectories crossing the contours upwards (curve 3) correspond to instability, as we now examine.
13.3.6
Instability Theorem
In the previous sections, we examined sufficient conditions for stability. We now consider a sufficient condition for instability. Let us start with the well known example of the Van der Pol oscillator x˙ 1 = x2 x˙ 2 = −x1 + μ(1 − x21 )x2
(13.24)
Taking the Lyapunov function candidate V (x1 , x2 ) = we have
x21 x2 + 2 >0 2 2
V˙ = x1 x˙ 1 + x2 x˙ 2 = μ(1 − x21 )x22
(13.25)
(13.26)
We see that, whenever |x1 | < 1, V˙ > 0. Thus, V˙ > 0 applies everywhere in a small set Ω containing the origin; this allows us to conclude that the system is unstable. In this example, V (x) is positive definite; in fact, instability can be concluded with a weaker statement: Theorem: If there exists a function V (x) continuously differentiable such that V˙ > 0 along every trajectory and V (x) > 0 for arbitrarily small values of x, the equilibrium x = 0 is unstable.
308
13 Stability
It can be further generalized as follows: Theorem: If there is a continuously differentiable function V (x) such that (i) in an arbitrary small neighborhood of the origin, there is a region Ω1 where V > 0 and V = 0 on its boundaries; (ii) at all points of Ω1 , V˙ > 0 along every trajectory and (iii) the origin is on the boundary of Ω1 ; then, the system is unstable.
A
Ω1 V=0
x0
0
V>0 V>0
V=0
Ω
B
Fig. 13.4 Definition of the domains Ω and Ω1 for the instability theorem.
The visual interpretation is shown in Fig.13.4: A trajectory starting at x0 within Ω1 will intersect the contours in the direction of increasing values of V , increasing the distance to the origin; it will never cross the lines OA and OB because this would require V˙ < 0.
13.4
Lyapunov Functions for Linear Systems
Consider the linear time-invariant system x˙ = Ax
(13.27)
We select the Lyapunov function candidate V (x) = xT P x
(13.28)
where the matrix P is symmetric positive definite. Its time derivative is V˙ (x) = x˙ T P x + xT P x˙ = xT (AT P + P A)x = −xT Qx
(13.29)
if P and Q satisfy the matrix equation AT P + P A = −Q
(13.30)
13.5 Lyapunov’s Indirect Method
309
This is the Lyapunov equation, that we already met several times. Thus, if we can find a pair of positive definite matrices P and Q satisfying Equ.(13.30), both V and −V˙ are positive definite functions and the system is asymptotically stable. Theorem: The following statements are equivalent for expressing asymptotic stability: 1. All the eigenvalues of A have negative real parts. 2. For some positive definite matrix Q, the Lyapunov equation has a unique solution P which is positive definite. 3. For every positive definite matrix Q, the Lyapunov equation has a unique solution P which is positive definite. Note that, in view of Lasalle’s theorem, Q can be semi-positive definite, provided that V˙ = −xT Qx = 0 on all nontrivial trajectories. The foregoing theorem states that if the system is asymptotically stable, for every Q ≥ 0 one can find a solution P > 0 to the Lyapunov equation. Note that the converse statement (for every P > 0, the corresponding Q is positive definite) is, in general, not true; this means that not every Lyapunov candidate is a Lyapunov function. The existence of a positive definite solution of the Lyapunov equation can be compared with the Routh-Hurwitz criterion, which allows us to determine whether or not all the eigenvalues of A have negative real parts without computing them.
13.5
Lyapunov’s Indirect Method
This method (also known as the first method), allows us to draw conclusions about the local stability of a nonlinear system from the analysis of its linearization about the equilibrium point. Consider the time-invariant nonlinear system x˙ = f (x) (13.31) Assume that f (x) is continuously differentiable and that f (0) = 0, so that x = 0 is an equilibrium point of the system. The Taylor’s series expansion of f (x) near x = 0 reads f (x) = f (0) + [
∂f ]0 x + f1 (x) ∂x
(13.32)
where f1 (x) = O(x2 ). Taking into account that f (0) = 0 and neglecting the second order term, we obtain the linearization around the equilibrium point x˙ = Ax
(13.33)
310
13 Stability
where A denotes the Jacobian matrix of f , at x = 0: A=[
∂f ]x=0 ∂x
(13.34)
Lyapunov’s indirect method assesses the local stability of the nonlinear system (13.31) from the eigenvalues of its linearization (13.33). Theorem: The nonlinear system (13.31) is asymptotically stable if the eigenvalues of A have negative real parts. Conversely, the nonlinear system is unstable if at least one eigenvalue of A has a positive real part. The method is inconclusive if some eigenvalues of A are purely imaginary. We shall restrict ourselves to the proof of the first part of the theorem. Assume that all the eigenvalues of A have negative real parts; then, we can find a symmetric positive definite matrix P solution of the Lyapunov equation AT P + P A = −I
(13.35)
Using V = xT P x as Lyapunov function candidate for the nonlinear system, we have V˙ = x˙ T P x + xT P x˙ = f T (x)P x + xT P f (x) Using the Taylor’s series expansion f (x) = Ax + f1 (x), we find V˙ = xT (AT P + P A)x + 2xT P f1 (x) Taking into account Equ.(13.35) and the fact that f1 (x) = O(x2 ), we obtain V˙ = −xT x + O(x3 )
(13.36)
Sufficiently near x = 0, V˙ is dominated by the quadratic term −xT x which is negative; V (x) is therefore a Lyapunov function for the system (13.31) which is asymptotically stable. We emphasize the fact that the conclusions based on the linearization are purely local; the global asymptotic stability of the nonlinear system can only be established by finding a global Lyapunov function.
13.6
An Application to Controller Design
Consider the asymptotically stable linear system x˙ = Ax + bu
(13.37)
with the scalar input u subject to the saturation constraint |u| ≤ u∗
(13.38)
13.7 Energy Absorbing Controls
311
If P is solution of the Lyapunov equation AT P + P A = −Q
(13.39)
with Q ≥ 0, V (x) = xT P x is a Lyapunov function of the system without control (u = 0). With control, we have V˙ = −xT Qx + 2xT P bu
(13.40)
u = −ψ(bT P x)
(13.41)
Any control where the scalar function ψ(.) is such that σ ψ(σ) > 0 will stabilize the system, because V˙ < 0. The following choice of u makes V˙ as negative as possible: u = −u∗ sign(bT P x) (13.42) This discontinuous control is often called bang-bang; it is likely to produce chattering near the equilibrium. The discontinuity can be removed by u = −u∗ sat(bT P x) where the saturation function is defined as ⎧ x>1 ⎨ 1 x |x| ≤ 1 sat(x) = ⎩ −1 x < −1
(13.43)
(13.44)
However, the practical implementation of this controller requires the knowledge of the full state x, which is usually not available in practice; asymptotic stability is no longer guaranteed if x is reconstructed from a state observer.
13.7
Energy Absorbing Controls
Consider a vibrating mechanical system, linear or not, stable in open-loop, and such that the conservation of the total energy (kinetic + potential) applies when the damping is neglected. Because there is always some natural damping in practice, the total energy E of the system actually decreases with time during its free response, E˙ < 0. Suppose that we add a control system using a collocated actuator/sensor pair; if we denote by D the power dissipated by damping (D < 0) and by W the power flow from the control system to the structure, we have E˙ = D + W (13.45) If we can develop a control strategy such that the power actually flows from the structure to the control system, W < 0 (the control system behaves like an energy sink), the closed-loop system will be asymptotically stable.
312
13 Stability u T
-g
F
u&
Force sensor (T)
(a) Velocity feedback
Piezoactuator (u)
(b) Force feedback
u = g ∫ T dt
F = − g u& E& = F u& = − g u& 2 < 0
E& = − T u& = − g T 2 < 0
Fig. 13.5 Energy absorbing controls.
Now, consider the situation depicted in Fig.13.5.a, where we use a point force actuator and a collocated velocity sensor. If a velocity feedback is used,
with g > 0, we have
F = −g u˙
(13.46)
W = F u˙ = −g u˙ 2 < 0
(13.47)
The stability of this Direct Velocity Feedback was already pointed out for linear system, in section 7.3. Here, it is generalized to nonlinear structures. Even more generally, any nonlinear control F = −ψ(u) ˙
(13.48)
where ψ(.) is such that σ ψ(σ) > 0 will be stabilizing.1 Next, consider the dual situation (Fig.13.5.b) where the actuator controls the relative position u of two points inside the structure, and the sensor output is the dynamic force T in the active member (T is collocated with u); this situation is that of the active truss considered in section 7.5. Again, the power flow into the structure is W = −T u˙ It follows that the positive Integral Force Feedback t u=g T (τ ) dτ
(13.49)
(13.50)
0 1
The above discussion applies also to any collocated dual actuator/sensor pair, as for example a torque actuator collocated with an angular velocity sensor.
13.8 Problems
313
with g > 0 will be stabilizing, because W = −T u˙ = −gT 2 < 0
(13.51)
The stability of the control law (13.50) was established for linear structures using the root locus technique. Here, we extend this result to nonlinear structures, assuming perfect actuator and sensor dynamics. Later in this book, we will apply this control law to the active damping of cable-structure systems. Because of their global asymptotic stability for arbitrary nonlinear structures, we shall refer to the controllers (13.48) and (13.50) as energy absorbing controllers. Note that, unlike those discussed in the previous section, these controllers do not require the knowledge of the states, and are ready for implementation; the stability of the closed-loop system relies very strongly on the collocation of the sensor and the actuator. Once again, we emphasize that we have assumed perfect sensor and actuator dynamics; finite actuator and sensor dynamics always have a detrimental effect on stability.
13.8
Problems
P.13.1 Show that the nonlinear oscillator m¨ x + cx˙ + k1 x − k2 x3 = 0 with m, c, k1 , k2 > 0, has three equilibrium points. Check them for stability. P.13.2 Consider the Van der Pol oscillator x ¨ − μ(1 − x2 )x˙ + x = 0 with μ > 0. Show that the trajectories converge towards a limit cycle (Fig.13.1.c) and that the system is unstable. P.13.3 Plot the phase portrait of the simple pendulum θ¨ + g/l sin θ = 0 P.13.4 Show that a linear system is externally (BIBO) stable if its impulse response satisfies the following inequality t |h(τ )|dτ ≤ β < ∞ 0
for all t > 0.
314
13 Stability
P.13.5 Show that a linear time-invariant system is asymptotically stable if its characteristic polynomial can be expanded into elementary polynomials (s + ai ) and (s2 + bi s + ci ) with all the coefficients ai , bi , ci positive. P.13.6 Examine the asymptotic stability of the systems with the following characteristic polynomials: (i) d1 (s) = s6 + 6s5 + 16s4 + 25s3 + 24s2 + 14s + 4 (ii) d2 (s) = s5 + 3s3 + 2s2 + s + 1 (iii) d3 (s) = s5 + 2s4 + 3s3 + 3s2 − s + 1 P.13.7 Examine the stability of the Rayleigh equation x ¨ + x = μ(x˙ −
x˙ 3 ) 3
with the direct method of Lyapunov. P.13.8 Examine the stability of the following equations: x ¨ + μx2 x˙ + x = 0 x ¨ + μ|x| ˙ x˙ + x +
(μ > 0)
x3 =0 3
(μ > 0)
P.13.9 (a) Show that, if A is asymptotically stable,
t
eA
S=
T
τ
M eAτ dτ
0
where M is a real symmetric matrix, satisfies the matrix differential equation S˙ = AT S + SA + M
[S(0) = 0]
(b) Show that the steady state value ∞ T S= eA τ M eAτ dτ 0
satisfies the Lyapunov equation AT S + SA + M = 0 P.13.10 Consider the free response of the asymptotically stable system x˙ = Ax from the initial state x0 . Show that, for any Q ≥ 0, the quadratic integral
13.8 Problems
∞
J=
315
xT Qx dt
0
is equal to J = xT0 P x0 where P is the solution of the Lyapunov equation AT P + P A + Q = 0 P.13.11 Consider the linear time invariant system x˙ = Ax + Bu Assume that the pair (A, B) is controllable and that the state feedback u = −Gx has been obtained according to the LQR methodology: G = R−1 B T P where P is the positive definite solution of the Riccati equation AT P + P A + Q − P BR−1 B T P = 0 with Q ≥ 0 and R > 0. Prove that the closed-loop system is asymptotically stable by showing that V (x) = xT P x is a Lyapunov function for the closedloop system. Note: From section 11.3, we readily see that V (x) is in fact the remaining cost to equilibrium: ∞ V (x) = (xT Qx + uT Ru)dτ t
P.13.12 Consider the bilinear single-input system x˙ = Ax + (N x + b)u where A is asymptotically stable (the system is linear in x and in u, but it is not jointly linear in x and u, because of the presence of the bilinear matrix N ). Show that the closed-loop system is globally asymptotically stable for the nonlinear state feedback u = −(N x + b)T P x where P is the solution of the Lyapunov equation AT P + P A + Q = 0
316
13 Stability
P.13.13 Consider the free response of a damped vibrating system Mx ¨ + C x˙ + Kx = 0 The total energy is E(x) =
1 T 1 x˙ M x˙ + xT Kx 2 2
(a) Show that its decay rate is ˙ E(x) = −x˙ T C x˙ (b) Show that if we normalize the mode shapes according to μ = 1 and if we use the state space representation (9.14), the total energy reads E(z) =
1 T z z 2
P.13.14 Consider a linear structure with a point force actuator collocated with a velocity sensor. Using the state space representation (9.14) and taking the total energy as Lyapunov function, show that the controller (13.41) is equivalent to (13.48).
14
Applications
After a brief overview of some critical aspects of digital control, this chapter applies the concepts developed in the foregoing chapters to a few applications; it is based on the work done at the Active Structures Laboratory of ULB before 2002. We believe that these early experiments have more than just an historical value. More applications will be considered in the next three chapters.
14.1
Digital Implementation
In recent years, low cost microprocessors have become widely available, and digital has tended to replace analog implementation. There are many reasons for this: digital controllers are more flexible (it is easy to change the coefficients of a programmable digital filter), they have good accuracy and a far better stability than analog devices which are prone to drift due to temperature and ageing. Digital controllers are available with several hardware architectures, including microcontrollers, PC boards, and digital signal processors (DSP). It appears that digital signal processors are especially efficient for structural control applications. Although most controller implementation is digital, current microprocessors are so fast that it is always more convenient, and sometimes wise, to perform a continuous design of the compensator and transform it into a digital controller as a second step, once a good continuous design has been achieved. This does not mean that the control designer may ignore digital control theory, because even though the conversion from continuous to digital is greatly facilitated by software tools for computer aided control engineering, there are a number of fundamental issues that have to be considered with care; they will be briefly mentioned below. For a deeper discussion, the reader may refer to the literature on digital control (e.g. Astr¨ om & Wittenmark; Franklin & Powell). A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 317–358. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
318
14 Applications
14.1.1
Sampling, Aliasing and Prefiltering
Since digital controllers operate on values of the process variables at discrete times, it is important to know under what conditions a continuous signal can be recovered from its discrete values only. The answer to this question is given by Shannon’s theorem (also called sampling theorem), which states that, to recover a band-limited signal with frequency content f < fb from its sampled values, it is necessary to sample at least at fs = 2fb . If a signal is sampled at fs , any frequency component above the limit frequency fs /2 will appear as a component at a frequency lower than fs /2. This phenomenon is called aliasing, and the limit frequency that can be theoretically recovered from a digital signal is often called Nyquist frequency, by reference to the exploratory work of Nyquist. Aliasing is of course not acceptable and it is therefore essential to place an analog low-pass filter at a frequency fc < fs /2 before the analog to digital converter (ADC), Fig.14.1. However analog prefilters have dynamics and, as we know from the first Bode integral, a sharp cut-off of the magnitude is always associated with a substantial phase lag at the cut-off frequency fc . As fc is related to fs , it is always a good idea to sample at a high rate and to make sure that the cut-off frequency of the prefilter is substantially higher than the crossover frequency of the control system. If the phase lag of the prefilter at crossover is significant, it is necessary to include the prefilter dynamics in the design (as a rule of thumb, the prefilter dynamics should be included in the design if the crossover frequency is higher than 0.1fc). A simple solution to prefiltering is to introduce an analog second order filter
Fig. 14.1 Prefiltering and A/D conversion.
14.1 Digital Implementation
Gf (s) =
s2
ωc2 + 2ξωc s + ωc2
319
(14.1)
which can be built fairly easily with an operational amplifier and a few passive components. A second order Butterworth filter corresponds to ξ = 0.71. Higher order filters are obtained by cascading first and second order systems; for example, a fourth order Butterworth filter is obtained by cascading two second order filters with the same cut-off frequency and ξ = 0.38 and ξ = 0.92, respectively (Problem 14.1).
14.1.2
Zero-Order Hold, Computational Delay
Sampling can be viewed as an impulse modulation converting the continuous signal x(t) into the impulse train ∗
x (t) =
∞
x(t)δ(t − kT )
(14.2)
k=−∞
where T is the sampling period (T = 1/fs ). The construction of a process which holds the sampled values x(kT ) constant during a sampling period is made by passing x∗ (t) through a zero-order hold which consists of a filter with impulse response (Fig.14.2) h(t) = 1(t) − 1(t − T ) where 1(t) is the Heaviside step function. It is easy to show that the corresponding transfer function is (Problem 14.2)
δ(t) 1
t
0
T
Zero-order hold
2T
t
0
0
T
T
t
2T
t
Fig. 14.2 The zero-order hold transforms an impulse into a rectangle of duration T , and an impulse train into a staircase function.
320
14 Applications
H0 (s) =
1 − e−sT s
(14.3)
and that it introduces a linear phase lag −ωT /2. Another effect of sampling is the computational delay which is always present between the access to the computer through the ADC and the output of the control law at the digital to analog converter DAC. This delay depends on the way the control algorithm is implemented; it may be fixed, equal to the sampling period T , or variable, depending on the length of the computations within the sampling period. A time delay T is characterized by the transfer function e−T s ; it introduces a linear phase lag −ωT . Rational approximations of the exponential by all-pass functions (Pad´e approximants) were discussed in Problem 10.8. The output of the DAC is also a staircase function; in some applications, it may be interesting to smooth the control output, to remove the high frequency components of the signal, which could possibly excite high frequency mechanical resonances. The use of such output filters, however, should be considered with care because they have the same detrimental effect on the phase of the control system as the prefilter at the input. In applications, it is advisable to use a sampling frequency at least 20 times, and preferably 100 times higher than the crossover frequency of the continuous design, to preserve the behavior of the continuous system to a reasonable degree.
