Vibration Control of Active Structures: An Introduction, 3rd Edition

Vibration Control of Active Structures SOLID MECHANICS AND ITS APPLICATIONS Volume 179 Series Editor: G.M.L. GLADWE...

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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.

For other titles published in this series, go to www.springer.com/series/6557

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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Preface to the Third Edition

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

X

Preface to the Second Edition

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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Preface to the First Edition

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

XIV

Contents

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

XV

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

XVI

Contents

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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XVII

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

XVIII Contents

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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XIX

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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Contents

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 << zm and that the strain level is uniform across the thickness of the patch, leading to S = −zm w according to the Bernoulli assumption. The electric charge is obtained by integrating over the area of the electrode x2 T 2 V Q= D dA = ε (1 − k ) dA − e31 zm bp w dx t A A x1 Q = (1 − k 2 )CV − e31 zm bp [w (x2 ) − w (x1 )] Thus,

n Q (e31 zm bp )2 Δθi2 = 1 + (1 − k 2 )CV (1 − k 2 )C μi (s2 + ωi2 )

(4.68)

i=1

or, after using (3.20) and C = εT bp l/t n 2 Q k 2 cE bp tzm Δθi2 1 = 1 + 2 2 2 (1 − k )CV 1 − k i=1 l μi ωi 1 + s2 /ωi2

(4.69)

If one notes that the average strain in the piezo patch is S¯ = −zm Δθ/l and bp lt is the volume of the patch, the strain energy in the patch when the structure vibrates according to mode i can be written approximately We

2 S¯2 .cE .bp lt cE zm Δθi2 bp t = 2 2l

(4.70)

Since μi ωi2 /2 is the total strain energy in the beam when it vibrates according to mode i, the residues in the modal expansion represent the fraction of modal strain energy in the piezo patch, for mode i:

82

4 Piezoelectric Beam, Plate and Truss

νi =

2 cE bp tzm Δθi2 Strain energy in the piezo patch ={ }i 2 l μi ωi Strain energy in the beam

(4.71)

and the relation between Q and V reads ﬁnally n Q k2 νi = 1 + (1 − k 2 )CV 1 − k 2 i=1 1 + s2 /ωi2

(4.72)

or equivalently, the dynamic capacitance reads n Q Ki2 = C(1 − k 2 )[1 + ] V 1 + s2 /ωi2 i=1

(4.73)

where

k 2 νi (4.74) 1 − k2 is known as the eﬀective electromechanical coupling factor for mode i. C(1 − k 2 ) is the blocked capacitance of the piezo. The static capacitance (at ω = 0) is given by Ki2 =

Q Cstat = { }ω=0 = C(1 − k 2 )(1 + Ki2 ) V i=1 n

(4.75)

The poles of the FRF are the natural frequencies ωi of the beam with shortcircuited electrodes, while the zeros correspond to the natural frequencies Ωi with open electrodes (Q = 0). Typically, Ωi is very close to ωi , so that in the vicinity of ω = Ωi , the modal expansion is dominated by the contribution of mode i. It follows that Ωi satisﬁes the equation 1+

Ki2 0 1 − Ωi2 /ωi2

or Ki2

Ωi2 − ωi2 ωi2

(4.76)

(4.77)

Thus, the eﬀective electromechanical coupling factor of all modes can be evaluated from a single admittance FRF test (the poles of which are ωi and the zeros are Ωi ). The foregoing formula is an extension of (3.38).2 Formula (4.73) can be written alternatively

Q (1 + s2 /Ωi2 ) = Cstat .

(4.78) V (1 + s2 /ωj2 ) 2

The presence of ωi instead of Ωi at the denominator of (4.77) is insigniﬁcant in practice.

4.8 Piezoelectric Laminate

4.8

83

Piezoelectric Laminate

In the ﬁrst part of this chapter, the partial diﬀerential equation governing the dynamics of a piezoelectric beam, and the equivalent piezoelectric loads were established from Hamilton’s principle. A similar approach can be used for piezoelectric laminates, but it is lengthy and cumbersome. The analytical expression for the equivalent piezoelectric loads and the sensor output can be obtained alternatively, as in the classical analysis of laminate composites, by using the appropriate constitutive equations; this is essentially the approach followed by (C.K. Lee, 1990).

4.8.1

Two Dimensional Constitutive Equations

Consider a two dimensional piezoelectric laminate in a plane (x, y): the poling direction z is normal to the laminate and the electric ﬁeld is also applied along z. In the piezoelectric orthotropy axes, the constitutive equations (4.1) (4.2) read ⎧ ⎫ ⎨ e31 ⎬ {T } = [c]{S} − e32 E3 (4.79) ⎩ ⎭ 0 D3 = {e31 e32 0}{S} + εE3

(4.80)

where ⎧ ⎫ ⎨ T11 ⎬ {T } = T22 ⎩ ⎭ T12

⎫ ⎧ ⎫ ⎧ ∂u/∂x ⎬ ⎨ S11 ⎬ ⎨ ∂v/∂y S22 {S} = = ⎭ ⎩ ⎭ ⎩ ∂u/∂y + ∂v/∂x 2S12

(4.81)

are the stress and strain vector, respectively, [c] is the matrix of elastic constants under constant electric ﬁeld, E3 is the component of the electric ﬁeld along z, D3 is the z component of the electric displacement and ε the dielectric constant under constant strain (εS ).

4.8.2

Kirchhoﬀ Theory

Following the Kirchhoﬀ theory (e.g. Agarwal and Broutman, 1990), we assume that a line originally straight and normal to the midplane remains so in the deformed state. This is equivalent to neglecting the shear deformations S23 and S31 . If the midplane undergoes a displacement u0 , v0 , w0 , a point located on the same normal at a distance z from the midplane undergoes the displacements (Fig.4.9)

84

4 Piezoelectric Beam, Plate and Truss

w B

u = u o à ëz ë = @w0=@x

A

B

z

wo

A

uo

x

Fig. 4.9 Kinematics of a Kirchhoﬀ plate.

∂w0 ∂x ∂w0 v = v0 − z ∂y w = w0

u = u0 − z

(4.82)

The corresponding strains are

where

{S} = {S 0 } + z{κ}

(4.83)

⎧ 0 ⎫ ⎧ ⎫ ∂u0 /∂x ⎨ S11 ⎬ ⎨ ⎬ 0 S22 ∂v0 /∂y {S 0 } = = ⎩ 0 ⎭ ⎩ ⎭ 2S12 ∂u0 /∂y + ∂v0 /∂x

(4.84)

are the midplane strains and ⎧ ⎫ ⎧ 2 ⎫ ⎨ κ11 ⎬ ⎨ ∂ w0 /∂x2 ⎬ ∂ 2 w0 /∂y 2 {κ} = κ22 = − ⎩ ⎭ ⎩ 2 ⎭ κ12 2∂ w0 /∂x∂y

(4.85)

are the curvatures (the third component represents twisting). The stresses in the laminate vary from layer to layer (because of varying stiﬀness properties) and it is convenient to integrate over the thickness to obtain an equivalent system of forces and moments acting on the cross sections: {N } =

h/2

−h/2

{T }dz

{M } =

h/2

−h/2

{T }z dz

(4.86)

The positive direction of the resultant forces and moments is given in Fig.4.10. {N } and {M } are respectively a force per unit length, and a moment per unit length.

4.8 Piezoelectric Laminate

z

Nxy

Nx

z

N xy

y

Ny

Mx

y

My

Mxy

Ny

Mx

N xy

x

M xy

Nx

Nxy

85

My

Mxy

x Fig. 4.10 Resultant forces and moments.

4.8.3

Stiﬀness Matrix of a Multi-layer Laminate

Before analyzing a piezoelectric laminate, let us recall the stiﬀness matrix of a multi-layer elastic laminate (Fig.4.11). If [c]k represents the stiﬀness matrix of the material of layer k, expressed in global coordinates, the constitutive equation within layer k is {T } = [c]k {S} = [c]k {S 0 } + z[c]k {κ}

(4.87)

Upon integrating over the thickness of the laminate, one gets 0 S A B N = κ B D M

(4.88)

z Layer

h n hk

k

h kà1 Mid plane

h0

Fig. 4.11 Geometry of a multilayered laminate.

x

86

4 Piezoelectric Beam, Plate and Truss

with A= B= D=

1 2 1 3

n k=1 n k=1 n

[c]k (hk − hk−1 ) [c]k (h2k − h2k−1 )

(4.89)

[c]k (h3k − h3k−1 )

k=1

where the sum extends over all the layers of the laminate; this is a classical result in laminate composites. A is the extensional stiﬀness matrix relating the in-plane resultant forces to the midplane strains; D is the bending stiﬀness matrix relating the moments to the curvatures, and B is the coupling stiﬀness matrix, which introduces coupling between bending and extension in a laminated plate; from (4.89), it is readily seen that B vanishes if the laminate is symmetric, because two symmetric layers contribute equally, but with opposite signs to the sum.

4.8.4

Multi-layer Laminate with a Piezoelectric Layer

Next, consider a multi-layer laminate with a single piezoelectric layer (Fig.4.12); the constitutive equations of the piezoelectric layer are (4.79) and (4.80). Upon integrating over the thickness of the laminate as in the previous section, assuming that the global axes coincide with orthotropy axes of the piezoelectric layer, one gets ⎧ ⎫ 0 ⎨ e31 ⎬ N A B S I3 e = + V (4.90) M B D κ zm I3 ⎩ 32 ⎭ 0 D3 = {e31 e32 0}[I3 zm I3 ]

S0 κ

− ε V /hp

(4.91)

where V is the diﬀerence of potential between the electrodes of the piezoelectric layer (E3 = −V /hp ), hp the thickness of the piezoelectric layer, zm the distance between the midplane of the piezoelectric layer and the midplane of the laminated; I3 is the unity matrix of rank 3 and A,B,D are given by (4.89), including the piezoelectric layer.3 In writing (4.91), it has been assumed that the thickness of the piezoelectric layer is small with respect to that of the laminate, so that the strain can be regarded as uniform across its thickness. 3

The piezoelectric layer contributes to A, B and D with the stiﬀness properties under constant electric ﬁeld.

4.8 Piezoelectric Laminate

87

z hp Piezo

h

Mid plane

zm

Fig. 4.12 Piezoelectric layer.

4.8.5

Equivalent Piezoelectric Loads

If there is no external load, {N } and {M } vanish and (4.90) can be rewritten ⎫ ⎧

⎨ e31 ⎬ 0

I3 S A B e =− V (4.92) κ zm I3 ⎩ 32 ⎭ B D 0 The right hand side represents the equivalent piezoelectric loads. If the material is isotropic, e31 = e32 , and the equivalent piezoelectric loads are hydrostatic (i.e. they are independent of the orientation of the facet within the part covered by the electrode). Overall, they consist of an in-plane force normal to the contour of the electrode, and a constant moment acting on the contour of the electrode (Fig.4.13); the force per unit length and moment per unit length are respectively Np = −e31 V

4.8.6

Mp = −e31zm V

(4.93)

Sensor Output

On the other hand, if the piezoelectric layer is used as a sensor and if its electrodes are connected to a charge ampliﬁer which enforces V ∼0, the sensor equation (4.91) becomes 0 S D3 = {e31 e32 0}[I3 zm I3 ] (4.94) κ

88

4 Piezoelectric Beam, Plate and Truss

piezoelectric patch Np Mp Electrode

Fig. 4.13 Equivalent piezoelectric loads (per unit length) for an isotropic piezoelectric actuator: Np = −e31 V , Mp = −e31 zm V .

Upon substituting the midplane strains and curvature from (4.84-85), and integrating over the electrode area, one gets ∂u0 ∂v0 ∂2w ∂ 2w Q= D3 dΩ = e31 + e32 − zm e31 2 + e32 2 dΩ ∂x ∂y ∂x ∂y Ω Ω (4.95)

The integral extends over the electrode area (the part of the piezo not covered by the electrode does not contribute to the signal). The ﬁrst part of the integral is the contribution of the membrane strains, while the second is due to bending. If the piezoelectric properties are isotropic (e31 = e32 ), the surface integral can be further transformed into a contour integral using the divergence theorem; the previous equation is rewritten w dΩ Q = e31 div u0 dΩ − e31 zm div. grad Ω Ω w dl = e31 n.u0 dl − e31 zm n. grad C

C

where n is the outward normal to the contour of the electrode in its plane. Alternatively, ∂w Q = e31 (u0 .n − zm )dl (4.96) ∂n C This integral extends over the contour of the electrode (Fig.4.14); the ﬁrst contribution is the component of the mid-plane, in-plane displacement normal to the contour and the second one is associated with the slope along the contour.

4.8 Piezoelectric Laminate

C

89

Ω

n ∂w/∂n

(u0 .n)n

Fig. 4.14 Contributions to the sensor output for an isotropic piezoelectric layer. Ω is the electrode area.

Once again, the duality between the equivalent piezoelectric loads generated by the transducer used as actuator, and the sensor output when the transducer is connected to a charge ampliﬁer must be pointed out.

4.8.7

Beam Model vs. Plate Model

In this chapter, we have analyzed successively the piezoelectric beam according to the assumption of Euler-Bernoulli, and piezoelectric laminate according to Kirchhoﬀ’s assumption. The corresponding piezoelectric loads have been illustrated in Fig.4.3 and 4.13, respectively; the sensor output, when the transducer is used in sensing mode, can be deduced by duality: a bending moment normal to the contour in actuation mode corresponds to the slope along the contour in sensing mode, and the in-plane force normal to the contour in actuation mode corresponds to the in-plane displacement normal to the contour in sensing mode. Figure 4.15 illustrates the equivalent piezoelectric loads according to both theories for a rectangular isotropic piezoceramic patch acting on a structure extending along one dimension: according to the beam theory, the equivalent piezoelectric loads consist of a pair of torques applied to the ends of the electrode (Fig.4.15.a), while according to the laminate theory, the torque is applied along the whole contour of the electrode and it is supplemented by an in-plane force Np normal to the contour (Fig.4.15.b). If the structure extends dominantly along one axis, and if one is interested in the structural response far away from the actuator (e.g. tip displacement), it is reasonable to think that the piezoelectric loads of the beam theory are indeed the dominant ones. However, in active vibration control, one is often interested in conﬁgurations where the dual actuator and sensor are close to being collocated, to warrant alternating poles and zeros in the open-loop FRF, for a wide frequency range (the perfectly dual and collocated case was considered in section 4.6); in this case, it turns out that the contributions to

90

4 Piezoelectric Beam, Plate and Truss

(b)

Np (a)

Mp

Mp

Mp

beam

axis

Fig. 4.15 Equivalent piezoelectric loads of a rectangular piezoelectric patch bonded on a beam: (a) beam theory, (b) laminate theory.

p2 p1

p4

Detail of the piezos

p3

450 mm

Test Structure

50mm

Fig. 4.16 Cantilever plate with piezoceramics.

the piezoelectric loading and to the sensor output which are ignored in the beam theory are signiﬁcant, and neglecting them usually leads to substantial errors in the open-loop zeros of the control system. This important issue will be addressed again in later chapters. To illustrate how a beam model and a plate model can be diﬀerent for nearly collocated systems, consider the cantilever plate of Fig.4.16; the steel plate is 0.5 mm thick and 4 piezoceramic strips of 250 μm thickness are bounded symmetrically as indicated in the ﬁgure. The size of the piezos is

4.8 Piezoelectric Laminate

Gain dB

91

p3 sensor p2 sensor

p4 sensor

Freq (Hz) Fig. 4.17 Cantilever plate with piezoceramics; experimental FRF for the three sensor conﬁgurations of Fig.4.16.

Gain dB

p3 sensor

p2 sensor

p4 sensor

Freq (Hz) Fig. 4.18 Cantilever plate with piezoceramics; Numerical (ﬁnite elements) FRF for the three sensor conﬁgurations of Fig.4.16.

respectively 55 mm × 25 mm for p1 and p3 , and 55 mm × 12.5 mm for p2 and p4 . p1 is used as actuator while the sensor is taken successively as p2 , p3 and p4 . Since they cover the same extension along the beam, the various sensor locations cannot be distinguished in the sense of the EulerBernoulli beam theory (except for the sign or a constant factor, because p2 and p4 are on opposite sides and the size of p3 is twice that of p2 and p4 ) and they should lead to the same transfer function. This is not the case in practice, as we can see in the experimental results shown in Fig.4.17. We see that the transmission zeros vary signiﬁcantly from one conﬁguration to

92

4 Piezoelectric Beam, Plate and Truss

the other. In fact, because of the nearness of the actuator to the sensor, and the small thickness of the plate, the membrane strains play an important role in the transmission of the strain from the actuator to the sensor and are responsible for the diﬀerences between the three sensor conﬁgurations. Figure 4.18 shows computed FRF based on the plate theory using Mindlintype ﬁnite elements (Piefort, 2001); one sees that the FRF curves behave in a way similar to the experimental ones and that the plate theory accounts for the experimental results. We shall see later that, because the closed-loop poles are located on branches going from the open-loop poles to the openloop zeros, overestimating the spacing between the poles and the zeros leads to overestimating the closed-loop performances of the system. A deeper discussion of the ﬁnite element formulation of multi-layer piezoelectric shells can be found in (Benjeddou, 2000, Garcia Lage et al., 2004, Heyliger et al, 1996, Piefort, 2001) and the literature quoted in these papers. The newly available PZT ﬁbers (with interdigitated electrodes or not), which are usually supplied in a soft polymer cladding, seem to be particularly diﬃcult to model accurately, due to the stiﬀness discrepancy between the supporting structure, the PZT ﬁbers and the soft polymer interface; this is the subject of on-going research.

4.8.8

Additional Remarks

1. Experiments conducted on a cantilever beam excited by a PZT patch on one side and covered with an isotropic PVDF ﬁlm on the other side, with an electrode shaped as a modal ﬁlter for the ﬁrst mode according to the theory of modal sensors developed in section 4.5, have revealed signiﬁcant discrepancies between the measured FRF and that predicted by the beam theory; however, the FRF could be predicted quite accurately by the laminated plate theory (Preumont et al., 2003). 2. For beams, modal ﬁltering has been achieved by shaping the width of the electrode. This concept cannot be directly transposed to plates. Spatial ﬁltering of two-dimensional structures will be examined in Chapter 14.

4.9

Active Truss

Figure 4.19 shows a truss structure where some of the bars have been replaced by active struts; each of them consists of a piezoelectric linear actuator colinear with a force transducer. Such an active truss can be used for vibration attenuation, or to improve the dimensional stability under thermal gradients. If the stiﬀness of the active struts matches that of the other bars in the truss, the passive behaviour of the truss is basically unchanged. Fig.4.20 shows an example of active truss equipped with two active struts (built in the late 80’s ).

4.9 Active Truss

p1 = K a δ 1

y 4

2

6

8

93

Active member 10

12

14

9

11

13

16

18

20

15

17

19

p1 θ

1

3

5

7

f

p2

x

p2 = K aδ 2

δ

Detail of an active member Force transducer

Piezoelectric linear actuator

Fig. 4.19 Active truss. The active struts consist of a piezoelectric linear actuator with a force sensor.

Fig. 4.20 ULB Active truss (1988).

94

4 Piezoelectric Beam, Plate and Truss

Consider a structure with a single discrete piezoelectric transducer (Fig.4.21); the transducer is governed by Equ.(3.22) C(1 − k2 ) nd33 Ka Q V = (4.97) f bT x −nd33 Ka Ka where Δ = bT x is the relative displacement at the extremities of the transducer. The dynamics of the structure is governed by Mx ¨ + K ∗ x = −bf

(4.98)

where K ∗ is the stiﬀness matrix of the structure without the transducer and b is the inﬂuence vector of the transducer in the global coordinate system of the structure. The non-zero components of b are the direction cosines of the active bar (Problem 4.5). The minus sign on the right hand side of the previous equation comes from the fact that the force acting on the structure is opposed to that acting on the transducer. Note that the same vector b appears in both equations because the relative displacement is measured along the direction of f . Substituting f from the constitutive equation into the second equation, one ﬁnds Mx ¨ + Kx = bKa δ (4.99) where K = K ∗ +bbT Ka is the global stiﬀness matrix of the structure including the piezoelectric transducer in short-circuited conditions (which contributes for bbT Ka ); δ = nd33 V is the free expansion of the transducer induced by a voltage V ; Ka δ is the equivalent piezoelectric loading: the eﬀect of the piezoelectric transducer on the structure consists of a pair of self-equilibrating forces applied axially to the ends of the transducer; as for thermal loads, their magnitude is equal to the product of the stiﬀness of the transducer (in

T

D=bx Structure

I V

f Piezoelectric Transducer

M,K *

Fig. 4.21 Structure with a piezoelectric transducer.

4.9 Active Truss

95

short-circuited conditions) by the unconstrained piezoelectric expansion; this is known as the thermal analogy. Let φi be the normal modes, solutions of the eigenvalue problem (K − ωi2 M )φi = 0

(4.100)

They satisfy the usual orthogonality conditions φTi M φj = μi δij

(4.101)

φTi Kφj = μi ωi2 δij

(4.102)

where ωi is the natural frequency when the transducer is short-circuited. If the global displacements are expanded into modal coordinates, x= zi φi (4.103) i

where zi are the modal amplitudes, (4.99) is easily transformed into μi (¨ zi + ωi2 zi ) = φTi bKa δ

(4.104)

Upon taking the Laplace transform, one easily gets x=

n i=1

φi φTi bKa δ μi (ωi2 + s2 )

(4.105)

and the transducer extension Δ = bT x =

n i=1

Ka (bT φi )2 δ μi ωi2 (1 + s2 /ωi2 )

(4.106)

From Equ.(4.102), μi ωi2 /2 is clearly the strain energy in the structure when it vibrates according to mode i, and Ka (bT φi )2 /2 represents the strain energy in the transducer when the structure vibrates according to mode i. Thus, the ratio Ka(bT φi )2 νi = (4.107) μi ωi2 is readily interpreted as the fraction of modal strain energy in the transducer for mode i. With this notation, the previous equation is rewritten Δ = bT x =

n i=1

νi δ (1 + s2 /ωi2 )

(4.108)

96

4 Piezoelectric Beam, Plate and Truss

Fig. 4.22 (a) Open-loop FRF of the active strut mounted in the structure (undamped). (b) Admittance of the transducer mounted in the structure.

4.9.1

Open-Loop Transfer Function

From the second constitutive equation (4.97), one readily establish the openloop transfer function between the free expansion δ of the transducer (proportional to the applied voltage) and the force f in the active strut n f νi = Ka [ − 1] δ (1 + s2 /ωi2 ) i=1

(4.109)

All the residues being positive, there will be alternating poles and zeros along the imaginary axis. Note the presence of a feedthrough in the transfer function. Figure 4.22.a shows the open-loop FRF in the undamped case; as expected the poles at ±jωi are interlaced with the zeros at ±zi . As usual, the transfer function can be truncated after m modes:

4.10 Finite Element Formulation m n f νi = Ka [ + νi − 1] δ (1 + s2 /ωi2 ) i=m+1 i=1

4.9.2

97

(4.110)

Admittance Function

According to the ﬁrst constitutive equation (4.97), Q = C(1 − k 2 )V + nd33 Ka bT x Using (4.108), 2

Q = C(1 − k )V +

n2 d233 Ka

n i=1

νi V (1 + s2 /ωi2 )

(4.111)

and, taking into account (3.23), one ﬁnds the dynamic capacitance n Q k2 νi = C(1 − k 2 )[1 + ] V 1 − k 2 i=1 (1 + s2 /ωi2 )

(4.112)

(the admittance is related to the dynamic capacitance by I/V = sQ/V ), Q Ki2 = C(1 − k 2 )[1 + ] V (1 + s2 /ωi2 ) n

(4.113)

i=1

where Ki2 =

k 2 νi 1 − k2

(4.114)

is once again the eﬀective electromechanical coupling factor for mode i. This equation is identical to (4.73). The corresponding FRF is represented in Fig.4.22(b). The zeros of the admittance (or the dynamic capacitance) function correspond to the natural frequencies Ωi with open electrodes and equations (4.77) and (4.78) apply also for this conﬁguration, Ωi2 − ωi2 ωi2

n (1 + s2 /Ωi2 ) Q = Cstat . i=1 n 2 2 V j=1 (1 + s /ωj ) Ki2

4.10

(4.115)

(4.116)

Finite Element Formulation

The ﬁnite element formulation of a piezoelectric continuum can be derived from Hamilton’s principle (section 4.2). The Lagrangian of a structure

98

4 Piezoelectric Beam, Plate and Truss

involving a ﬁnite number of discrete piezoelectric transducers can be written in the general form L = T ∗ + We∗ =

1 T 1 1 x˙ M x˙ − xT Kxx x + φT Cφφ φ + φT Kφx x 2 2 2

(4.117)

In this equation, M is the mass matrix, Kxx is the stiﬀness matrix (including the mechanical part of the transducers with short circuited electrical boundary conditions), Cφφ is the matrix of capacitance of the transducers (for ﬁxed displacements) and Kφx is the coupling matrix of piezoelectric properties, relating the mechanical and electrical variables. The resulting dynamic equations read T Mx ¨ + Kxx x − Kφx φ=f

(4.118)

Kφx x + Cφφ φ = Q

(4.119)

where Q is the vector of electric charges appearing on the electrodes. For voltage driven electrodes, the electric potential φ is controlled and Equ.(4.118) can be rewritten (assuming no external load, f = 0) T Mx ¨ + Kxx x = Kφx φ

(4.120)

where the right hand side represents the self-equilibrated piezoelectric loads associated with the voltage distribution φ. Note that the dynamics of the system with short-circuited electrodes (φ = 0) is the same as if there were no piezoelectric electromechanical coupling. Conversely, open electrodes correspond to a charge condition Q = 0; in this case, one can substitute φ from Equ.(4.119) into Equ.(4.118) −1 T Mx ¨ + (Kxx + Kφx Cφφ Kφx )x = f

(4.121)

This equation shows that the piezoelectric coupling tends to increase the global stiﬀness of the system if the electrodes are left open. The natural frequencies of the system with open electrodes are larger than those with short-circuited electrodes, as we have discussed extensively earlier in this chapter. If the electrodes are connected to a passive electrical network, the relationship between φ and Q is ﬁxed by the network, as discussed in the following chapter.

4.11

Problems

P.4.1 Consider a simply supported beam with collocated piezoelectric d31 rectangular actuator and sensor extending longitudinally from x = a to x = b. Using the results of section 2.6, show that the expression Δθi appearing at the numerator of Equ.(4.56) can be written analytically

4.11 Problems

Δθi = 2

99

iπ iπ a + b iπ a − b {sin ( ). sin ( )} l l 2 l 2

P.4.2 From the result of the previous problem, sketch the actuator (and the sensor) which maximizes the response of mode 1, mode 2 and mode 3, respectively. P.4.3 Consider the active cantilever beam of transfer function (4.56). Assuming that G(0) is available from static calculations or from an experiment at low frequency, show that the truncated modal expansion including a quasistatic correction can be written G(ω) = G(0) + ga gs

m Δθi2 (ω 2 − 2jξi ωi ω) 2 2 μi ωi (ωi − ω 2 + 2jξi ωi ω) i=1

P.4.4 Consider a free-free beam covered with a piezoelectric layer with the various electrodes proﬁles of Fig.4.23. For each of them, sketch the equivalent piezoelectric loading.

l b parabola

a

a parabola

Fig. 4.23 Free-free beam covered with a piezoelectric layer with various electrode proﬁles.

P.4.5 Consider the active truss of Fig.4.19; the coordinates of the ﬁnite element model are the cartesian coordinates of the nodes (xi , yi ). (a) For each active strut, write the inﬂuence vector b of the piezoelectric loads in global coordinates.

100

4 Piezoelectric Beam, Plate and Truss

(b) Assuming small displacements, check that the projection of the diﬀerential displacements of the end nodes of the active struts are given by bT x where b is the result of (a). P.4.6 Consider a nine bay planar truss similar to that of Fig.4.19. Each bay is square with one diagonal; all the bars have the same cross section. For the following boundary conditions, use a ﬁnite element program to calculate the ﬁrst three ﬂexible modes of the truss. Suggest two reasonable locations of an active strut to control these modes. (a) Free-free boundary conditions. (b) Clamped-free boundary conditions. [Hint: Use the modal fraction of strain energy νi as index in the selection of the active strut location.] P.4.7 Consider a general piezoelectric structure governed by Equ.(4.118-119) (assume f = 0). Write the general form of the admittance matrix Y : sQ = Y φ. Compare it to Equ.(4.113) and interpret the meaning of the various terms. P.4.8 The principle of Rosen’s piezoelectric transformer4 is shown in Fig.4.24; the left side is the driving section; the input a.c. voltage generates an axial vibration thanks to the d31 coeﬃcient. The axial vibration is transmitted to the power generating section which is polarized in the axial direction and generates the output voltage thanks to the d33 coeﬃcient. The system is supposed to work at the second axial resonance of the free-free mechanical system, with a mode shape φ(x) = φ0 cos (πx/l). Show that the voltage ampliﬁcation ratio is given by r=

Vout 4 k33 k31 l = 2 Qm 2 V π (1 − k33 /2) t

(4.122)

where l/t is the length to thickness ratio, k33 and k31 are the respective electromechanical coupling factors of the two sections and Qm = 1/2ξ is the mechanical quality factor of the axial vibration.

4

Piezoelectric transformers were introduced by Rosen in 1956; they have been very successful for low power applications such as power supply of laptop computers. Due to the high energy density of piezoelectric materials, the high electromechanical coupling factors and the high quality factor of the mechanical resonance (low damping), they tend to be lighter and more eﬃcient than wire wound transformers whose eﬃciency tends to decrease rapidly as the size is reduced. Besides, they are free from electromagnetic interference and the solid-state nature of piezoelectric transformers is the key to mass production.

4.11 Problems

101

3 2 1

V

Vibration direction

t Driving section

Power generating section

Fig. 4.24 Rosen’s piezoelectric transformer. P indicates the direction of polarization; the grey areas indicate the electrodes. The system vibrates at the second axial resonance of the free-free mechanical system, with a mode shape φ(x) = φ0 cos (πx/l).

5

Passive Damping with Piezoelectric Transducers

5.1

Introduction

In this chapter, we examine the basics of the passive damping with shunted piezoelectric transducers. We have seen in the previous chapters that a ﬂexible structure (assumed undamped) with embedded piezoelectric transducers (Fig.5.1) is characterized by an admittance function of the form

n (1 + s2 /Ωi2 ) I sQ = = s.Cstat . i=1 (5.1) n 2 2 V V j=1 (1 + s /ωj ) or alternatively n I sQ Ki2 = = sC(1 − k 2 )[1 + ] V V 1 + s2 /ωi2 i=1

(5.2)

where C(1 − k 2 ) is the blocked capacitance1 of the transducer and Ki2 are the modal eﬀective electromechanical coupling factors Ki2 =

k 2 νi Ωi2 − ωi2 = 1 − k2 ωi2

(5.3)

with ωi being the natural frequencies of the structure when the piezoelectric transducer has short-circuited electrodes and Ωi when the electrodes are open. It is assumed that the structure has well separated modes, and that the transducer produces only a minor perturbation to the original structure: the mode shapes are not aﬀected signiﬁcantly by the electrical boundary conditions and ωi and Ωi are very close to each other. 1

Comparing the two foregoing'equations, one ﬁnds that the static capacitance reads Cstat = C(1 − k2 )(1 + Ki2 ).

A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 103–115. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

104

5 Passive Damping with Piezoelectric Transducers

Structure I

I PZT Transducer

V

PZT patch

V

(b) I

(a)

RL-shunt R

I R-shunt V

R

V L

(c)

(d)

Fig. 5.1 Structure with a piezoelectric transducer (a) in d33 mode (b) in d31 mode (c) R shunt (d) RL shunt.

According to the foregoing formula, there are two contributions to Ki2 : (i) the electromechanical coupling factor k 2 which is a material property. If the material is used in the extension (d33 ) mode, the value k33 should be used, and if it is used in shear mode (d13 ), k13 applies. Typical values are given in Table 4.1; note that k13 is signiﬁcantly smaller than k33 . (ii) the modal fraction of strain energy νi which depends on the size as well as the location of the transducer inside the structure. It varies from mode to mode. The transducer should be located where the strain energy is large for the targeted mode(s). Note that the deﬁnition (5.3) of Ki2 does not imply that Ki2 = k2 if νi = 1. The classical deﬁnition is Ki2 = (Ωi2 − ωi2 )/Ωi2 but, Ωi and ωi are in general very close and the diﬀerence is insigniﬁcant in most cases.

5.2

Resistive Shunting

Using the same positive signs for V and I as for the structure (Fig.5.1.c), the admittance of the shunt is I/V = −1/R. The characteristic equation of the system is obtained by expressing the equality between the admittance of the structure and that of the passive shunt:

5.2 Resistive Shunting n 1 Ki2 2 − = sC(1 − k )[1 + ] R 1 + s2 /ωi2 i=1

or

K 2ω2 1 i i = 1 + sRC(1 − k 2 ) s2 + ωi2

105

(5.4)

n

−

(5.5)

i=1

In the vicinity of ±jωi , the sum is dominated by the contribution of mode i and the other terms can be neglected; deﬁning γ = [RC(1 − k 2 )]−1 , the equation may be simpliﬁed as −

γ K 2ω2 = 1 + 2 i i2 s s + ωi

which in turn can be rewritten 1+γ

s2 + ωi2 =0 s(s2 + Ωi2 )

(5.6)

Fig. 5.2 Resistive shunt. Evolution of the poles of the system as γ = [RC(1−k2 )]−1 goes from 0 to ∞ (the diagram is symmetrical with respect to the real axis, only the upper half is shown).

This form of the characteristic equation is that of a root locus in automatic control (Fig.5.2);2 where the parameter γ acts as the feedback gain in classical root locus plots. For γ = 0 (R = ∞), the poles are purely imaginary, ±jΩi , corresponding to the natural frequency of the system with open electrodes; the system is undamped. As the resistance decreases (γ increases), the poles 2

This root locus will be met many times in the following chapters.

106

5 Passive Damping with Piezoelectric Transducers

move to the left and some damping appears in the system; the maximum damping is achieved for γ = Ωi Ωi /ωi Ωi and is (Problem 5.1) ξimax =

5.3

Ωi − ω i Ω2 − ω2 K2 i 2 i = i 2ωi 4ωi 4

(5.7)

Inductive Shunting

We proceed in the same way as for the R-shunt (Fig.5.1.d); the admittance of the shunt is now I/V = −1/(R + Ls). The characteristic equation is obtained by expressing the equality between the admittance of the structure and that of the passive shunt: −

n 1 Ki2 = sC(1 − k 2 )[1 + ] R + Ls 1 + s2 /ωi2 i=1

(5.8)

K 2ω2 1 i i = 1 + (R + Ls)sC(1 − k 2 ) s2 + ωi2

(5.9)

or

n

−

i=1

Once again, in the vicinity of ±jωi , the sum is dominated by the contribution of mode i and the equation is simpliﬁed as −

1 K 2ω2 = 1 + 2 i i2 2 (R + Ls)sC(1 − k ) s + ωi

(5.10)

Deﬁning the electrical frequency ωe2 =

1 LC(1 − k 2 )

(5.11)

and the electrical damping 2ξe ωe =

R L

(5.12)

Equ.(5.10) is rewritten − or

ωe2 Ki2 ωi2 s2 + Ωi2 = 1 + = 2ξe ωe s + s2 s2 + ωi2 s2 + ωi2

s4 + 2ξe ωe s3 + (Ωi2 + ωe2 )s2 + 2Ωi2 ξe ωe s + ωi2 ωe2 = 0

(5.13)

(5.14)

This can be rewritten in a root locus form 1 + 2ξe ωe

s4

s(s2 + Ωi2 ) =0 + (Ωi2 + ωe2 )s2 + ωi2 ωe2

(5.15)

5.3 Inductive Shunting

107

In this formulation, 2ξe ωe = R/L plays the role of the gain in a classical root locus. Note that, for large R, the poles tend to ±jΩi , as expected. For R = 0 (i.e. ξe = 0), they are the solutions p1 and p2 of the characteristic equation s4 + (Ωi2 + ωe2 )s2 + ωi2 ωe2 = 0 which accounts for the classical double peak of resonant dampers, with p1 above jΩi and p2 below jωi . Figure 5.3 shows the root locus for a ﬁxed value of ωi /Ωi and various values of the electrical tuning, expressed by the ratio αe =

ω e ωi Ωi2

(5.16)

The locus consists of two loops, starting respectively from p1 and p2 ; one of them goes to jΩi and the other goes to the real axis, near −Ωi . If αe > 1 (Fig.5.3.a), the upper loop starting from p1 goes to the real axis, and that

p1

(a)

p1

(b)

ëe < 1

ëe > 1

jÒ i j!i p2

jÒ i j!i

p2

Resistive shunting

à Òi

à Òi

Im(s)

p1 (d)

(c)

ëe = 1

Q

jÒ i j!i

Q

jÒ i

p2

Optimal Damping

à Òi

à Òi

Re(s)

Fig. 5.3 Root locus plot for inductive shunting (only the upper half is shown). The optimum damping at Q is achieved for αe = 1 and ξe = Ki ; the maximum modal damping is ξi Ki /2.

108

5 Passive Damping with Piezoelectric Transducers

starting from p2 goes to jΩi , and the upper pole is always more heavily damped than the lower one (note that, if ωe → ∞, p1 → ∞ and p2 → jωi ; the lower branch of the root locus becomes that of the resistive shunting). The opposite situation occurs if αe < 1 (Fig.5.3.b): the upper loop goes from p1 to jΩi and the lower one goes from p2 to the real axis; the lower pole is always more heavily damped. If αe = 1 (Fig.5.3.c), the two poles are always equally damped until the two branches touch each other in Q. This double root is achieved for αe =

ω e ωi =1 Ωi2

,

ξe2 = 1 −

ωi2 Ki2 Ωi2

(5.17)

This can be regarded as the optimum tuning of the inductive shunting. The corresponding eigenvalues satisfy s2 + Ωi2 + Ωi (

Ωi2 − 1)1/2 s = 0 ωi2

(5.18)

For various values of ωi /Ωi (or Ki ), the optimum poles at Q move along a circle of radius Ωi (Fig.5.3.d). The corresponding damping ratio can be obtained easily by identifying the previous equation with the classical form of the damped oscillator, s2 + 2ξi Ωi s + Ωi2 = 0, leading to ξi =

1 Ωi2 Ki 1 k 2 νi 1/2 ( 2 − 1)1/2 = = ( ) 2 ωi 2 2 1 − k2

(5.19)

This value is signiﬁcantly higher than that achieved with purely resistive shunting [it is exactly the square-root of (5.7)]. Note, however, that it is much more sensitive to the tuning of the electrical parameters on the targeted modes. This is illustrated in Fig.5.4, which displays the evolution of the damping ratio ξi when the actual natural frequency ωi moves away from the nominal frequency ωi for which the shunt has been optimized (the damping ratio associated with p1 and p2 is plotted in dotted lines; the ratio ωi /Ωi is kept constant in all cases). One sees that the performance of the inductive shunting drops rapidly below that of the resistive shunting when the de-tuning increases. Note that, for low frequency modes, the optimum inductance value can be very large; such large inductors can be synthesized electronically. The multimodal passive damping via resonant shunt has been investigated by (Hollkamp, 1994). All the dissipation mechanisms considered in this chapter are based on linear time-invariant ﬁlters; an alternative approach based on switching the transducer periodically on a small inductor is examined in the next section.

5.4 Switched Shunt

109

øi 0.3

p1

p2 0.25 0.2 0.15

Inductive shunting

0.1

Resistive shunting

0.05 0 0.1

0.2

0.5

1

2

5

10

!0i=!i

Frequency ratio

Fig. 5.4 Evolution of the damping ratio of the inductive and resistive shunting with the de-tuning of the structural mode. ωi is the natural frequency for which the shunt has been optimized, ωi is the actual value (k = 0.5, νi = 0.3).

5.4

Switched Shunt

Consider a piezoelectric transducer connected to a RL shunt with an electrical circuit equipped with a switch (Fig.5.5). The governing equations of the system are obtained by substituting f = −M x ¨ in the constitutive equations of the transducer:

Ka V 1/Ka −nd33 Q = (5.20) −M x ¨ C x C(1 − k 2 ) −nd33 When the switch is closed, the voltage V and the charge Q are related through the impedance of the RL circuit, V = −(Ls + R)I = −(Ls + R)sQ, leading to:

M

Switch i R V L

Fig. 5.5 Piezoelectric transducer with a switched RL shunt.

110

5 Passive Damping with Piezoelectric Transducers

¨ = nd33 Ka x Q + RC(1 − k 2 )Q˙ + LC(1 − k 2 )Q

(5.21)

and, using the deﬁnitions (5.11) and (5.12), Q+

2ξe ˙ 1 ¨ Q+ 2Q = nd33 Ka x ωe ωe

(5.22)

It is assumed that the electrical circuit is such that the electrical resonance frequency is signiﬁcantly larger than the mechanical resonance,3 ωe ωn , in such a way that the displacement may be regarded as constant over one half period π/ωe , when the switch is closed, and the charge Q essentially evolves as the step response of a second order system (Fig.5.6). The √ overshoot α is −ξe π/ 1−ξe2 related to the electrical damping according to α = e .

1+ a

Q n d33 Ka x 1

Q0 = 0 t = p/we

t

switch closed

√ 2 Fig. 5.6 Overshoot after closing the RL shunt. α = e−ξe π/ 1−ξe . The switch remains closed exactly one half of the electrical period, and then opens again.

The control strategy is the following: The electrical switch is closed when x˙ = 0 (at the extrema), and remains closed exactly one half of the electrical period, τ = π/ωe , and then opens again; this technique is known as Synchronized Switch Damping on Inductor (SSDI). Assuming that the piezoelectric transducer is initially not charged and that it starts from non-zero initial conditions, the electric charge after the ﬁrst switch is (Fig.5.7) Q0 = (1 + α)nd33 Ka x0 When the switch is open, the charge remains the same until the next extremum, at x1 −x0 . Repeating the switching sequence, Q1 = nd33 Ka x1 + α(nd33 Ka x1 − Q0 ) 3

ωn2 = Ka /M (1 − k 2 ).

5.4 Switched Shunt

x0

111

x2 Q2

Q0

0

t Q1 x1

V

0

t

Fig. 5.7 Impulse response of the piezoelectric transducer of Fig.5.5. The upper ﬁgure shows the displacement x and the electric charge Q; the voltage V between the electrodes of the piezoelectric transducer is shown in the lower ﬁgure (k = 0.03). For low k, xi+1 −xi .

The ﬁrst contribution in the right hand side is the forcing term of the charge equation (5.22), and the second one is the overshoot associated with the diﬀerence with respect to the previous value. Combining with the previous equation and taking into account that x1 −x0 , Q1 nd33 Ka (1 + α)(1 + α)x1 Similarly, Q2 = nd33 Ka x2 + α(nd33 Ka x2 − Q1 ) Q2 nd33 Ka (1 + α)x2 − nd33 Ka α(1 + α)(1 + α)x1 Q2 nd33 Ka (1 + α)x2 [1 + α(1 + α)] etc...

Qn nd33 Ka (1 + α)xn (1 + α + α2 + . . . + αn )

112

5 Passive Damping with Piezoelectric Transducers

Since 0 < α < 1, the asymptotic value is Qn 1+α nd33 Ka ( ) xn 1−α

(5.23)

Thus, the ratio between the electric charge and the vibration amplitude tends to stabilize to a constant value which corresponds to the static response of (5.22) ampliﬁed by (1 + α)/(1 − α), where α is the overshoot in the electric charge following the closure of the switch. Note that Qn has always a sign opposed to that of the velocity in the following half-cycle, which means that it works as dry friction. According to (5.23), the friction force is maximized when α is close to one; however, small values of electrical damping ξe lead to beat (Fig.5.8).

x

øe = 15% øe = 0:4%

0

t

Fig. 5.8 Impulse response of the piezoelectric transducer for two values of the electrical damping ξe (k = 0.1). For small values of ξe , beat is observed in the damped response.

5.4.1

Equivalent Damping Ratio

The dynamics of the mass M is governed by the second equation (5.20) Mx ¨+

Ka nd33 Ka x= Q 2 1−k C(1 − k 2 )

(5.24)

where the force in the right hand side is opposing the velocity and has a constant value during every half-cycle separating two extrema (where the switch occurs); Q is given by Equ.(5.23). The equivalent damping ratio may be evaluated by comparing the energy loss in one cycle to that of an equivalent linear viscous damper. The free response of a linear viscous damper is x = x0 e−ξωn t . In one cycle, the amplitude is reduced by x1 = x0 e−2πξ and the strain energy loss4 is 4

The strain energy is proportional to the square of the displacement.

5.4 Switched Shunt

ΔV V(x0 ) − V(x1 ) = = 1 − e−4πξ 4πξ V(x0 ) V(x0 )

113

(5.25)

Similarly, the energy loss in one cycle associated with the friction damping is the work of the (constant) friction force F0 ; it is obtained from Equ.(5.23-24) ΔV = 4x0 F0 = 4

(nd33 Ka )2 1 + α 2 Ka 1 + α 2 ( )x0 = 4k 2 ( )x 2 C(1 − k ) 1 − α 1 − k2 1 − α 0

after using Equ.(3.23); ΔV 1+α = 8k 2 ( ) V(x0 ) 1−α

(5.26)

Comparing with (5.25), one gets the equivalent viscous damping ξS =

2 1+α 2 ( )k π 1−α

(5.27)

This result must be compared with ξR = k 2 /4 for a purely resistive shunting and ξRL = k/2 for a tuned RL shunt, Fig.5.9. One observes that the performance of the SSDI depends on the electrical damping ξe ; smaller values of ξe lead to larger equivalent mechanical damping, but a lower limit exists, corresponding to the appearance of beat. It is indicated by a triangle in Fig.5.9. Figure 5.10 shows the lower limit of the electrical damping as a function of the electromechanical coupling factor. Note that Fig. 5.9 and 5.10 have

10

ø% RL shunt

1

øe =

0,1

0,01

1% 2% 4% 10% 0% 15% 2

SSDI

R shunt

0,001

K2 -5

10

-4

10

-3

10

-2

10

Fig. 5.9 Equivalent damping ratio ξ as a function of the electromechanical coupling factor k 2 . Comparison of the resistive shunting (R), inductive shunting (RL) and synchronized switch shunting SSDI, for various values of the electrical damping ξe . The triangle indicates the limit value before beat.

114

5 Passive Damping with Piezoelectric Transducers

øe

%

10

No beat

Beat K2

1 -4

10

-3

10

-2

10

Fig. 5.10 Limit value of the electrical damping ξe under which the beat occurs.

been obtained by considering a discrete transducer where the piezoelectric material makes up the entire transducer; for a more complicated system, the modal electromechanical coupling factor Ki should be used instead of k.

5.5

Problems

P.5.1 Show that the maximum damping achievable with a resistive shunt is given by Equ.(5.7). [Hint: The use of a symbolic calculation software is recommended.] P.5.2 Consider a beam equipped with a rectangular piezoelectric transducer extending from a to b, and a collocated actuator-sensor pair at x = l (Fig.5.11). The natural frequencies and the mode shapes with short-circuited electrodes are respectively ωk and φk (x). This system is intended to be equipped with various forms of shunt damping or energy harvesting devices. The input-output relationship of this system can be written in the form f G11 (s) G12 (s) w = V i G21 (s) G22 (s) Write the analytical form of the various transfer functions Gkj (s) involved in this expression.

5.5 Problems

115

l b

a

w x

i

V

f

shunt Fig. 5.11 Beam equipped with a piezoelectric transducer extending from a to b, and a collocated actuator sensor pair at x = l.