14.1.3
Quantization
After prefiltering at a frequency fc below the Nyquist frequency fs /2, the signal is passed into the ADC for sampling and conversion into a digital signal of finite word length (typically N =14 or 16 bits) representing the total range of the analog signal. Because of the finite number of quantization levels, there is always a roundoff error which represents 2−N times the full range of the signal; the quantization error can be regarded as a random noise. The signal to noise ratio is of the order 2N provided that the signal is properly scaled to use the full range of the ADC. Near the equilibrium point, only a small part of the dynamic range is used by the signal, and the signal to noise ratio drops substantially. The quantization error is also present at the output of the DAC; the finite word length of the digital output is responsible for a finite resolution in the analog output signal; the resolution of the output is δ = R/2M , where R is the dynamic range of the output and M the number of bits of the DAC. To appreciate the limitations associated with this formula, consider a positioning problem with a range of R = 10 mm and a DAC of 16 bits; the resolution on the output will be limited to δ = 10/216 = 0.15 μm. Quantization errors may be responsible for limit cycle oscillations.
14.1 Digital Implementation
321
Let us briefly mention that the finite word length arithmetic in the digital controller is another source of error, because finite word length operations are no longer associative or distributive, due to rounding. We shall not pursue this matter which is closer to the hardware (e.g. Jackson).
14.1.4
Discretization of a Continuous Controller
Although all the design methods exist in discrete form, it is quite common to perform a continuous design, and to discretize it in a second step. This procedure works quite well if the sampling rate fs is much higher than the crossover frequency fc of the control system (in structural control, it is quite customary to have fs /fc 100). Assume that the compensator transfer function has been obtained in the form U (s) b1 sn−1 + . . . + bn = H(s) = n (14.4) Y (s) s + a1 sn−1 + . . . + an For digital implementation, it must be transformed to the form of a difference equation u(k) =
n
αi u(k − i) +
i=1
m
βj y(k − j)
(14.5)
j=0
The corresponding z-domain transfer function is 'm −j U (z) j=0 βj z ' = H(z) = n Y (z) 1 − i=1 αi z −i
(14.6)
where z −1 is the delay operator. The coefficients αi and βj of H(z) can be obtained from those of H(s) following Tustin’s method: H(z) and H(s) are related by the bilinear transform s=
2(z − 1) T (z + 1)
or
z=
1 + T s/2 1 − T s/2
(14.7)
where T is the sampling period. This transformation maps the left half splane into the interior of the unit circle in the z-plane, and the imaginary axis from ω = 0 to ∞ into the upper half of the unit circle from z = 1 to z = −1 (e.g. see Franklin & Powell or Oppenheim & Schafer). Tustin’s method can be applied to multivariable systems written in state variables; for the continuous system described by x˙ = Ac x + Bc u
(14.8)
y = Cx + Du
(14.9)
322
14 Applications
the corresponding discrete system resulting from the bilinear transform (14.7) reads x(k + 1) = Ax(k) + B1 u(k + 1) + B0 u(k) (14.10) y = Cx + Duk
(14.11)
with
T −1 T ] [I + Ac ] 2 2 T T B1 = B0 = [I − Ac ]−1 Bc 2 2 The proof is left as an exercise (Problem 14.3). A = [I − Ac
14.2
(14.12) (14.13)
Active Damping of a Truss Structure
One of the earliest active damping experiments that we performed at ULB is that with the truss of Fig.4.20, built in 1989. It consists of 12 bays of 14 cm each, made of steel bars of 4 mm diameter; it is clamped at the bottom, and two active struts are located in the lower bay. Similar studies were performed at other places at about the same time (Fanson et al., Chen et al., Peterson et al.). The distinctive feature of this work was that the active strut was built with low cost commercial components (Philips linear piezoelectric actuator and Bruel & Kjaer piezoelectric force sensor). The design was such that the length and the stiffness of the active strut almost exactly matched that of one bar; in this way, the insertion of the active element did not change the stiffness of the structure. Because of the high-pass nature of the piezoelectric force sensors (and electronics), only the dynamic component of the force is measured by the force sensor. Other types of active members with built-in viscous damping have been developed (Hyde & Anderson). The mathematical modelling of an active truss was examined in section 4.9 and the active damping with Integral Force Feedback was investigated in section 7.5. It was found that the closed-loop poles of an active truss provided with a single active element follow the root locus defined by Equ.(7.31).
14.2.1
Actuator Placement
More than any specific control law, the location of the active member is the most important factor affecting the performance of the control system. The active element should be placed where its authority over the modes it is intended to control is the largest. According to Equ.(7.35), the control authority is proportional to the fraction of modal strain energy in the active element, νi . It follows that the active struts should be located in order to maximize νi in the active members for the critical vibration modes. The
14.2 Active Damping of a Truss Structure
323
Mode 1 (8.8 Hz) z y 10
9
x
98 2
1
Mode 2 (10.5 Hz) z 2
x y
Element #
ν1 (%)
ν2 (%)
98
15.7
0.1
1
4.1
11.2
2
3.1
11.4
9 10
3.4 2.6
9.4 9.4
Fig. 14.3 Finite element model of the truss of Fig.4.20, mode shapes and fraction of modal strain energy in selected elements; the active members have been placed in elements No 2 and 98.
324
14 Applications
search for candidate locations where active struts can be placed is greatly assisted by the examination of the map of the fraction of strain energy in the structural elements, which is directly available in commercial finite element packages. Such a map is presented in Fig.14.3 ; one sees that substituting the active member for the bar No 98 provides a strong control on mode 1 [ν1 = 0.157](in-plane bending mode), but no control on mode 2 (out-of-plane bending mode), which is almost uncontrollable from an active member placed in bar No 98 [ν2 = 0.001]. By contrast, an active member substituted for the bar No 2 offers a reasonable control on mode 1 [ν1 = .031] and excellent control on mode 2 [ν2 = 0.114]; these two locations were selected in the design. The fraction of modal strain energy is well adapted to optimization techniques for actuator placement.
14.2.2
Implementation, Experimental Results
1
Using the bilinear transform (14.7), we can readily transform the integral control law (IFF) g δ= y (14.14) s into the difference equation T δi+1 = δi + g (yi+1 + yi ) 2
(14.15)
which we recognize as the trapezoid rule for integration. In order to avoid saturation, it is wise to slightly modify this relation according to T δi+1 = αδi + g (yi+1 + yi ) 2
(14.16)
where α is a forgetting factor slightly lower than 1. α depends on the sampling frequency; it can either be tuned experimentally or obtained from a modified compensator g δ= y (14.17) s+a where the breakpoint frequency a is such that a ω1 (the first natural frequency), to produce a phase of 90o for the first mode and above (Problem 14.5). Note that, for a fast sampling rate, the backward difference rule δi+1 = αδi + gT yi+1 1
(14.18)
The implementation of the IFF controller presented here is that done at the time of this experiment. Other aspects of the control implementation, particularly concerning the recovery of the static stiffness of the uncontrolled structure, will be addressed in section 15.5.
14.2 Active Damping of a Truss Structure
325
Free response
Force # 1
Impulse Force # 2
Control on at this time Fig. 14.4 Force signal from the two active members during the free response after an impulsive load (experimental results).
B
without control A
with control
8.8
10.5
Hz
Fig. 14.5 FRF between A and B, with and without control (experimental results, linear scale).
326
14 Applications
works just as well as (14.16). In our experiment, the two active members operated independently in a decentralized manner with fs =1000 Hz. Figure 14.4 shows the force signal in the active members during the free response after an impulsive load, first without, and then with control. Figure 14.5 shows the frequency response between a point force applied at A along the truss and an accelerometer located at B, at the top of the truss. A damping ratio larger than 0.1 was obtained for the first two modes. Finally, it is worth pointing out that : (i) The dynamics of the charge amplifier does not influence the result appreciably, provided that the corner frequency of the high-pass filter is significantly lower than the natural frequency of the targeted mode. (ii) In this application as in all applications involving active damping with piezo struts, no attempt was made to correct for the large hysteresis of the piezotranslator; it was found that the hysteresis does not deteriorate the closed-loop response significantly, as compared to the linear predictions.
14.3
Active Damping Generic Interface
The active strut discussed in the previous section can be integrated into a generic 6 d.o.f. interface connecting arbitrary substructures. Such an interface is shown in Fig.14.6.a and b (the diameter of the base plates is 250 mm); it consists of a Stewart platform with a cubic architecture [this provides a uniform control capability and uniform stiffness in all directions, and minimizes the cross-coupling thanks to mutually orthogonal actuators (Geng & Haynes)]. However, unlike in section 8.5.2 where each leg consists of a single d.o.f. soft isolator, every leg consists of an active strut including a piezoelectric actuator, a force sensor and two flexible tips.
14.3.1
Active Damping
The control is a decentralized IFF with the same gain for all loops. Let Mx ¨ + Kx = 0
(14.19)
be the dynamic equation of the passive structure (including the interface). According to section 4.9, the dynamics of the active structure is governed by Mx ¨ + Kx = BKa δ
(14.20)
where the right hand side represents the equivalent piezoelectric loads : δ = (δ1 , ..., δ6 )T is the vector of piezoelectric extensions, Ka is the stiffness of one strut and B is the influence matrix of the interface in global coordinates. The output y = (y1 , ..., y6 )T consists of the six force sensor signals which are proportional to the elastic extension of the active struts y = Ka (q − δ)
(14.21)
14.3 Active Damping Generic Interface
327
Fig. 14.6 Stewart platform with piezoelectric legs as generic active damping interface. (a) General view. (b) With the upper base plate removed. (c) Interface acting as a support of a truss.
where q = (q1 , ..., q6 )T is the vector of global leg extensions, related to the global coordinates by q = BT x (14.22) The same matrix appears in Equ.(14.20) and (14.22) because the actuators and sensors are collocated. Using a decentralized IFF with constant gain on the elastic extension, g δ= y (14.23) Ka s
328
14 Applications
the closed-loop characteristic equation is obtained by combining Equ.(14.20) to (14.23): g [M s2 + K − BKa B T ]x = 0 (14.24) s+g In this equation, the stiffness matrix K refers to the complete structure, including the full contribution of the Stewart platform (with the piezoelectric actuators with short-circuited electrodes). The open-loop poles are ±jΩi where Ωi are the natural frequencies of the complete structure. The openloop zeros are the asymptotic values of the eigenvalues of Equ.(14.24) when g −→ ∞; they are solution of [M s2 + K − BKa B T ]x = 0
(14.25)
The corresponding stiffness matrix is K −BKa B T where the axial stiffness of the legs of the Stewart platform has been removed from K. Without bending stiffness in the legs, this matrix is singular and the transmission zeros include the rigid body modes (at s = 0) of the structure where the piezo actuators have been removed. However, the flexible tips are responsible for a non-zero bending stiffness of the legs and the eigenvalues of Equ.(14.25) are located at ±jωi , at some distance from the origin along the imaginary axis. Upon transforming into modal coordinates, x = Φz and assuming that the normal modes are normalized according to ΦT M Φ = I, we get [s2 + Ω 2 − where
g ΦT BKa B T Φ]z = 0 s+g
Ω 2 = diag(Ωi2 ) = ΦT KΦ T
(14.26)
(14.27)
T
As in section 7.5, the matrix Φ BKa B Φ is, in general, fully populated; assuming it is diagonally dominant and neglecting the off diagonal terms, we can rewrite it ΦT BKaB T Φ diag(νi Ωi2 ) (14.28) where νi is the fraction of modal strain energy in the active damping interface, that is the fraction of the strain energy concentrated in the legs of the Stewart platform when the structure vibrates according to the global mode i. From the definition of the open-loop transmission zeros, ±jωi , we also have ω 2 diag(ωi2 ) = ΦT (K − B T Ka B)Φ = diag[Ωi2 (1 − νi )]
(14.29)
and the characteristic equation (14.26) can be rewritten as a set of uncoupled equations g s2 + Ωi2 − (Ω 2 − ωi2 ) = 0 (14.30) s+g i or s2 + ωi2 1+g 2 =0 (14.31) s(s + Ωi2 )
14.3 Active Damping Generic Interface
329
This equation is identical to Equ.(7.31) and all the results of section 7.5 apply. Note that, in this section, the previous results have been extended to a multi-loop decentralized controller with the same gain for all loops.
14.3.2
Experiment
The test set-up is shown in Fig.14.6.c; the interface is used as a support for the truss discussed in the previous section (used as a passive truss in this case). The six independent controllers have been implemented on a DSP board; the feedback gain is the same for all the loops. Figure 14.7 shows some typical experimental results; the time response shows the signal from one of the force sensors of the Stewart platform when the truss is subjected to an impulse at mid height from the base, first without, and then with control. The FRFs (with and without control) are obtained between a disturbance applied to the piezoactuator in one leg and its collocated force sensor. One sees that fairly high damping ratios can be achieved for the low frequency modes (4 − 5Hz) but also significant damping in the high frequency modes (40 − 90Hz). The experimental root locus of the first two modes is shown in Fig.14.8; it is compared to the analytical prediction of Equ.(14.31). In drawing Fig.14.8, the transmission zeros ±jωi are taken as the asymptotic natural frequencies of the system as g → ∞.
Fig. 14.7 Impulse response and FRF of the truss mounted on the active interface (experimental results, Abu-Hanieh).
330
14 Applications
Fig. 14.8 Poles, zeros and experimental root locus for the truss mounted on the active interface. The continuous lines are the root locus predictions from Equ.(14.31).
14.3.3
Pointing and Position Control
As a closing remark, we wish to emphasize the potential of the stiff Stewart platform described here for precision pointing and precision control. With piezoceramic actuators of 50 μm stroke, the overall stroke of the platform is 90, 103 and 95 μm along the x, y and z directions (in the payload plate axis of Fig.8.30) and 1300, 1050 and 700 μrad around the x, y and z axes, respectively. Embedding active damping in a precision pointing or position control loop can be done with the HAC/LAC strategy discussed in section 14.6.
14.4
Active Damping of a Plate
In 1993, at the request of ESA, we developed a laboratory demonstration model of an active plate controlled by PZT piezoceramics; it was later transformed into a flight model (to be flown in a canister) by our industrial partner SPACEBEL and the experiment (named CFIE: Control-Flexibility Interaction Experiment), was successfully flown by NASA in the space shuttle in September 1995 (flight STS-69).
14.4 Active Damping of a Plate
331
Support Structure Charge amplifier DSP
PZT piezoceramic
Voltage amplifier
g Laser mirror Additional Masses
Fig. 14.9 Laboratory demonstration model of the CFIE experiment.
According to the specifications, the experiment should fit into a “GAS” canister (cylinder of 50 cm diameter and 80 cm high), demonstrate significant gravity effects, and use the piezoelectric technology. We settled on a very flexible steel plate of 0.5 mm thickness hanging from a support as shown in Fig.14.9; two additional masses were mounted, as indicated in the figure, to lower the natural frequencies of the system. The first mode is in bending and the second one is in torsion. Because of the additional masses, the structure has a significant geometric stiffness due to the gravity loads, which is responsible for a rise of the first natural frequency from f1 =0.5Hz in zero gravity to 0.9Hz with gravity. The finite element model of the structure in the gravity field could be updated to match the experimental results on the ground, but the in-orbit natural frequencies could only be predicted numerically and were therefore subject to uncertainties. As we know from the previous chapters, in order to achieve active damping, it is preferable to adopt a collocated actuator/sensor configuration. In principle, a strictly collocated configuration can be achieved with self-sensing actuators (Dosch et al.), but from our own experience, these systems do not work well, mainly because the piezoceramic does not behave exactly like a capacitance as assumed in the self-sensing electronics. As a result, self-sensing was ruled out and we decided to adopt a nearly collocated configuration, which is quite sufficient to guarantee alternating poles and zeros at low frequencies. However, as we saw in section 4.8.7, nearly collocated piezoelectric plates are not trivial to model, because of the importance of the membrane strains in the input-output relationship; this project was at the origin of our work on the finite element modelling of piezoelectric plates and shells (Piefort).
332
14 Applications
14.4.1
Control Design
According to section 7.4, achieving a large active damping with a Positive Position Feedback (PPF) and strain actuator and sensor pairs relies on two conditions: (i) obtaining a precise tuning of the controller natural frequency on the targeted mode and (ii) using an actuator/sensor configuration leading to sufficient spacing between the poles and the zeros, so that wide loops can be obtained. We will discuss the tuning issue a little later; for nearly collocated systems, the distance between the poles and the zeros depends strongly on local effects in the strain transmission. In the CFIE experiment, the control system consists of two independent control loops with actuator/sensor pairs placed as indicated in Fig.14.9; finite element calculations confirmed that the spacing between the poles and the zeros was acceptable. The controller consists of two independent PPF loops, each of them targeted at modes 1 and 2 of the structure, respectively at f1 =0.86Hz and f2 =3.01Hz with gravity and f1 =0.47Hz and f2 =2.90Hz in zero gravity (predicted from finite element calculations). The compensator reads D(s) =
g1 ωf2 1 g2 ωf2 2 + s2 + 2ξf 1 ωf 1 s + ωf2 1 s2 + 2ξf 2 ωf 2 s + ωf2 2
(14.32)
The determination of the gains g1 and g2 requires some trial and error; as already mentioned, it is generally simpler to adjust the gain of the filter of higher frequency first, because the roll-off of the second order filter reduces the influence from the filter tuned on a lower frequency. Note that, although its stability is guaranteed for moderate values of g1 and g2 , the performance of the PPF depends heavily on the tuning of the filter frequencies ωf 1 and ωf 2 on the targeted modes ω1 and ω2 . It is therefore essential to predict the natural frequencies accurately.
ξ 1 (% )
14 12 10 8 6 4
Δωf (%) ωf
2 0 -20
-10
0
10
20
30
Fig. 14.10 Sensitivity of the performance to the tuning of the controller.
14.4 Active Damping of a Plate
control off
0
2
4
333
control on
6
8
10
12
14
Time (s) Fig. 14.11 Free response after some disturbance (laser sensor).