6

Collocated versus Non-collocated Control

6.1

Introduction

In the foregoing chapters, we have seen that the use of collocated actuator and sensor pairs, for a lightly damped ﬂexible structure, always leads to alternating poles and zeros near the imaginary axis, Fig.6.1.a. In this chapter, using the root locus technique, we show that this property guarantees the asymptotic stability of a wide class of single-input single-output (SISO) control systems, even if the system parameters are subject to large perturbations. This is because the root locus plot keeps the same general shape, and remains entirely within the left half plane when the system parameters are changed from their nominal values. Such a control system is said to be robust with respect to stability. The use of collocated actuator/sensor pairs is recommended whenever it is possible. This interlacing property of the poles and zeros no longer holds for a noncollocated control, and the root locus plot may experience severe alterations for small parameter changes. This is especially true when the sequence of

Im(s) Pole / Zero flipping

x x

x

x x

x

Re(s) xx Fig. 6.1 (a) Alternating pole-zero pattern of a lightly damped ﬂexible structure with collocated actuator and sensor. (b) Pole-zero ﬂipping for a non-collocated system. A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 117–130. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

118

6 Collocated versus Non-collocated Control

poles and zeros along the imaginary axis is reversed as in Fig.6.1.b. This situation is called a pole-zero ﬂipping. It is responsible for a phase uncertainty of 3600, and the only protection against instability is provided by the damping (systems which are prone to such a huge phase uncertainty can only be gainstabilized).

6.2

Pole-Zero Flipping

Recall that the root locus shows, in a graphical form, the evolution of the poles of the closed-loop system as a function of the scalar gain g applied to the compensator. The open-loop transfer function GH includes the structure, the compensator, and possibly the actuator and sensor dynamics, if necessary. The root locus is the locus of the solution s of the closed-loop characteristic equation 1+gGH(s) = 0 when the real parameter g goes from zero to inﬁnity. If the open-loop transfer function is written

m (s − zi )

GH(s) = k ni=1 (6.1) (s − pi ) i=1 the locus goes from the poles pi (for g = 0) to the zeros zi (as g → ∞) of the open-loop system, and any point P on the locus is such that m i=1

φi −

n

ψi = 1800 + l 3600

(6.2)

i=1

where φi are the phase angles of the vectors ai joining the zeros zi to P and ψi are the phase angles of the vector bi joining the poles pi to P (see Fig.6.8). Since n ≥ m, there are n − m branches of the locus going asymptotically to inﬁnity as g increases. Consider the departure angle from a pole and the arrival angle at the zero when they experience a pole-zero ﬂipping; since the contribution of the far away poles and zeros remains essentially unchanged, the diﬀerence φi − ψi must remain constant after ﬂipping. As a result, a nice stabilizing loop before ﬂipping is converted into a destabilizing one after ﬂipping (Fig.6.2). If the system has some damping, the control system is still able to operate with a small gain after ﬂipping. Since the root locus technique does not distinguish between the system and the compensator, the pole-zero ﬂipping may occur in two diﬀerent ways: • There are compensator zeros near system poles (this is called a notch ﬁlter). If the actual poles of the system are diﬀerent from those assumed in the compensator design, the notch ﬁlter may become ineﬃcient (if the pole moves away from the zero), or worse, a pole-zero ﬂipping may occur. This is why notch ﬁlters have to be used with extreme care. As we shall see in later chapters, notch ﬁlters are generated by optimum feedback

6.3 The Two-Mass Problem

Stable

Unstable

Stable

119

Unstable

ψi φi

φi

Root locus

Root locus

ψi

Fig. 6.2 Detail of a root locus showing the eﬀect of the pole-zero ﬂipping on the departure angle from a pole. Since the contribution of the far away poles and zeros is unchanged, that of the pole and the nearby zero, φi − ψi must also remain unchanged.

compensators and this may lead to serious robustness questions if the parameter uncertainty is large. • Some actuator/sensor conﬁgurations may produce pole-zero ﬂipping within the system alone, for small parameter changes. These situations are often associated with a pole-zero (near) cancellation due to a deﬁciency in the controllability or the observability of the system (typically, when the actuator or the sensor is close to a nodal point in the mode shapes). In some cases, however, especially if the damping is extremely light, instability may occur. No pole-zero ﬂipping can occur within the structure if the actuator and sensor are collocated. The following sections provide examples illustrating these points.

6.3

The Two-Mass Problem

Consider the two-mass problem of Fig.6.3. The system has a rigid body mode along the x axis; it is controlled by a force f applied to the main body M . A ﬂexible appendage m is connected to the main body by a spring k

y

f

d k

m

M b

Fig. 6.3 Two-mass problem.

x

120

6 Collocated versus Non-collocated Control

and a damper b. First, a position control system will be designed, using a sensor placed on the main body (collocated); a sensor attached to the ﬂexible appendage will be considered in the next section. With f representing the control torque and y and d being the attitude angles, this problem is representative of the single-axis attitude control of a satellite, with M representing the main body, and the other inertia representing either a ﬂexible appendage like a solar panel (in which case the sensor can be on the main body, i.e. collocated), or a scientiﬁc instrument like a telescope which must be accurately pointed towards a target (now the sensor has to be part of the secondary structure; i.e. non-collocated). A more elaborate single-axis model of a spacecraft is considered in Problem 2.8. The system equations are: ˙ + (y − d)k = f M y¨ + (y˙ − d)b

(6.3)

md¨ + (d˙ − y)b ˙ + (d − y)k = 0

(6.4)

With the notations ωo2 = k/m,

μ = m/M,

2ξωo = b/m

(6.5)

the transfer functions between the input force f and y and d are respectively: G1 (s) =

Y (s) s2 + 2ξωo s + ωo2 = 2 2 F (s) M s [s + (1 + μ) (2ξωo s + ωo2 )]

(6.6)

G2 (s) =

D(s) 2ξω0 s + ω02 = F (s) M s2 [s2 + (1 + μ) (2ξωo s + ωo2 )]

(6.7)

ω02 M s2 [s2 + (1 + μ) (2ξωo s + ωo2 )]

(6.8)

G2 (s)

Approximation (6.8) recognizes the fact that, for low damping (ξ 1), the far away zero will not inﬂuence the closed-loop response. There are two poles near the imaginary axis. In G1 (s), which refers to the collocated sensor, there are two zeros also near the imaginary axis, at (−ξω0 ± jω0 ). As observed earlier, these zeros are identical to the poles of the modiﬁed system where the main body has been blocked (i.e. constrained mode of the ﬂexible appendage). When the mass ratio μ is small, the polynomials in the numerator and denominator are almost equal, and there is a pole-zero cancellation.

6.3.1

Collocated Control

Let us consider a lead compensator H(s) = g

Ts + 1 αT s + 1

(α < 1)

(6.9)

6.3 The Two-Mass Problem

121

Im(s)

x

x

xx

G1 ( s ) =

s 2 + 0.04 s + 1 s 2 ( s 2 + 0.044 s + 11 .)

H (s) =

10 s + 1 0.04 s + 1

Re(s)

Fig. 6.4 Two-mass problem, root locus plot for the collocated control with a lead compensator (the plot is symmetrical with respect to the real axis, only the upper part is shown).

It includes one pole and one zero located on the negative real axis; the pole is to the left of the zero. Figure 6.4 shows a typical root locus plot for the collocated case when ω0 = 1, M = 1, ξ = 0.02 and μ = 0.1. The parameters of the compensator are T = 10 and α = 0.004. Since there are two more poles than zeros (n − m = 2), the root locus has two asymptotes at ±900 . One observes that the system is stable for every value of the gain, and that the bandwidth of the control system can be a substantial part of ω0 . The lead compensator always increases the damping of the ﬂexible mode. If there are not one, but several ﬂexible modes, there are as many pole-zero pairs and the number of poles in excess of zeros remains the same (n − m = 2 in this case), so that the angles of the asymptotes remain ±900 and the root locus never leaves the stable region. The lead compensator increases the damping ratio of all the ﬂexible modes, but especially those having their natural frequency between the pole and the zero of the compensator. Of course, we have assumed that the sensor and the actuator have perfect dynamics; if this is not the case, the foregoing conclusions may be considerably modiﬁed, especially for large gains.

6.3.2

Non-collocated Control

Figure 6.5 shows the root locus plot for the lead compensator applied to the non-collocated open-loop system characterized by the transfer function G2 (s), Equ.(6.7), with the following numerical data: ω0 = 1, M = 1, μ = 0.1, ξ = 0.02. The excess number of poles is in this case n − m = 3 so that, for

122

6 Collocated versus Non-collocated Control

Im(s)

x

x

xx

G1 ( s ) =

s 2 + 0.04 s + 1 s ( s 2 + 0.044 s + 11 .)

H (s) =

10 s + 1 0.04 s + 1

2

Re(s)

Fig. 6.5 Two-mass problem, root locus plot for the non-collocated control with a lead compensator.

gGH 0 dB

GM =161 .

Bandwidth ωc=0.056

1

ω

-90°

φ(gGH ) PM = 29°

-180°

ω Fig. 6.6 Two-mass problem, Bode plots of the non-collocated control for g = 0.003.

6.4 Notch Filter

123

large gains, the ﬂexible modes are heading towards the asymptotes at ±600 , in the right half plane. For a gain g = 0.003, the closed-loop poles are located at −0.0136 ± 0.0505j and −0.0084 ± 1.0467j (these locations are not shown in Fig.6.5 for clarity: the poles of the rigid body mode are close to the origin and those of the ﬂexible mode are located between the open-loop poles and the imaginary axis). The corresponding Bode plots are shown in Fig.6.6; the phase and gain margins are indicated. One observes that even with this small bandwidth (crossover frequency ωc = 0.056), the gain margin is extremely small. A slightly lower value of the damping ratio would make the closed-loop system unstable (Problem 6.1).

6.4

Notch Filter

A classical way of alleviating the eﬀect of the ﬂexible modes in non-collocated control is to supplement the lead compensator with a notch ﬁlter with two zeros located near the ﬂexible poles: H(s) = g .

T s + 1 s2 /ω12 + 1 . αT s + 1 (s/a + 1)2

(6.10)

The zeros of the notch ﬁlter, at s = ±jω1 , are selected right below the ﬂexible poles. The double pole at −a aims at keeping the compensator proper (i.e. the degree of the numerator not larger than that of the denominator); it can, for example, be selected far enough along the negative real axis. The corresponding root locus is represented in Fig.6.7.a for ω1 = 0.9 and a = 10. This compensator allows a larger bandwidth than the lead compensator alone (Problem 6.2). To be eﬀective, a notch ﬁlter must be closely tuned to the ﬂexible mode that we want to attenuate. However, as we already mentioned, the notch ﬁlter suﬀers from a lack of robustness and should not be used if the uncertainty in the system properties is large. To illustrate this, Fig.6.7.b shows a detail of the root locus near the notch, when the natural frequency of the system is smaller than expected (in the example, ω0 is reduced from 1 rad/s to 0.8 rad/s; the other data are identical to that of Fig.6.5, while the notch ﬁlter is kept the same (being implemented in the computer, the notch ﬁlter is not subject to parameter uncertainty). The rest of the root locus is only slightly aﬀected by the change. Because the open-loop poles of the ﬂexible mode move from above to below the zero of the notch ﬁlter (from ±j to ±0.8j with the zeros at ±0.9j in the example), there is a pole-zero ﬂipping, with the consequence that the branch of the root locus connecting the pole to the zero rapidly becomes unstable. This example emphasizes the fact that notch ﬁlters should be used with extreme care, especially for systems where the uncertainty is large (Problem 6.3).

124

6 Collocated versus Non-collocated Control Im(s)

(a)

(b) notch at ω1 = 0.9 j

x

x ω0 = 0.8 rad/s

xx

xx

Re(s)

Fig. 6.7 Two-mass problem, non-collocated control; (a) Lead compensator plus notch ﬁlter (b) Detail of the root locus near the notch for oﬀ-nominal open-loop system (ω0 reduced from 1 to 0.8).

6.5

Eﬀect of Pole-Zero Flipping on the Bode Plots

From Equ.(6.1),

m (jω − zi ) GH(jω) = k ni=1 (jω − pi ) i=1

(6.11)

the phase of GH(jω) for a speciﬁc value jω is given by m i=1

φi −

n

ψi

(6.12)

i=1

where φi is the phase angle of the vector ai joining the zero zi to jω and ψi is the phase angle of the vector bi joining the pole pi to jω (Fig.6.8). Accordingly, an imaginary zero at jω0 produces a phase lead of 1800 for ω > ω0 and an imaginary pole produces similarly a phase lag of 1800 . Therefore, a pole-zero ﬂipping near the imaginary axis produces a phase uncertainty of 3600 in the frequency range between the pole and the zero. It appears that the only way the closed-loop stability can be guaranteed in the vicinity of a pole-zero ﬂipping is to have the open-loop system gain-stabilized (i.e. such that |gHG| < 1) in that frequency range.

6.6

Nearly Collocated Control System

In many cases, the actuator and sensor pair are close to each other without being strictly collocated. This situation is examined here. Consider the undamped system of Fig.6.9 where the actuator input u is applied at a and the sensor y is located at s. The open-loop FRF of the system is given by n y φi (a)φi (s) G(ω) = = (6.13) u μi (ωi2 − ω 2 ) i=1

6.6 Nearly Collocated Control System

pk

r b r k ai

φi

zi

ψk

jω

r bi

ψ i (−)

φ i (+ )

φ i (+ )

ψ i (−)

φ i − ψ i = 180°

ψi

125

φ i − ψ i = −180°

pi Fig. 6.8 Eﬀect of the pole-zero ﬂipping on the phase diagram.

u s a

y

Fig. 6.9 Structure with nearly collocated actuator/sensor pair.

where φi (a) and φi (s) are the modal amplitudes at the actuator and the sensor locations, respectively (the sum includes all the normal modes in this case). The residues of (6.13) are no longer guaranteed to be positive; however, if the actuator location a is close to the sensor location s, the modal amplitudes φi (a) and φi (s) will be close to each other, at least for the low frequency modes, and the corresponding residues will again be positive. The following result can be established in this case : If two neighboring modes are such that their residues φi (a)φi (s) and φi+1 (a)φi+1 (s) have the same sign, there is always an imaginary zero between the two poles (Martin, 1978). Since G(ω) is continuous between ωi and ωi+1 , this result will be established if one proves that the sign of G(ω) near ωi is opposite to that near ωi+1 . At ω = ωi + δω, G(ω) is dominated by the contribution of mode i and its sign is sign[

φi (a)φi (s) ] = −sign[φi (a)φi (s)] ωi2 − ω 2

(6.14)

At ω = ωi+1 − δω, G(ω) is dominated by the contribution of mode i + 1 and its sign is sign[

φi+1 (a)φi+1 (s) ] = sign[φi+1 (a)φi+1 (s)] 2 ωi+1 − ω2

(6.15)

126

6 Collocated versus Non-collocated Control

− Thus, if the two residues have the same sign, the sign of G(ω) near ωi+1 + is opposite to that near ωi . By continuity, G(ω) must vanishes somewhere − in between, at zi such that ωi+ < zi < ωi+1 . Note, however, that when the residues of the expansion (6.13) are not all positive, there is no guarantee that G(ω) is an increasing function of ω, and one can ﬁnd situations where there are more than one zero between two neighboring poles.

6.7

Non-collocated Control Systems

Since the low frequency modes vary slowly in space, the sign of φi (a)φi (s) tends to be positive for low frequency modes when the actuator and sensor are close to each other, and the interlacing of the poles and zeros is maintained at low frequency. This is illustrated in the following example: Consider a simply supported uniform beam of mass per unit length m and bending stiﬀness EI. The natural frequencies and mode shapes are respectively ωi2 = (iπ)4

EI ml4

(6.16)

Sensor motion

u

s>a

y

a = l=10

Mode 1 sin(ùx=l)

Mode 2 sin(2ùx=l) l=2

2l=3

l=3

l=4

Mode 3 sin(3ùx=l)

l=2

3l=4

Mode 4 sin(4ùx=l)

Fig. 6.10 Uniform beam with non-collocated actuator/sensor pair. Mode shapes.

6.7 Non-collocated Control Systems

127

and

iπx (6.17) l (the generalized mass is μi = ml/2). Note that the natural frequency increases as the square of the mode order. We assume that a force actuator is placed at a = 0.1 l and we examine the evolution of the open-loop zeros as a displacement sensor is moved to the right from s = a (collocated), towards the end of the beam (Fig 6.10). The evolution of the open-loop zeros with the sensor location along the beam is shown in Fig 6.11; the plot shows the ratio zi /ω1 , so that the openloop poles (independent of the actuator/sensor conﬁguration) are at 1, 4, 9, 25, etc.... For s = a = 0.1 l, the open-loop zeros are represented by ◦; they alternate with the poles. Another position of the actuator/sensor pair along the beam would lead to a diﬀerent position of the zeros, but always alternating with the poles. As the sensor is displaced from the actuator, s > a, the zeros tend to increase in magnitude as shown in Fig.6.11, but the low frequency ones still alternate. When s = 0.2 l, z4 becomes equal to ω5 and there is no zero any longer between ω4 and ω5 when s exceeds 0.2 l. Thus, a pole/zero ﬂipping occurs. Similarly, z3 ﬂips with ω4 for s = l/4, z2 ﬂips with ω3 for s = l/3 and z1 ﬂips with ω2 for s = l/2. Examining the mode shapes, one notices that φi (x) = sin

! !1

Actuator

25

z4

Mode 5

s 0.2= l

z3

16

Mode 4

z2

0.25

z1 Mode 3

9

0.33 Mode 2

4

0.5

1 0.1

0.2 0.3 0.4 0.5

0.6

Mode 1 0.7 0.8 0.9 1.0

s l

Fig. 6.11 Evolution of the imaginary zeros when the sensor moves away from the actuator along a simply supported beam (the actuator is at 0.1 l). The abscissa is the sensor location, the ordinate is the frequency of the transmission zero.

128

6 Collocated versus Non-collocated Control

the pole-zero ﬂipping always occurs at a node of one of the mode shapes, and this corresponds to a change of sign in φi (a)φi (s), as discussed above. This simple example conﬁrms the behavior of the pole/zero pattern for nearly collocated control systems: the poles and zeros are still interlacing at low frequency, but not at higher frequency, and the frequency where the interlacing stops decreases as the distance between the actuator and sensor increases. A more accurate analysis (Spector & Flashner, 1989; Miu, 1993) shows that: For structures such as bars in extension, shafts in torsion or simply connected spring-mass systems (non dispersive), when the sensor is displaced from the actuator, the zeros migrate along the imaginary axis towards inﬁnity. The imaginary zeros are the resonance frequencies of the two substructures formed by constraining the structure at the actuator and sensor (this generalizes the result of chapter 2). For beams with speciﬁc boundary conditions, the imaginary zeros still migrate along the imaginary axis, but every pair of zeros that disappears at inﬁnity reappears symmetrically at inﬁnity on the real axis and moves towards the origin (Fig 6.12). Systems with right half plane zeros are called non-minimum phase. Thus, non collocated control systems are always nonminimum phase, but this does not cause diﬃculties if the right half plane zeros lie well outside the desired bandwidth of the closed-loop system. When they interfere with the bandwidth, they put severe restrictions on the control system, by reducing signiﬁcantly the phase margin; this point will be discussed later in chapter 10.

Imaginary zeros migrate towards

j1

Real zeros come from 1

Fig. 6.12 Evolution of the zeros of a beam when the sensor moves away from the actuator. Every pair of imaginary zeros which disappears at inﬁnity reappears on the real axis.

6.9 Problems

6.8

129

The Role of Damping

To conclude this chapter, we would like to insist on the role of the damping for non-collocated control systems. We have seen that the imaginary zeros provide the necessary phase lead to compensate the undesirable phase lag caused by the poles. Whenever a ﬂexible pole is not associated with a zero, it produces a net phase lag of 1800 . According to the stability criterion, the amplitude of the open-loop transfer function must satisfy |gGH| < 1 whenever the phase lag exceeds 1800. Since the amplitude of gGH in the rolloﬀ region is dominated by the resonant peaks of G, it is clear that the damping of the ﬂexible modes is essential for non-collocated systems (Problem 6.1). Damping augmentation can be achieved by passive as well as active means. For spacecraft applications, the former often use constrained layers of high damping elastomers placed at appropriate locations in the structure (e.g. Johnson 1981 or Ikegami, 1986). More varied ways are regularly used in civil engineering applications, such as tuned-mass dampers, tuned liquid dampers, chain dampers, etc... Active damping is the subject of next chapter.

6.9

Problems

P.6.1 Consider the lead compensator for the non-collocated control of the two-mass system. (a) Determine the value of the damping ratio ξ which would reduce the gain margin to zero. (b) What would be the gain margin if ξ = 0.04 instead of ξ = 0.02. P.6.2 Consider the lead compensator plus notch ﬁlter for the non-collocated control of the two-mass system (section 6.4). Draw the corresponding Bode plots. Select a reasonable value of the gain g and compare the bandwidth, the gain and phase margins with those of the lead compensator of Fig.6.6. P.6.3 (a) Repeat the previous problem when the frequency of the appendage is lower than that of the notch ﬁlter (ω0 =0.8 rad/sec); compare the Bode plots and comment on the role of the damping. (b) Same as (a) with the frequency of the appendage moving away from the notch ﬁlter (ω0 =1.1 rad/sec). Comment on the importance of tuning the notch ﬁlter. P.6.4 Consider the PD regulator H(s) = g(T s + 1) applied to the open-loop structure G(s) =

∞ φi (a)φi (s) i=1

s2 + ωi2

130

6 Collocated versus Non-collocated Control

Assuming that the modes are well separated, show that, for small gain g, the closed-loop damping ratio of mode i is ξi = gT

φi (a)φi (s) 2ωi

Conclude on the stability condition (Gevarter, 1970). [Hint: Use a perturbation method, s = ωi [−ξ + j(1 + δ)] in the vicinity of jωi , and write the closed-loop characteristic equation.] P.6.5 Consider a simply supported uniform beam with a point force actuator and a displacement sensor. Based on the result of the previous problem, sketch a non-collocated actuator and sensor conﬁguration such that a PD regulator is stabilizing for the ﬁrst three modes. P.6.6 Consider a system of n identical masses M simply connected with n+1 springs of stiﬀness k; assume that a point force is applied on mass i and a displacement sensor is connected to mass j(> i). Show that the zeros of the transfer function are the resonance frequencies of the two substructures (from 1 to i and from j + 1 to n ), formed by constraining the masses i and j (Miu, 1991). P.6.7 Consider the non-collocated control of the two-mass problem [the system transfer function is given by Equ.(6.7)] with M = 1. For various values of the mass ratio μ = 0.1, 0.01, 0.001, assuming a lead compensator (6.9), draw a diagram of the bandwidth of the control system, ωc /ω0 as a function of the damping ratio ξ for the limit of stability (GM = 0).

7

Active Damping with Collocated System

7.1

Introduction

The use of collocated (and dual) actuator and sensor pairs, for a lightly damped ﬂexible structure, always leads to alternating poles and zeros near the imaginary axis, Fig.7.1. In this chapter, we use this interlacing property to develop Single Input-Single Output (SISO) active damping schemes with guaranteed stability. By active damping, we mean that the primary objective of the controller is simply to increase the negative real part of the system poles, while maintaining the natural frequencies essentially unchanged. This simply attenuates the resonance peak in the dynamic ampliﬁcation (Fig 7.2). Recall that the relationship between the damping ratio ξ and the angle φ with respect to the imaginary axis is sin φ = ξ, and that the dynamic ampliﬁcation at resonance is 1/2ξ. Note that for typical damping values encountered in practice, the values of φ are very close to 0; this is why in most of the root locus plots shown in this chapter, diﬀerent scales are used for the real and the imaginary axes, leading to a distortion of the angles.

Im(s)

(a)

jzi j!i

Im(s)

x

x

x

x

x

Re(s)

x

(b)

Re(s)

Fig. 7.1 Pole/Zero pattern of a structure with collocated (dual) actuator and sensor; (a) undamped; (b) lightly damped.

A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 131–154. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

132

7 Active Damping with Collocated System Im(s)

(a)

Dynamic Amplification (dB)

(b)

Open-loop

þ

Damping

0

sinþ = ø

!

Re(s)

Fig. 7.2 Role of damping (a) System poles. (b) Dynamic ampliﬁcation (1/2ξ).

Active damping requires relatively little control eﬀort; this is why it is also called Low Authority Control (LAC), by contrast with other control strategies which fully relocate the closed-loop poles (natural frequency and damping) and are called High Authority control (HAC). A remarkable feature of the LAC controllers discussed here is that the control law requires very little knowledge of the system (at most the knowledge of the natural frequencies). However, guaranteed stability does not mean guaranteed performance; good performance does require information on the system as well as on the disturbance applied to it, for appropriate actuator/sensor placement, actuator sizing, sensor selection and controller tuning. Actuator placement means good controllability of the modes dominating the response (note that controllability and observability go together when the sensor is collocated and dual); this will be reﬂected by well separated poles and zeros, leading to wide loops in the root-locus plots. In order to keep the formal complexity to a minimum, we assume no structural damping and perfect actuator and sensor dynamics throughout most of this chapter. The impact of the actuator and sensor dynamics on stability, and the beneﬁcial eﬀect of passive damping is discussed at the end. Consider an undamped structure with a collocated, dual actuator/sensor pair (typically, a point force actuator collocated with a displacement sensor, or a torque actuator collocated with an angular sensor). The actuator/sensor position is deﬁned by the inﬂuence vector b.1 We assume that the open-loop FRF G(ω) does not have any feedthrough (constant) component, so that G(ω) decays at high frequency as s−2 ; the roll-oﬀ (high frequency decay rate) is −40 dB/decade in this case. The open-loop transfer function of such a system, expressed in modal coordinates, reads 1

Appearing in the equations M x ¨ + Kx = bu and y = bT x.

7.2 Lead Control

G(s) =

n i=1

(bT φi )2 μi (s2 + ωi2 )

133

(7.1)

where bT φi is the modal amplitude at the actuator/sensor location. The polezero pattern is that of Fig 7.3 (where 3 modes have been assumed); there are two structural poles in excess of zeros, which provide a roll-oﬀ rate s−2 (a feedthrough component would introduce an additional pair of zeros).

Im(s)

jz i

j! i

àp

Lead

àz

Structure

Re(s)

Fig. 7.3 Open-loop pole/zero pattern and root locus of the lead compensator applied to a structure with collocated actuator/sensor (open-loop transfer function with two poles in excess of zeros). Diﬀerent scales are used on the real and imaginary axes and only the upper half of the plot is shown.

7.2

Lead Control

This system can be damped with a lead compensator : H(s) = g

s+z s+p

(p z)

(7.2)

The block diagram of the control system is shown in Fig 7.4. This controller takes its name from the fact that it produces a phase lead in the frequency band between z and p, bringing active damping to all the modes belonging to z < ωi < p. Figure 7.3 also shows the root locus of the closed-loop poles when the gain g is varied from 0 to ∞. The closed-loop poles which remain at ﬁnite distance start at the open-loop poles for g = 0 and eventually go to the open-loop zeros for g → ∞. Since there are two poles more than zeros, two branches go to inﬁnity (at ±900 ). The controller does not have any roll-oﬀ,

134

7 Active Damping with Collocated System

but the roll-oﬀ of the structure is enough to guarantee gain stability at high frequency. Note that the asymptotic values of the closed-loop poles for large gains being the open-loop zeros zi , which are the natural frequencies of the constrained system, they are therefore independent of the lead controller parameters z and p. For a structure with well separated modes, the individual loops in the root-locus (Fig 7.3) are to a large extent independent of each other, and the root-locus for a single mode can be drawn from the lead controller and the asymptotic values ωi and zi of that mode only (Fig 7.5). The characteristic equation for this simpliﬁed system can be written from the pole-zero pattern: 1+α

(s2 + zi2 )(s + z) =0 (s2 + ωi2 )(s + p)

(7.3)

where α is the variable parameter going from α = 0 (open-loop) to inﬁnity. This can be written alternatively

+

g s+z s+p

à

u

G(s) =

P i

(b Tþ i) 2 ö i(s 2+! 2i )

y

Fig. 7.4 Block diagram of the lead compensator applied to a structure with collocated actuator/sensor (open-loop transfer function G(s) with two poles in excess of zeros).

Im(s)

jz i ømax

j! i

Re(s)

àp

àz

Fig. 7.5 Structure with well separated modes and lead compensator, root-locus of a single mode.

7.3 Direct Velocity Feedback (DVF)

1+

135

1 (s2 + ωi2 )(s + p) =0 α (s2 + zi2 )(s + z)

If z and p have been chosen in such a way that z ωi < zi p, this can be approximated in the vicinity of jωi by 1+

p (s2 + ωi2 ) =0 α s(s2 + zi2 )

(7.4)

This characteristic equation turns out to be the same as that of the Integral Force Feedback (IFF) controller discussed a little later in this chapter, Equ.(7.31); it follows that the maximum achievable modal damping is given by zi − ω i ξmax (ωi > zi /3) (7.5) 2ωi Note that the maximum achievable damping is controlled by the separation between the open-loop pole ωi and the nearby zero zi .

7.3

Direct Velocity Feedback (DVF)

The Direct Velocity Feedback (DVF) is the particular case of the lead controller as z → 0 and p → ∞. Returning to the basic equations: Structure: Mx ¨ + Kx = bu (7.6) Output (velocity sensor): y = bT x˙

(7.7)

u = −gy

(7.8)

Control : one ﬁnds easily the closed-loop equation Mx ¨ + gbbT x˙ + Kx = 0

(7.9)

Upon transforming into modal coordinates, x = Φz and taking into account the orthogonality conditions, one gets diag(μi )¨ z + gΦT bbT Φ z˙ + diag(μi ωi2 )z = 0

(7.10)

where z is the vector of modal amplitudes. The matrix ΦT bbT Φ is in general fully populated. For small gains, one may assume that it is diagonally dominant, diag(bT φi )2 . This assumption leads to a set of decoupled equations. Mode i is governed by μi z¨i + g(bT φi )2 z˙i + μi ωi2 zi = 0

(7.11)

136

7 Active Damping with Collocated System

By analogy with a single d.o.f. oscillator, one ﬁnds that the active modal damping ξi is given by 2ξi μi ωi = g(bT φi )2 (7.12) or ξi =

g(bT φi )2 2μi ωi

(7.13)

Thus, for small gains, the closed-loop poles sensitivity to the gain (i.e. the departure rate from the open-loop poles) is controlled by (bT φi )2 , the square of the modal amplitude at the actuator/sensor location. Now, let us examine the asymptotic behavior for large gains. For all g, the closed-loop eigenvalue problem (7.9) is (M s2 + gbbT s + K)x = 0

(7.14)

Except for the presence of s in the middle term, this equation is close to Equ.(2.45); proceeding as we did in chapter 2, it follows that x = −(K + M s2 )−1 gsbbT x or

bT x = −gsbT (K + M s2 )−1 bbT x

(7.15)

T

Since b x is a scalar, one must have sbT (K + M s2 )−1 b = −

1 g

(7.16)

and taking the limit for g → ∞ sbT (K + M s2 )−1 b = 0

(7.17)

The solutions of this equation are s = 0 and the solutions of (2.49), that are the eigenvalues of the constrained system. The fact that the eigenvalues are purely imaginary, s = ±jω0 , stems from the fact that K and M are symmetric and semi-positive deﬁnite. Typical root locus plots for a lead controller and a DVF controller are compared in Fig.7.6. As for the lead controller, for well separated modes, those which are far enough from the origin can be analyzed independently of each other. In this way, the characteristic equation for mode i is approximated by 1+g

s(s2 + zi2 ) =0 (s2 + ω12 )(s2 + ωi2 )

(besides the poles at ±jωi and the zeros at ±jzi , we include the zero at s = 0 and the poles at ±jω1 ) which in turn, if ωi > zi ω1 , may be approximated by

7.4 Positive Position Feedback (PPF) Im(s)

g

(a)

137

Im(s)

g

(b)

Structure

Re(s)

Structure

Re(s)

Lead

DVF

Fig. 7.6 Collocated control system. (a) Root locus for a lead controller. (b) DVF controller.

1+g

s2 + zi2 =0 s(s2 + ωi2 )

(7.18)

in the vicinity of mode i. This root locus is essentially the same as in the previous section (with zi appearing in the numerator and ωi in the denominator), and the formula for the maximum modal damping (7.32) applies ξmax

7.4

ωi − z i 2zi

(zi > ωi /3)

(7.19)

Positive Position Feedback (PPF)

There are frequent situations where the open-loop FRF does not exhibit a roll-oﬀ of −40 dB/decade as in the previous section. In fact, a feedthrough component may arise from the truncation of the high frequency dynamics, as in (2.34), or because of the physical nature of the system (e.g. beams or plates covered with collocated piezoelectric patches, Fig.4.8). In these situations, the degree of the numerator of G(s) is the same as that of the denominator and the open-loop pole-zero pattern has an additional pair of zeros at high frequency. Since the overall degree of the denominator of H(s)G(s) must exceed the degree of the numerator, the controller H(s) must have more poles than zeros. The Positive Position Feedback was proposed to solve this problem (Goh & Caughey). The second-order PPF controller consists of a second order ﬁlter H(s) =

−g s2 + 2ξf ωf s + ωf2

(7.20)

where the damping ξf is usually rather high (0.5 to 0.7), and the ﬁlter frequency ωf is adapted to target a speciﬁc mode. The block diagram of the control system is shown in Fig.7.7; the negative sign in H(s), which produces a positive feedback, is the origin of the name of this controller.

138

7 Active Damping with Collocated System

Fig. 7.7 Block diagram of the second-order PPF controller applied to a structure with collocated actuator and sensor (the open-loop transfer function has the same number of poles and zeros).

Im(s)

Im(s)

jz i

Structure

j! i PPF (mode 1)

PPF (mode 2)

Re(s) Stability limit

Fig. 7.8 Root locus of the PPF controller applied to a structure with collocated actuator and sensor (the open-loop transfer function has the same number of poles and zeros). (a) Targeted at mode 1. (b) Targeted at mode 2. (For clarity, diﬀerent scales are used for the real and the imaginary axes).

Figure 7.8 shows typical root loci when the PPF poles are targeted to mode 1 and mode 2, respectively (i.e. ωf close to ω1 or ω2 , respectively). One sees that the whole locus is contained in the left half plane, except one branch on the positive real axis, but this part of the locus is reached only for large values of g, which are not used in practice. Observe that the closed-loop poles with a natural frequency close to ωf move much faster than the others. The stability condition can be established as follows: the characteristic equation of the closed-loop system reads

ψ(s) = 1 + gH(s)G(s) = 1 −

m g bT φi φTi b { }=0 s2 + 2ξf ωf s + ωf2 i=1 μi (s2 + ωi 2 )

7.4 Positive Position Feedback (PPF)

or ψ(s) = s2 + 2ξf ωf s + ωf2 − g{

m i=1

139

bT φi φTi b }=0 μi (s2 + ωi 2 )

According to the Routh-Hurwitz criterion for stability (see chapter 13), if one of the coeﬃcients of the power expansion of the characteristic equation becomes negative, the system is unstable. It is not possible to write the power expansion ψ(s) explicitly for an arbitrary value of m, however, one can see easily that the constant term (in s0 ) is an = ψ(0) =

ωf2

−g

m bT φi φT b i

i=1

μi ωi 2

In this case, an becomes negative when the static loop gain gG(0)H(0) becomes larger than 1. The stability condition is therefore gG(0)H(0) =

m g bT φi φTi b { }<1 ωf2 i=1 μi ωi 2

(7.21)

Note that it is independent of the structural damping in the system. Since the instability occurs for large gains which are not used in practice, the PPF can be regarded as unconditionally stable. Unlike the lead controller of the previous section which controls all the modes which belong to z < ωi < p and even beyond, the PPF ﬁlter must be tuned on the targeted mode (it is therefore essential to know the natural frequency accurately), and the authority on the modes with very diﬀerent frequencies is substantially reduced. Several PPF ﬁlters can be used in parallel, to target several modes simultaneously, but they must be tuned with care, because of the cross coupling between the various loops. An application will be considered in section 14.4. The following ﬁrst-order PPF controller is an alternative to the second order controller: −g H(s) = (7.22) 1 + τs A typical root locus is shown in Fig.7.9. As compared to the second-order controller, this one does not have to be tuned on targeted modes; the roll-oﬀ is reduced to −20 dB/decade instead of −40 dB/decade with the second order controller. The part of the locus on the real axis also becomes unstable for large gains. Proceeding exactly as we did for the second order controller, one ﬁnds easily that the stability condition is, once again, that the static loop gain must be lower than 1: gG(0) < 1 (7.23) Thus the gain margin is GM = [gG(0)]−1 . The stability condition corresponds to the negative stiﬀness of the controller overcoming that of the structure. Because of this negative stiﬀness, the root-locus does not leave the open-loop poles orthogonally to the imaginary axis; this is responsible

140

7 Active Damping with Collocated System

Im(s)

jz i

j! i PPF (1st order)

Structure

Re(s)

s = à 1=ü Fig. 7.9 Root locus of the ﬁrst-order PPF controller (the scale on the real axis has been magniﬁed for clarity).

for larger control eﬀorts, as compared to the other strategies considered before, which may be a serious drawback in applications where the control eﬀort is an important issue.

7.5

Integral Force Feedback(IFF)

So far, all the collocated system that we have considered exhibit alternating poles and zeros, starting with a pole at low frequency (ω1 < z1 < ω2 < z2 < ω3 ....). This corresponds to an important class of actuator/sensor pairs, including (force actuator/displacement or velocity sensor), (torque actuator/angular position or angular velocity sensor), (piezoelectric patches used as actuator and sensor). In this section, we discuss an other interlacing polezero conﬁguration but starting with a zero (z1 < ω1 < z2 < ω2 ...). This situation arises, typically, in a actuator/sensor pair made of a displacement actuator and a force sensor, such as that of the piezoelectric active truss of Fig 7.10 already considered in section 4.9. The governing equation has been found to be (7.24) Mx ¨ + (K ∗ + bbT Ka )x = bKa δ where K ∗ is the stiﬀness matrix of the structure without the active strut and K ∗ + bbT Ka is the global stiﬀness matrix, including the active strut. Note that this equation applies to linear actuators of many types, piezoelectric, magnetostrictive, thermal, ball-screw,..., provided that the constitutive equation of the actuator is

7.5 Integral Force Feedback(IFF)

141

Fig. 7.10 Active truss with an active strut consisting of a displacement actuator and a force sensor. δ = ga u is the unconstrained extension of the strut induced by the control u. The passive stiﬀness of the strut is Ka and f /Ka is the elastic extension. Δ = bT x is the total extension of the active strut.

Δ = δ + f /Ka or equivalently y = f = Ka (bT x − δ)

(7.25)

where Δ = bT x is the total extension of the active strut, δ its free extension (control), f is the force sensor output, and Ka is the strut stiﬀness. The open-loop transfer function has been found to be G(s) =

m n y νi = Ka [ + νi − 1] δ s2 /ωi2 + 1 i=1

(7.26)

i=m+1

where the residues νi are the fraction of modal strain energy in the active strut. The FRF exhibit alternating poles and zeros on the imaginary axis, beginning with a zero, Fig.7.11. The feedback control law is in this case a positive Integral Force Feedback δ=

g y Ka s

(7.27)

(the Ka at the denominator is for normalization purpose; y/Ka is the elastic extension of the strut). The block diagram of the system is represented in Fig.7.12. The pole-zero pattern of the system is shown in Fig 7.13. It consists of interlacing pole-zero pairs on the imaginary axis (z1 < ω1 < z2 < ω2 ...) and the pole at s = 0 from the controller. The root locus plot consists of the negative real axis and a set of loops going from the open-loop poles

142

7 Active Damping with Collocated System

FRF

!1

z1

!3

!2

z2

z3

!

Fig. 7.11 Open-loop FRF of an active truss. The active strut consists of a piezoelectric actuator and a collocated force sensor.

Unconstrained piezo expansion

gD(s) + à

àg K as

î

Ka

Force

G0(s) ún P ÷i i=1

û à1 1+s2=! 2

y

i

Fig. 7.12 Block diagram of the IFF control.

±jωi to the open-loop zeros ±jzi . All the loops are entirely contained in the left half plane, so that the closed-loop system is unconditionally stable, for all values of the gain g. Note also that the root-locus plot does not change signiﬁcantly if the pole at the origin is moved slightly in the left half plane, to avoid saturation (which is often associated with integral control). In fact, piezoelectric force sensors have a built-in high-pass ﬁlter and cannot measure a static component.2 2

This issue and the static behavior of the active truss will be reexamined later in in section 15.5 where the “Beta controller” will be introduced; see also Problem 7.5.

7.5 Integral Force Feedback(IFF)

143

Im(s)

j! i

Structure

jz i

Re(s) control Fig. 7.13 Pole-zero pattern of the active strut and root-locus of the IFF.

Equations (7.25) and (7.27) can be combined, leading to δ=

g T b x s+g

(7.28)

and upon substituting into (7.24), one gets the closed-loop characteristic equation g [M s2 + (K ∗ + bbT Ka ) − bbT Ka ]x = 0 (7.29) s+g For g = 0, the eigenvalues are indeed the open-loop poles, ±jωi . Asymptotically, for g → ∞, the eigenvalue problem becomes [M s2 + K ∗ ]x = 0

(7.30)

where K ∗ is the stiﬀness matrix of the structure without the active strut. Thus, the open-loop zeros ±jzi are the natural frequencies of the truss after removing the active strut. This situation should be compared to that discussed earlier (section 2.5.1) in connection with a displacement sensor. In that case, the open-loop zeros were found to be the natural frequencies of the constrained system where the d.o.f. along which the actuator and sensor operate is blocked (i.e. the sensor output is cancelled). In the present case, if the sensor output is zero, the force carried by the active strut vanishes and it can be removed. For well separated modes, the individual loops in the root-locus of Fig 7.13 are, to a large extent, independent of each other, and the root locus

144

7 Active Damping with Collocated System

IFF

a)

b)

Fig. 7.14 (a) IFF root locus of a single mode. (b) Evolution of the root locus as zi moves away from ωi .

of a single mode can be drawn from the asymptotic values ±jωi and ±jzi only (Fig 7.14.a). The corresponding characteristic equation is 3 1+g

(s2 + zi2 ) =0 s(s2 + ωi2 )

(7.31)

The actual root-locus, Fig.7.13, which includes the inﬂuence of the other modes, is only slightly diﬀerent from that of Fig 7.14.a, with the same asymptotic values at ±jωi and ±jzi . It can be shown (Problem 7.3) that the maximum modal damping for mode i is given by ξimax =

ω i − zi 2zi

(zi ≥ ωi /3)

(7.32)

It is achieved for g = ωi ωi /zi. This result applies only for ξi ≤ 1, that is for zi ≥ ωi /3. For zi = ωi /3, the locus touches the real axis as indicated in Fig 7.14.b, and for even larger diﬀerences ωi − zi , the root locus includes part of the real axis, which means that one can achieve enormous damping values. Equation (7.32) relates clearly the maximum achievable damping and the distance between the pole and the zero. Note, however, that if the system has several modes, there is a single tuning parameter g, and the various loops are travelled at diﬀerent speeds. As a result, the optimal value of g for one mode will not be optimal for another one, and a compromise must be found. Equation (7.29) can be transformed into modal coordinates. Using the orthogonality conditions, one ﬁnds 3

Note the similarity with (7.4) for the lead controller and with (7.18) for the DVF.

7.6 Duality between the Lead and the IFF Controllers

[diag(μi )s2 + diag(μi ωi2 ) − ΦT bbT ΦKa

g ]z = 0 s+g

145

(7.33)

For small g, the equations are nearly decoupled: [s2 + ωi2 − νi ωi2

g ]zi = 0 s+g

(7.34)

after using the deﬁnition of νi , Equ.(4.107). Since the root locus plot leaves the open-loop poles orthogonally to the imaginary axis, for small gain, one can assume a solution of the form s = ωi (−ξi ± j). Substituting into (7.34), one ﬁnds easily gνi ξi = (7.35) 2ωi Thus, for small gains, the closed-loop poles sensitivity to the gain, i.e. the departure rate from the open-loop poles, is controlled by the fraction of modal strain energy in the active element. This result is very useful for the design of active trusses. The active strut should be located to maximize νi for the critical modes of the structure. Note that νi is readily available from ﬁnite element softwares; an example is analyzed in section 14.2. In summary, the active strut placement can be made from the inspection of the map of modal strain energy in the ﬁnite element model, once the active strut location has been selected, a modal analysis of the truss including the active strut gives the open-loop poles ±jωi , and a modal analysis after removing the active strut gives the open-loop zeros ±jzi. Then, the rootlocus plot can be drawn. The case of a truss involving several active struts controlled in a decentralized manner is examined in section 14.3.

7.6

Duality between the Lead and the IFF Controllers

In both cases, if the modes are well separated, they behave essentially independently, and their closed-loop behavior may be analyzed as a single mode, without considering the interaction with the other modes.

7.6.1

Root-Locus of a Single Mode

Figure 7.15 illustrates the duality between the IFF and the Lead controllers. If the pole and the zero of the lead controller (7.2) are such that z ωi < zi p, the root-locus plots of every mode turn out to be very similar (Fig.7.15.c and d). The DVF can be looked at as the limit of the Lead controller as z → 0 and p → ∞; however, depending on the situation, the individual loops starting from an open-loop pole may go either to a zero with higher frequency (similar to Fig.7.15.d) or to a zero with lower frequency (similar to Fig.7.15.c).

146

7 Active Damping with Collocated System

Lead

f f

-

î +

IFF

É

f

(a)

(b)

Im(s)

Im(s)

j!i

jz i

jz i

j!i

s 2+z 2i

1+

1 + g s(s2+!2) = 0 i

2 2 p s +!i g s(s2+z2)

=0

i

Re(s) IFF (c)

Re(s) Lead (d)

Fig. 7.15 Duality between the IFF and the Lead (DVF) control conﬁgurations. (a) IFF architecture with displacement actuator, force sensor and positive integral force feedback. (b) Force actuator and collocated displacement transducer and (negative) Lead controller (DVF is a particular case). (c) IFF control: Root locus for a single mode. (d) Lead control: Root locus for a single mode (z ωi < zi p). The loops of the DVF can be approximated by one conﬁguration or the other, depending on the relative value of ωi and zi .

7.7 Actuator and Sensor Dynamics

7.6.2

147

Open-Loop Poles and Zeros

For the IFF (Fig.7.15.a), the open-loop poles jωi are the natural frequencies of the structure with the active element working passively (contributing with its own stiﬀness Ka ), while the open-loop zeros, jzi , are the natural frequencies when the force f in the active strut is zero, that is when the active element is removed. On the contrary, in the control conﬁguration of Fig.7.15.b, the open-loop poles are the natural frequencies when the active element produces no force (same as the zeros in the previous case), and the open-loop zeros are the natural frequencies with the d.o.f. along the actuator blocked (Δ = 0). Note that these are larger than the open-loop poles in the IFF case, because the stiﬀness Ka of the active strut is ﬁnite.