To illustrate the degradation of the performance when the controller is not tuned properly, Fig.14.10 shows the sensitivity of the performance, taken as the maximum closed-loop modal damping of the first mode, as a function of the relative error in the frequency of the PPF filter; Δωf = 0 corresponds to the optimally tuned filter, leading to a modal damping ξ1 over 0.13. We see that even small tuning errors can significantly reduce the performance, and that a 20 % error makes the control system almost ineffective. This problem was particularly important in this experiment where the first natural frequency could not be checked from tests. Fig.14.11 illustrates the performance of the control system on the laboratory demonstration model; it shows the free response measured by laser of one of the additional mass after some disturbance, with and without control, when the tuning is optimal. The laboratory demonstration model shown in Fig.14.9 is very flimsy and would not withstand the environmental loads (static and dynamics) during the launch of the spacecraft; the test plate would even buckle under its own weight if turned upside down. As a result, the flight model was substantially reinforced with a strong supporting structure, and a latching mechanism was introduced to hold the plate during the launch. The flight model successfully underwent the vibration tests before launch, but the charge amplifiers were destroyed (!), because the amount of electric charge generated during the qualification tests was several orders of magnitude larger than the level expected during the in-orbit experiment; the problem was solved by changing the electronic design, to include low leakage diodes with appropriate threshold at the input of the charge amplifiers. No problem occurred during the flight.
334
14.5
14 Applications
Active Damping of a Stiff Beam
We begin with a few words about the background in which this problem was brought to our attention in the early 90’s. Optical instruments for space applications require an accuracy on the wave front in the range of 10 nm to 50 nm. The ultimate performance of the instruments must be evaluated on earth, before launch, in a simulated space environment. This is done on sophisticated test benches resting on huge seismically isolated slabs and placed in a thermal vacuum chamber. Because of the constraint on accuracy, the amplitude of the microvibrations must remain below 1 nm or, equivalently, if the first natural frequency of the supporting structure is around 60Hz, the acceleration must remain below 10−5 g. This limit is fairly easy to exceed, even under such apparently harmless excitations as the noise generated by the air conditioning of the clean room. Beyond the specific problem that we have just mentioned, the damping of microvibrations is a fairly generic problem which has many applications in other fields of precision engineering, such as machine tools, electronic circuit lithography, etc...
14.5.1
System Design
A simple active damping device has been developed, based on the following premises: (i) The control system should use an accelerometer which is more appropriate than a displacement or a velocity sensor for this problem (an acceleration of 10−5 g can be measured with a commercial accelerometer, while a displacement of 1 nm requires a sophisticated laser interferometer). (ii) The structures considered here are fairly stiff and well suited to the use of a proof-mass actuator without excessive stroke (section 3.2.1). (iii) The sensor and the actuator should be collocated, in order to benefit from guaranteed stability. The test structure is represented in Fig.14.12; it consists of a 40 kg steel beam of 4.7 m, mounted on three supports located at the nodes of the second free-free mode, to minimize the natural damping. The first natural frequency of the beam is f1 = 68Hz. The proof-mass actuator consists of a standard electrodynamic shaker (Bruel & Kjaer 4810) fitted with an extra mass of 500 gr, to lower its natural frequency to about 20Hz. In this way, the amplitude diagram of the frequency response F/i is nearly constant for f >40Hz, indicating that the proof-mass actuator behaves nearly as an ideal force generator (section 3.2.1). The phase diagram is also nearly flat above 40Hz, but contains a linear phase due to the digital controller. The control law can be either g/s, leading to a Direct Velocity Feedback, or the set of second order filters as discussed in Problem 7.2. Both have guaranteed stability (assuming perfect actuator and sensor dynamics). In choosing between the two alternatives, we must take the following aspects into account: (i) Since the transfer function of the structure does not have any roll-off, the roll-off of the open-loop system is entirely controlled by the compensator.
14.6 The HAC/LAC Strategy
335
Fig. 14.12 Test structure and impulse response, with and without control.
(ii) The Direct Velocity Feedback is wide band, while the acceleration feedback, based on second order filters, must be tuned on the targeted modes. (iii) In theory, the phase margin of the Direct Velocity Feedback is 900 for all modes, but its roll-off is only −20 dB/decade. The acceleration feedback has a roll-off of −40 dB/decade, but the phase margin gradually vanishes for the modes which are above the frequency appearing in the filter of the compensator (Problem 14.6). Based on the foregoing facts and depending on the structure considered, one alternative may be more effective than the other in not destabilizing the high frequency dynamics, which is more sensitive to the finite dynamics of the actuator and sensor, delays, etc... For the test structure of Fig.14.12, which is fairly simple and does not involve closely spaced modes, both compensators have been found very effective; the damping ratio of the first mode has been increased from ξ1 =0.002 to ξ1 =0.04.
14.6
The HAC/LAC Strategy
In active structures for precision engineering applications, the control system is used to reduce the effect of transient and steady state disturbances on the controlled variables. Active damping is very effective in reducing the settling time of transient disturbances and the effect of steady state disturbances near
336
14 Applications
the resonance frequencies of the system; however, away from the resonances, the active damping is completely ineffective and leaves the closed-loop response essentially unchanged. Such low gain controllers are often called Low Authority Controllers (LAC), because they modify the poles of the system only slightly (Aubrun). To attenuate wide-band disturbances, the controller needs larger gains, in order to cause more substantial modifications to the poles of the openloop system; this is the reason why they are often called High Authority Controllers (HAC). Their design requires a model of the structure and, as we saw in chapter 10, there is a trade-off between the conflicting requirements of performance-bandwidth and stability in the face of parametric uncertainty and unmodelled dynamics. The parametric uncertainty results from a lack of knowledge of the structure (which could be reduced by identification) or from changing environmental conditions, such as the exposure of a spacecraft to the sun. Unmodelled dynamics include all the high frequency modes which cannot be predicted properly, but are candidates for spillover instability.
LAC : collocated active damping
g D(s) r
- e +
H(s)
u
HAC compensator (model-based)
Go(s) G(s,g)
y
Structure
Fig. 14.13 Principle of the dual loop HAC/LAC control.
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts (Benhabib et al.), but in many situations, collocated pairs are not feasible for HAC; we know from chapter 6 that such configurations do not have a fixed pole-zero pattern and are much more sensitive to parametric uncertainty. LQG controllers are an example of HAC; their lack of robustness with respect to the parametric uncertainty was pointed out in section 9.10. The situation is even worse for the unmodelled dynamics, particularly for very flexible structures which have a high modal density, because there are always flexible modes near the crossover frequency. Without frequency shaping, LQG methods often require an accurate modelling for approximately two decades beyond the bandwidth of the closed-loop system, which is unrealistic in most
14.6 The HAC/LAC Strategy
337
practical situations. The HAC/LAC approach originated at Lockheed in the early 80’s; it consists of combining the two approaches in a dual loop control as shown in Fig.14.13. The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. This approach has the following advantages: • The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outside the bandwidth. • The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the outer loop (improved gain margin). • The larger damping of the modes within the controller bandwidth makes them more robust to the parametric uncertainty (improved phase margin). Singular value robustness measures generalize the phase and gain margin for MIMO systems; some of these tests are discussed in section 10.9 (see also Kosut et al., or Mukhopadhyay & Newsom).
14.6.1
Wide-Band Position Control
In order to illustrate the HAC/LAC strategy for a non-collocated system, let us consider once again the active truss of Fig.4.20. The objective is to design a wide-band controller using one of the piezo actuators to control the tip displacement y along one coordinate axis (Fig.14.14), measured by a laser interferometer. The compensator should have some integral action at low frequency, to compensate the thermal perturbations and avoid steady state errors; the targeted bandwidth of 100 rad/s includes the first two vibration modes. Note that the actuator and the displacement sensor are located at opposite ends of the structure, so that the actuator action cannot be transmitted to the sensor without exciting the entire truss. The LAC consists of the active damping discussed in section 14.2; the transfer function G(ω, g) between the input voltage of the actuator and the tip displacement y is shown in Fig.14.15 for various values of the gain g of the active damping. One observes that the active damping works very much like passive damping, affecting only the frequency range near the natural frequencies. Below 100 rad/s, the behavior of the system is dominated by the first mode; the second mode does not substantially affect the amplitude of G(ω, g), and the phase lag associated with the second mode is compensated by the phase lead of a zero at a frequency slightly lower than ω2 (although not shown, the general shape of the phase diagram can be easily drawn from the amplitude plot). From these observations, we conclude that mode 2, which is close to mode 1, will be phase-stabilized with mode 1 and, as a result, the compensator design can be based on a model including a single vibration mode; the active damping can be closely approximated by passive damping. Thus, the compensator design is based on the very simple model of a damped oscillator.
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14 Applications
y
Laser Interferometer
HAC Position Control
LAC Active Damping
+ +
Active Damping
u Fig. 14.14 Wide-band position control of the truss. The objective is to control the tip displacement y with one of the piezo actuators; the HAC/LAC controller involves an inner active damping loop with collocated actuator/sensor pairs. 40 g=0
dB 30 20
g = 0 .5
10 g =1
0 -10 -20 -30 10
ω (rad / s)
100
Fig. 14.15 FRF y/u of the structure for various values of the gain g of the active damping (experimental results).
14.6 The HAC/LAC Strategy
14.6.2
339
Compensator Design
The compensator should be designed to achieve integral action at low frequency and to have enough roll off at high frequency to avoid spillover instability. The standard LQG is not well suited to these requirements, because the quadratic performance index puts an equal weight on all frequencies; the design objectives require larger weights on the control at high frequency to avoid spillover, and larger weights on the states at low frequency to achieve integral action; both can be achieved by the frequency-shaped LQG as explained in section 11.16. The penalty on the high frequency components of the control u is obtained by passing the control through a low-pass filter (selected as a second order Butterworth filter in this case) and the P+I action is achieved by passing the output y (which is also the control variable z) through a first order system as indicated in Fig.11.9. The state feedback is obtained by solving the LQR problem for the augmented system with the quadratic performance index E[z1T z1 + uT1 u1 ]
(14.33)
The structure of the compensator is that of Fig.11.10; the frequency distribution of the weights for the original problem is shown in Fig.11.8 ; the large weights Q(ω) on the states at low frequency correspond to the integral action, and the large penalty R(ω) on the control at high frequency aims at reducing the spillover. The states of the structure (only two in this case) must be reconstructed with an observer; in this case, a Kalman filter is used; the noise intensity matrices have been selected to achieve the appropriate dynamics.
14.6.3
Results
The Bode plots of the compensator are shown in Fig.14.16; it behaves like an integrator at low frequency, provides some phase lead near the flexible mode and crossover, and roll-off at high frequency. The open-loop transfer function of the control system, GH, is shown in Fig.14.17 (G corresponds to the model); the bandwidth is 100 rad/s and the phase margin is P M = 38o . The effect of this compensator on the actual structure G∗ can be assessed from Fig.14.18. As expected, the second flexible mode is phase stabilized and does not cause any trouble. On the other hand, we observe several peaks corresponding to higher frequency modes in the roll-off region; some of these peaks exceed 0 dB and their stability must be assessed from the Nyquist plot, which is also represented in Fig.14.18. We see that the first peak exceeding 1 in the roll-off region (noted 1 in Fig.14.18) is indeed stable (it corresponds to the wide loop in the right side of the Nyquist plot). The second peak in the roll-off region (noted 2) is slightly unstable for the nominal gain of the compensator; some reduction of the gain is necessary to achieve stability (small loop near -1 in the Nyquist plot); this reduces the bandwidth to about 70 rad/s. A detailed
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H
ω ( rad / s )
φ
ω ( rad / s ) Fig. 14.16 Bode plots of the compensator H(ω).
GH
Integral effect Increased Roll-off
dB
ω ( rad / s )
φ
PM=38.8°
ω ( rad / s ) Fig. 14.17 Bode plots of the simplified model G(ω, g)H(ω).
14.6 The HAC/LAC Strategy
341
20
|G*H|
(1) (2)
0
-20
w (rad/s)
-40 10
f
1
10
2
10
3
0°
-500°
-1000°
w (rad/s)
-1500° 10
1
10
2
10
3
(1)
(-1,0) (2)
Fig. 14.18 Open-loop transfer function G∗ (ω, g)H(ω) of the actual control system, Bode plots and Nyquist plot demonstrating the stability (with experimental G∗ ).
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examination showed that the potentially unstable mode corresponds to a local mode of the support of the mirror of the displacement measurement system. This local mode is not influenced by the active damping; the situation could be improved by a redesign of the support for more stiffness and more damping (e.g. passive damping locally applied). This controller has been implemented digitally on a DSP processor with a sampling frequency of 1000 Hz. Figure 14.19 compares the predicted step response with the experimental one. The settling time is reduced to 0.2 s, about 10 times faster than what would be achievable with a PID compensator.
1.4 Experiment
1.2 1 0.8
Simulation
0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
Time (sec) Fig. 14.19 Step response of the control system, comparison between predictions and experimental results of the top displacement.
14.7
Vibroacoustics: Volume Displacement Sensors
The general problem of Active Structural Acoustic Control (ASAC) of a baffled plate is represented in Fig.14.20. The performance objective is to minimize the far field radiated noise. The control system consists of one or several actuators acting on the baffled plate itself and a structural sensor measuring in real time the sound power radiated by the plate. This section is focused on the construction of a volume velocity (or displacement) sensor. The volume velocity V˙ of a vibrating plate is defined as w˙ dS (14.34) V˙ = S
where w is the transverse displacement of the plate and the integral extends over the entire plate area. It is a fairly important quantity in vibroacoustics, because it is strongly correlated to the sound power radiated by the
14.7 Vibroacoustics: Volume Displacement Sensors
Baffled plate
343
Transmitted noise
Rigid wall
~ = Sound power (at low frequencies)
Acoustic disturbance
Structural sensor
Actuator
Feedback controller
Fig. 14.20 Active Structural Acoustic Control (ASAC) of a baffled plate.
plate (Johnson & Elliott), and the modes which do not contribute to the net volume velocity (anti-symmetric modes for a symmetric plate) are poor contributors to the sound power radiations at low frequency (Fahy). In this section, we discuss the sensing of the volume velocity with an arrangement of piezoelectric sensors; note that the same sensor arrangement can be used to measure the volume displacement V by using a charge amplifier instead of a current amplifier as we discussed in Fig.4.5, so that the two quantities are fully equivalent from a sensor design viewpoint. This section examines three totally different concepts for sensing the volume displacement with piezoelectric sensors; the first one is based on a distributed sensor initially developed for beams, and extended to plates by discretizing them into narrow strips; it is biased, due to the inability of the beam theory to represent two-dimensional structures. The second is based on a discrete array sensor connected to a linear combiner; it is subjected to spatial aliasing. The third concept is based on a porous electrode design which allows to tailor the effective piezoelectric properties of piezoelectric films.
14.7.1
QWSIS Sensor
The Quadratically Weighted Strain Integrator Sensor (QWSIS ) is a distributed sensor which applies to any plate without rigid body mode (Rex & Elliott).
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Fig. 14.21 Beam covered with a PVDF sensor with a parabolic electrode.
Beam Consider a beam fixed at both ends: w(0) = w(a) = 0; it is covered with a piezoelectric film sensor (e.g. PVDF) with a parabolic electrode, as indicated in Fig.14.21; the profile of the electrode is defined by x x bp (x) = 4Δ (1 − ) a a
(14.35)
According to Equ.(4.32), if the electrode is connected to a charge amplifier, the sensor output is a
v0 ∼
0
bp w dx
(14.36)
Upon integrating by parts twice, and taking into account the boundary conditions bp (0) = bp (a) = 0 and w(0) = w(a) = 0, we get a v0 ∼ bp w dx (14.37) 0
Since the width of an electrode of parabolic shape has a constant second derivative with respect to the space coordinate, the output of the sensor is proportional to the volume displacement: a v0 ∼ w dx (14.38) 0
Plate In the QWSIS, the plate is discretized in a set of narrow strips (Fig.14.22) which are provided with parabolic electrodes connected in series; if we consider the elementary strips as beams, the total amount of electric charge is proportional to the volume displacement of the plate. The QWSIS sensor is based on the beam theory, but the actual behavior of the plate produces curvatures in two directions; assuming that the
14.7 Vibroacoustics: Volume Displacement Sensors
345
Fig. 14.22 QWSIS sensor.
piezoelectric orthotropy axis 1 of the sensor coincides with the x axis of the strip, the amount of electric charges generated by the sensor can be obtained by integrating Equ.(4.80) over the electrode area Ω, with the electrical boundary condition E = 0 enforced by the charge amplifier: Q= D dS = (e31 S1 + e32 S2 ) dS (14.39) Ω
Ω
where S1 and S2 are the strain components along the orthotropy axes in the mid-plane of the sensor. If the membrane strains in the plate are small as compared to the bending strains, ∂ 2w ∂2w Q = −zm (e31 2 + e32 2 ) dS (14.40) ∂x ∂y Ω where zm is the distance between the mid-plane of the sensor and the midplane of the baffle plate [see Equ.(4.95)]. If e32 = 0, this equation is reduced to that of a beam, which means that the convergence of the sensor is guaranteed when the number of strips increases. However, although strongly anisotropic, PVDF exhibits a piezoelectric coefficient e32 which is at least 20% of e31 , which introduces a bias in the sensor output.
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14 Applications
Dual actuator A piezoelectric strip can be used either as a sensor or as an actuator; in the latter case, according to the beam theory, a distributed actuator of width bp (x) produces a distributed load proportional to the second derivative of the width of the electrode, bp (x). Accordingly, if the QWSIS is used as an actuator, it is equivalent to a uniform pressure actuator (Fig.4.3.c). This led to the idea of building a collocated active structural acoustic (ASAC) plate with one side covered with a QWSIS volume displacement sensor and the opposite side covered with the dual actuator (Gardonio et al.). Unfortunately, such an arrangement performs poorly, because the input-output relationship between the strain actuator and the strain sensor is dominated by the membrane strains in the plate, which have been ignored in the theory, and are not related to the transverse displacements w (w is the useful output of the system). The anisotropy of PVDF can be exploited to improve the situation, by placing the strips of the actuator and the sensor orthogonal to each other (Piefort, p.91).