7.7

Actuator and Sensor Dynamics

Throughout this chapter, it has been assumed that the actuator and sensor have perfect dynamics. As a result, the active damping algorithms are stable for all gains g (except the PPF). In practice, however, the open-loop transfer function becomes g A(s) H(s) G(s) (7.36) where A(s) includes the sensor, actuator and the digital controller dynamics. The low frequency behavior of the proof-mass actuator and the charge ampliﬁer can both be approximated by a second order high-pass ﬁlter such as (3.8). The high frequency behavior of the actuators and sensors can often be represented by a second order low-pass ﬁlter A(s) =

ωa2 s2 + 2ξa ωa s + ωa2

(7.37)

It is easy to see that the two extra poles of A(s) bring the asymptotes of the root locus inside the right half plane and substantially alter its shape for ω > ωa . Figure 7.16 shows the eﬀect of the second order low-pass ﬁlter on the root locus plot of the lead compensator (we have assumed ξa = 0.5 and ω2 < ωa < ω3 ). The active damping is no longer unconditionally stable and always has some destabilizing inﬂuence on the modes with natural frequencies beyond ωa . Fortunately, in practice, the modes of the structure are not exactly on the imaginary axis, because of the structural damping (Fig.7.17); this allows us to operate the controller with small gains. The control system becomes insensitive to the actuator dynamics if ωa is far beyond the crossover frequency of gH(s)G(s). The eﬀect of the low frequency dynamics of a proof-mass actuator is left as a problem (Problem 7.2).

148

(a)

7 Active Damping with Collocated System

Im(s)

Im(s)

(b)

4

Actuator

3

2

1

Re(s)

Re(s)

Fig. 7.16 Eﬀect of the actuator dynamics on the Lead compensator. (a) With perfect actuator. (b) Including the actuator dynamics A(s) given by (7.37) (the corner frequency of the actuator is such that ω2 < ωa < ω3 ).

Im(s)

Actuator

Re(s)

Fig. 7.17 Stabilizing eﬀect of the structural damping on the actuator dynamics.

7.8 Decentralized Control with Collocated Pairs

7.8

149

Decentralized Control with Collocated Pairs

7.8.1

Cross Talk

Consider the Multi-Input Multi-Output (MIMO) control of a structure with two independent control loops using collocated pairs. The input-output relationship for this system can be written in compact form ⎛ ⎞ ⎡ ⎤⎛ ⎞ y1 u1 G11 G12 G1w ⎝ y2 ⎠ = ⎣ G21 G22 G2w ⎦ ⎝ u2 ⎠ (7.38) z Gz1 Gz2 Gzw w where w is the disturbance and z is the performance metrics. One sees that the output y1 responds to u2 through G12 and y2 responds to u1 through G21 , respectively. These terms are called cross-talk, and are responsible for interactions between the two loops.

7.8.2

Force Actuator and Displacement Sensor

Consider a control system with m collocated force actuator/displacement sensor pairs. The control is governed by the following equations: Structure: Mx ¨ + Kx = Bu

(7.39)

y = BT x

(7.40)

Output: where B deﬁnes the topology of the actuator/sensor pairs (the size of the vectors u and y is equal to the number m of collocated pairs). Control: u = −gH(s)y (7.41) where H(s) is a square matrix and g is a scalar parameter (the discussion is not restricted to decentralized control; H(s) is diagonal if the control is decentralized). The closed-loop eigenvalue problem is obtained by combining the three equations above: [M s2 + K + gBH(s)B T ]x = 0

(7.42)

One can show that the asymptotic values of the ﬁnite eigenvalues of this equation as g → ∞ are independent of H(s) (Davison & Wang); therefore, they can be computed with H(s) = I: lim [M s2 + K + gBB T ]x = 0

g→∞

(7.43)

The asymptotic solutions of this equation are the transmission zeros of the MIMO system. The matrix gBB T is the contribution to the global stiﬀness

150

7 Active Damping with Collocated System

matrix of a set of springs of stiﬀness g connected to all the d.o.f. involved in the control. Asymptotically, when g → ∞, the additional springs act as supports restraining the motion along the controlled d.o.f.. Thus, the transmission zeros are the poles (natural frequencies) of the constrained system where the d.o.f. involved in the control are blocked. Since all the matrices involved in (7.43) are symmetrical and positive semi-deﬁnite, the transmission zeros are purely imaginary; since blocking the controlled d.o.f. reduces the total number of d.o.f. by the number m of control loops, the number of zeros is 2m less than the number of poles.

7.8.3

Displacement Actuator and Force Sensor

The equations are in this case: Structure: Mx ¨ + Kx = BKa δ

(7.44)

y = Ka (B T x − δ)

(7.45)

δ = gH(s)y

(7.46)

Output: Control: where B deﬁnes the topology of the active members within the structure, assumed of equal stiﬀness Ka , H(s) is a square matrix and g is a scalar gain (a positive feedback is assumed as in the IFF controller). The closed-loop eigenvalues are solutions of [M s2 + K − gBKaH(I + gKa H)−1 Ka B T ]x = 0

(7.47)

The asymptotic values are respectively, for g = 0, the open-loop poles (natural frequencies of the system including the active members) and, for g → ∞,4 they are solution of (M s2 + K − BKa B T )x = 0 (7.48) Thus, asymptotically, as g → ∞, the ﬁnite eigenvalues coincide with the transmission zeros which are the poles (natural frequencies) of the system where the contribution of the active members to the stiﬀness matrix has been removed. This result applies in particular for independent IFF loops.

7.9

Problems

P.7.1 Compare the following implementations of the Lead and the Direct Velocity Feedback compensators: 4

Because limg→∞ (I + gKa H) ∼ gKa H.

7.9 Problems

151

H(s) = s H(s) = 1 + T s s H(s) = s+a Ts + 1 H(s) = (α < 1) αT s + 1 H(s) =

ωf2 s s2 + 2ξf ωf s + ωf2

Discuss the conditions under which these compensators would be applicable for active damping. P.7.2 Consider a vibrating structure with a point force actuator collocated with an accelerometer. Consider the two compensators: H(s) = g/s H(s) =

g s2 + 2ξf ωf s + ωf2

Draw the block diagram of the control system, examine the stability and performance. What would be the eﬀect of the dynamics of a charge ampliﬁer, represented as a second order high-pass ﬁlter of corner frequency ωp and damping ξp = 0.7, assuming that its corner frequency satisﬁes ωp ω1 . P.7.3 Show that the maximum damping achievable in an active truss with the integral force feedback is ξimax =

ω i − zi 2zi

(zi ≥ ωi /3)

where ωi is the natural frequency of the truss including the active strut as a passive element and zi is the natural frequency when the active strut has been removed, and that it is achieved for g = ωi ωi /zi . [Hint: The use of a symbolic calculation software is recommended.] P.7.4 Consider the plane truss of Fig.4.19; the coordinates of the ﬁnite element model are the cartesian coordinates of the nodes (xi , yi ). For every active strut shown in the ﬁgure, write the inﬂuence vector b appearing in Equ.(7.24).

152

7 Active Damping with Collocated System

P.7.5 To avoid the saturation associated with integral control, the IFF controller (7.27) may be replaced by the “Beta controller” δ gs = H(s) = y Ka (s + β)2 where β ω1 (de Marneﬀe). See section 15.5. (a) Compare the root locus and the damping performance of the two controllers. (b) Show that, on the contrary to (7.27), this control law does not reduce the static stiﬀness of the structure. P.7.6 Consider the seven-story shear frame of Fig.7.18. It is controlled in a decentralized manner with two independent and identical feedback loops. Every actuator ui applies a pair of forces equal and opposite between ﬂoor i and ﬂoor i − 1,5 while the sensor yi = xi − xi−1 measures the relative displacement between the same ﬂoors. The mass, stiﬀness and B matrices are respectively M = mI7 ,

x7 x6 x5 x4

a)

x3

! k = 2Ò sin[ù(2kà1) 2(14+1) ]

b)

k = 1; . . .; 7 ù(2là1) z l = 2Ò sin[2(10+1) ]

l = 1; . . .; 5

u2

u1

x2 y2 = x2 à x1

x1

y1 =x1

Fig. 7.18 (a) Shear frame with two independent control loops (displacement sensor and force actuator). (b) Conﬁguration corresponding to transmission zeros.

5

The height of a story is assumed h = 0 in the model, so that the control force may be assumed to act in the horizontal direction.

7.9 Problems

⎛

2 ⎜ −1 ⎜ ⎜ 0 K = k⎜ ⎜ ⎜ ⎝ 0 0

−1 0 2 −1 −1 2 ... 0 −1 ... 0

⎞ ... 0 ... 0 ⎟ ⎟ ... 0 ⎟ ⎟, ⎟ ⎟ 2 −1 ⎠ −1 1

⎛

1 ⎜0 ⎜ ⎜0 ⎜ B=⎜. ⎜ .. ⎜ ⎝0 0

⎞ −1 1 ⎟ ⎟ 0 ⎟ ⎟ .. ⎟ . ⎟ ⎟ 0 ⎠ 0

153

(7.49)

(m is the mass and k the stiﬀness of a single ﬂoor). The natural frequency of mode l is given by ( k π (2l − 1) sin[ ] , l = 1, · · · , n (7.50) ωl = 2 m 2 (2n + 1) where n is the number of storeys (for the calculations, one can normalize according to Ω = k/m = 1). Consider the lead compensator H(s) = g

1 + Ts 1 + αT s

(α < 1)

(7.51)

and select the parameters T and α to control properly at least the ﬁrst 3 modes of the system. (a) Consider a single control loop in the ﬁrst ﬂoor. Compute the transmission zeros, draw the root locus and select a reasonable value of the gain; evaluate the active damping obtained in the 3 targeted modes. (b) Consider the decentralized control with two independent loops with the same gain. Compute the transmission zeros, draw the root locus and compare it to the previous one; select a reasonable value of the gain and evaluate the active damping obtained for the 3 targeted modes. (c) For the single loop controller, evaluate the eﬀect of a band limited actuator by including a low-pass ﬁlter (7.37) with a corner frequency ωa = ω5 and ξa = 0.7. Comment on the feasibility of such a control system. P.7.7 Consider again the seven-story shear frame of Fig.7.18 with one or two active struts in the ﬁrst two ﬂoors; the stiﬀness Ka of the strut is such that Ka /k = 5. The active struts are equipped with a displacement actuator and a force sensor pair and are controlled in a decentralized manner with the IFF controller: H(s) = g/s (7.52) (a) Consider a single control loop in the ﬁrst ﬂoor. Compute the open-loop poles and the transmission zeros, draw the root locus and select a reasonable value of the gain; evaluate the active damping obtained in the 3 targeted modes. (b) Consider the decentralized control with two independent loops with the same gain. Compute the open-loop poles and the transmission zeros, draw

154

7 Active Damping with Collocated System

the root locus and compare it to the previous one; select a reasonable value of the gain and evaluate the active damping obtained for the 3 targeted modes. (c) For the single loop controller, evaluate the eﬀect of a band limited actuator by including a low-pass ﬁlter (7.37) with a corner frequency ωa = ω5 and ξa = 0.7. Comment on the feasibility of such a control system and compare to the previous problem.

8

Vibration Isolation

8.1

Introduction

There are two broad classes of problems in which vibration isolation is necessary: (i) Operating equipments generate oscillatory forces which can propagate into the supporting structure (Fig.8.1.a). This situation corresponds to that of an engine in a car. (ii) Sensitive equipments may be supported by a structure which vibrates appreciably (Fig.8.1.b); in this case, it is the support motion which constitutes the source of excitation; this situation corresponds to, for example, a telescope in a spacecraft, a wafer stepper or a precision machine tool in a workshop, or a passenger seated in a car. The disturbance may be either deterministic, such as the unbalance of a motor, or random as in a passenger car riding on a rough road. For deterministic sources of excitation which can be measured, such as a rotating unbalance, feedforward control can be very eﬀective (see for example chapter 7 of Fuller’s book).1 However, the present chapter is focused on the feedback strategies for active isolation; they apply to both deterministic and random disturbances, and they do not need a direct measurement of the disturbance. Let us begin with the system depicted in Fig.8.1.a, excited by a disturbance force fd . If the support is ﬁxed, the governing equation is: Mx ¨ + cx˙ + kx = fd

(8.1)

The force transmitted to the support is given by fs = kx + cx˙ 1

(8.2)

In feedforward control, it is not necessary to measure directly the disturbance force, but rather a signal which is correlated to it, such as the rotation velocity, if the disturbance results from a rotating unbalance.

A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 155–186. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

156

8 Vibration Isolation

fd

(a) M k

(c)

(b) M

x c

k

M

xc k

c xd

c

xc fa

xd

fs Fig. 8.1 (a) Operating equipment generating a disturbance force fd . (b) Equipment subjected to a support excitation xd . (c) Active isolation device.

In the Laplace domain, Fd (s) M (s2 + 2ξωn s + ωn2 )

(8.3)

Fs (s) = M (ωn2 + 2ξωn s)X(s)

(8.4)

X(s) =

where X(s), Fd (s) and Fs (s) stand for the Laplace transform of respectively x(t), fd (t) and fs (t), and with the usual notations ωn2 = k/M and 2ξωn = c/M . The transmissibility of the support is deﬁned in this case as the transfer function between the disturbance force fd applied to the mass and the force fs transmitted to the support structure; combining the foregoing equations, we get Fs (s) 1 + 2ξs/ωn = (8.5) Fd (s) 1 + 2ξs/ωn + s2 /ωn2 Next, consider the second situation illustrated in Fig.8.1.b; the disturbance is the displacement xd of the supporting structure and the system output is the displacement xc of the sensitive equipment. Proceeding in a similar way, it is easily established that the transmissibility of this isolation system, deﬁned in this case as the transfer function between the support displacement and the absolute displacement of the mass M , is given by (Problem 8.1) Xc (s) 1 + 2ξs/ωn = Xd (s) 1 + 2ξs/ωn + s2 /ωn2

(8.6)

which is identical to the previous one; the two isolation problems can therefore be treated in parallel. The amplitude of the corresponding FRF, for s = jω, is represented in Fig.8.2 for various values of the damping ratio ξ. We observe √ that: (i) All the curves are larger than 1 for ω < 2 ω and become smaller n √ √ than 1 for ω > 2 ωn . Thus the critical frequency 2 ωn separates the domains of ampliﬁcation and attenuation of the isolator. (ii) When ξ = 0,

8.2 Relaxation Isolator

dB

x c (j xd (j

disturbance source ) )

157

sensitive equipment

xc

xd C

=0 m

M

k

10

0 Objective of the active isolation

-10

2

>

1

1

1/s

-20

1 / s²

0.1

1

2

10 n

Fig. 8.2 Transmissibility of the passive isolator for various values of the damping ratio ξ. The performance objectives of active isolation are a high frequency decay like s−2 together with no overshoot at resonance.

the high frequency decay rate is s−2 , that is -40 dB/decade, while very large amplitudes occur near the corner frequency ωn (the natural frequency of the spring-mass system). Figure 8.2 illustrates the trade-oﬀ in passive isolator design: large damping is desirable at low frequency to reduce the resonant peak while low damping is needed at high frequency to maximize the isolation. One may already observe that if the disturbance is generated by a rotating unbalance of a motor, there is an obvious beneﬁt to use a damper with variable damping characteristics which can be adjusted according to the rotation velocity: high when ω < √ √ 2ωn and low when ω > 2ωn . Such variable (adaptive) devices will be discussed in chapter 17. Figure 8.2 also shows the target of an active isolation system which combines a decay rate of -40 dB/decade with no overshoot at resonance. Before addressing the active isolation, the following section discusses one way of improving the high frequency isolation in a passive way.

8.2

Relaxation Isolator

In the relaxation isolator, the viscous damper c is replaced by a Maxwell unit consisting of a damper c and a spring k1 in series (Fig.8.3.a). The governing equations are Mx ¨ + k(x − x0 ) + c(x˙ − x˙ 1 ) = 0 (8.7)

158

8 Vibration Isolation

Fig. 8.3 (a) Relaxation isolator. (b) Electromagnetic realization.

c(x˙ − x˙ 1 ) = k1 (x1 − x0 )

(8.8)

or, in matrix form using the Laplace variable s,

M s2 + cs + k −cs x k = x0 −cs k1 + cs x1 k1

(8.9)

It follows that the transmissibility reads x (k1 + cs)k + k1 cs (k1 + cs)k + k1 cs = = (8.10) x0 (M s2 + cs + k)(k1 + cs) − c2 s2 (M s2 + k)(k1 + cs) + k1 cs One sees that the asymptotic decay rate for large frequencies is in s−2 , that is -40 dB/decade. Physically, this corresponds to the fact that, at high frequency, the viscous damper tends to be blocked, and the system behaves like an undamped isolator with two springs acting in parallel. Figure 8.4 compares the transmissibility curves for given values of k and k1 and various values of c. For c = 0, the relaxation isolator behaves like an undamped isolator of natural frequency ωn = (k/M )1/2 . Likewise, for c → ∞, it behaves like an undamped isolator of frequency Ωn = [(k + k1 )/M ]1/2 . In between, the poles of the system are solution of the characteristic equation (M s2 + k)(k1 + cs) + k1 cs = (M s2 + k)k1 + cs(M s2 + k + k1 ) = 0 which can be rewritten in root locus form 1+

k1 s2 + ωn2 =0 c s(s2 + Ωn2 )

(8.11)

8.2 Relaxation Isolator

x x0

dB

159

c!1

c=0

c opt A

0

-20

-40 1

!=! n

10

Fig. 8.4 Transmissibility of the relaxation oscillator for ﬁxed values of k and k1 and various values of c. The ﬁrst peak corresponds to ω = ωn ; the second one corresponds to ω = Ωn . All the curves cross each other at A and have an asymptotic decay rate of -40 dB/decade. The curve corresponding to copt is nearly maximum at A.

Fig. 8.5 Root locus of the solutions of Equ.(8.11) as c goes from zero to inﬁnity. 3/2 −1/2 The maximum damping is achieved for k1 /c = Ωn ωn .

160

8 Vibration Isolation

It is very similar to (7.31); it is represented in Fig.8.5 when c varies from 0 to ∞; using the results of section 7.5, the maximum damping ratio is achieved for 3/2 k1 Ωn = 1/2 (8.12) c ωn and the corresponding damper constant is copt =

k1 ωn 1/2 k1 k1 k1 k1 ( ) = (1 + )−1/4 = (1 + )−3/4 Ωn Ωn Ωn k ωn k

(8.13)

The transmissibility corresponding to copt is also represented in Fig.8.4; it is nearly maximum at A.

8.2.1

Electromagnetic Realization

The principle of the relaxation isolator is simple and it can be realized with viscoelastic materials. However, it may be diﬃcult to integrate in the system, and also to achieve thermal stability. In some circumstances, especially when thermal stability is critical, it may be more convenient to achieve the same eﬀect through an electromechanical converter which consists of a voice coil transducer, an inductor L and a resistor R. A voice coil transducer is an energy converter transforming mechanical energy into electrical energy and vice-versa; its constitutive equations are given by Equ.(3.1) to (3.3). Referring to Fig.8.3.b, the governing equations of the system are Mx ¨ + k(x − x0 ) − T i = 0

(8.14)

di + T (x˙ − x˙ 0 ) + Ri = 0 (8.15) dt where T is the transducer constant; in matrix form, using the Laplace variable,

M s2 + k −T x k = x0 (8.16) Ts Ls + R i Ts L

It follows that the transmissibility reads x (Ls + R)k + T 2 s = 2 x0 (M s + k)(Ls + R) + T 2 s

(8.17)

Comparing with Equ.(8.10), one sees that the electromechanical isolator behaves exactly like a relaxation isolator provided that Ls + R cs + k1 = 2 T k1 c

(8.18)

8.3 Active Isolation

161

or

T2 T2 c= (8.19) L R These are the two relationships between the three parameters T , L and R so that the transmissibility of the electromechanical system of Fig.8.3.b is the same as that of Fig.8.3.a. k1 =

8.3

Active Isolation

Consider the single-axis isolator connecting a disturbance source m to a payload M (Fig.8.6). It consists of a soft spring k in parallel with a force actuator fa ; the objective is to isolate the motion xc of the payload M from the motion xd of m due to the disturbance load fd . The governing equations are Mx ¨c + k(xc − xd ) = fa

(8.20)

m¨ xd + k(xd − xc ) = fd − fa

(8.21)

or, in matrix form using the Laplace variable s,

M s2 + k −k Xc Fa = −k ms2 + k Xd Fd − Fa

(8.22)

Upon inverting this equation, one gets Xc =

kFd mFa + s2 [M ms2 + (M + m)k] M ms2 + (M + m)k

(8.23)

The ﬁrst term of this expression describes the payload response to the disturbance load while the second term is the payload response to the actuator. If an accelerometer or a geophone is attached to the payload, measuring the absolute acceleration x ¨c or the absolute velocity x˙ c , the open-loop transfer function is

disturbance source m

xc sensitive

xd

equipment

Accelerometer

Fa

Fd

M

k

Fa Fk

Fig. 8.6 Single-axis active isolator.

162

8 Vibration Isolation

G(s) =

s2 X c ms2 = 2 Fa M ms + (M + m)k

(8.24)

Consider the closed-loop response to a general feedback law based on the absolute velocity x˙ c : Fa = −H(s)sXc (s) (8.25) Introducing this into (8.22), one gets

M s2 + H(s)s + k −k Xc 0 = −k − H(s)s ms2 + k Xd Fd

(8.26)

Upon considering the ﬁrst line of this equation, one ﬁnds easily that the closed-loop transmissibility is Xc k = 2 Xd M s + H(s)s + k

8.3.1

(8.27)

Sky-Hook Damper

Equation (8.27) shows the inﬂuence of the feedback control law on the transmissibility; it shows that a simple velocity feedback, H(s) = g leads to the transmissibility Xc k 1 = = 2 2 2 Xd M s + gs + k s /ωn + gs/k + 1

(8.28)

which complies with the objectives of active isolation stated in Fig.8.2, because the asymptotic decay rate is in s−2 (i.e. -40 dB/decade) and the overshoot at resonance may be controlled by adjusting the gain g of the controller to achieve critical damping. This control law is called sky-hook, because the control force fa = −g x˙ c is identical to that of a viscous damper of constant g attached to the payload and a ﬁxed point in space (the sky), Fig.8.7.

(b)

(a)

m

disturbance source

g

“sky”

F a = - g s Xc Fa

Accelerometer

k

M

m

M

k

sensitive equipment

Fig. 8.7 (a) Isolator based on an acceleration or an absolute velocity sensor (geophone). (b) Equivalent sky-hook damper.

8.3 Active Isolation

163

The open-loop transfer function between the input force fa and the output acceleration x ¨c is given by (8.24); it has a pair of imaginary poles at ( (M + m)k pi = ±j (8.29) Mm and a pair of zeros at the origin. The root-locus of the sky-hook is shown in Fig.8.8; it is entirely contained in the left-half plane, which means that the sky-hook damper is unconditionally stable (inﬁnite gain margin).

Im

G(s)

H(s) g

s

m s2

Fa

s²Xc

m M s + k (m+M) 2

Re (2 zeros + 1 pole)

Fig. 8.8 Root-locus of the sky-hook damper (acceleration sensor).

8.3.2

Integral Force Feedback

We have just seen that the sky-hook damper based on the absolute velocity of the payload is unconditionally stable for a rigid body. However, this is no longer true if the payload cannot be regarded as a rigid body, situation which is frequently met in space applications. Since the absolute acceleration of a rigid body is proportional to the force applied to it, F = M x ¨c , the acceleration feedback of Fig.8.7 may be replaced by a force feedback as shown in Fig.8.9. Note that, besides the advantage of achieving alternating poles and zeros discussed here, a force sensor may be more sensitive than an accelerometer in low frequency applications; for example, a force sensor with a sensitivity of 10−3 N is common place; for a mass M of 1000 kg (e.g. a space telescope), this corresponds to an acceleration of 10−6 m/s2 ; such a sensitivity is more diﬃcult to achieve. Force sensing is especially attractive in micro-gravity where one does not have to consider the dead loads of a structure. The open-loop transfer function is, in this case G(s) =

F M ms2 = Fa M ms2 + (M + m)k

(8.30)

164

8 Vibration Isolation

which has the same pole/zero pattern and the same root-locus as Fig.8.8. However, when the payload is ﬂexible, the force applied and the acceleration are no longer proportional and the pole/zero pattern may diﬀer signiﬁcantly. It can be observed that the feedback based on the acceleration still leads to alternating poles and zeros in the open-loop transfer function when the ﬂexible modes are signiﬁcantly above the suspension mode, but they do not alternate any more when the ﬂexible modes interact with the suspension mode. On the contrary, if two arbitrary undamped ﬂexible bodies are connected by a single-axis isolator with force feedback (Fig.8.10), the poles and zeros of the open-loop transfer function F/Fa always alternate on the imaginary axis (Fig.8.11). This result is not obvious, because the actuator Fa and the sensor F , if collocated, are not dual as requested for alternating poles and zeros (as emphasized in previous chapters); it can be demonstrated as follows: The system with input Fa and output Δx = x1 − x2 , the relative displacement between the two bodies, is collocated and dual ; therefore, the FRF (which is purely real in the undamped case) exhibits alternating poles and zeros (full line in Fig.8.11). On the other hand, the control force Fa , the relative displacement Δx and the output (total) force F are related by F = kΔx − Fa

disturbance source

xd

(8.31)

xc sensitive equipment

Fa = - g/s F

Fa

m

M F

k Fig. 8.9 Sky-hook based on a force sensor (F is taken positive when it is acting in the direction of xc on mass M ).

k disturbance source

F

x1 sensitive equipment

Fa x2

h(s)

Fig. 8.10 Arbitrary ﬂexible structures connected by a single-axis isolator with force feedback.