14.7.2
Discrete Array Sensor
In this section, we discuss an alternative set-up using a discrete array of n strain sensors bonded on the plate according to a regular mesh (Fig.14.23). The strain sensors consist of piezo patches connected to individual charge amplifiers with output Qi ; they are connected to a linear combiner, the output of which is n y= αi Qi (14.41) i=1
The coefficients of the linear combiner can be adjusted by software in order that the sensor output y be as close as possible (in some sense) of a desired quantity such as a modal amplitude, or, in this case, the volume displacement. The electric charges Qi generated by each strain sensor is a linear combination of the modal amplitude zj : qij zj (14.42) Qi = j
where qij is the electric charge generated on sensor i by a unit amplitude of mode j. The volume displacement V is also a linear combination of the modal amplitude, m V = Vj zj (14.43) j=1
where Vj is the modal volume displacement of mode j. At low frequency, V is dominated by the contribution of the first few modes and therefore only
14.7 Vibroacoustics: Volume Displacement Sensors
347
Fig. 14.23 Principle of the discrete array sensor of n patches.
these modal amplitudes, zj , j = 1, ..., m, have to be reconstructed from the electric charges Qi produced by a redundant set of piezoelectric strain sensors (n > m), leading to aji Qi (14.44) zj = i
(where the coefficients aij are unknown at this stage). Combining with Equ.(14.43), we find V = Vj aji Qi = αi Qi (14.45) j
i
where αi =
i
Vj aji
(14.46)
j
Equation (14.45) has the form of a linear combiner with constant coefficients αi ; it can be rewritten in the frequency domain V (ω) =
n
αi Qi (ω)
(14.47)
i=1
where V (ω) is the FRF between a disturbance applied to the baffled plate and the volume displacement, measured with a laser scanner vibrometer, Qi (ω) is the FRF between the same disturbance and the electric charge on sensor i in the array. If this equation is written at a set of l discrete frequencies (l > n) regularly distributed over the frequency band of interest, it can be transformed into a redundant system of linear equations,
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14 Applications
⎛
Q1 (ω1 ) ⎜ Q1 (ω2 ) ⎜ ⎝ Q1 (ωl )
⎞⎛ ⎞ ⎛ ⎞ ... Qn (ω1 ) α1 V (ω1 ) ⎜ ⎟ ⎜ ⎟ ... Qn (ω2 ) ⎟ ⎟ ⎜ α2 ⎟ = ⎜ V (ω2 ) ⎟ ⎠ ⎝ ⎠ ⎝ ... ... ... ⎠ ... Qn (ωl ) αn V (ωl )
(14.48)
or, in matrix form, Qα = V
(14.49)
where Q is a complex valued rectangular matrix (l×n), V is a complex-valued vector and α is the vector of linear combiner coefficients (real). Since the FRFs Q and V are determined experimentally, the solution of this redundant system of equations requires some care to eliminate the effect of noise; the coefficients resulting from the pseudo-inverse in the mean-square sense α = Q+ V
(14.50)
are highly irregular and highly sensitive to the disturbance source. This difficulty can be overcome by using a singular value decomposition of Q, Q = U1 ΣU2H
(14.51)
where U1 and U2 are unitary matrices containing the eigenvectors of QQH and QH Q, respectively (the superscript H stands for the Hermitian, that is the conjugate transpose), and Σ is the rectangular matrix of dimension (l × n) with the singular values σi on the diagonal (equal to the square root of the eigenvalues of QQH and QH Q). If ui are the column vectors of U1 and vi are the column vectors of U2 , Equ.(14.51) reads Q=
n
σi ui viH
(14.52)
i=1
and the pseudo-inverse is Q+ =
n 1 vi uH i σi
(14.53)
i=1
This equation shows clearly that, because of the presence of 1/σi , the lowest singular values tend to dominate the pseudo-inverse; this is responsible for the high variability of the coefficients αi resulting from Equ.(14.50). The problem can be solved by truncating the singular value expansion (14.53) and deleting the contribution relative to smaller singular values which are dominated by the noise. Without noise, the number of singular values which are significant (i.e. the rank of the system) is equal to the number of modes responding significantly in the frequency band of interest (assuming this number smaller than the number n of sensors in the array); with noise, the selection is slightly more difficult, because the gap in magnitude between significant and
14.7 Vibroacoustics: Volume Displacement Sensors
349
insignificant singular values disappears; some trial and error is needed to identify the optimum number of singular values in the truncated expansion (Fran¸cois et al.). Figure 14.24 shows typical results obtained with a glass plate covered with an array of 4 × 8 PZT patches.
14.7.3
Spatial Aliasing
The volume displacement sensor of Fig.14.24 is intended to be part of a control system to reduce the sound transmission through a baffled plate in the low frequency range (below 250Hz ), where the correlation between the volume velocity and the sound power radiation is high. Figure 14.24.c shows that the output of the array sensor follows closely the volume displacement below 400Hz. However, in order to be included in a feedback control loop, the quality of the sensor must be guaranteed at least one decade above the intended bandwidth of the control system. Figure 14.25 shows a numerical simulation of the open-loop FRF of a SISO system where the input consists of 4 point force actuators controlled with the same input current and the output is the volume displacement of the 4 × 8 array sensor. The comparison of the sensor output with the actual volume displacement reveals substantial differences at higher frequency, the amplitude of the sensor output being much
Fig. 14.24 (a) Experimental set-up: glass plate covered with an array of 4 × 8 PZT patches. (b) Coefficients of the linear combiner to reconstruct the volume displacement. (c) Comparison of the volume displacement FRF obtained with the array sensor and a laser scanner vibrometer.
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(a)
(b)
x 10
4
6 5 4 3 2 1 0 1 2 3 4
point force actuators PZT patches -50
(c)
Sensor output
dB
-100
-150
Volume displacement
-200
-250 1 10
10
2
10
3
10
4
Phase (deg)
0 -1000 -2000 -3000 1 10
10
2
10 Frequency (Hz)
3
10
4
Fig. 14.25 (a) Geometry of the 4 × 8 array sensor and the 4 point force actuators (controlled with the same input current), (b) Weighting coefficients αi of the linear combiner, (c) Comparison of the FRF between the actuators and the volume displacement, and the sensor output (numerical simulation).
14.7 Vibroacoustics: Volume Displacement Sensors
351
larger than the actual volume displacement, which is not acceptable from a control point of view, for reasons which have been discussed extensively in chapter 10. This is due to spatial aliasing, as explained in Fig.14.26. The left part of the figure shows the shape of mode (1,1) and mode (1,15); the diagrams on the right show the electric charges Qi generated by the corresponding mode shape on the PZT patches. We observe that the electric charges generated by mode (1,15) have the same shape as those generated by mode (1,1). Thus, at the frequency 1494.4Hz, the plate vibrates according to mode (1,15) which contributes only little to the volume displacement; however the output of the array sensor is the same as that of mode (1,1) which contributes a lot to the volume displacement; this explains why the high frequency amplitude of the FRF are much larger than expected. Note that it is a typical property of aliasing that a higher frequency component is aliased into a lower frequency component symmetrical with respect to the Nyquist frequency. In this case, the number of patches in the array being 8 along the length of the plate, mode (1,15) is aliased into the symmetrical one with respect to 8, that is into mode (1,1); similarly, mode (1,13) would be aliased into mode (1,3). The most obvious way to alleviate aliasing is to increase the sampling rate, that is, in this case, to increase the size of the array; this is illustrated in Fig.14.27 where one can see that an array of 16 × 32 gives a good agreement up to 5000Hz. However, dealing with such big arrays brings practical problems with the need for independent conditioning electronics (charge amplifier) prior to the linear combiner. If one accepts to give up the programmability of the linear combiner, the coefficients αi can be incorporated into the size of the electrodes, leading to the design of Fig.14.28, which requires only a single charge amplifier. The shape of this sensor has some similarity with the QWSIS.2
14.7.4
Distributed Sensor
For the design of Fig.14.28 involving an electrode connecting 16 × 32 patches of variable size, the spatial aliasing still occurs above 10000 Hz; it can be pushed even further by increasing the number of patches. This suggests the distributed sensor with a single “porous” electrode shown in Fig.14.29. The electrode is full in the center of the plate and becomes gradually porous as one moves towards the edge of the plate, to achieve an electrode density which produces the desired weighting coefficient α(x, y). This pattern can be placed on one side or on the two sides of the piezo material; for manufacturing, it seems simpler to apply the pattern on one side only, with a continuous electrode on the other side. 2
The electrode shape in Fig.14.28 is nearly that obtained by cutting parabolic strips in two orthogonal directions.
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14 Applications
Fig. 14.26 Modes shapes (1,1) and (1,15) and electric charges Qi generated by mode (1,1) and mode (1,15).
The amount of electrical charges generated on the electrode is given by Equ.(14.39) where the integral extends over the area of the electrode; it follows that tailoring the porosity of the electrode (i.e. Ω) is equivalent to tailoring the piezoelectric constants of the material, e31 and e32 . Equation (14.39) assumes that the size of the electrode is much larger than its thickness. However, when the pattern of the electrode becomes small, tridimensional (edge) effects start to appear and the relationship between the porosity and the equivalent piezoelectric property is no longer linear.
14.7 Vibroacoustics: Volume Displacement Sensors
353
Fig. 14.27 Effect of the size of the array on the open-loop FRF (a) 8 × 16, (b) 16 × 32 (simulations).
Fig. 14.28 Variable size array with 16 × 32 patches interconnected (simulations).
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Fig. 14.29 (a) Porous electrode, (b) detail of the pattern with variable porosity, (c) double sided pattern (fraction of electrode area = 50 %), (d) single sided pattern (the other electrode is continuous).
14.7 Vibroacoustics: Volume Displacement Sensors
355
Fig. 14.30 Tridimensional finite element analysis; the sample is strained in the direction S1 , while V = 0 is enforced between the electrodes. The equipotential surfaces show clearly the edge effect. (a) Two-sided electrode. (b) One-sided electrode.
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Fig. 14.31 Effective piezoelectric coefficient vs. fraction of electrode area, for PVDF films of 10 μm and 100 μm thickness.
The exact relationship between the porosity and the equivalent piezoelectric coefficients can be explored with a tridimensional finite element analysis. Figure 14.30 shows the equipotential surfaces for the two electrode configurations when a small sample (1 mm × 1 mm × 100 μm) is subjected to a strain along the x axis and a potential difference V = 0 is enforced between the electrodes; the material assumed in this study is isotropic PVDF polarized in the direction perpendicular to the electrodes; the edge effects appear clearly in the figures. For this sample, Fig.14.31 shows the relationship between the effective piezoelectric coefficient and the fraction of electrode area;
14.8 Problems
357
the two electrode configurations are considered for two sample thicknesses (10 μm and 100 μm); we observe that for a very thin sensor, the two electrode configurations produce the same results and the relationship is almost linear. The potential of this concept of “porous” electrode for shaping the effective piezoelectric properties of the material for two-dimensional structures is far beyond the design of a volume displacement sensor. Modal filtering is another obvious application. A transparent sensor has been realized for window applications (Preumont et al., 2005).
14.8
Problems
P.14.1 An anti-aliasing filter of bandwidth ωc can be obtained by cascading second order filters of the form (s/ωc
)2
ω2 + 2ξω(s/ωc ) + ω 2
The Butterworth filters correspond to order 2: ω = 1, ξ = 0.71 order 4: ω = 1, ξ = 0.38 ω = 1, ξ = 0.92 order 6: ω = 1, ξ = 0.26 ω = 1, ξ = 0.71 ω = 1, ξ = 0.97 The Bessel filters correspond to order 2: ω = 1.27, ξ = 0.87 order 4: ω = 1.60, ξ = 0.62 ω = 1.43, ξ = 0.96 order 6: ω = 1.90, ξ = 0.49 ω = 1.69, ξ = 0.82 ω = 1.61, ξ = 0.98 (a) Compare the Bode plots of the various filters and, for each of them, evaluate the phase lag for 0.1ωc and 0.2ωc. (b) Show that the poles of the Butterworth filter are located on a circle of radius ωc according to the configurations depicted in Fig.11.1. (c) Show that, at low frequency, the Bessel filter has a linear phase, and can be approximated with a time delay (˚ Astr¨ om & Wittenmark). P.14.2 Show that the transfer function of the zero-order hold is H0 (s) =
1 − e−sT s
Show that the frequency response function is H0 (ω) =
2 ωT −jωT /2 sin .e ω 2
Draw the amplitude and phase plots. P.14.3 Using the bilinear transform, show that the discrete equivalent of Equ.(14.8) (14.9) is given by Equ.(14.10)-(14.13). P.14.4 Consider a truss structure with several identical active members controlled with the same control law (IFF) and the same gain. Making the proper
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assumptions, show that each closed-loop pole follows a root locus defined by Equ.(7.31), where the natural frequency ωi is that of the open-loop structure and the zero zi is that of the structure where the active members have been removed. P.14.5 For the active truss of section 14.2, show that the compensator δ=
g y s+a
is equivalent to δ = g/s provided that the breakpoint frequency a is such that a ω1 . Show that its digital counterpart is δi+1 =
2 − Ta gT δi + (yi+1 + yi ) 2 + Ta 2 + Ta
P.14.6 Consider a simply supported beam with a point force actuator and a collocated accelerometer at x = l/6. Assume that EI = 1 N m2 , m = 1 kg/m and l = 1 m. Design a compensator to achieve a closed-loop modal damping ξi > 0.1 for i = 1 and 2, using the Direct Velocity Feedback and a second order filter (see Problem 7.2). Draw the Bode plots for the two compensators and compare the phase margins. For both cases, check the effect of the delay corresponding to a sampling frequency 100 times larger than the first natural frequency of the system (ωs = 100 ω1 ) and that of the actuator dynamics, assuming that the force actuator is a proof-mass with a natural frequency ωp = ω1 /3 (assume ξp = 0.5).
15
Tendon Control of Cable Structures
15.1
Introduction
Cable structures are used extensively in civil engineering: suspended bridges, cable-stayed bridges, guyed towers, roofs in large public buildings and stadiums. The main span of current cable-stayed bridges (Fig.15.1) can reach more than 850 m (e.g. Normandy bridge, near Le Havre, in France). These structures are very flexible, because the strength of high performance materials increases faster than their stiffness; as a result, they become more sensitive to wind and traffic induced vibrations. Large bridges are also sensitive to flutter which, in most cases, is associated with the aeroelastic damping coefficient in torsion becoming negative above a critical velocity (Scanlan, 1974). The situation can be improved either by changing the aerodynamic shape of the deck, or by increasing the stiffness and damping in the system; the difficulty in active damping of cable structures lies in the strongly nonlinear behavior of the cables, particularly when the gravity loads introduce some sag (typical sag to length ratio is 0.5% for a cable-stayed bridge). The structure and the cables interact with linear terms (at the natural frequency of the cable ωi ) and quadratic terms resulting from stretching (at 2ωi ); the latter may produce parametric resonance if some tuning conditions are satisfied (parametric excitation has indeed been identified as the source of vibration in several existing cable-stayed bridges). Cable structures are not restricted to civil engineering applications; the use of cables to achieve lightweight spacecrafts was recommended in Herman Oberth’s visionary books on astronautics. Tension truss structures have already been used for large deployable mesh antennas. The use of guy cables is probably the most efficient way to stiffen a structure in terms of weight; in addition, if the structure is deployable and if the guy cables have been properly designed, they may be used to prestress the structure, to eliminate the geometric uncertainty due to the gaps. This chapter examines the possibility of connecting guy cables to active tendons to bring active damping into cable structures; the same strategy A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 359–383. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
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15 Tendon Control of Cable Structures
Fig. 15.1 Cable-stayed bridge and conceptual design of an active tendon.
applies to large space structures and to cable-stayed bridges and other civil engineering structures; however, the technology used to implement the control strategy is vastly different (piezoelectric actuators for space and hydraulic actuators for bridges).
15.2
Tendon Control of Strings and Cables
The mechanism by which an active tendon can extract energy from a string or a cable is explained in Fig.15.2 with a simplified model assuming only one mode (Rayleigh-Ritz) and for situations of increasing complexity. The simplest case is that of a linear string with constant tension To (Fig.15.2.a); the equation becomes nonlinear when the effect of stretching is added (cubic nonlinearity). In Fig.15.2.b, a moving support is added; the input u of this active tendon produces a parametric excitation,1 which is the only way one can control a string with this type of actuator. The difference between a string and a cable is the effect of gravity, which produces sag (Fig.15.2.c). In this case, the equations of motion in the gravity plane and in the plane orthogonal to it are no longer the same, and they 1
The excitation u appears as a parameter in the differential equation.
15.3 Active Damping Strategy
361
Fig. 15.2 Mechanism of active tendon control of strings and cables.
are coupled. In the gravity plane (z coordinate), the active tendon control u still appears explicitly as a parametric excitation, but also as an inertia term −αc u¨ whose coefficient αc depends on the sag of the cable; even for cables with moderate sag (say sag to length ratio of 1% or more), this contribution becomes significant and constitutes the dominant control term of the equation. On the contrary, in the out-of-plane equation (y coordinate), the tendon control u appears explicitly only through the parametric excitation, as for the string.
15.3
Active Damping Strategy
Figure 15.3 shows a schematic view of a cable-structure system, where the control u is the support displacement, T is the tension in the cable, z the transverse vibration of the cable and q the vibration of the structure; we seek a control strategy for moving the active tendon u to achieve active damping in the structure and the cable. Any control law based on the non-collocated measurements of the cable and structure vibration u = Ψ (z, z, ˙ q, q) ˙
(15.1)
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15 Tendon Control of Cable Structures
Cable
Active tendon
u
Structure
q z
K M
T 1
s
Fig. 15.3 Cable-structure system with an active tendon.
must, at some stage, rely on a simplified model of the system; as a result, it is sensitive to parametric variations and to spillover. Such control laws have been investigated by (Chen, 1984) and (Fujino & coworkers) with very limited success; it turned out that the control laws work in specific conditions, when the vibration is dominated by a single mode, but they become unstable when the interaction between the structure and the cable is strong, which removes a lot of their practical value. By contrast, we saw in section 13.7 that, if a force sensor measuring the tension T in the cable is collocated with the active tendon, the positive Integral Force Feedback t
u=g
T (τ ) dτ
(15.2)
0
produces an energy absorbing controller, which can only extract energy from the system. However, for cable-structure applications, a high-pass filter is necessary to eliminate the static tension in the cable.2 2
To establish the vibration absorbing properties of Equ.(15.2) when T is the dynamic component of the tension in the cable, one can show that the dynamic contribution to the total energy, resulting from the vibration around the static equilibrium position, is a Lyapunov function. Thus, the stability is guaranteed if we assume perfect sensor and actuator dynamics. Note that the fact that the global stability is guaranteed does not imply that all the vibration modes are effectively damped. In fact, from a detailed examination of the dynamic equations (e.g. Fujino or Achkire), it appears that not all the cable modes are controllable with this actuator and sensor configuration. The odd numbered in-plane modes (in the gravity plane) can be damped substantially because they are linearly controllable by the active tendon (inertia term in Fig.15.2.c) and linearly observable from the tension in the cable; all the other cable modes are controllable only through active stiffness variation (parametric excitation in Fig.15.2), and observable from quadratic terms due to cable stretching. However, these weakly controllable modes are never destabilized by the control system, even at the parametric resonance, when the natural frequency of the structure is twice that of the cable.
15.4 Basic Experiment
15.4
363
Basic Experiment
Figure 15.4.a shows the test structure that was built to represent the ideal situation of Fig.15.3; the cable is a 2 m long stainless steel wire of 0.196 mm2 cross section provided with additional lumped masses at regular intervals, to achieve a sag to span ratio comparable to actual bridges; the active tendon is materialized by a piezoelectric linear actuator acting on the support point with a lever arm, to amplify the actuator displacement by a factor 3.4; this produces a maximum axial displacement of 150 μm for the moving support. A piezoelectric force sensor is colinear with the actuator; because of the Mass for frequency adjustment Lumped mass
Actuator
Accelerometer
Force Sensor
Force sensor
Shaker Charge amplifier Current amplifier Strain gauge
Mass for initial tension adjustment
Amplifier
DSP Microprocessor
Wave generator
Fig. 15.4 (a) Cable-structure laboratory model. (b) Experimental frequency response between the shaker force and the accelerometer, and free response of the structure, with and without control.
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15 Tendon Control of Cable Structures
high-pass behavior of this type of sensor, it measures only the dynamic component of the tension in the cable. The spring-mass system (in black on the figure) has an adjustable mass whereby the natural frequency can be tuned; a shaker and an accelerometer are attached to it, to evaluate the performance of the control system. In addition, a non-contact laser measurement system was developed to measure the cable vibration (Achkire). Figure 15.4.b shows the effect of the control system on the structure; we see that the controller brings a substantial amount of damping to the system. As far as the cable modes are concerned, the out-of-plane modes and the anti-symmetric in-plane modes are not affected by the controller (except for large amplitudes where the cable stretching becomes significant); the amount of active damping brought into the symmetric in-plane modes depends very much on the sag to span ratio. The control system behaves nicely, even at the parametric resonance, when the natural frequency of the structure is exactly twice that of the cable. This experiment was the first demonstration of robust active damping of a cable-structure; it demonstrates that active damping can be achieved without fear of destabilizing the cables, in spite of their complex dynamics; it also suggests that a simple treatment of the cables is acceptable in the design of the control system.
15.5
Linear Theory of Decentralized Active Damping
In this section, we follow an approach similar to that of section 7.5 to predict the closed-loop poles of the cable-structure system. Each active tendon consists of a displacement piezoelectric actuator co-linear with a force sensor. Ti is the tension in the active cable i, measured by the sensor integrated in the active tendon, and δi is the free extension of the actuator, the variable used to control the system. ki is the combined axial stiffness of the cable and the active tendon. The control is decentralized, so that each control loop operates independently. We assume that the dynamics of the active cables can be neglected and that their interaction with the structure is restricted to the tension Ti in the active cables (Fig.15.5). Accordingly, the governing equation is Mx ¨ + Kx = −BT + f (15.3) where x is the vector of global coordinates of the finite element model, M and K are respectively the mass and stiffness matrices of the passive structure (including a linear model of the passive cables, if any, but excluding the active cables). The right hand side represents the external forces applied to the system; f is the vector of external disturbances (expressed in global coordinates), T = (T1 , . . . , Ti , . . .)T is the vector of tension in the active cables and B is the influence matrix of the cable forces, projecting the cable forces in the global coordinate system (the columns of B contain the direction cosines of the various active cables).
15.5 Linear Theory of Decentralized Active Damping
365
f
f Active tendon
e r rc ce Fo sdu n
tra
Ti
(s) g.h
Ti
di
Ti
ric ct r le ato o e tu e z ac Pi ear lin
ki di Active tendon
Fig. 15.5 Left: Cable-structure system with active tendons and decentralized control. Center: Active tendon. Right: Passive structure. Ti is the tension in the active cable i of axial stiffness ki and free active displacement δi .
If we neglect the cable dynamics, the active cables behave like (massless) bars. If δ = (δ1 , . . . , δi , . . .)T is the vector of unconstrained active displacements of the active tendons acting along the cables, the tension in the cables are given by T = Kc (B T x − δ) (15.4) where Kc = diag(ki ) is the stiffness matrix of the cables, B T x are the relative displacements of the end points of the cables projected along the chord lines. This equation expresses that the tension in the cable is associated with the elastic extension of the cable. Combining Equ.(15.3) and (15.4), we get Mx ¨ + (K + BKc B T )x = BKc δ + f
(15.5)
This equation indicates that K + BKc B T is the stiffness matrix of the structure including all the guy cables (passive + active). Next, we assume that all the active cables are controlled according to the force feedback law: δ = gh(s).Kc−1T
(15.6)
where gh(s) is the scalar control law applied to all control channels. (note that Kc−1 T represents the elastic extension of the active cables). Combining Equ.(15.4) to (15.6), the closed-loop equation is
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15 Tendon Control of Cable Structures
[M s2 + K +
1 .BKc B T ]x = f 1 + gh(s)
(15.7)
It is readily observed that the open-loop poles, solutions of the characteristic equation for g = 0, satisfy [M s2 + K + BKc B T ]x = 0
(15.8)
(the solutions are the eigenvalues of the structure with all cables), while the zeros, solutions of Equ.(15.7) for g −→ ∞, satisfy [M s2 + K]x = 0
(15.9)
which is the eigenvalue problem for the open-loop structure where the active cables have been removed. If a IFF controller is used, h(s) = s−1 and the closed-loop equation becomes s [M s2 + K + BKc B T ]x = f (15.10) s+g which indicates that the closed-loop static stiffness matrix lim [M s2 + K + s=0
s BKc B T ] = K s+g
(15.11)
This means that the active cables do not contribute to the static stiffness and this may be problematic in some applications, especially in presence of gravity loads. However, if the control is slightly changed into the “Beta controller ” gh(s) =
gs (s + β)2
(15.12)
where β is small and positive (the influence of β will be discussed later), the closed-loop equation becomes [M s2 + K +
(s + β)2 BKc B T ]x = f gs + (s + β)2
(15.13)
and the closed-loop static stiffness matrix becomes lim [M s2 + K + s=0
(s + β)2 BKc B T ] = K + BKc B T gs + (s + β)2
(15.14)
which indicates that the active cables have a full contribution to the static stiffness. Next, let us project the characteristic equation on the normal modes of the structure with all the cables, x = Φz, which are normalized according to ΦT M Φ = 1. According to the orthogonality condition, ΦT (K + BKc B T )Φ = Ω 2 = diag(Ωi2 )
(15.15)
15.5 Linear Theory of Decentralized Active Damping
367
where Ωi are the natural frequencies of the complete structure. In order to derive a simple and powerful result about the way each mode evolves with g, let us assume that the mode shapes are little changed by the active cables, so that we can write ΦT KΦ = ω 2 = diag(ωi2 ) (15.16) where ωi are the natural frequencies of the structure where the active cables have been removed. It follows that the fraction of modal strain energy is given by φT BKc B T φi Ωi2 − ωi2 νi = T i = (15.17) Ωi2 φi (K + BKc B T )φi Considering the IFF controller, the closed-loop characteristic equation (15.10) can be projected into modal coordinates, leading to (s2 + Ωi2 ) − or 1+g
g (Ω 2 − ωi2 ) = 0 g+s i
s2 + ωi2 =0 s(s2 + Ωi2 )
(15.18)
which is identical to (7.31). This result indicates that the closed-loop poles can be predicted by performing two modal analyzes (Fig.15.6), one with all the cables, leading to the open-loop poles ±jΩi , and one with only the passive cables, leading to the open-loop zeros ±jωi, and drawing the independent root loci (15.18). As in section 7.5, the maximum modal damping is given by ξimax =
Ωi − ωi 2ωi
(15.19)
and it is achieved for g = Ωi Ωi /ωi . This equation relates directly the maximum achievable modal damping with the spacing between the pole Ωi and the zero ωi , which is essentially controlled by the fraction of modal strain energy in the active cables, as expressed by Equ.(15.17). The foregoing results are very easy to use in design. Although they are based on several assumptions (namely that the dynamics of the active cables can be neglected, the passive cables behave linearly and that the mode shapes are unchanged), they are in good agreement with experiments as shown below. If, instead of the IFF controller, the Beta controller is used, Equ.(15.12), the closed-loop characteristic equation projected into modal coordinates reads (s2 + Ωi2 ) − or 1+g
gs (Ω 2 − ωi2 ) = 0 gs + (s + β)2 i
s(s2 + ωi2 ) =0 (s + β)2 (s2 + Ωi2 )
(15.20)
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15 Tendon Control of Cable Structures
natural frequency with the active cables
active cables removed
Fig. 15.6 Root locus of the closed-loop poles with an IFF controller.
Thus, as compared to the IFF controller, the pole at the origin has been replaced by a zero at the origin and a pair of poles at −β on the real axis. The effect of this change on the root locus is shown in Fig.15.7. When β = 0, there is a pole-zero cancellation and the control is reduced to the IFF. As β increases, the root locus has two branches on the real axis, starting from s = −β in opposite directions; one of the closed-loop poles remains trapped between 0 and −β; the loop still go from ±jΩi to ±jωi , but they tend to be smaller, leading to less active damping; this is the price to pay for recovering the static stiffness of the active cables. Analyzing the root locus in detail, one can show that the system is unconditionally stable (for all modes) provided that β < ω1 .
15.6
Guyed Truss Experiment
This experiment aims at comparing the closed-loop predictions of the linear model with experiments. The test structure consists of the active truss
15.6 Guyed Truss Experiment
369
Im(s)
Wi wi
b/wi = 0 0.25
0.5
Wi b/wi = 1 b/wi = 0.5
-b
wi
Re(s)
Fig. 15.7 Root locus of the closed-loop poles with the Beta controller gs/(s + β)2 , for various values of the ratio β/ωi . The IFF controller corresponds to β = 0. The locus is always stable for β < ωi ; for β = ωi , it is tangent to the imaginary axis at the zero ±jωi .
Fig. 15.8 (a) Guyed truss. (b) Design of the active tendon.
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15 Tendon Control of Cable Structures
Fig. 15.9 Experimental vs. analytical closed-loop poles.
of Fig.4.20 equipped with three identical cables made of synthetic fiber “Dynema” of 1 mm diameter (Fig.15.8.a); the tension in the cables is not important provided that the effective Young modulus (due to sag) is close to the actual one; in this experiment, the tension is such that the cable frequency is above 500 rad/sec. The design of the active tendon is shown in Fig.15.8.b (a better design is shown in Fig.15.13); the amplification ratio of the lever arm is 3, leading to a maximum stroke of 150 μm. The natural frequencies with and without the active cables are respectively Ω1 = 67.9 rad/s, ω2 = 53.8 rad/s, Ω2 = 78.9 rad/s, ω2 = 66 rad/s. Figure 15.9 shows the root locus predicted by the linear model together with the experimental results for various values of the gain; only the upper part of the loops is available experimentally because the control gain is limited by the saturation due to the finite stroke of the actuators. The agreement between the experimental results and the linear predictions of Equ.(15.18) is quite good.
15.7
Micro Precision Interferometer Testbed
To illustrate further the application of the control strategy to the damping of large space trusses, let us consider a numerical model of the microprecision interferometer (MPI) testbed used at NASA Jet Propulsion Laboratory (JPL) to develop the technology of precision structures for future interferometric missions (Neat et al.). The first three flexible modes are displayed in Fig.15.10. We investigate the possibility of stiffness augmentation and active
15.7 Micro Precision Interferometer Testbed
371
Fig. 15.10 JPL-MPI testbed, shape of the first three flexible modes (by courtesy of R. Laskin-JPL).
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15 Tendon Control of Cable Structures
damping of these modes with a set of three active tendons acting on Kevlar cables of 2 mm diameter, connected as indicated in Fig.15.11 (Kevlar properties : E = 130 GP a, ρ = 1500 kg/m3 , tensile strength σy = 2.8 GP a). The global added mass for the three cables is only 110 gr (not including the active tendons and the control system). The natural frequencies of the first three modes, with and without the cables, are reported in Table 15.1; the root locus of the three global flexible modes as functions of the control gain g are represented in Fig.15.12; for g = 116 rad/s, the modal damping ratios are ξ7 = 0.21, ξ8 = 0.16, ξ9 = 0.14.
Fig. 15.11 Proposed location of the active cables in the JPL-MPI testbed. Table 15.1 Natural frequencies (rad/s) of the first flexible modes of the JPL-MPI testbed, with and without cables. i 7 8 9
15.8
ωi 51.4 76.4 83.3
Ωi 74.6 101 106.4
ξimax 0.23 0.16 0.14
Free Floating Truss Experiment
In order to confirm the spectacular analytical predictions obtained with the numerical model of the JPL-MPI testbed, a similar structure (although smaller) was built and tested (Fig.15.13); the free-floating condition was simulated by hanging the structure with soft springs. The active tendon consists of a APA 100 M amplified actuator from CEDRAT Recherche together with a B&K 8200 force sensor and flexible tips (this design is much simpler than that used earlier, Fig.15.8). The stroke is 110 μm and the total weight of the tendon is 55 gr; the cable is made of Dynema with axial stiffness EA = 19000 N .
15.8 Free Floating Truss Experiment
373
Fig. 15.12 Analytical prediction of the closed-loop poles. Table 15.2 Natural frequencies (rad/s) of the free floating truss, with and without cables. i 1 2 3 4
ωi 119.4 157.1 165.7 208.1
Ωi 146.1 173.6 205.8 220.7
The natural frequencies of the first flexible modes, with and without cables, are reported in Table 15.2. Figure 15.14 compares the analytical predictions of the linear model and the experiments.3 3
All the results discussed above have been obtained for vibrations in a range going from millimeter to micron; in order to apply this technology to future large space platforms for interferometric missions, it is essential that these results be confirmed for microvibrations. In fact, it could well be that, for very small amplitudes, the behavior of the control system be dominated by the nonlinearity of the actuator (hysteresis of the piezo) or the noise in the sensor or in the voltage amplifier. Tests have been conducted for vibrations of decreasing amplitudes, and the influence of the various hardware components has been analyzed (Bossens), these tests indicate that active damping is feasible at the nanometric level, provided that adequately sensitive components are used.
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15 Tendon Control of Cable Structures
Fig. 15.13 ULB free floating truss test structure and detail of the active tendon.
Fig. 15.14 ULB free floating truss test structure: Comparison between the analytical predictions of the linear model and the experiments. The numbers correspond to equal values of the gain.
15.11 Control of Parametric Resonance
15.9
375
Application to Cable-Stayed Bridges
In what follows, we summarize some of the findings of a research project called “ACE” which was funded by the EU in the framework of the BriteEuram program, between 1997 and 2000, and involved several academic and industrial partners. The overall objective of the project was to demonstrate the use of the active control in civil engineering. Several experiments were conducted, on different scales; the main results are explained below.
15.10
Laboratory Experiment
The test structure is a laboratory model of a cable-stayed bridge during its construction phase, which is amongst the most critical from the point of view of the wind response. The structure consists of two half decks mounted symmetrically with respect to a central column of about 2 m high (Fig.15.15); each side is supported by 4 cables, two of which are equipped with active piezoelectric tendons identical to those of Fig.15.8.b. The cables are provided with lumped masses at regular intervals, so as to match the sag to length ratio of actual stay cables [a discussion of the similarity aspects can be found in (Warnitchai et al.)]. Figure 15.16 compares the evolution of the first bending and torsion closed-loop poles of the deck with the analytical predictions of the linear theory. The agreement is good for small gains, when the modal damping is smaller than 20%.
15.11
Control of Parametric Resonance
In this experiment, the bridge deck is excited harmonically with an electrodynamic shaker at a frequency f close to the first torsion mode, and the tension in the two passive cables on one side is chosen in such a way that the first in-plane mode of one of them is tuned on the excitation frequency f , while the other is tuned on f /2, to experience the parametric resonance when the deck vibrates (Fig.15.17). This tuning is achieved by monitoring the cable vibration with a specially developed non-contact optical measurement system (Achkire). Figure 15.18 shows the vibration amplitude of the deck and the transverse amplitude of the in-plane mode of the two passive cables when the deck is excited at resonance; the excitation starts at t = 5 sec and the control is turned on after t = 30 sec. We note that: 1. The amplitude of the cable vibration is hundred times larger than that of the deck vibration. 2. The parametric resonance is established after some transient period in which the cable vibration changes from frequency f to f /2. The detail of the transition to parametric resonance is shown clearly in the central part of Fig.15.19 which shows a detail of Fig.15.18 in the range (10 < t < 14 sec).
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15 Tendon Control of Cable Structures
Fig. 15.15 Test structure used at ULB to demonstrate the control of parametric resonance. Above left, the Skarnsund cable-stayed bridge during construction (Norway). Im (s)
x=
0.5
0.2
deck torsion mode
5
90
x=
100
80 70 60 50
deck st 1 bending mode
Analytical prediction
40 30 20 10 0 -50
-40
-30
-20
-10
0 Re(s)
Fig. 15.16 Evolution of the first bending and torsion poles of the deck with the control gain.
15.12 Large Scale Experiment
377
Fig. 15.17 Set-up of the experiment of parametric resonance.
3. The control brings a rapid reduction of the deck amplitude (due to active damping) and a slower reduction of the amplitude of the cable at resonance f (due only to the reduced excitation from the deck, since no active damping is applied to this cable). 4. The control suppresses entirely the vibration of the cable at parametric resonance f /2. This confirms that a minimum deck amplitude is necessary to trigger the parametric resonance.
15.12
Large Scale Experiment
Although appropriate to demonstrate control concepts in labs, the piezoelectric actuators are inadequate for large scale applications. For cable-stayed bridges, the active tendon must simultaneously sustain the high static load (up to 400 t) and produce the dynamic load which is at least one order of magnitude lower than the static one (< ±10%). This has led to an active tendon design consisting of two cylinders working together: one cylinder pressurized by an accumulator compensates for the static load, and a smaller double rod cylinder drives the cable dynamically to achieve the control law. The two functions are integrated in a single cylinder, as illustrated in Fig.15.20; the double rod part of the cylinder is achieved by a “rod in rod” design; this solution saves hydraulic energy and reduces the size of the hydraulic components.
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15 Tendon Control of Cable Structures
0.04
Deck
mm
0.02 0 -0.02
Control on
-0.04
mm
4
Cable at parametric resonance (f /2)
2 0 -2 -4 4
mm
Cable at resonance (f)
2 0 -2 -4
0
10
20
30
40
Time (sec)
Fig. 15.18 Vibration amplitude under harmonic excitation of the bridge deck at f : deck, passive cable at parametric resonance f /2, passive cable at resonance f ; the active damping of the four control cables is switched on at t = 30 sec. The control suppresses entirely the vibration of the cable at parametric resonance.