8.4 Flexible Body

165

1 0 1

Fig. 8.11 FRFs of the single-axis oscillator connecting two arbitrary ﬂexible structures. The full line corresponds to kΔX/Fa and the dotted line to F/Fa ; it is obtained by vertical translation of the full line by -1.

(in this equation, F is assumed positive in traction while Fa is positive when it tends to separate the two bodies). It follows that the FRF F/Fa and ΔX/Fa are related by F kΔX = −1 (8.32) Fa Fa This equation states that the FRF with force sensor, F/Fa , can be obtained from that with relative displacement sensor, kΔX/Fa by a simple vertical translation bringing the amplitude to 0 at ω = 0 (from the full line to the dotted line). This changes the locations of the zeros Zi but the continuity of the FRF curve between two resonances guarantees that there is a zero between two consecutive poles (natural frequencies): ωi < Zi < ωi+1

8.4

(8.33)

Flexible Body

When the payload is ﬂexible, the behavior of the acceleration feedback and the force feedback are no longer the same, due to diﬀerent poles/zeros conﬁgurations of the two control strategies. In fact, diﬀerent sensor conﬁgurations correspond to diﬀerent locations of the zeros in the s-plane. To analyze this situation, consider the payload with a ﬂexible appendage of Fig.8.12; the nominal numerical values used in the calculations are m = 1.1kg, M = 1.7kg,

166

8 Vibration Isolation

k = k1 = 12000N/m, c1 = 0; the mass m1 of the ﬂexible appendage is taken as a parameter to analyze the interaction between the ﬂexible appendage and the isolation system. When m1 is small, the ﬂexible appendage is much more rigid than the isolation system and the situation is not much diﬀerent from that of a rigid body. Figure 8.13 shows the root locus plots for m1 = 0.5kg; the acceleration feedback and the force feedback have similar root locus plots, with a new pole/zero pair appearing higher on the imaginary axis; the poles and zeros still alternate on the imaginary axis and the only diﬀerence between the two plots is the distance between the pole and the zero which is larger for the acceleration feedback; as a result, the acceleration feedback produces a larger damping of the higher mode. By contrast, when m1 is large, the root locus plots are reorganized as shown in Fig.8.14 for m1 = 3.5kg. For force feedback, the poles and zeros still alternate on the imaginary axis, leading to a stable root locus; this property is lost for the acceleration feedback, leading to an unstable loop for the lower mode. In practice, however, this loop is moved slightly to the left by the

xd

flexible appendage

xc

[ c1

Fa m

F

m1

M

k

k1

disturbance source

sensitive equipment

x1

Fig. 8.12 Payload with a ﬂexible appendage.

(a)

(b)

Im

Re

Im

Re

Fig. 8.13 Root locus of the isolation system with a light ﬂexible appendage (m1 = 0.5kg). (a) Force feedback. (b) Acceleration feedback.

8.4 Flexible Body

(a)

(b)

Im

Re

167

Im

Re

Fig. 8.14 Root locus of the isolation system with a heavy ﬂexible appendage (m1 = 3.5kg). (a) Force feedback. (b) Acceleration feedback.

structural damping, and the control system can still operate with small gains (conditionally stable).

8.4.1

Free-Free Beam with Isolator

To analyze a little further the situation when the payload is ﬂexible, consider the vertical isolation of a free-free continuous beam from the disturbance of a body of mass m, Fig.8.15.a. This situation is representative of a large space structure with its attitude control system attached to m (the disturbance is created by the unbalance of the rotating wheels). Note that the rigid body modes are uncontrollable from the internal force Fa . In the numerical example described below, the length of the beam is l = 5m, the mass per unit length is = 2kg/m, the stiﬀness of the isolator is k = 1N/m and the mass where the disturbance is applied is m = 1kg; the stiﬀness EI of the beam is taken as a parameter. Let Ωi be the natural frequencies of the ﬂexible modes of the free-free beam alone, Fig.8.15.b and Zi be the transmission zeros corresponding to a force excitation and a collocated displacement sensor (or equivalently acceleration). According to what we have seen in the previous chapters, Zi are the natural frequencies of the system with an additional restraint at the connecting degree of freedom of the isolator. Because of the collocation, the poles and zeros are alternating on the imaginary axis, so that Ωi and Zi satisfy Zi < Ωi < Zi+1

(8.34)

Next, consider the complete system (beam + mass m) and let ωi be its natural frequencies (ﬂexible mode only, because the rigid body modes are not controllable from the internal force Fa ). If the control system uses a force

168

8 Vibration Isolation

Fig. 8.15 (a) Free-free beam and single axis isolator. The other ﬁgures illustrate the various situations and the boundary conditions corresponding to the transmission zeros. (b) Free-free beam with displacement sensor and point force actuator. (c) Free-free beam and sky-hook isolator. (d) Free-free beam and isolator with force feedback.

sensor, Fig.8.15.d, the transmission zeros, obtained by enforcing a zero force at the connecting d.o.f., are identical to the natural frequencies of the system when the isolator is disconnected from the beam; which are identical to the natural frequencies Ωi of the free-free beam. The conﬁguration is exactly that of Fig.8.10 and, accordingly, the open-loop FRF has alternating poles and zeros and the following relation holds: ωi < Ωi < ωi+1

(8.35)

This condition guarantees the stability of the closed-loop system when a force feedback is used. With an acceleration feedback (sky-hook damper, Fig.8.15.c, the poles are still ±jωi while the zeros, obtained by enforcing a zero acceleration at

8.4 Flexible Body

169

Fig. 8.16 Flexible beam with an isolator; evolution of ωi ,Zi and ωi with the ﬂexibility of the beam.

the connecting d.o.f. are ±jZi , as for the free-free beam of Fig.8.15.b. This actuator/sensor conﬁguration is no longer collocated, so that no condition similar to (8.34) or (8.35) holds between ωi and Zi . When the beam is stiﬀ, the interlacing property ωi < Zi < ωi+1 is satisﬁed and the stability is guaranteed, but as the beam becomes more ﬂexible, the values of ωi and Zi decrease at diﬀerent rates and a pole/zero ﬂipping occurswhen they both become equal to the natural frequency of the isolator (ω ∗ = k/m), Fig.8.16. As a result, the system is no longer unconditionally stable when the ﬂexibility is such that ω1 = Z1 = ω ∗ = k/m, and above. As a conclusion to this section, it seems that the sky-hook damper implementation (acceleration feedback) is preferable when the payload is fairly stiﬀ as compared to the isolator corner frequency (e.g. car suspension), to beneﬁt from the better active damping properties of the ﬂexible modes (Fig.8.13). On the contrary, the force feedback implementation is preferable when the payload is very ﬂexible (e.g. space structure), to beneﬁt from the interlacing of the poles and zeros, leading to guaranteed stability.

170

8 Vibration Isolation

8.5

Payload Isolation in Spacecraft

Space telescopes and precision payloads are subject to jitter due to the unbalanced masses of the attitude control reaction wheels or gyros. The performance of the instruments may be improved by inserting one or several isolators in the transmission path between the disturbance source and the payload. If the isolator is designed in such a way that its transmissibility exhibits a decay rate of −40dB/decade, the jitter can be reduced by a factor 100 by selecting the isolator corner frequency, f0 , one decade lower than the ﬁrst ﬂexible mode of the payload, fn (Fig.8.17). Extremely sensitive payloads may even involve several isolation layers.2

Fig. 8.17 Eﬀect of the isolator on the transmissibility between the spacecraft bus and the telescope.

8.5.1

Interaction Isolator/Attitude Control

There are several possible locations for the isolator, depending on the spacecraft architecture (Fig.8.18). If the attitude control wheels are packed in a single assembly (RWA), the isolator may be placed between the RWA and the spacecraft bus, Fig.8.18.b. Another option consists in placing the isolator 2

The future James Webb Space Telescope, JWST will involve two isolation layers, (i) the wheel isolator supporting six reaction wheels, with corner frequencies at 7 Hz for rocking and 12 Hz for translation and (ii) a 1 Hz passive isolator at the interface between the telescope deployment tower and the spacecraft bus (Bronowicki).

8.5 Payload Isolation in Spacecraft

171

Fig. 8.18 Spacecraft architecture. (a) Without isolator. (b) Isolator placed between the Reaction Wheel Assembly (RWA) and the spacecraft bus. (c) Isolator between the spacecraft bus and the telescope.

between the spacecraft bus and the instrument, Fig.8.18.c; in this alternative, the rotating wheels are rigidly attached to the spacecraft bus. The additional compliance introduced by the vibration isolator has a major impact on the low frequency dynamics of the system and its interaction with the attitude control system must be taken into account. The most favorable situation is that where the attitude control actuators and the attitude sensors (star trackers) are both rigidly attached to the spacecraft bus (collocated). For non-collocated situations, the stability of the control system requires that the corner frequency f0 of the isolator be one decade above the attitude control bandwidth, fc ; altogether, fc ∼ 0.1f0 ∼ 0.01fn

8.5.2

(8.36)

Gough-Stewart Platform

To fully isolate two rigid bodies with respect to each other, six single-axis isolators judiciously placed are needed. For a number of space applications, generic multi-purpose isolators have been developed with a standard GoughStewart platform architecture, in which every leg of the platform consists of a single-axis active isolator, connected to the base plates by spherical joints. In the cubic architecture (Fig.8.19), the legs are mutually orthogonal, which

172

8 Vibration Isolation

Fig. 8.19 Multi-purpose soft (relaxation) isolator based on a Gough-Stewart platform with cubic architecture (ULB).

minimizes the cross coupling between them. This conﬁguration is particularly attractive, because it also has uniform stiﬀness properties and uniform control capability, and it has been adopted in most of the projects.

8.6

Six-Axis Isolator

Let us consider a payload isolated by six identical isolators (Fig.8.20); if the isolators consist of simple springs of stiﬀness k, the six suspension modes are solution of an eigenvalue problem (M s2 + K)x = 0

(8.37)

where x is a vector of 6 coordinates describing the position of the payload, e.g. x = (x, y, z, θx , θy , θz )T . The mass matrix M can be obtained by writing the kinetic energy in terms of x. ˙ Similarly, the stiﬀness matrix is obtained by writing the strain energy in terms of x. The strain energy in the system is V = 12 k q T q, where q = (q1 , . . . , q6 )T is the vector of the spring extensions in the isolator and k is the stiﬀness common to all springs. If J is the Jacobian

8.6 Six-Axis Isolator

173

Fig. 8.20 Six-axis isolator (only four legs are represented). The connection between the leg and the support as well as the payload is done with spherical joints. The coordinates of the payload x = (x, y, z, θx , θy , θz )T and the leg extensions q = (q1 , . . . , q6 )T are related by q = Jx where J is the Jacobian of the isolator.

matrix connecting the spring extensions q to the coordinates x (J depends on the topology of the isolator), q = Jx

(8.38)

one gets that 1 T 1 k q q = k xT J T Jx 2 2 which means that the stiﬀness matrix is V =

K = kJ T J

8.6.1

(8.39)

(8.40)

Relaxation Isolator

If the linear spring is replaced by a relaxation isolator, the common stiﬀness k must be replaced by the appropriate relationship between the spring force F and the spring extension x − x0 . From the constitutive equations of the isolator, Equ.(8.9), one ﬁnds that the dynamic stiﬀness of the relaxation isolator is (Problem 8.5): F k1 cs = k(s) = k[1 + ] x − x0 k(k1 + cs)

(8.41)

(the stiﬀness is k at low frequency and k1 at high frequency). Thus, the (frequency-dependent) stiﬀness matrix of the six-axis relaxation isolator reads K(s) = J T J k[1 +

k1 cs k1 cs ] = K[1 + ] k(k1 + cs) k(k1 + cs)

(8.42)

174

8 Vibration Isolation

and the eigenvalue problem (8.37) becomes {M s2 + K[1 +

k1 cs ]}x = 0 k(k1 + cs)

(8.43)

If ωi and Φ = (φ1 , . . . , φ6 ) are the solution of the eigenvalue problem (8.37), normalized according to ΦT M Φ = I, one can transform (8.43) into modal coordinates, x = Φz; using the orthogonality conditions, one ﬁnds a set of decoupled equations s2 + ωi2 [1 +

k1 cs ]=0 k(k1 + cs)

(8.44)

k1 ) k

(8.45)

Upon introducing Ωi2 = ωi2 (1 + the previous equation may be rewritten k1 2 (s + ωi2 ) + s(s2 + Ωi2 ) = 0 c or 1+

k1 s2 + ωi2 =0 c s(s2 + Ωi2 )

(8.46)

which is identical to (8.11). Thus, according to the foregoing equation, the six suspension modes follow independent root-loci connecting ωi and Ωi (Fig.8.5). However, k1 /c being a single scalar parameter, the optimal damping cannot be reached simultaneously in the six modes, because of the modal spread (ω1 < ω6 ).

8.6.2

Integral Force Feedback

If the linear springs are substituted with identical active isolators such as in Fig.8.10, the dynamics of the isolator is governed by Mx ¨ + Kx = Bu

(8.47)

where the right hand side is the projection of the six actuator forces, u = (u1 , . . . , u6 )T in the global coordinate system attached to the payload. The control forces ui act in the direction where the leg extension qi is measured; from the principle of virtual work, (Bu)T δx = uT δq

−→

uT B T δx = uT Jδx

which implies B = JT

and

K = kBB T

(8.48)

8.6 Six-Axis Isolator

175

The force sensor equation is the same as (8.31): y = kq − u

(8.49)

where y = (y1 , . . . , y6 )T is the output vector of the six force sensors; the IFF feedback law is g u= y (8.50) s where an equal gain is assumed for the six independent loops. Combining Equ.(8.47), (8.49) et (8.50), one gets the closed-loop equation (M s2 + K)x = or

(M s2 + K

g kBB T x s+g

s )x = 0 s+g

(8.51)

If we transform into modal coordinates, x = Φz, and take into account the orthogonality relationships, the characteristic equation is reduced to a set of uncoupled equations ! 2 s " s + Ωi2 zi = 0 (8.52) s+g Thus, every mode follows the characteristic equation s2 + Ωi2

s =0 s+g

1+g

s =0 + Ωi2

or s2

(8.53)

where Ωi are the natural frequencies of the six suspension modes. The corresponding root locus is shown in Fig.8.21.a. It is identical to Fig.8.8 for a single-axis isolator; however, unless the 6 natural frequencies are identical, a given value of the gain g will lead to diﬀerent pole locations for the various modes and it will not be possible to achieve the same damping for all modes. This is why it is recommended to locate the payload in such a way that the spread of the modal frequencies is minimized.

8.6.3

Spherical Joints, Modal Spread

The foregoing results have been obtained with the assumptions that the connections are made by perfect spherical joints, so that the only contribution to the stiﬀness matrix come from the axial stiﬀness of the legs, K = kBB T . However perfect spherical joints do not exist and they have friction and backlash. Backlash is not acceptable in precision engineering and the spherical joints are replaced by ﬂexible connections with large longitudinal and shear stiﬀness and low bending stiﬀness, such as the one shown in Fig.8.22. This

176

8 Vibration Isolation

(a)

(b)

Im

jÒ i

Im

jÒ i

jzi Re

Re

Fig. 8.21 Six-axis active isolator with independent IFF loops: root locus of individual modes. (a) with perfect spherical joints. (b) with ﬂexible joints.

F.E. mesh (Samcef field)

Fig. 8.22 Typical ﬂexible joint for the connections of a six-axis isolator. Its behavior is close to a universal joint, with low bending stiﬀness while the other d.o.f. are rather stiﬀ.

is responsible for an additional contribution Ke to the stiﬀness matrix. The global stiﬀness matrix is kBB T + Ke and the closed-loop equation of the suspension with the IFF controller becomes (M s2 + Ke + kBB T

s )x = 0 s+g

(8.54)

According to this equation, the transmission zeros, which are the asymptotic solutions as g → ∞, are no longer at the origin (s = 0), but are solutions of the eigenvalue problem

8.7 Active vs. Passive

[M s2 + Ke ]x = 0

177

(8.55)

It follows that the zeros are shifted along the imaginary axis, leading to the root locus of Fig.8.21.b, which reduces the performance of the suspension system.

Im

Ò5; Ò6 Ò4

Ò3

Ò1; Ò2 Re Fig. 8.23 Typical root locus of a complete isolation system with real joints. The bullets indicate the location of the closed-loop poles for the adopted value of the gain g (from Preumont et al., 2007).

As mentioned before, the six suspension modes have diﬀerent natural frequencies and the decentralized IFF controller has a single gain g which has to be adjusted to achieve a good compromise in the suspension performance for the six modes. The best performance is achieved if the suspension is designed in such a way that the modal spread, Ω6 /Ω1 , is minimized. The combined eﬀect of the modal spread and the joint stiﬀness is illustrated in Fig.8.23; there are only 4 diﬀerent curves because of the symmetry of the system. The bullets correspond to the closed-loop poles for a ﬁxed value of g; they illustrate the fact that the various loops are travelled at diﬀerent speeds as g increases. How this impacts the transmissibility is examined below.

8.7

Active vs. Passive

Figure 8.24 compares the components involved in the passive relaxation isolator and the active one. The active isolation requires conditioning electronics for the force sensor and power electronics for the voice coil actuator. The relaxation isolator requires only a passive RL circuit but also requires a bigger

178

8 Vibration Isolation

transducer (with a larger transducer constant T ). Also, it does not have a force sensor, which makes it lighter. In fact, the legs have their own local dynamics which interfere with that of the isolator and impact signiﬁcantly the transmissibility in the vicinity of the resonance frequency of the local modes and beyond. Maximizing the natural frequency of the local modes of the legs is a major challenge in the design of a six-axis isolator with broadband isolation capability. This is achieved through careful design of all the components of the isolator. Figure 8.25 shows the leg of a passive relaxation isolator; the exploded view of the transducer shows the membrane which acts as an axial spring and also plays the role of spherical joint, the magnetic circuit and the voice coil, and its connection to the stinger, made of CFRP to minimize its weight. From the comparison of the root locus plots, of Fig.8.5 and 8.21.b, one would expect that the active isolator would have less overshoot near the resonance frequencies of the suspension. On the other hand, the passive isolator does not need a force sensor, which makes the leg a little lighter and improves the high frequency behavior of the isolator. Fig.8.26 compares the transmissibility of the active (IFF) and the passive (relaxation) isolator. The dotted line refer to the transmissibility of the passive isolator when the RL circuit is open. As expected, the overshoot of the active one is a little lower; both have a decay rate of -40 dB/decade in the intermediate frequency range, and the high frequency behavior is dominated by the local modes; the passive isolator behaves better in high frequency, because the local modes have higher frequencies.

quiet side F

Force Sensor

fc

k

noisy side fd

quiet side

Electric circuit

I

I Controller

Electromagnetic transducer

fc

k

V

noisy side fd

Fig. 8.24 Comparison of the active isolator (left) with the passive isolator (right); if a RL electrical circuit is used, the passive isolator is a relaxation isolator; a purely resistive circuit produces a linear viscous isolator.

8.8 Car Suspension

179

Fig. 8.25 Leg of a passive relaxation isolator; conceptual design and exploded view of the transducer showing the membrane, the magnetic circuit, the voice coil and its connection with the stinger.

8.8

Car Suspension

Ride comfort requires good vibration isolation; it is usually measured by the car body acceleration, or sometimes its derivative called jerk. Vehicle handling requires good road holding, which is measured by the tyre deﬂection. In addition to the car body acceleration and the tyre deﬂection, there are design constraints on the suspension travel, that is the relative displacement between the car body and the wheel. Figure 8.27 shows a quarter-car model of a vehicle equipped with a fully active suspension (Fig.8.27.a) or a passive suspension with an active, skyhook damper (Fig.8.27.b); ms refers to the sprung mass, equal to a quarter of the car body mass, and mus is the unsprung mass (wheel); kt is the stiﬀness of the tyre; k and c are the stiﬀness and damping of the passive suspension. The ﬁgure also shows the deﬁnition of the state variables used to model the systems: x1 = xs − xus is the relative displacement of the sprung mass with respect to the wheel (suspension travel), x2 = x˙ s is the absolute velocity of the car body, x3 = xus − w is the tyre deﬂection, and x4 = x˙ us is the absolute wheel velocity. With this deﬁnition of the state variables, the dynamics of the fully active suspension (Fig.8.27.a) reads

180

8 Vibration Isolation

Active control

Fig. 8.26 Vertical transmissibility of a six-axis isolator: comparison of the Openloop (dotted line), closed-loop (IFF) active isolator and passive relaxation isolator with a RL shunt (from de Marneﬀe et al.).

Fig. 8.27 Quarter-car, two-d.o.f. models. (a) Fully active suspension. (b) Passive suspension with an added sky-hook active control.

8.8 Car Suspension

181

ms x˙ 2 = f mus x˙ 4 = −f − kt x3 x˙ 1 = x2 − x4 x˙ 3 = x4 − v where v = w˙ is the road velocity. Deﬁning the force per unit sprung mass u = f /ms , the unsprung mass ratio μ = mus /ms and the tyre frequency ωt = (kt /mus )1/2 , this system is rewritten in matrix form, ⎛ ⎞ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x˙ 1 0 1 0 −1 x1 0 0 ⎜ x˙ 2 ⎟ ⎢ 0 0 0 0 ⎥ ⎜ x2 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ ⎟=⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (8.56) ⎝ x˙ 3 ⎠ ⎣ 0 0 0 1 ⎦ ⎝ x3 ⎠ + ⎝ 0 ⎠ u + ⎝ −1 ⎠ v 2 x˙ 4 0 0 −ωt 0 x4 −1/μ 0 With the same state-variables, the dynamics of the system of Fig.8.27.b (without control) is governed by ms x˙ 2 = f − kx1 + c(x4 − x2 ) mus x˙ 4 = −f − kt x3 + kx1 + c(x2 − x4 ) x˙ 1 = x2 − x4 x˙ 3 = x4 − v ωn2

and, upon deﬁning = k/ms (ωn is the body resonance), c/ms = 2ξωn , they are rewritten in matrix form ⎛

⎞ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 1 0 −1 x˙ 1 x1 0 0 2 ⎜ x˙ 2 ⎟ ⎢ −ωn −2ξωn 0 2ξωn ⎥ ⎜ x2 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ ⎟=⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0 1 ⎦ ⎝ x3 ⎠ + ⎝ 0 ⎠ u + ⎝ −v ⎠ ⎝ x˙ 3 ⎠ ⎣ 0 2 ωn 2ξωn n x˙ 4 x4 −1/μ 0 −ωt2 −2ξω μ μ μ (8.57) Passive Suspension The trade-oﬀ in the design of a passive suspension can be illustrated with the following example taken from Chalasani: The nominal values of the passive suspension are kt = 160000 N/m (tyre stiﬀness), k = 16000 N/m (suspension spring stiﬀness), ms = 240 kg (car body), mus = 36 kg (wheel). Figure 8.28.a shows the transmissibility Tx¨s v between the road velocity v = w˙ and the body absolute acceleration x˙ 2 = x¨s for three values of the damping constant, c = 200 Ns/m, 980 Ns/m, 4000 Ns/m. For the smallest value of c, one sees clearly the two peaks associated with the body resonance (sprung mass) and the tyre resonance (unsprung mass); the body resonance is at

182

8 Vibration Isolation

Fig. 8.28 Behavior of the passive suspension for various values of the damping constant: c = 200 Ns/m, 980 Ns/m, 4000 Ns/m: (a) Transmissibility Tx¨s v between the road velocity v = w˙ and the body absolute acceleration x˙ 2 = x ¨s . (b) Cumulative RMS value of the sprung mass acceleration σx¨s .

7.8 rad/s and the tyre resonance is at 69.5 rad/s, respectively very close to ωn = k/ms = 8.16 rad/s and ωt = kt /mus = 66.7 rad/s. As the damping increases, the amplitude of the two peaks is reduced; one sees clearly that the passive damping cannot control the body resonance without deteriorating the isolation at higher frequency. The cumulative RMS value of the body acceleration is deﬁned by the integral ω σx¨s (ω) = [ |Tx¨s v |2 dν]1/2 (8.58) 0

it is represented in Fig.8.28.b; since the road velocity is approximately a white noise, σx¨s describes how the various frequencies contribute to the RMS of the body acceleration (in relative terms). When the damping increases, the RMS body acceleration initially decreases, and increases again for larger values of c. Active Suspension A partial state feedback is added to the passive suspension; it consists of a sky-hook damper, f = −g x˙ s as shown in Fig.8.27.b. Figure 8.29.a shows the impact of the control gain on the transmissibility Tx¨s v between the road

8.8 Car Suspension

183

Fig. 8.29 Active suspension for various values of the control gain, g = 0, 1000 and 2000. The damping of the shock absorber is c = 200 Ns/m: (a) and (b) Sprung mass acceleration (c) and (d) Tyre deﬂection.

184

8 Vibration Isolation

velocity v = w˙ and the body absolute acceleration x ¨s ; the cumulative RMS value of the sprung mass acceleration σx¨s is shown in Fig.8.29.b. One can see that the active control acts very eﬀectively on the body resonance and that the attenuation is achieved without deteriorating the high frequency isolation. However, the active control is unable to reduce the wheel resonance. The active control produces a signiﬁcant reduction of the RMS sprung mass acceleration but the control system fails to reduce the wheel resonance, Fig.8.29.c. The RMS tyre deﬂection is dominated by the wheel resonance and is not much reduced by the sky-hook control, Fig.8.29.d.

8.9

Problems

P.8.1 Consider the passive isolator of Fig.8.1.b. Find the transmissibility Xc (s)/Xd (s) of the isolation system. P.8.2 Consider the active isolator of Fig.8.7.a with a sky-hook controller. Analyze the eﬀect of the passive damping on the transmissibility. Is the damping beneﬁcial or detrimental ? P.8.3 Write the diﬀerential equations governing the system of Fig.8.12 in state variable form. Using the following values of the parameters: m = 1.1kg, M = 1.7kg, k = k1 = 1.2 104 N/m, m1 = 0.5kg, c1 = 0N s/m. Write the open-loop frequency response for the acceleration feedback (x¨c ) and force feedback (f ) conﬁgurations and draw the corresponding poles/zeros pattern. In both cases draw the root locus for an integral controller. Do the same for m1 = 3.5kg; investigate the eﬀect of structural damping in the ﬂexible appendage. P.8.4 Consider the modal expansion of the open-loop FRF (F/Fa ) of the system of Fig.8.15.d. Show that the residues are all positive and that this results in alternating poles and zeros. P.8.5 Show that the dynamic stiﬀness of the relaxation isolator (Fig.8.3.a) is given by F k1 cs = k(s) = k[1 + ] x − x0 k(k1 + cs) P.8.6 Consider the Gough-Stewart platform with cubic architecture of Fig.8.30 (Spanos et al.); the basic frame {xb , yb , zb } has its origin at node 0; the reference (or payload) frame {xr , yr , zr } has its origin at the geometrical center of the hexapod, noted as node 8, and zr is perpendicular to the payload plate; the orientation of xr and yr is shown in the ﬁgure. The small displacements of the system are described by the coordinates x = (xr , yr , zr , θx , θy , θz )T (a) Show that the control inﬂuence matrix appearing in Equ.(8.47) reads

8.9 Problems

zb

185

zb 5

5

4 4

6

4

4 5

3 5

yb

6

6 8

3

0

7

6

2

xb

xr

1

1

xb

node 0 : (0,0,0) node 4 : (0,L,L) node 1 : (L,0,0) node 5 : (0,0,L) node 2 : (L,L,0) node 6 : (L,0,L) node 3 : (0,L,0) node 7 : (L,L,L) node 8 : (L/2, L/2, L/2)

3

yr

8+

1

2

1

Payload Plate

2

Base

Plate

2

3

yb

z r station of payload plate = L/2 3 zr station of base plate = - L/2 3

Fig. 8.30 Geometry and coordinate systems for the cubic hexapod isolator. Numbers in bold indicate the active struts.

⎛

1 √1 √ ⎜ 3 − ⎜ √ √3 1 ⎜ 2 2 B=√ ⎜ ⎜ −L/2 L/2 6⎜ √ √ ⎝ −L 3/2 −L 3/2 √ √ L 2 −L 2

−2 √0 2 L 0 √ L 2

√1 √3 2 L/2 √ L 3/2 √ −L 2

⎞ 1 −2 √ −√ 3 √0 ⎟ ⎟ 2 2 ⎟ ⎟ −L/2 −L ⎟ ⎟ √ L √ 3/2 0√ ⎠ L 2 −L 2

(b) If the base is ﬁxed and the payload is an axisymmetrical rigid body of mass m with the principal axes of inertia aligned with {xr , yr , zr }, principal moment of inertia Ix = Iy = mRx2 , Iz = mRz2 , and with the center of mass located at an oﬀset distance Zc from the geometrical center, along the vertical axis zr , show that the mass and stiﬀness matrices are respectively ⎛ ⎞ 1 0 0 0 Zc 0 ⎜ 0 1 0 −Zc 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 1 0 0 0 ⎟ ⎜ ⎟ M = m⎜ 2 2 ⎟ ⎜ 0 −Zc 0 (Rx + Zc ) 2 0 2 0 ⎟ ⎝ Zc 0 0 0 (Rx + Zc ) 0 ⎠ 0 0 0 0 0 Rz2 K = kBB T = k diag(2, 2, 2, 0.5L2, 0.5L2 , 2L2 )

186

8 Vibration Isolation

where k is the stiﬀness of one strut. Observe that the translational stiﬀness is uniform in all directions and that the two bending stiﬀnesses are identical. (c) Consider the natural frequencies of the isolator, solutions of the eigenvalue problem (8.37). Show that the the z-translation or “bounce” mode and the z-rotation or “torsional” mode are decoupled, with natural frequencies given by √ √ 2 Ω3 = 2 Ω0 Ω6 = Ω0 ρz where Ω0 = k/m and ρz = Rz /L is the z-axis radius of gyration normalized to the strut length (for most cases, ρz < 1 and Ω6 > Ω3 ). Show that the remaining four modes are lateral bending coupled with shear; their natural frequencies occur in two identical pairs, solutions of the characteristic equation 2 Ω2 1 Ω2 2Ω 2− 2 − ρx 2 − 2ρ2c 2 = 0 Ω0 2 Ω0 Ω0 where ρx = Rx /L is the x-axis radius of gyration normalized to the strut length and ρc = Zc /L is the center of mass oﬀset normalized to the strut length. Show that if the center of mass is at the geometric center (ρc = 0) and if ρx = 12 and ρz = 1, the hexapod will have 6 identical natural frequencies, all equal to Ω3 . P.8.7 Consider the fully active suspension of Fig.8.27.a where v = w˙ is assumed to be a white noise process. Find the Linear Quadratic Regulator (LQR) minimizing the performance index J = E[1 x21 + 2 x22 + u2 ] and J = E[1 x21 + 2 x22 + 3 x23 + u2 ] Discuss the meaning of the various terms in the performance index (The solution of this problem requires a prior reading of Chapters 9 an 11). P.8.8 Consider the active suspension of Figure 8.27.b with kt = 160000 N/m (tyre stiﬀness), k = 16000 N/m (suspension spring stiﬀness), ms = 240 kg (car body), mus = 36 kg (wheel). For the three values of the control gain, g = 0, 1000 and 2000, plot the transmissibility between the road velocity v and the body absolute velocity x˙ s and between the road velocity and the force in the dash-pot c. Compare their frequency content.

9

State Space Approach

9.1

Introduction

The methods based on transfer functions are often referred to as Classical Methods; they are quite suﬃcient for the design of single-input single-output (SISO) control systems, but they become diﬃcult to apply to multi-input multi-output (MIMO) systems. By contrast, the design methods based on the state space approach, often called Modern Methods, start from a system description using ﬁrst order diﬀerential equations governing the state variables. The formalism applies equally to SISO and MIMO systems, which considerably simpliﬁes the design of multivariable control systems. Although it is formally the same for SISO and MIMO systems, we shall ﬁrst study the state space methods for SISO systems. In this case, it is possible to draw a clear parallel with the frequency domain approach, and it is possible to solve the problems of the optimum state regulator and the optimum state observer on a purely geometric basis, replacing the solution of the Riccati equation by the use of the symmetric root locus. The optimal control of MIMO systems will be considered in chapter 11. In the state variable form, Fig.9.1, a linear system is described by a set of ﬁrst order linear diﬀerential equations x˙ = Ax + Bu + Ew1

(9.1)

y = Cx + Du + w2

(9.2)

with the following notations: x = state vector, u = input vector, y = output vector, w1 = system noise, w2 = measurement noise, A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 187–213. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

188

9 State Space Approach

A = system matrix, B = input matrix, C = output matrix, D = feedthrough matrix, E= system noise input matrix.

D u

y

u

x

C (a)

+ +

y

(b) Fig. 9.1 (a) Transfer function approach; (b) State variable form.

For SISO systems, both u and y are scalar functions. Equation (9.1) provides a complete description of the internal dynamics of the system, which may be partially hidden in the transfer function (the internal dynamics may include uncontrollable and unobservable modes which do not appear in the input/output relationship). The feedthrough matrix D is often omitted in control textbooks; however, in earlier chapters, we saw several examples where a feedthrough component does occur as a result of the sensor type (e.g. accelerometer) or location (when collocated with the actuator), or as a result of the modal truncation (residual mode); the impact of a feedthrough component on the compensator design was also emphasized. The system noise w1 may include environmental loads, modelling errors, unmodelled dynamics (including that of the actuator and sensor), nonlinearities, and the noise in the input vector. The system noise input matrix E is in general diﬀerent from the input matrix B. The measurement noise w2 includes the sensor noise and the modelling errors. It must be emphasized that the choice of the state variables is not unique. In principle, their number is independent of the speciﬁc choice (it is equal to the order of the system), but this is not true in practice, because a model can only be correct over a limited bandwidth; if the state variables can be selected in such a way that some of them do not respond dynamically within the bandwidth of interest, they can be eliminated from the dynamic model and treated as singular perturbations (i.e. quasi-static). In structural control, the modal coordinates often lead to the minimum number of state variables.

9.2 State Space Description

9.2

189

State Space Description

The state space equations of the car suspension was already considered at the end of the previous chapter; in this section we review the state variable form of the dynamic equations for a number of mechanical systems that will be used later in this chapter.

9.2.1

Single Degree of Freedom Oscillator

First, consider the familiar second order diﬀerential equation of a single degree of freedom oscillator: f (9.3) m This second order equation implies that there will be two state variables; they can be selected as x ¨ + 2ξωn x˙ + ωn2 x =

x1 = x,

x2 = x˙

(9.4)

With this choice, Equ.(9.3) can be rewritten as a set of two ﬁrst order equations x˙ 1 = x2 x˙ 2 = −2ξωn x2 − ωn2 x1 + f /m or

x˙ 1 x˙ 2

=

0 1 −ωn2 −2ξωn

x1 x2

+

(9.5) 0

1 m

f

(9.6)

This equation explicitly shows the system and input matrices A and B. If one measures the displacement, y = x1 , the output matrix is C = (1 0) while if one uses a velocity sensor, y = x2 and C = (0 1). An accelerometer can only be accounted for by using a feedthrough matrix D in addition to C. In fact, from Equ.(9.5), one gets ! " x1 1 2 y=x ¨ = x˙ 2 = −ωn −2ξωn + f (9.7) x2 m The output and feedthrough matrices are respectively C = (−ωn2 − 2ξωn ) and D = 1/m. The choice (9.4) of the state variables is not unique and it may be convenient to make another choice: x1 = ωn x, x2 = x˙

(9.8)

The advantage of this representation is that both state variables express a velocity, and that the free response trajectories in the phase plane (x1 , x2 )

190

9 State Space Approach

are slowly decaying spirals (Problem 9.1). With this choice, Equ.(9.3) can be rewritten 0 x˙ 1 0 ωn x1 = + 1 f (9.9) x˙ 2 −ωn −2ξωn x2 m This form is sometimes preferred to (9.6) because the system matrix is dimensionally homogeneous.

9.2.2

Flexible Structure

Next, consider the multi degree of freedom vibrating system governed by the set of second order diﬀerential equations Mx ¨ + C x˙ + Kx = Lu f

(9.10)

where Lu is the input force inﬂuence matrix, indicating the way the input forces act on the structure. Equation (9.10) is expressed in physical coordinates. If one deﬁnes the state vector as z = (xT x˙ T )T , it can be rewritten in state space form as 0 I 0 z˙ = z + f (9.11) −M −1 K −M −1 C M −1 Lu The foregoing state variable form is acceptable if M is invertible. Calculating M −1 is straightforward for a lumped mass system. However, Equ.(9.11) may not be practical because the size of the state vector (which is twice the number of degrees of freedom of the system) may be too large. This is why it is customary to transform Equ.(9.10) into modal coordinates before deﬁning the state vector. In this way, the state equation can be restricted to the few structural modes which describe the main part of the system dynamics in the frequency range of interest. Neglecting the high frequency dynamics of the system in the compensator design, however, must be done with extreme care, because the interaction between the neglected modes and the controller may lead to stability problems (spillover instability). This will be of prime concern in the compensator design. Equation (9.10) can be transformed into modal coordinates following the procedure described in chapter 2. If we perform the change of variables x = Φη, the governing equations in modal coordinates read η¨ + 2ξΩ η˙ + Ω 2 η = μ−1 ΦTu f

(9.12)

with the usual notations μ = diag(μi ) (modal mass), ξ = diag(ξi ) (modal damping) and Ω = diag(ωi ) (modal frequencies). In this equation, Φu = LTu Φ

(9.13)

9.2 State Space Description

191

where Φ is the matrix of the mode shapes and Lu is the input force inﬂuence matrix. The columns of Φu contain the modal amplitudes at the actuators location, so that the vector ΦTu f represents the work of the input forces f on the various mode shapes. Here, exactly as for the single degree of freedom oscillator, one can deﬁne the state variable as either Ωη η or z = z= η˙ η˙ In the latter case, the state space equation reads 0 0 Ω z+ z˙ = f μ−1 ΦTu −Ω −2ξΩ

(9.14)

This form is similar to Equ.(9.9), except that diagonal matrices are substituted for scalar quantities. Note that, in most cases, the size of the state vector in Equ(9.14) (which is twice the number of modes included in the model) is considerably smaller than in Equ.(9.11). The output equations for a set of sensors distributed according to the inﬂuence matrix LTy are as follows • displacement sensors

• velocity sensors

y = LTy x = LTy Φη = Φy η = (Φy Ω −1 0)z

(9.15)

y = LTy x˙ = Φy η˙ = (0 Φy )z

(9.16)

• accelerometers y = LTy x ¨ = Φy η¨ = (−Φy Ω

− 2Φy ξΩ)z + Φy μ−1 ΦTu f

(9.17)

where the columns of Φy represent the modal amplitudes at the sensor locations. As for the single degree of freedom oscillator, there is a feedthrough component in the output equation for acceleration measurements. For collocated actuators and sensors, the input and output inﬂuence matrices are the same : Lu = Ly and Φu = Φy .

9.2.3

Inverted Pendulum

As another example of state space description, consider the inverted pendulum with a point mass at a distance l from the support, as represented in Fig.9.2.a. The horizontal displacement u of the support O is the input of the system. The diﬀerential equation governing the motion is u¨ cos θ + lθ¨ = g sin θ

(9.18)

192

9 State Space Approach

This equation also applies to more complicated situations where the pendulum consists of an arbitrary rigid body (Fig.9.2.b) if l is taken as the equivalent length of the pendulum, deﬁned as l=

J + mL2 mL

(9.19)

where J is the moment of inertia about the center of mass and L is the distance between the center of mass C and the hinge O ( J +mL2 is the moment of inertia about O). This can be readily established using the Lagrange or the Newton/Euler equations (Problem 9.2). Equation (9.18) can be linearized about θ = 0 as u ¨ + lθ¨ = gθ (9.20) When the displacement u of the hinge is the input, it is convenient to make a change of variable and introduce the absolute position x of the point mass, rather than the angle θ: x = u + θl (9.21) With this transformation, Equ.(9.20) becomes x ¨ = ω02 x − ω02 u

(9.22)

where ω0 = g/l is the natural frequency of the pendulum. Note that, in general, the coordinate x is diﬀerent from that of the center of mass C in Fig.9.2.b. Using the state variables x1 = x, x2 = x, ˙ we ﬁnd the state space equation x˙ 1 0 1 x1 0 = + u (9.23) x˙ 2 ω02 0 x2 −ω02 If the output of the system is the angle of the pendulum, y = θ = x1 /l − u/l, the output and feedthrough matrices are C = (1/l 0),

(a)

x

(9.24)

(b)

m θ

D = −1/l

g

J

c

l u

o

o

L

Fig. 9.2 Inverted pendulum: (a) Point mass. (b) Arbitrary rigid body.

9.3 System Transfer Function

9.3

193

System Transfer Function

In this section, the input-output transfer function is derived from the state space equations. The relationship is formally the same for the scalar and multidimensional cases. The noise terms are deleted in Equ.(9.1) and (9.2), because they have nothing to do with the present discussion. Laplace transforming the state equation x˙ = Ax + Bu gives sX(s) − x(0) = AX(s) + BU (s) and, assuming zero initial conditions, X(s) = (sI − A)−1 B U (s)

(9.25)

and, from the output equation (9.2), we have Y (s) = [C(sI − A)−1 B + D] U (s)

(9.26)

Y (s) = G(s)U (s)

(9.27)

G(s) = C(sI − A)−1 B + D

(9.28)

or with

In the scalar (SISO) case, G(s) is the transfer function of the system. For MIMO systems, it is the transfer matrix. It is rectangular, with as many rows as there are outputs and as many columns as there are inputs to the system. As an example, consider the single degree of freedom oscillator described by Equ.(9.6), in which the output is the displacement (y = x1 ). We readily obtain s −1 sI − A = ωn2 s + 2ξωn 1 s + 2ξωn 1 (sI − A)−1 = 2 −ωn2 s s + 2ξωn s + ωn2 G(s) = C(sI − A)−1 B =

1 m(s2 + 2ξωn s + ωn2 )

(9.29)

This result could have been obtained more easily by Laplace transforming Equ.(9.3). Similarly, applying Equ.(9.28) to the inverted pendulum, using (9.23) and (9.24), we ﬁnd G(s) = C(sI − A)−1 B + D = −

1 s2 l s2 − ω02

Once again, this result is straightforward from Equ.(9.20).

(9.30)

194

9 State Space Approach

Before discussing the poles and zeros, an important theorem in Matrix Analysis must be established: If a matrix is partitioned into blocks, the following identity applies I 0 A B A B = −CA−1 I C D 0 D − CA−1 B The ﬁrst and third matrices involved in this identity are block triangular, and the determinant of a block triangular matrix is equal to the product of the determinants of the diagonal blocks ; it follows that: A B det = det A. det[D − CA−1 B] (9.31) C D Using this theorem with sI − A instead of A, and −B instead of B, we get sI − A −B det = det(sI − A). det[D + C(sI − A)−1 B] C D Comparing with Equ.(9.28), we see that the transfer function of a SISO system can be rewritten 1 sI − A −B −1 G(s) = C(sI − A) B + D = det (9.32) C D det(sI − A) Now, we demonstrate that, as in the classical pole-zero representation of transfer functions, the denominator and the numerator of Equ.(9.32) explicitly show the poles and zeros of the system.

9.3.1

Poles and Zeros

The poles are the values si such that the free response of the system from some initial state x0 has the form x = x0 esi t . Substituting this in the free response equation, x˙ = Ax, one gets si x0 esi t = Ax0 esi t or (si I − A)x0 = 0

(9.33)

This equation shows that the poles are the eigenvalues of the system equation A (and the initial conditions are the eigenvectors). The poles are solutions of det(sI − A) = 0, which is the denominator of (9.32). The zeros of the system are the values of s such that an input u = u0 est

(9.34)

9.4 Pole Placement by State Feedback

195

applied from appropriate initial conditions x0 , produces a zero output, y = 0. The state vector has the form x = x0 est

(9.35)

if the following condition is fulﬁlled: sx0 est = Ax0 est + Bu0 est

(9.36)

(sI − A)x0 = Bu0

(9.37)

that is, if Under this condition, the output is y = Cx + Du = (Cx0 + Du0 )est

(9.38)

Therefore, having y = 0 requires that Cx0 + Du0 = 0

(9.39)

The two conditions (9.37) and (9.39), can be cast in compact form as x0 sI − A −B =0 (9.40) C D u0 The values of s for which this system of homogeneous equations has a nontrivial solution are the zeros of the system. They are solution of sI − A −B =0 (9.41) det C D which is the numerator of (9.32). From Equ.(9.40), we observe that, to achieve a zero output, the system needs to be excited at the frequency of a transmission zero, and also must start from the appropriate initial conditions x0 ; other initial conditions would produce a transient output that would gradually disappear if the system is asymptotically stable.

9.4

Pole Placement by State Feedback

The idea in the state space approach is to synthesize a full state feedback u = −Gx

(9.42)

where the gain vector G (we deal with SISO systems ﬁrst) is selected to achieve desirable properties of the closed-loop system. The problem is that, in most practical cases, the state vector is not known. Therefore, there must be an additional step of state reconstruction. One of the nice features of the

196

9 State Space Approach

state feedback approach is that the two steps can be performed independently; this property is called the separation principle. We shall address the design of the state feedback ﬁrst (Fig.9.3), and leave the state reconstruction until later.

D x

u

C

-

+ +

y

G Fig. 9.3 Principle of the state feedback (the state is assumed known).

A system of the nth order involves n state variables. Therefore there are n feedback gains gi that can be adjusted independently. Since there are n poles in the system, either real or complex conjugate, they can be assigned arbitrarily by proper choice of the gi . This is true, in principle, if the system is controllable, but it may not always be practical, because the control eﬀort may be too large for the actuators, or the large values of the feedback gains may unduly increase the bandwidth of the control system and lead to noise problems. Substituting the feedback law (9.42) into the system equation (9.1) yields x˙ = Ax − BGx + Ew1 = (A − BG)x + Ew1

(9.43)

The closed-loop system matrix is A − BG. Its eigenvalues are the closed-loop poles; they determine the natural behavior of the closed-loop system and are solutions of the characteristic equation det[sI − (A − BG)] = 0

(9.44)

The state feedback design consists of selecting the gains gi so that the roots of (9.44) are at desirable locations. If the system is controllable, arbitrary pole locations s1 , s2 , ..., sn (either real or complex conjugate pairs) can be enforced by choosing the gi in such a way that Equ.(9.44) is identical to αc (s) = (s − s1 )(s − s2 )...(s − sn ) = 0

(9.45)

The gain values gi are achieved by matching the coeﬃcients of (9.44) and (9.45).