The cylinder is position controlled; the long term changes of the static loads as well as the temperature differences require adaptation of the hydraulic conditions of the accumulator. The mock-up (Fig.15.21 and 15.22) was designed and manufactured by Bouygues in the framework of the ACE project; it has been installed on the reaction wall of the ELSA facility at the Joint Research Center in Ispra. It consists of a cantilever beam (l = 30 m) supported by 8 stay cables (d = 13 mm); the stay cables are provided with additional masses to achieve a representative sag-to-length ratio (the overall mass per unit length is 15 kg/m). An intermediate support can also be placed along the deck to tune the first global mode and the cable frequencies. Because of the actuator dynamics and the presence of a static load, the implementation of the control requires some alterations from the basic idea of Fig.15.4: (i) A high-pass filter must be included after the force sensor to eliminate the static load in the active cables. (ii) In hydraulics, the flow rate is directly related to the valve position which is the control element; it is therefore more natural to control the actuator velocity than its position. In addition, a proportional controller acting on the
15.12 Large Scale Experiment
379
Fig. 15.19 Detail of Fig.15.18 in the range (10 < t < 14 sec) showing the transition from the forced response at f to the parametric resonance at f /2.
Fig. 15.20 Conceptual design of the two-stage hydraulic actuator (by courtesy of Mannesmann Rexroth).
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15 Tendon Control of Cable Structures
actuator velocity is equivalent to an integral controller acting on the actuator displacement. The actual implementation of the control is shown in Fig.15.23. The overall controller includes a high-pass filter with a corner frequency at 0.1Hz (to eliminate the static load), an integrator (1/s), and a low-pass filter with corner at 20Hz, to eliminate the internal resonance of the hydraulic actuator. The overall FRF (u/T ) of the active control device is represented in Fig.15.24. The dotted line refers to the digital controller alone (between 1 and 3 in Fig.15.23) while the full line includes the actuator dynamics (between 1 and 2 in Fig.15.23). Notice that (i) the controller behavior follows closely a pure integrator in the frequency range of interest (0.5Hz − 2Hz) and (ii) the actuator dynamics introduces a significant phase lag above the dominant modes of the bridge. Figure 15.25 shows the envelope of the time response of the bridge deck displacement near the tip when a sweep sine input is applied to a proof-mass actuator (MOOG, max. inertial force 40 kN ) located off axis near the end of the deck (Fig.15.22). The sweep rate is very slow (from 0.5 Hz to 2 Hz in 1000 sec). The three curves correspond to various values of the gain of the decentralized controller when the two active tendons are in operation (g = 0 corresponds to the open-loop response). The instantaneous frequency of the input signal is also indicated on the time axis, to allow the
Fig. 15.21 (a) Global view of the large scale mock-up in the ELSA facility (JRC Ispra), (b) detail of the hydraulic actuator.
15.12 Large Scale Experiment
381
Fig. 15.22 Schematic view of the mock-up and location of the main components.
identification of the main contributions to the response. The numerous peaks in the envelope indicate a complex dynamics of the cable-deck system. One sees that the active tendon control brings a substantial reduction in the vibration amplitude of all modes, and especially the first global bending mode. Using a band-limited white noise excitation and a specially developed identification technique based on the spectral moments of the power spectral density of the bridge response, M. Auperin succeeded in isolating the first global mode of the bridge. Figure 15.26 compares the experimental root-locus with the predictions of the linear approximation; the agreement is surprisingly good, especially if one thinks of the simplifying assumptions leading to Equation (15.18) . The marks on the experimental and theoretical curves indicate the fraction of optimum gain g/g ∗ , where g ∗corresponds to the largest modal damping ratio (theoretical value g ∗ = Ωi Ωi /ωi ). Note that the maximum damping ratio is close to 17%.
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15 Tendon Control of Cable Structures
Fig. 15.23 Actual control implementation. Both position and velocity constitute the inputs of the local actuator controller.
Fig. 15.24 FRF between T and u (between 1 and 2 in Fig.15.23). The dotted line does not include the actuator dynamics (between 1 and 3 in Fig.15.23).
15.12 Large Scale Experiment
383
150
Deck displacement envelope (mm) 100
First bending mode g=0 g=2 g=15
Active cables resonance Mixed lateral & torsion modes
50
0
-50
-100
Time (sec) -150
0
100
200
300
0.5
0.65
0.8
0.95 1.1 1.25 1.4 1.55 Excitation instantaneous frequency (Hz)
400
500
600
700
800
900
1000
1.7
1.85
2
Fig. 15.25 Envelope of the time response of the bridge deck displacement when a sweep sine input (from 0.5 Hz to 2 Hz in 1000 sec) is applied to the proof-mass actuator, for various values of the control gain.
Fig. 15.26 Comparison of experimental and analytical root-locus of the first bending mode.
16
Active Control of Large Telescopes
16.1
Introduction
A reflective telescope (Fig.16.1) consists of a set of reflectors which transform a plane wavefront into a convergent spherical wavefront, in such a way that a point source at infinity forms a point image in the focal plane. However, because of diffraction, the image of a point object of a perfect (circular) telescope is not a point, but an area of finite size resulting from the spreading of the light energy, called Airy disk. Its size is directly related to the wavelength λ of the light observed. More generally, the image of a point object is called the Point Spread Function (PSF) of the instrument. The theoretical limit resolution of a telescope is proportional to the ratio λ/D between the wavelength and the diameter of the primary mirror M1. Increasing the
Point source
...
Secondary mirror (M2)
Spherical wavefront (divergent) Plane wavefront
Primary Mirror (M1)
Image
Spherical wavefront (convergent)
Fig. 16.1 Principle of a reflective telescope with “Alt-Az” mount (only the primary and the secondary mirrors are considered). The plane wave front is transformed into a convergent spherical wave front. A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 385–401. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
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16 Active Control of Large Telescopes
diameter D has two beneficial effects: (i) gathering more light (more photons), which allows the observation of fainter objects, and (ii) increasing the limit resolution of the telescope, which allows to distinguish finer details. Earth based telescopes are subjected to two broad classes of aberrations: (i) Atmospheric turbulence produces a temporal and spatial random variation of the refraction index of the air, which distorts the wavefront of incoming plane waves (Fig.16.2). Atmospheric turbulence can be cured by Adaptive Optics. (ii) Manufacturing errors, thermal gradients and the variations of the gravity loads as the telescope moves to follow a fixed target in the sky while the earth rotates, are responsible of low frequency but significant structural deformations, which can be compensated by Active Optics.
8
An optical system is considered as diffraction limited if the RMS wavefront error is less than λ/14 (0.4 μm < λ < 0.8 μm in the visible spectrum and 0.8 μm < λ < 5 μm in the near infrared); thus, more accuracy is required for shorter wavelength.
Plane wavefront
Atmospheric turbulence
Deformed wavefront
Telescope
Instrument Deformable mirror
Controller Wavefront analyser (Shack-Hartmann sensor)
Fig. 16.2 Principle of adaptive optics to correct atmospheric turbulence. A small deformable mirror is controlled in real time to compensate the wavefront aberrations, measured with a wavefront sensor.
16.2 Adaptive Optics
16.2
387
Adaptive Optics
The principle of adaptive optics (AO) is shown in Fig.16.2; a small deformable mirror is inserted in the optical train to counteract in real time the aberration of the wavefront. The most popular wavefront sensor is the so-called “Shack-Hartmann”; a beam splitter deviates part of the incoming light towards an array of micro lenses which measure the normals to the wavefront of a reference star at an array of discrete points in the telescope aperture. The wavefront aberrations are often expressed in a set of orthogonal functions called the Zernike polynomials (Fig.16.3).The design of an AO system depends on many parameters such as the wavelength, the field of view and the size of the primary mirror. Typically, the amplitude of the corrected shape is a few microns, and the bandwidth is in the range 50-100Hz, depending on the wavelength; it is essential that the natural frequency of the mirror be significantly higher than the bandwidth, to operate the deformable mirror in the quasi-static mode. The number of degrees of freedom (of independent Azimuthal Order -4
-3
-2
-1
0
1
2
3
4
0 Piston
1 Tilt
Radial order
Tilt
2 Astigmatism
Defocus
Astigmatism
3 Trefoil
Coma
Coma
Trefoil
4 Tetrafoil
Spherical Aberration
Fig. 16.3 Optical aberrations: low order Zernike polynomials.
Tetrafoil
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16 Active Control of Large Telescopes
actuators) also depends on the wavelength and on the size of the primary mirror; approaching the diffraction limit in the visible with a large telescope may require thousands of independent actuators. Figure 16.4 shows a bimorph deformable mirror made of a silicon wafer of 150 mm diameter and 800 μm thickness, covered with a screen printed thick film of PZT of 70 μm (Rodrigues); the honeycomb actuator array consists of 91 independent electrodes, the voltage of which can be adjusted independently between 0 and 160V.1 Fig. 16.5 shows examples of experimentally corrected aberrations and the corresponding voltage distribution. Let s be the output vector of the wavefront sensor (of dimension n) and v the vector (of dimension m) of input voltages applied to the independent electrodes. Assuming a linear relationship between s and v, one has s = Dv
(16.1)
where the (n × m) matrix D is the Jacobian of the system, which can be constructed column by column during the calibration phase of the AO system, by applying a given voltage to every input channel, one after the other. n may be larger, equal or smaller than m. Once the matrix D is available, the voltage v necessary to correct the wavefront corresponding to a sensor output s is obtained by performing a Singular Value Decomposition (SVD) of the Jacobian matrix according to D = U ΣV T
(16.2)
where the columns of U are the orthonormalized sensor modes, the columns of V are the orthonormalized actuator modes and Σ is a (n × m) rectangular matrix which contains the singular values σi on its diagonal. The number of non zero singular values is equal to the rank of the matrix D. The control voltage v necessary to correct the sensor error s is given by v = D+ s = V Σ −1 U T s
(16.3)
where Σ −1 is (m × n) with σi−1 on its diagonal. The rectangular matrix D+ is called the pseudo-inverse of the Jacobian matrix. The solution of the foregoing equation may have different meaning, depending on the relative size of n and m. If the sensor output vector is larger than the voltage input vector (n > m), the solution (16.3) is that minimizing the quadratic norm
1
An offset of 80V is applied to allow corrections of ±80V .
16.2 Adaptive Optics
389
Fig. 16.4 Deformable mirror made of a 150 mm silicon wafer covered on its back side with an array of screen printed PZT actuators with honeycomb electrodes (developed jointly with the Fraunhofer IKTS).
Defocus
Astigmatism
5µm
Tetrafoil
4.5µm
3µm
Fig. 16.5 Deformable mirror: typical corrected aberrations with the corresponding voltage distribution within the honeycomb electrodes.
390
16 Active Control of Large Telescopes
||s − Dv||; if m > n, the solution is that of minimum norm ||v|| satisfying the equality constraint (16.1).2
16.3
Active Optics
Spatial frequency
Adaptive optics is intended to correct the wavefront errors introduced by the atmospheric turbulence, which exist even in a perfect telescope operating on earth. However (Fig.16.2), the wavefront sensor cannot separate the wavefront error due to atmospheric turbulence from that due to the telescope imperfections and the adaptive optics will attempt to correct them as well. For large telescopes, however the telescope deformations due to manufacturing errors, thermal gradients and gravity loads are several orders of magnitude larger and cannot be fully corrected by the adaptive optics; they are alleviated with a specific control system called active optics.
M1 Shape
Adaptive Optics
100
3
M2 Rigid Body
2
Main axes 0.01
0.1
1
10
100
Bandwidth [Hz]
Fig. 16.6 Spatial and temporal frequency distribution of the various active control layers of extremely large telescopes (adapted from Angeli et al.). Adaptive optics has amplitudes of a few microns. The shape control of M1 involves much larger amplitudes. 2
The SVD controller will be investigated below in relation with active optics. Note that the tracking errors involved in adaptive optics are typically of the order of ∼ 10λ. Since an optical system is considered as diffraction limited if the RMS wavefront error is less than λ/14, the control objectives would be to reduce the RMS surface error to below λ/28 (the optical path difference is twice the surface error). Overall, this leads to a gain ∼ 50dB. Another issue is the control-structure interaction when the control bandwidth interferes with the vibration modes of the flexible mirror.
16.3 Active Optics
391
Figure 16.6 shows the various control layers of an extremely large telescope and their spatial and temporal frequency distribution. Adaptive optics covers a wide band of temporal frequencies as well as spatial frequencies (Zernike modes of higher orders), but with small amplitudes of a few microns. On the contrary, the control of the rigid body motion of the secondary mirror M2 and the active shape control of the primary mirror M1 must counteract disturbances of very low frequency (changes in gravity loads take place at one cycle per day, that is 1.16 10−5 Hz ), but with much larger amplitudes, in millimeters (>> 100λ); this requires larger gains than adaptive optics, and a different technology. We will examine successively the shape control of monolithic and segmented primary mirrors.
16.3.1
Monolithic Primary Mirror
Figure 16.7 shows the principle of the active optics for a monolithic primary mirror as it was first implemented on the ESO-NTT telescope (Wilson, 1987). The primary mirror consists of a thin deformable meniscus equipped with an array of force actuators on its back3 and the secondary mirror can also be actuated to correct defocus and coma.
M2 Beam splitters
Controller CCD ( f camera c ~ 0.1Hz)
Science instrument
ShackHartmann
Alt-az axes
M1
Controller ( fc = 0.03Hz) Fig. 16.7 Active optics of a monolithic primary mirror (ESO-NTT).
The control system uses a Shack-Hartmann sensor and the signal is averaged over a long period (30 s) to eliminate the effect of atmospheric 3
For the NTT (located in La Silla), the diameter of M1 is 3.6 m, the thickness is 0.24 m and there are 78 actuators; For the VLT (located in Paranal), the diameter of M1 is 8.2 m, the thickness is 0.17 m and there are 150 actuators.
392
16 Active Control of Large Telescopes
turbulence. The system is essentially described by Equ.(16.1), where v is the vector of control inputs (including the forces acting at the back of M1 and the position of M2) and s is the vector of sensor outputs. The Jacobian is once again determined column by column by analyzing the impact of every actuator on the wavefront sensor.
16.3.2
Segmented Primary Mirror
Monolithic mirrors are limited to a maximum size of about 8 m; larger mirrors must be segmented. The Keck (located in Hawa¨ı) and the Gran TeCan (located in the Cannary Islands) are the largest optical telescopes in operation, with a diameter of the primary mirror (M1) a little larger than 10 m; their primary mirror consists of 36 segments of hexagonal shape. Extremely Large Telescopes (ELTs), with a primary mirror of diameter up to 42 m (E-ELT will have 984 segments!), are currently in their design phase. Since the sensitivity to disturbance increases with the size of the telescope and the wavefront error cannot exceed a fraction of the wavelength of the incoming light, larger telescopes will rely more and more on active control, with higher gains, leading to wider bandwidth. On the other hand, since the natural frequencies of the structural system tend to be lower for larger telescopes, control-structure interaction becomes a central issue in the control system design. Figure 16.8 shows a schematic view of a segmented primary mirror, with the supporting truss. Every segment can be regarded as a rigid body; it is supported by 3 position actuators via a whiffle tree. A set of six edge sensors monitor the position of every segment with respect to its neighbors (overall, there will be 2952 position actuators and 5604 edge sensors for E-ELT); the edge sensors play the key role of co-phasing the various segments (i.e. making them work as a single, monolithic mirror). If the supporting truss is assumed
Wind
Segments
Edge sensors (y1)
y1 =Je .a Position Controller actuators (a)
Supporting truss fi Fig. 16.8 Schematic view of a segmented primary mirror, with the supporting truss, the position actuators and the edge sensors. The quasi-static behavior of the reflector follows y1 = Je a, where a is the control input (position of the actuators), y1 is the edge sensor output (relative displacement between segments) and Je is the Jacobian of the segmented mirror.
16.3 Active Optics n +
Normal to segment
y2
Atmospheric turbulence Edge Sensors
d (gravity, wind)
393
y1
Normal to wavefront (Shack-Hartmann)
m Segment
a ka ca
Fa
Position Actuators
H( s)
Supporting truss fi
Fig. 16.9 Active optics control flow for a large segmented mirror. The axial d.o.f. at both ends of the actuators are retained in the reduced dynamical model. Controlstructure interaction may arise from the force actuator Fa exciting the resonances fi of the supporting truss or the local modes of the segments.
infinitely stiff, the quasi-static behavior of the system is governed by a purely kinematic relationship y1 = Je .a (16.4) where Je is the Jacobian of the edge sensors. Since the lower optical modes (piston, tilt, defocus) are not observable from the edge sensors, another set of sensors measuring the normal to the segments (e.g. one per segment, or just a few for the entire mirror) may be used, leading to y2 = Jn .a
(16.5)
where Jn is the Jacobian of the normal sensors. Note, however, that the output signal y2 is also affected by atmospheric turbulence, which is eliminated by time averaging, just as for monolithic mirrors. The active optics control flow is shown in Fig.16.9. The mirror segments are represented by rigid bodies. In order to include the flexibility of the whiffle tree into the model, the position actuator can be modelled with a force actuator Fa acting in parallel on a spring ka and dash-pot ca ; the stiffness ka is selected to account for the local modes of the segments and ca to provide the appropriate damping. the force is related to the unconstrained displacement a by Fa = a.ka . The position actuators rest on the supporting truss carrying the whole mirror. The disturbances d acting on the system come from thermal gradients, changing gravity vector with the elevation of the telescope, and wind. Controlstructure interaction may arise from the force actuator Fa exciting the resonances fi of the supporting truss or the local modes of the segments. Before
394
16 Active Control of Large Telescopes
addressing the dynamic response of the mirror, let us discuss the quasi-static shape control of a deformable mirror.
16.4
SVD Controller
The quasi-static behavior of adaptive optics mirrors, or of deformable primary mirrors (whether monolithic or segmented) is described by an equation Je a = J.a (16.6) y= Jn where y is the output of some set of sensors and a is the input of some set of actuators and J is the Jacobian of the system. The block diagram of the SVD controller is shown in Fig.16.10; V Σ −1 U T is the inverse of the plant; the diagonal gains σi−1 provide equal authority on all singular value modes; only the modes with non-zero singular values are considered in the control block. H(s) is a diagonal matrix of filters intended to supply appropriate disturbance rejection and stability margin. The distribution of the singular values depends on the sensor configuration, as illustrated in Fig.16.11 which compares the use of edge sensors alone with the option combining edge sensors and normal sensors (the conditioning of the Jacobian is much better in this case).
16.4.1
Loop Shaping of the SVD Controller
The open loop transfer function of the nominal plant is G0 (s) = J = U ΣV T and the controller is K(s) = V H(s)Σ −1 U T . If one assumes the same loop shape for all singular value modes, H(s) = h(s)I and K(s) = h(s)V Σ −1 U T
(16.7)
essentially inverts the Jacobian of the mirror, leading to the open-loop transfer matrix G0 (s)K(s) being diagonal, with all non-zero singular values being equal to Primary mirror + Position actuators
a
d y
J = US V T
V
H(s)
Edge sensors
S -1
UT
Fig. 16.10 Block diagram of the SVD controller.
16.5 Dynamics of a Segmented Mirror
10 0
395
Trifoil, Defocus Trifoil, Coma
Astigmatism Tilt 10
-2
10 -15 Piston 0
Coma Trifoil Astigmatism Defocus Tilt Piston
edge sensors edge + normal
25
50 250
275
Index
Fig. 16.11 Distribution of the singular values of a segmented mirror for two sensor configurations (edge sensor measurements and actuator displacements are expressed in meters, tilt angles in radians).
σ(G0 K) = |h(jω)|
(16.8)
Thus, the loop shaping can be done as for a SISO controller, according to the techniques developed in Chapter 10. The control objective is to maximize the loop gain in the frequency band where the disturbance has a significant energy content while keeping the roll-off slow enough near crossover to achieve a good phase margin. An integral component is necessary to eliminate the static error in the mirror shape. The gain at the earth rotation frequency must be large enough to compensate the gravity sag. Figure 16.12 shows typical Bode plots and the corresponding Nichols chart; the compensator consists of an integrator, a lag filter, followed by a lead and a second order Butterworth filter. The crossover frequency is fc = 0.25Hz and the amplitude at the earth rotation frequency is 125dB. The robustness margins of this quasi-static controller are clearly visible on the Nichols chart [the exclusion zone around the critical point (−1800, 0dB ) corresponds to (PM= ±450 , GM= ±10dB )]. Note that the Nichols chart is invariant with respect to a shift of the Bode plots along the frequency axis, which gives a simple way to adjust the control bandwidth to achieve the low frequency specification. However, it is important to point out that the robustness margins displayed by the Nichols chart do not tell anything about the control-structure interaction, since our analysis has been based on a quasi-static model of the plant, ignoring the dynamics of the supporting truss.
16.5
Dynamics of a Segmented Mirror
The dynamics of the mirror consist of global modes involving the supporting truss and the segments, and local modes involving the segments alone. The
396
16 Active Control of Large Telescopes [dB] [dB]
100
f
125dB GM = 25dB 0
45° 0 dB
fc = 0.25Hz
Earth rotation (1.16 10 -5 Hz)
-100
10
-5
10
-4
10
-3
10
-2
[Hz]
10 dB [°]
10
-1
10
0
10
1
10
2
[°] PM = 70°
-100 -150 -200 -180°
-250
10
-5
10
-4
10
-3
10
-2
[Hz]
10
-1
10
0
10
1
10
2
Fig. 16.12 Compensator h(s) common to all loops of the SVD controller. Left: Nichols chart. Right: Bode plots showing the gain of 125dB at the earth rotation frequency and a cut off frequency of fc = 0.25Hz.
global modes are critical for the control-structure interaction; the segments are normally designed in such a way that their local modes have natural frequencies far above the critical frequency range, but their quasi-static response (to the actuator as well as to gravity and wind disturbances) must be dealt with accurately in the model. In order to handle large optical configurations, it is important to reduce the model as much as possible, without losing the features mentioned before. A model of minimum size can be constructed using a Craig-Bampton reduction (section 2.8), where the master d.o.f. consist of the axial d.o.f. at both ends of the position actuators (represented by circles in Fig.16.9) which are necessary to describe the kinematics of the system, supplemented by an appropriate set of fixed boundary modes (usually a small number) which take care of possible internal modes of the supporting truss. Figure 16.13 shows the eigenfrequency distribution of a typical segmented mirror; the first 20 modes or so are global modes, with mode shapes combining optical aberration modes of low order; they are followed by local modes of the segments (tilt near 75Hz and piston near 100Hz ). If the system is properly designed, only the low frequency modes can potentially jeopardize the system stability and, provided that the static behavior is not altered, the reduced model can be truncated as shown in Fig.16.14 (in the figure, Fm is the quasistatic response of the flexible modes included in the residual response, which has already been included in the Jacobian; the matrices Sy1 and Sy2 describe the sensor topology and Sa describes the actuator topology). The global input-output relationship is written in the form G(s) = G0 (s) + GR (s)
(16.9)
16.6 Control-Structure Interaction
Piston
Tilt
200
397
Eigen frequency [Hz]
150
100
50 global modes
0
50
100
150
200
250
300
Mode index
Fig. 16.13 Eigenfrequency distribution of a typical segmented mirror.
where the nominal plant G0 (s) = J has been taken into account in the controller design (primary response), and GR (s) is the dynamic deviation (residual response), which is considered as an additive uncertainty.
16.6
Control-Structure Interaction
The controller transfer matrix is essentially the inverse of the quasi-static response of the mirror. However, because the response of the mirror includes a dynamic contribution at the frequency of the lowest structural modes and above, the system behaves according to Fig.16.14 and the robustness with respect to control-structure interaction must be examined with care. The structure of the control system is that of Fig.16.15.a, where the primary response G0 (s) corresponds to the quasi-static response described earlier and the residual response GR (s) is the deviation resulting from the dynamic amplification of the flexible modes; K(s) is the controller, given by Equ.(16.7). The control-structure interaction may be addressed with the general robustness theory of multivariable feedback systems (section 10.9); the residual response being considered as uncertainty.
16.6.1
Multiplicative Uncertainty
According to section 10.9.3, if one assumes a multiplicative uncertainty, the standard structure of Fig.16.15.b applies, and one can check that Fig.16.15.a
398
16 Active Control of Large Telescopes Flexible modes i = 1,…,m
GR
a
+
...
y
(Residual response)
fiiT 2 s + 2xii wii s + wii2
+
GO
Fm
...
ka Sa
Sy1
+
y1
-
Sy2
Fm
y2
Jacobian
a
J=
(( Je Jn
(Primary response)
y1 y2
+ +
y
Fig. 16.14 Input-output relationship of the segmented mirror. The nominal plant G0 (s) = J accounts for the quasi-static response (primary response) and the dynamic deviation GR (s) (residual response) is regarded as an additive uncertainty. b) +
Residual
a)
GR a + -
+ Primary
GO K
y
-
I +L
G = KGO
G
-1
L = GO GR
+ c) + -
L G
+
G = GO K
+
L = GR K
Control
Fig. 16.15 Block diagram of the control system (a) Mirror represented by its primary and residual dynamics. (b) Multiplicative uncertainty. (c) Additive uncertainty. 4 and b are equivalent with G(s) = K(s)G0 (s) and L = G−1 0 GR . A sufficient condition for stability is given by (10.64):
σ ¯ [L(jω)] < σ[I + G−1 (jω)],
ω>0
(16.10)
(¯ σ and σ stand respectively for the maximum and the minimum singular value), which is transformed here into −1 σ ¯ [G−1 (jω)], 0 GR (jω)] < σ[I + (KG0 )
ω>0
(16.11)
This test is quite meaningful; the left hand side represents an upper bound to the relative magnitude of the uncertainty; it is independent of the controller; it starts from 0 at low frequency where the residual dynamics is negligible and increases gradually when the frequency approaches the flexible modes of 4
In all this section, the inverse of rectangular matrices should be understood in the sense of pseudo-inverse.
16.6 Control-Structure Interaction
399
GM
10
A
s [ I + (KG0) -1 ] Structural uncertainty
1
0.1
0.01 0.01
-1 s [ G0 GR ]
f1 0.1
1 [Hz]
10
100
Fig. 16.16 Robustness test assuming multiplicative uncertainty. σ[I + (KG0 )−1 ] refers to the nominal system used in the controller design. σ ¯ [G−1 0 GR ] is an upper bound to the relative magnitude of the residual dynamics. The critical point A corresponds to the closest distance between these curves. The vertical distance between A and the upper curve has the meaning of a gain margin.
the mirror structure, which are not included in the nominal model G0 ; the amplitude is maximum at the resonance frequencies where it is only limited by the structural damping of the flexible modes. The right hand side starts from unity at low frequency where | KG0 | 1 (KG0 controls the performance of the control system) and grows larger than 1 outside the bandwidth of the control system where the system rolls off (| KG0 | 1). A typical robustness test plot is represented in Fig.16.16; the critical point A corresponds to the closest distance between these curves. The vertical distance between A and the upper curve has the meaning of a gain margin GM (if the gain of all control channels is multiplied by a scalar g, the high frequency part of the upper curve will be lowered by g). When the natural frequency of the structure changes from f1 to f1∗ , point A moves horizontally according to the ratio f1∗ /f1 (increasing the frequency will move A to the right). Similarly, changing the damping ratio from ξ1 to ξ1∗ will change the amplitude according to ξ1 /ξ1∗ (increasing the damping will decrease the amplitude of A).
16.6.2
Additive Uncertainty
Alternatively, if one assumes an additive uncertainty, the standard structure of Fig.16.15.c applies, and one can check that Fig.16.15.a and c are equivalent with G = G0 K and L = GR K; a sufficient condition for stability is given by (10.62): σ ¯ [L(jω)] < σ[I + G(jω)],
ω>0
(16.12)
400
16 Active Control of Large Telescopes
s [ I + (G0K) ]
10
1
GM A
0.1
0.01 0.01
Structural uncertainty
s [ GR K ] f1 0.1
1 [Hz]
10
100
Fig. 16.17 Robustness test assuming additive uncertainty. σ[I + G0 K(jω)] refers to the nominal system used in the controller design. σ ¯ [GR K(jω)] is an upper bound to the effect of the controller on the residual dynamics. The critical point A corresponds to the closest distance between these curves. The vertical distance between A and the upper curve has the meaning of a gain margin.
which is translated into σ ¯ [GR K(jω)] < σ[I + G0 K(jω)],
ω>0
(16.13)
Again, the smallest distance between these two curves has the meaning of a gain margin (if the gain of all control channels is multiplied by a scalar g, the lower curve, GR K, will be multiplied by g). Note that the stability conditions (16.11) and (16.13) come from the small gain theorem; being sufficient conditions, they are both conservative and one may be more conservative than the other.
16.6.3
Discussion
As the telescopes increase in size, so does the gravity sag, requiring higher control gains to maintain the right shape, and increasing the control bandwidth fc ; this means that the curve σ[I + (KG0 )−1 ] in Fig.16.16 is moving to the right. At the same time, the natural frequencies fi of the flexible modes decrease when the size of the structure increases, which means that the curve σ ¯ [G−1 0 GR ] is moving to the left. The robustness with respect to the controlstructure interaction tends to be controlled by the ratio fi /fc between the natural frequency of the critical mode (not necessarily the first)5 and the control bandwidth. For a given telescope design, using the foregoing robustness tests, it is possible to plot the evolution of the gain margin with the 5
For the Keck telescope, the critical mode turned out to be a local mode of the segments with a frequency of ∼ 25Hz.
16.6 Control-Structure Interaction
80
x 5 0.0 2 0.0 1 0.0 05 0.0
60
GM [dB]
401
40 20
GM=10dB
0 UNSTABLE -20 10
1
10
2
f1/fc Fig. 16.18 Evolution of the gain margin with the frequency ratio f1 /fc for various values of the damping ratio ξi .
frequency ratio f1 /fc ; this curve depends strongly on the structural damping ratio, since the amplitude of the various resonance peaks is proportional to ξ −1 . Figure 16.18 shows a typical plot; one sees that if the critical structural mode of the supporting structure has a damping ratio of 1%, a gain margin GM=10 requires a frequency separation f1 /fc significantly larger than one decade. This condition may be more and more difficult to fulfill as the size of the telescope grows. The situation can be improved by increasing the structural damping of the supporting truss, possibly actively.
17
Semi-active Control
17.1
Introduction
Active control systems rely entirely on external power to operate the actuators and supply the control forces. In many applications, such systems require a large power source, which makes them costly (this is why there has been very few cars equipped with fully active suspensions) and vulnerable to power failure (this is why the civil engineering community is reluctant to use active control devices for earthquake protection). Semi-active devices require a lot less energy than active devices; and the energy can often be stored locally, in a battery, thus rendering the semi-active device independent of any external power supply. Another critical issue with active control is the stability robustness with respect to sensor failure; this problem is especially difficult when centralized controllers are used. On the contrary, semi-active control devices are essentially passive devices where properties (stiffness, damping,...) can be adjusted in real time, but they cannot input energy directly in the system being controlled. Note however that since semi-active devices behave nonlinearly; they can transfer energy from one frequency to another. The variable resistance law can be achieved in a wide variety of forms, as for example position controlled valves, rheological fluids, or piezoelectrically actuated friction joints. Over the past few years, semi-active control has found its way in many vibration control applications, for large and medium amplitudes, (particularly vehicle suspension, but also earthquake protection,...). However, it should be kept in mind that, in most cases, semi-active devices are designed to operate in the “post yield” region, when the stress exceeds some controllable threshold; this makes them inappropriate for vibrations of small amplitude where the stress remains below the minimum controllable threshold in the device. It should also be pointed out that, in many applications (e.g. domestic appliances), the cost of the control system is a critical issue (it is much more important than the optimality of the performances); this often leads to simplified control architectures with extremely simple sensing devices. A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 403–416. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com
404
17 Semi-active Control
Magneto-rheological fluids exhibit very fast switching (of the order of millisecond) with a substantial yield strength; this makes them excellent contenders for semi-active devices, particularly for small and medium-size devices, and justifies their extensive discussion. This chapter begins with a review of magneto-rheological (MR) fluids and a brief overview of their applications to date. Next, some semi-active control strategies are discussed.
17.2
Magneto-Rheological Fluids
In 1947, W.Winslow observed a large rheological effect (apparent change of viscosity) induced by the application of an electric field to collo¨ıdal fluids (insulating oil) containing micron-sized particles; such fluids are called electro-rheological (ER) fluids. The discovery of MR fluid was made in 1951 by J.Rabinow, who observed similar rheological effects by application of a magnetic field to a fluid containing magnetizable particles. In both cases, the particles create columnar structures parallel to the applied field (Fig.17.1) and these chain-like structures restrict the flow of the fluid, requiring a minimum shear stress for the flow to be initiated. This phenomenon is reversible, very fast (response time of the order of milliseconds) and consumes very little energy. When no field is applied, the rheological fluids exhibit a Newtonian behavior. Typical values of the maximum achievable yield strength τ are given in Table 17.1. ER fluids performances are generally limited by the electric field breakdown strength of the fluid while MR fluids performances are limited by the magnetic saturation of the particles. Iron particles have the highest saturation magnetization. In Table 17.1, we note that the yield stress of MR fluids is 20 to 50 times larger than that of ER fluids. This justifies why most practical applications use MR fluids. Typical particle sizes are 0.1 to 10μm and typical particle volume fractions are between 0.1 and 0.5; the carrier fluids are selected on the basis of their tribology properties and thermal stability
Applied field No field
Fig. 17.1 Chain-like structure formation under the applied external field.
17.2 Magneto-Rheological Fluids
405
Table 17.1 Comparison of typical ER and MR fluid properties.
Property
ER fluid
Yield Strength τ Max. field Viscosity η (at 25o C under no field) Density Response time
MR fluid
2 − 5 kPa 50 − 100 kPa 3 − 5 kV/mm 150 − 250 kA/m 0.2 − 0.3 Pa.s 0.2 − 0.3 Pa.s 1 − 2 g/cm 3 ms
3 − 4 g/cm 3 ms
(the operable temperature range of MR fluids is −400 C< T 0 < 1500C); they also include additives that inhibit sedimentation and aggregation. (b)
(a)
y (H)
y (H)
(c) H max
(d)
operating range
H=0
Fig. 17.2 (a) and (b) Bingham plastic model consisting of a constant viscous damper in parallel with a variable friction device. (c) Operating range. (d) Hysteretic behavior observed.
The behavior of MR fluids is often represented as a Bingham plastic model with a variable yield strength τy depending on the applied magnetic field H, Fig.17.2. The flow is governed by the equation
406
17 Semi-active Control
τ = τy (H) + η γ˙
,
τ > τy (H)
(17.1)
where τ is the shear stress, γ is the shear strain and η is the viscosity of the fluid. The operating range is the shaded area in Fig.17.2.c. Below the yield stress (at strains of order 10−3 ), the material behaves viscoelastically: τ =Gγ
,
τ < τy (H)
(17.2)
where G is the complex material modulus. Bingham’s plastic model is also a good approximation for MR devices (with appropriate definitions for τ , γ and η). However, the actual behavior is more complicated and includes stiction and hysteresis such as shown in Fig.17.2.d; more elaborate models attempting to account for the hysteresis are available in the literature, Fig.17.3, but Bingham’s model is sufficient for most design work.
17.3
MR Devices
Figure 17.4 shows the four operating modes of controllable fluids: valve mode, direct shear mode, squeeze mode and pinch mode. The valve mode is the normal operating mode of MR shock absorbers (Fig.17.5); the control variable is the current through the coil, which controls the magnetic field in the active part of the fluid and, as a result, creates the variable yield force in the device. The direct shear mode is that of clutches and brakes (Fig.17.6).
17.4
Semi-active Suspension
A semi-active suspension consists of a classical suspension provided with a controllable shock absorber, capable of changing its characteristics in realtime with a small amount of energy. The device remains essentially passive and can only dissipate energy, that is to produce a force opposing the motion applied to the device. In general the term semi-active suspension refers to a suspension provided with a controllable shock absorber capable of changing its characteristics in wide-band; this requires a fast responding controllable device. Adaptive suspensions involve controllable shock absorbers with lowfrequency capability, allowing the damper characteristics to be adapted to optimize ride comfort and road holding for the current road roughness and driving conditions; such a system is available on many cars, with various degrees of sophistication; they offer new capabilities to enhance the vehicle dynamics, in connection with the so-called “ESP” system.
17.4.1
Semi-active Devices
Two of the most frequently used semi-active devices are illustrated in Fig.17.7, with their respective operating range. The first one (left) consists of
17.4 Semi-active Suspension
407
1500
x
a)
model experiment
Force [N]
c0 F
0
-1500 -12
0
12
Velocity [cm/s]
x2
x1 c0
1500
x3
c1 F k2
Force [N]
b)
model experiment
0
k1 -1500 -12
0
12
Velocity [cm/s]
Bouc-Wen
k0
x
1500 model experiment
Force [N]
c)
F
0
c0 -1500 -12
0
12
Velocity [cm/s]
d)
Bouc-Wen
x
1500
c1
k0 c0 k1
F
Force [N]
model experiment
0
-1500 -12
0
12
Velocity [cm/s]
Fig. 17.3 MR fluid and MR damper phenomenological models: (a) Bingham model. (b) Gamota and Filisko (c) Bouc-Wen. (d) Spencer et al. Forces-velocity curves are adapted from (Spencer et al.).