9.4 Pole Placement by State Feedback

9.4.1

197

Example: Oscillator

As an example, consider the single degree of freedom oscillator of Equ.(9.6) x˙ 1 0 1 x1 0 = + 1 f (9.46) x˙ 2 −ωn2 −2ξωn x2 m The poles of the open-loop system are represented in Fig.9.4.a. We want to design a state feedback u = −g1 x1 − g2 x2 (9.47) such that the poles of the closed-loop system are moved to the locations indicated in Fig.9.4.b. From Equ.(9.43), the closed-loop system matrix, A − BG, reads " 0 1 0 ! 0 1 g1 g2 = − 1 −ωn2 −2ξωn −ωn2 − gm1 −2ξωn − gm2 m and the characteristic equation det[sI − (A − BG)] = s2 + s(2ξωn +

g2 g1 ) + (ωn2 + ) = 0 m m

(9.48)

The desired characteristic equation is αc (s) = s2 + 2ξ0 ω0 s + ω02 = 0

(9.49)

Comparing Equ.(9.48) and (9.49), one sees that the desired behavior will be achieved if the gains are selected according to

Fig. 9.4 Single degree of freedom oscillator: (a)open-loop system, (b)closed-loop poles.

198

9 State Space Approach

ω02 = ωn2 +

g1 m

(9.50)

g2 (9.51) m These equations indicate that the controller will act as a spring (of stiﬀness g1 ), and damper (of damping constant g2 ). This is a general observation about the state feedback which consists of a generalization of the PD control. Matching the coeﬃcients of the characteristic equation to the desired ones rapidly becomes tedious as the size of the system increases. It becomes straightforward if the state equation is written in control canonical form. We shall not expand on these aspects because the techniques for pole placement are automated in most control design softwares. The fact that the poles of the closed-loop system can be located arbitrarily in the complex plane is remarkable. Two questions immediately arise: 2ξ0 ω0 = 2ξωn +

• Can it always be done ? • Is it always practical to do it ? The answer to the ﬁrst question is yes if the system is controllable (controllability is discussed in chapter 12). However, even if the system complies with the controllability condition, it may not be practical, because the control eﬀort involved would be too large, or because the large values of the feedback gains would render the system oversensitive to noise or to modelling errors, when the control law is implemented on the reconstructed state from an observer. These robustness aspects are extremely important and actually dominate the controller design. One reasonable way of selecting the closedloop poles is discussed in the next section.

9.5

Linear Quadratic Regulator

One eﬀective way of designing a full state feedback in terms of a single scalar parameter is to use the Optimal Linear Quadratic Regulator (LQR). In this section, we analyze this problem for SISO systems; a more general formulation will be examined in chapter 11. We seek a state feedback (9.42) minimizing the performance index ∞ J= [z 2 (t) + u2 (t)]dt (9.52) 0

where z(t) is the controlled variable, deﬁned by z = Hx

(9.53)

The performance index has two contributions; the ﬁrst one reﬂects the desire of bringing the controlled variable to zero (minimizing the error) while the second one that of keeping the control input as small as possible. is a

9.5 Linear Quadratic Regulator

199

scalar weighting factor used as a parameter in the design : large values of correspond to more emphasis being placed on the control cost than on the tracking error. Note that the control variable, z, may or may not be the actual output of the system. If this is the case, H = C.

9.5.1

Symmetric Root Locus

The solution of the LQR is independent of the initial conditions; for a SISO system, it can be shown (e.g. Kailath, p.226) that the closed-loop poles are the stable roots (i.e. those in the left half plane) of the characteristic equation 1 + −1 G0 (s)G0 (−s) = 0 where G0 (s) =

Z(s) = H(sI − A)−1 B U (s)

(9.54)

(9.55)

is the open-loop transfer function between the input and the controlled variable. Equation (9.54) deﬁnes a root locus problem in terms of the scalar parameter , the weighting factor of the control cost in the performance index. Note that s and −s aﬀect Equ.(9.54) in an identical manner. As a result, the root locus will be symmetric with respect to the imaginary axis, in addition to being symmetric with respect to the real axis. This is why it is called a symmetric root locus. Note that, since only the left part of the root locus must be considered, the LQR approach is guaranteed to be stable. The regulator design proceeds according to the following steps • Select the row vector H which deﬁnes the controlled variable appearing in Equ.(9.53). • Draw the symmetric root locus in terms of the parameter −1 and consider only the part of the locus in the left half plane. • Select a value of −1 that provides the appropriate balance between the tracking error and the control eﬀort. Once the closed-loop poles have been selected, the state feedback gains can be computed as for pole placement (a more eﬃcient way of computing the control gains will be discussed in chapter 11). • Check with simulations that the corresponding control eﬀort is compatible with the actuator. If not, return to the previous step.

9.5.2

Inverted Pendulum

To illustrate the procedure, consider the inverted pendulum described by the state space equation (9.23). If we adopt the absolute displacement x as the controlled variable, H = (1 0); the transfer function (9.55) reads

200

9 State Space Approach

G0 (s) = H(sI − A)−1 B = − G0 (s)G0 (−s) =

s2

ω02 − ω02

ω04 (s2 − ω02 )2

(9.56)

(9.57)

The symmetric root locus is represented in Fig.9.5. The closed-loop poles consist of a complex conjugate pair on the left branch of the locus, s = −ξωn ± jωd . Note that the transfer function G0 (s) used in the regulator design is diﬀerent from the open-loop transfer function (9.30) between the input and the output variable (the latter is the tilt angle θ rather than the absolute position x). Using x instead of θ in the performance index will not only keep the pendulum vertical, but will also prevent it from drifting away from its initial position.

-w0

w0

Fig. 9.5 Symmetric root locus of the inverted pendulum. G0 (s)G0 (−s) has double poles at s = ω0 and s = −ω0 .

Once appropriate pole locations have been selected, the control gains G can be calculated as indicated in the previous section, by matching the closedloop characteristic equation (9.44) to the desired one (9.45): det[sI − (A − BG)] = s2 − sω02 g2 − ω02 (1 + g1 ) = 0

(9.58)

αc (s) = s2 + 2ξωn s + ωn2 = 0

(9.59)

Equating the coeﬃcients of the various powers in s provides g1 = −(1 +

ωn2 ) ω02

(9.60)

9.6 Observer Design

g2 = −

2ξωn ω02

201

(9.61)

The open-loop system has a stable pole at −ω0 and an unstable one at +ω0 . The cheapest optimal state feedback which stabilizes the system, obtained for ρ−1 = 0 on the root locus, simply relocates the unstable pole at −ω0 (thus, the closed-loop system has two poles at −ω0 ). The corresponding characteristic equation (9.59) is (s + ω0 )2 = s2 + 2sω0 + ω02 = 0 and the optimal gains are g1 = −2

9.6

,

g2 = −

2 ω0

Observer Design

The state feedback (9.42) assumes that the state vector is available at all times. This is not the case in general, because it would require too many sensors, and in many applications, some of the states would simply not be available, for physical reasons. The aim of the observer is to reconstruct the state vector from a model of the system and the output measurement y. In this way, the state feedback can be implemented on the reconstructed state, x ˆ, rather than on the actual state x. It must be emphasized that the state reconstruction requires a model of the system. We shall assume that an accurate model is available. To begin with, consider the case where the noise and feedthrough terms are removed from the system and output equations (9.1) and (9.2). x˙ = Ax + Bu

(9.62)

y = Cx

(9.63)

The following form is assumed for the full state observer (also called Luenberger observer). x ˆ˙ = Aˆ x + Bu + K(y − C x ˆ) (9.64) x ˆ(0) = 0 The ﬁrst part of Equ.(9.64) simulates the system and the second contribution (innovation) uses the information contained in the sensor signal; y − C x ˆ is the diﬀerence between the actual and the predicted output; the gain matrix K is chosen in such a way that the error between the true state and the reconstructed one, e = x − x ˆ, converges to zero. Combining Equ.(9.62) to (9.64), we ﬁnd the error equation e˙ = (A − KC)e

(9.65)

202

9 State Space Approach

with the initial condition e(0) = x(0). This equation shows that the error goes to zero if the eigenvalues of A − KC (the observer poles) have negative real parts (i.e. if A − KC is asymptotically stable). In a manner parallel to the state feedback design, if the system is observable, the n observer poles can be assigned arbitrarily in the complex plane by choosing the values of the n independent gains ki (observability will be discussed in chapter 12). The procedure for pole assignment is identical to that used for the regulator design : If a set of poles si has been selected, the gains ki can be computed by matching the coeﬃcients of the characteristic equation det[sI − (A − KC)] = 0

(9.66)

αo (s) = (s − s1 )(s − s2 )...(s − sn ) = 0

(9.67)

to that of the desired one

(once again, the poles can be only real or complex conjugate pairs). Note that the nice form of the error equation is due to the fact that the same matrices A, B, C have been used in the system equations (9.62), (9.63) and the observer equation (9.64). This amounts to assuming perfect modelling. In practice, however, the actual system will be substituted for Equ.(9.62) and (9.63) while the observer equation (9.64) will be integrated numerically in a computer. As a result, there will always be an error associated with the modelling of the system. If a feedthrough term appears in the output equation, the innovation term in Equ.(9.64) must be taken as y − (C x ˆ + Du) instead of y − C x ˆ; this leads to the same error equation (9.65). If noise terms are included as in Equ.(9.1) and (9.2), the error equation becomes e˙ = (A − KC)e + Ew1 + Kw2

(9.68)

The plant noise Ew1 appears as an excitation; so does the measurement noise w2 , after being ampliﬁed by the observer gain K. If one wants the regulator poles to dominate the closed-loop response, the observer poles should be faster than (i.e. to the left of) the regulator poles. This will ensure that the estimation error decays faster than the desired dynamics, and the reconstructed state follows closely the actual one (at least without noise and modelling error). As a rule of thumb, the observer poles should be 2 to 6 times faster than the regulator poles. With noisy measurements, one may wish to decrease the bandwidth of the observer by having the observer poles closer to those of the regulator. This produces some ﬁltering of the measurement noise. In this case, however, the observer poles have a signiﬁcant inﬂuence on the closed-loop response. Notice, from Equ.(9.65), that if the open-loop system is stable (and the model accurate), the error will converge to zero even if the gain K is zero.

9.7 Kalman Filter

203

The observer equation (9.64) is based partly on the system model, Aˆ x+Bu, and partly on the measurement error, y − C x ˆ, the latter being ampliﬁed by the observer gain K. The relative importance which is given to the model and the measurement contributions should depend on their respective quality. If the sensor noise is low, more weight can be placed on the measurement error (large gains ki ), while noisy measurements do require lower gains. The minimum variance observer is that minimizing the variance of the measurement error. If one assumes that the plant noise and measurement noise are white noise processes (i.e. with uniform power distribution over the whole frequency range), the minimum variance observer is known as the Kalman Filter.

9.7

Kalman Filter

The Kalman Bucy Filter (KBF) will be studied in detail in chapter 11. Here, exactly as we used the LQR as a sensible pole placement strategy for the regulator design of SISO systems (in terms of a single scalar parameter), we consider the particular case where the plant noise and the measurement noise are scalar white noise processes. In this case, it is, once again, possible to draw a symmetric root locus plot in terms of a single parameter expressing the relative intensity of the plant and measurement noise. Let x˙ = Ax + Bu + Ew

(9.69)

y = Cx + v

(9.70)

be the system and output equations, where w and v are scalar white noise processes. It can be demonstrated that the optimal pole locations corresponding to the KBF are the stable roots of the symmetric root locus deﬁned by the equation 1 + qGe (s)Ge (−s) = 0 (9.71) where Ge (s) is the transfer function between the plant noise w and the output y Ge (s) = C(sI − A)−1 E (9.72) and the parameter q is the ratio between the intensity of the plant noise w and that of the measurement noise v (noisy measurements correspond to small q). The observer design proceeds exactly as for the regulator: Selecting a noise input matrix E, we draw the symmetric root locus as a function of q. When proper pole locations have been selected on the left half part of the locus, the gains ki can be computed by matching the coeﬃcients of the characteristic equation (9.66) to that of the desired one (9.67). If none of the values of q provides a desirable set of poles, another E matrix can be selected and the procedure repeated.

204

9 State Space Approach

In the LQR approach, the regulator design involves the symmetric root locus of the open-loop transfer function between the input u and the controlled variable z Go (s) = H(sI − A)−1 B (9.73) while in the KBF approach, the observer design is based on the transfer function between the system noise w and the output y. Ge (s) = C(sI − A)−1 E

(9.74)

If one assumes that the plant noise w enters the system at the input (E = B) and the controlled variable is the output variable (H = C), these transfer functions become identical to the open-loop transfer function G(s) = C(sI − A)−1 B

(9.75)

In this case, the regulator and the observer can both be designed from a single symmetric root locus.

9.7.1

Inverted Pendulum

To illustrate the procedure, consider again the inverted pendulum and assume that the noise enters the system at the input (E = B). From Equ.(9.23) and (9.24), the transfer function between the noise w and the output y, Equ.(9.74), is 1 ω2 Ge (s) = − 2 0 2 (9.76) l s − ω0 Thus, although the output variable θ is diﬀerent from the controlled variable x, Ge (s) is proportional to Go (s) used in the regulator design [Equ.(9.56)]; this is because there is no feedthrough component in Ge (s) as there is in G(s) [Equ.(9.30)]. As a result, the root locus used in the regulator design can also be used for the observer design (Fig.9.6). How far the observer poles should be located on the locus really depends on how noisy the measurements are. Note that when the regulator poles are near the asymptotes, the KBF obtained here is very close to the design consisting of assigning the observer poles by scaling the regulator poles ( ωn , ξ ) to ( αωn , ξ ) where 2 < α < 6, according to the rule of thumb mentioned before (Problem 9.5). Finally, assume that the noise enters the system according to E = (1 a)T . From Equ.(9.23) and (9.24), the transfer function between the noise and the output is readily obtained as Ge (s) =

s+a l(s2 − ωo2 )

(9.77)

9.7 Kalman Filter

205

Im(s)

Observer poles

Regulator poles

-w0

Re(s)

w0

Fig. 9.6 Inverted pendulum. The same symmetric root locus is used for the design of the regulator and the full state observer (E = B).

-ω0 -a

ω0 a

Fig. 9.7 Full state observer of the inverted pendulum. Symmetric root locus when the noise enters the system according to E = (1 a)T .

The corresponding symmetric root locus is shown in Fig.9.7, assuming a > ω0 . One notices that one of the poles goes to the zero at s = −a as q increases. This is the optimum for the assumed distribution of the plant noise, but the observer obtained in this way may be sluggish, which may not be advisable for reasons mentioned before.

206

9 State Space Approach

9.8

Reduced Order Observer

With the full state observer, the complete state vector is reconstructed from the output variable y. When the number of state variables is small and the output consists of one of the states, it may be attractive to restrict the state reconstruction to the missing state variables, so that the measured state variable is not aﬀected by the modelling error. This is nice, but on the other hand, by building the state feedback partly on the output measurement without prior ﬁltering, there is a feedthrough component from the sensor noise to the control input, which increases the sensitivity to noise. In this section, we restrict ourselves to the frequent case of reconstructing the velocity from a displacement measurement for a second order mechanical system. This is probably the most frequent situation where a reduced order observer is used. In this case, the reduced observer is of the ﬁrst order while the full state observer is of the second order. Extensions to more general situations can be found in the literature (e.g. Luenberger, 1971).

9.8.1

Oscillator

Consider the single degree of freedom oscillator governed by Equ.(9.6). The output measurement is the displacement (y = x). The velocity equation is (v = x) ˙ v˙ = −2ξωn v − ωn2 x + u/m (9.78) We seek a ﬁrst order observer governed by the following equations z˙ = −az − bx + u/m

(9.79)

vˆ = z + cx

(9.80)

where vˆ is the estimated velocity, z is an internal variable, and a, b and c are free parameters; they are selected in such a way that the error equation governing the behavior of e = vˆ − v is asymptotically stable, with an appropriate decay rate a: e˙ = −ae (9.81) Equations (9.78)-(9.80) give the error equation e˙ = −aˆ v + (c + 2ξωn )v + (ωn2 + ac − b)x

(9.82)

It can be made identical to Equ.(9.81) if the coeﬃcients are selected according to (9.83) a = c + 2ξωn b = ωn2 + ac

(9.84)

From Equ.(9.81), a is the eigenvalue of the observer; it can be chosen arbitrarily.

9.8 Reduced Order Observer

9.8.2

207

Inverted Pendulum

The same procedure can be applied to the inverted pendulum described by Equ.(9.23) and (9.24). The displacement is obtained from the output measurement (y = θ) and the input as x = ly + u

(9.85)

v˙ = ω02 x − ω02 u

(9.86)

The velocity equation is (v = x) ˙

The velocity observer has again the form z˙ = −az − bx − ω02 u

(9.87)

vˆ = z + cx

(9.88)

and the error equation (e = vˆ − v ) is readily written as e˙ = −aˆ v + cv + (−ω02 + ac − b)x

(9.89)

It becomes identical to (9.81) if the coeﬃcients satisfy a=c

(9.90)

b = c2 − ω02

(9.91)

The reduced order observer is therefore z˙ = −cz − (c2 − ω02 )x − ω02 u

(9.92)

vˆ = z + cx

(9.93)

where the only remaining parameter c is the eigenvalue of the observer, which can be selected arbitrarily. Note that c appears also as the feedthrough component of the measured variable in Equ.(9.93), leading to a direct eﬀect of the measurement noise on the reconstructed velocity (ampliﬁed by cl). Because of the existence of a feedthrough component from the measurement to the reconstructed states, and therefore to the control, the bandwidth of a compensator based on a the reduced observer is much wider than that of a compensator based on a full state observer. The simplicity of the observer structure must be weighed against the higher sensitivity to sensor noise. If the latter is signiﬁcant, the reduced order observer becomes less attractive than the full state observer.

208

9 State Space Approach

9.9

Separation Principle

Figure 9.8 shows the complete picture of the state feedback regulator implemented on the reconstructed states, obtained from a full state observer (the case without feedthrough component is represented for simplicity). The closed-loop equations are: x˙ = Ax + Bu y = Cx u = −Gˆ x ˙x ˆ = Aˆ x + Bu + K(y − C x ˆ)

(9.94)

The complete system has 2n state variables. If one uses the reconstruction error, e = x − x ˆ, as state variable instead of the reconstructed state vector x ˆ, one can write the closed-loop system equation as x A − BG BG x˙ (9.95) = e 0 A − KC e˙ It is block triangular and, as a result, the eigenvalues of the closed-loop system are those of the diagonal blocks A − BG and A − KC. Thus, the poles of the closed-loop system consist of the poles of the regulator and those of the observer. This means that the eigenvalues of the regulator and the observer are not changed when the two subsystems are put together. Therefore, the design of the regulator and of the observer can be carried out independently; this is known as the separation principle.

u

C

-

y

^

G

^

^

^

Compensator Fig. 9.8 State feedback with full state observer.

9.10

Transfer Function of the Compensator

The transfer function of the compensator can be obtained from Equ.(9.94). It reads

9.10 Transfer Function of the Compensator

H(s) =

U (s) = −G(sI − A + KC + BG)−1 K Y (s)

209

(9.96)

The poles of the compensator are solutions of the characteristic equation det(sI − A + KC + BG) = 0

(9.97)

Note that they have not been speciﬁed anywhere in the design, and that they are not guaranteed to be stable, even though the closed-loop system is. Working with an unstable compensator (which fortunately, is stabilized by the plant!) may bring practical diﬃculties as, for example, that the open-loop frequency response of the compensator cannot be checked experimentally. Figure 9.9 represents the compensator and the system in the standard unity feedback form used in classical methods (root locus, Bode, Nyquist). There is a major diﬀerence between the state feedback design and the classical methods: in classical methods, the structure of the compensator is selected to achieve desired closed-loop properties; in the state feedback design, the structure of the compensator is never directly addressed because the attention is focused on the closed-loop properties; the compensator is always of the same order as the system.

Compensator H(s) -

G (sI − A + K C + BG ) −1 K

System G(s) u

C (sI − A ) −1 B

y

Fig. 9.9 State feedback with full state observer. Unity feedback form of the closedloop system.

9.10.1

The Two-Mass Problem

Let us illustrate this point on the two-mass problem of Fig.6.3. Deﬁning the ˙ y, y) state vector x = (d, d, ˙ T , Equ.(6.3) and (6.4) can be written as ⎛ ⎞ ⎛ ⎞ 0 1 0 0 0 ⎜ −ω02 −2ξω0 ⎜ ⎟ ω02 2ξω0 ⎟ ⎟x +⎜ 0 ⎟f x˙ = ⎜ (9.98) ⎝ 0 ⎠ ⎝ 0 0 1 0 ⎠ μω02 2μξω0 −μω02 −2μξω0 1/M For a non-collocated displacement sensor, d = (1 0 0 0) x

210

9 State Space Approach

The open-loop transfer function is given by Equ.(6.7). If μ = 0.1, M = 1 and the open-loop system has a pair of poles at s = −0.02 ± j1 (ﬂexible mode) the open-loop transfer function is G(s) =

D(s) 0.036(s + 25) = 2 F (s) s (s + 0.02 ± j1)

(9.99)

where the compact notation s+0.02±j1 is used for (s+0.02+j1)(s+0.02−j1).

Im(s) Observer poles Regulator poles

2

Re(s) -2

-1

-2

Fig. 9.10 Symmetric root locus for the two-mass system.

Using the output d as the controlled variable and assuming that the system noise enters at the input, we can design the regulator and the observer using the same symmetric root locus based on G(s). It is shown in Fig.9.10. Once the regulator and observer poles have been selected on the left side of the root locus, the corresponding gains can be calculated by matching the coeﬃcients of the characteristic equations (9.44) and (9.66) to the desired ones; for the pole locations indicated in the ﬁgure, we get G = (−0.558 0.403 1.364 1.651) K = (5.78 16.71 34.35 31.26)T The resulting compensator is H(s) =

−98.4(s + 0.28)(s + 0.056 ± j0.89) (s + 0.68 ± j2.90)(s + 3.06 ± j1.54)

(9.100)

9.10 Transfer Function of the Compensator

211

Im(s)

Compensator

Flexible mode Notch filter

Re(s)

XX

Rigid body mode

Fig. 9.11 Root locus plot, 1 + gHG = 0, for the optimal control of the two-mass system with nominal values of the parameters. The triangles indicate the location of the closed-loop poles for g = 1 (only the upper half is shown).

It is of the fourth order, like the system. The numerator consists of a PD plus a pair of zeros near the ﬂexible poles of G(s), to produce a notch ﬁlter. With this nominal compensator, it is possible to draw a conventional root locus plot for gHG (Fig.9.11), which describes the evolution of the poles of the closed-loop system when the scalar gain g varies from 0 to ∞. The pole locations on this root locus for g = 1 (represented by triangles) coincide with those selected on the symmetric root locus (Fig.9.10). To assess the robustness of the control system, Fig.9.12 shows the root locus plot when the natural frequency is changed from ωn = 1 rad/s to ωn = 2 rad/s. One observes that the notch ﬁlter does not operate properly any more (the pair of zeros of the compensator no longer attracts the ﬂexible poles of the structure) and the closed-loop system soon becomes unstable. The situation is even worse if the natural frequency is reduced to ωn = 0.8 rad/s. In this case, a pole-zero ﬂipping occurs between the pole of the ﬂexible mode and the zero of the notch ﬁlter (Fig.9.13). As we already stressed in chapter 6, the pole-zero ﬂipping changes the departure angles from the poles and the arrival angles at the zeros by 1800 , transforming the stabilizing loop of Fig.9.11 into a destabilizing one as in Fig.9.13. This lack of robustness is typical of state feedback and notch ﬁlters.

212

9 State Space Approach

Im(s)

Re(s) XX

Fig. 9.12 Root locus plot for oﬀ-nominal parameters ( ωn = 2 instead of 1).

Im(s)

Pole-zero flipping Re(s) XX

Fig. 9.13 