408
17 Semi-active Control
L
a)
b)
A
flow
g
pressure
A
force
w
speed
g
MR-fluid
MR-fluid
applied magnetic field
applied magnetic field
pressure
d)
c) force
displacement non-magnetic spacer
MR-fluid
MR-fluid
applied magnetic field
flow
applied magnetic field Fig. 17.4 Operating modes of controllable fluids: (a) valve mode, (b) direct shear mode, (c) squeeze mode and (d) pinch mode.
a)
b) Fig. 17.5 MR shock absorber (adapted from Carlson, 2007).
17.4 Semi-active Suspension
409
Fig. 17.6 Various MR brake designs: (a) drum, (b) inverted drum, (c) T-shaped rotor, (d) disk, (e) multiple disks (from Avraam).
410
17 Semi-active Control
(a)
f
(b)
v f = c(u)v
f
v
(u)
c
f = c v + (u) f
f
c max
max
c min
v
v
Fig. 17.7 Semi-active devices and their operating range. (a) Viscous damper with variable damping coefficient. (b) MR fluid device and its Bingham model.
a classical viscous damper with a variable damping coefficient c(u) obtained by controlling the size of the opening of an orifice between the two chambers of the damper (e.g. with an electromagnet). The operating range is the shaded area between two lines corresponding to the minimum and maximum damping coefficients, cmin < c(u) < cmax . The second one (right) consists of a MR fluid damper similar to that of Fig.17.5; its behavior is represented by its Bingham model (Fig.17.7.b).
17.5
Narrow-Band Disturbance
Referring to the transmissibility of a passive isolator (Fig.8.2), when the dis√ turbance frequency is ω < 2ωn ,√the overshoot is minimized by setting a high damping constant, while above 2ωn , the damping should be minimum to enjoy the maximum roll-off rate. This suggests the following control strategy according to the disturbance frequency ω: √ If ω ≤ 2ωn then c = cmax (17.3) √ If ω > 2ωn then c = cmin
17.5 Narrow-Band Disturbance
411
Fig. 17.8 (a) MR fluid engine mount and (b) vibration isolation performances (adapted from Carlson, 2007).
If cmax is large enough and cmin is small enough, the transmissibility achieved in this way fits closely that of the objective of active isolation in Fig.8.2. Thus, the semi-active isolation is optimal in this case. One must be careful, however, because the behavior a semi-active isolation device, as any nonlinear system, depends strongly on the excitation, and what is optimal for an harmonic excitation is not for a wide-band excitation. The MR engine mount of Fig.17.8 is an example of adaptive suspension; the activation of the device allows to go from a low stiffness state (MR valve open, allowing the flow between the upper and the lower sides of the MR fluid chamber) to a high stiffness state (MR valve closed). If the disturbance is narrow band with a variable frequency (typically the rotation speed of the engine) and, if the MR device is activated properly, the overall isolation is the lower bound of the two curves of Fig.17.8.b.
17.5.1
Quarter-Car Semi-active Suspension
The principle of the semi-active suspension is illustrated in Fig.17.9 (compare with Fig.8.27.b). The semi-active control unit activates the controllable device to achieve the variable control force fc subject to the constraint imposed by the passivity of the device1 fc .(x˙ s − x˙ us ) ≤ 0 1
(17.4)
fc = −c(u)(x˙ s − x˙ us ) is the force applied by the shock absorber to the sprung mass ms . More complex situations may also be considered, in which the spring stiffness is also variable.
412
17 Semi-active Control
As a nonlinear device, the response of a controllable shock absorber depends on the excitation amplitude and on its frequency content, and it has the capability to transfer energy from one frequency to another.
Fig. 17.9 Principle of the semi-active suspension. One or several sensors monitor the state of the suspension and a semi-active control unit controls the shock absorber constant c(u).
The semi-active sky-hook consists of trying to emulate the sky-hook control with the controllable shock absorber, by producing the best possible approximation fc = −c(u)(x˙ s − x˙ us ) ≈ −bx˙ s (17.5) Because of the passivity constraint (17.4), this is possible only if the sprung mass velocity and the relative velocity have the same sign x˙ s .(x˙ s − x˙ us ) ≥ 0
(17.6)
and if the magnitude of the requested control force belongs to the operating range of the controllable shock absorber, cmin ≤
|bx˙ s | ≤ cmax |x˙ s − x˙ us |
The damping constant which fits best the requested (sky-hook) control force is bx˙ s c(u) = max{cmin , min[ , cmax ]} (17.7) x˙ s − x˙ us
17.5 Narrow-Band Disturbance
413
However, the sprung mass velocity x˙ s and the suspension relative velocity x˙ s − x˙ us have widely different frequency contents, and the foregoing strategy tends to produce a fast switching control force fc , as illustrated below. The above strategy requires a fast, calibrated, proportional valve; an alternative on/off implementation is c(u) = cmax c(u) = cmin
If If
x˙ s .(x˙ s − x˙ us ) ≥ 0 x˙ s .(x˙ s − x˙ us ) < 0
(17.8)
Although simpler, this strategy is likely to produce even sharper changes in the control force. The following example illustrates the energy transfer from low frequency to high frequency associated with the semi-active sky-hook control. The system of Fig.17.9 is modelled using the same state variables as for the passive suspension of Fig.8.27.b, x1 = xs − xus , x2 = x˙ s , x3 = xus − w, x4 = x˙ us , the set of governing equations is identical to that of the passive suspension, except that the damping coefficient c(u) of the shock absorber depends on the control variable u: ms x˙ 2 = −kx1 + c(u)(x4 − x2 ) mus x˙ 4 = −kt x3 + kx1 + c(u)(x2 − x4 ) x˙ 1 = x2 − x4
(17.9)
x˙ 3 = x4 − v where v = w˙ is the road velocity. Time domain simulations have been conducted with the same numerical data as the passive suspension analyzed earlier: ms = 240 kg, mus = 36 kg, k = 16000 N/m, kt = 160000 N/m, b = 2000 N s/m (gain of the sky-hook control). The shock absorber constant is supposed to vary between cmin = 100 N s/m and cmax = 2000 N s/m. The body resonance and the tyre resonance are respectively ωn = (k/ms )1/2 ∼ 8 rad/s and ωt = (kt /mus )1/2 ∼ 70 rad/s. The road velocity v is assumed to be a white noise; the control law is (17.7). Figure 17.10 shows various time-histories of the quarter-car response, respectively the tyre force kt x3 , the body velocity x˙ s = x2 , the relative velocity x˙ 1 = x˙ s − x˙ us , the requested (sky-hook) force f = −bx˙ s and the actual control force fc = −c(u)(x˙ s − x˙ us ), and finally the damper constant c(u). Note that the relative velocity oscillates much faster (at 70 rad/s) than the body velocity, resulting in sharp changes in the control force fc . Figure 17.11.a compares the transmissibility between the road velocity and the body acceleration, Tx¨s v of, respectively the passive suspension (c = 200 N s/m), the sky-hook control (c = 200 N s/m and b = 2000 N s/m), and the semi-active sky-hook (17.7) with cmin = 100 N s/m and cmax = 2000 N s/m. The first two curves are the same as in Fig.8.29.a; The semi-active control is successful in reducing the body resonance, and the
414
17 Semi-active Control
Fig. 17.10 Quarter-car model with continuous semi-active sky-hook control: (a) tyre force kt x3 , (b) body velocity x˙ s = x2 , (c) relative velocity x˙ 1 = x˙ s − x˙ us , (d) sky-hook force f = −bx˙ s and control force fc = −c(u)(x˙ s − x˙ us ), (e) damper constant c(u) obtained from (17.7).
transmissibility of the body acceleration is comparable to that of the active control with b 1000 N s/m at low frequency; however, a significant amplification occurs at the wheel resonance, ωt = 70 rad/s, and above ωt , the transmissibility rolls off much slower than in the previous cases. Besides, one
17.5 Narrow-Band Disturbance
415
Fig. 17.11 Quarter-car model with continuous semi-active sky-hook control. (a) Transmissibility between the road velocity and the body acceleration, Tx¨s v ; passive suspension (c = 200 N s/m), sky-hook controller (b = 2000 N s/m), semi-active skyhook (17.7) with cmin = 100 N s/m and cmax = 2000 N s/m. (b) Transmissibility between the road velocity and the tyre deflection, Tx3 v .
observes peaks at various harmonics of the wheel resonance,2 which are likely to excite flexible modes of the vehicle if nothing is done to attenuate them. Figure 17.11.b shows the transmissibility between the road velocity and the tyre deflection; an amplification at the wheel resonance is also observed, but no spurious high frequency components appear. The transmissibility diagrams of Fig.17.11 have been obtained from timehistories with cross power spectra and auto power spectra estimates: Tyx =
Φyx E[Y (ω)X ∗ (ω)] = Φxx E[X(ω)X ∗ (ω)]
(17.10)
Further evidence of the nonlinear energy transfer from low to high frequencies can be obtained from the coherence function between the road velocity and the body acceleration, 2
The two peaks at 132 rad/s and 148 rad/s seem to result from the modulation of the second harmonic of the wheel mode (2ωt = 140 rad/s) by the car body mode (ωn = 8 rad/s), producing frequency peaks at 2ωt − ωn and 2ωt + ωn .
416
17 Semi-active Control
Fig. 17.12 Quarter-car model with continuous semi-active sky-hook control. Coherence function γx2¨s v between the road velocity and the body acceleration.
γx¨2s v =
|Φx¨s v |2 ≤1 Φvv Φx¨s x¨s
(17.11)
γx¨2s v is equal to 1 for a perfect linear system; it measures the causality of the signal at every frequency; it is a standard tool to detect the presence of noise and nonlinearities. According to Fig.17.12, the coherence is very good up to the tyre mode, and falls rapidly to zero above 100 rad/s, which indicates that at those frequencies, the energy content of the body acceleration is not due to the road profile.
17.6
Problems
P.17.1 Consider a MR device operating according to the direct shear mode, Fig.17.4.b; the electrodes move with respect to each other with a relative velocity U . If A is the active area of the device, g the distance between the electrodes, the viscous components Fη and the field-induced yield stress components Fτ are respectively Fη = η
U A g
Fτ = τ A
From these equations, show that the minimum volume of active fluid to achieve a given control ratio Fτ /Fη for a specified maximum controlled force Fτ and a maximum relative velocity U reads η Fτ V = gA = Fτ U 2 τ Fη where η is the viscosity and τ is the maximum yield strength induced by the magnetic field. From this result, η/τ 2 can be regarded as a figure of merit of a controllable fluid (Coulter et al.). This explains the superiority of the MR fluids over ER fluids (Table 17.1).
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Index
Accelerometer Piezoelectric- 58 Active Optics 386, 390 Active Structural Acoustic Control (ASAC) 342 Active suspension 179 Active truss 92, 322 Adaptive Optics (AO) 1, 386 Adaptive suspension 406 Additive uncertainty 224, 241, 400 Admittance 57, 80, 97, 103 Aliasing 318 spatial- 349 All-pass function 233 Analog to digital converter (ADC) 318 Anti-resonance 28 Bandwidth 219 Bessel filter 357 Beta controller 152, 366, 367 Bilinear transform 321 Bingham model 406 Bode gain-phase relationships 228 Ideal Cutoff 231 Integrals 228 Butterworth filter 273, 319, 357 pattern 254 Cable-stayed bridge 359, 375 Cable Structures 359 Cauchy’s principle 216
Cayley-Hamilton theorem 277 Charge amplifier 71 dynamics 73 Coenergy density function 64 function 54 Collocated control 27, 76, 117, 131, 191 Constitutive equations piezoelectric laminate 83 piezoelectric material 61 piezoelectric transducer 51 Constrained system 30 Control budget 14 Control canonical form 282 Control-structure interaction 397, 400 Controllability 275 matrix 276 Covariance intensity matrix 250 Craig-Bampton reduction 36, 396 Cross talk 149 Crossover frequency 215, 228 Cubic architecture 172, 184, 326 Cumulative MS response 14 Curie temperature 50 Current amplifier 71 Decentralized control 149, 364 Digital to analog converter (DAC) 320 Direct piezoelectric effect 50 Direct Velocity Feedback (DVF) 135, 312, 335
430
Index
Discrete array sensor 346 Distributed sensor 71, 343, 351 Duality actuator-sensor 46, 89, 346 Lead-IFF 145 LQR-KBF 260 Dynamic amplification 22, 79, 131 Dynamic capacitance 80 Dynamic flexibility matrix 21 Dynamics (Actuator and sensor-) 147 Dynamic Vibration Absorbers (DVA) 5 E-ELT 392 Electro-rheological (ER) fluid 404 Electrode shape 69 Electromechanical converter 43 coupling factor 52, 55, 82, 100, 103 transducer 46 Energy absorbing control 311 Energy density function 64 Energy transformer 41 Error budget 14 Faraday’s law 42 Feedforward control 9 Feedthrough 23, 79, 80, 189 Flutter 359 Fraction of modal strain energy 81, 95, 328, 367 Frequency shaped LQG 266, 339 Frequency shaping 266 Gain margin (GM) 215, 224, 255 Gain stability 134, 226 Geophone 45 Gough-Stewart platform 171 Gramians Controllability- 288 Observability- 288 Guyan reduction 34 Guyed truss 368 Gyrostabilizer 48 HAC/LAC 335 Hamilton’s principle 66 Hankel singular values 291 High Authority Control (HAC)
132
High-pass filter 73 Hydraulic actuator 378 Impedance (Piezoelectric transducer) 57 Integral control 265 Integral Force Feedback (IFF) 140, 163, 174, 312, 324, 366 Interlacing 28, 131 Internally balanced realization 292 Inverse piezoelectric effect 50 Inverted pendulum 191, 199, 204, 207 double 278, 279 Isolator Active- 161, 177 Passive- 156, 177 Relaxation- 157, 173 Kalman Bucy Filter (KBF) 203, 258 Kalman Filter, see Kalman Bucy Filter Kirchhoff plate theory 83 Kronecker delta 19, 74 Lag compensator 239 Laminar sensor 71 Lasalle’s theorem 305 Lead compensator 120, 133, 237 Lead-Zirconate-Titanate 50 Legendre transformation 54, 64 Linear Quadratic Gaussian, see LQG Linear Quadratic Regulator, see LQR Loop shaping 394 Loop Transfer Recovery (LTR) 264 Lorentz force 42 Low Authority Control (LAC) 132 LQG 259 LQR Deterministic- 198, 248 Stochastic 251 Luenberger observer 201 Lyapunov direct method 303 equation 247, 251, 288, 309 function 248, 303, 306, 308 indirect method 309 Magneto-rheological, see MR Magnetostrictive materials 7
Index Micro Precision Interferometer 370 MIMO 187, 239, 247 Minimum phase 233 Minimum realization 291 Modal damping 20 filter 73 mass 19 spread 175 truncation 79 Mode shape 19, 33 MR clutch and brake 406 engine mount 411 fluid 404 shock absorber 406 Multi-functional materials 50 Multi-Input Multi-Output, see MIMO Multi-layer laminate 85 Multiplicative uncertainty 224, 242, 397 Natural frequency 19 Nichols chart 221 Non-minimum phase 128, 233 Notch filter 123, 211 Nyquist frequency 318 plot 30, 257 stability criterion 217 Observability 275 matrix 276 Observer 201, 257 Operational amplifier 71 Orthogonality conditions 19, 33, 74 Pad´e approximants 235, 246 Parametric excitation 360 resonance 359, 375 Parseval’s theorem 266 Payload isolation 170 PD compensator 237 Performance index 251, 252 Phase margin (PM) 215, 224, 255 Phase portrait 300 PI compensator 238 PID compensator 239
431
Piezoelectric beam 66, 90 coenergy 55 constants 61 energy 54 laminate 83, 86 loads 68, 87 material 50, 61 transducer 51, 56, 94 transformer (Rosen’s) 100 Pole 194 Pole placement 195 Pole-zero flipping 118, 211 Pole-zero pattern 28, 78 Poling 50 Polyvinylidene fluoride 50 Popov-Belevitch-Hautus (PBH) test 285 Positive Position Feedback (PPF) 137, 332 Power Spectral Density (PSD) 14, 250 Prescribed degree of stability 254 Proof-mass actuator 43, 59 PVDF 50 properties 65 Pyroelectric effect 50 PZT 50 properties 65 Quality factor 22, 100 Quantization 320 Quarter-car model 179 Quasi-static correction 79 QWSIS sensor 343 Rayleigh damping 18 Reaction wheel 48 Reduced order observer 206 Relaxation isolator, see Isolator Residual dynamics 243 Residual mode 23, 80 Residual modes (spillover) 261 Return difference 222 Riccati equation 249, 252 Rigid body mode 23 Robust performance 225 stability 225, 241
432
Index
Robustness test 241 Roll-off 79, 132 Rosen’s piezoelectric transformer 100 Routh-Hurwitz criterion 139, 302 Sampling 318 Segmented mirror 392 Self-equilibrating forces 69 Self-sensing 47 Semi-active control 403 sky-hook 412 suspension 406 Sensitivity function 222 Separation principle 196, 208, 257 Shape Memory Alloys (SMA) 6 Shunting inductive (RL) 106 resistive (R) 104 switched (SSDI) 109 Singular Value 240 controller 394 Decomposition (SVD) 348, 388 SISO 187, 222 Six-axis isolator 172 Sky-hook damper 162, 182 Small gain theorem 241 Smart materials 50 Sound power 343 Spillover 9, 76, 190, 261 Stability 299 asymptotic- 300, 305
BIBO- 300 in the sense of Lyapunov 300 Stability robustness 225, 241 State feedback 195 State space 189 Stewart platform, see GoughStewart, 326 Symmetric root locus 199 Synchronized Switch Damping 110 System type 235 Telescope 385 Tendon Control 359 Thermal analogy 95 Time delay 235, 246 Tracking error 223 Transmissibility 156 Tustin’s method 321 Unstructured uncertainty
224
Vandermonde matrix 285 Van der Pol oscillator 300, 307 Vibration isolation 155 Vibroacoustics 342 Voice coil transducer 41 Volume displacement sensor 73, 342 White noise
250
Zernike polynomials 387 Zero (transmission-) 30, 150, 194 Zero-order hold 319
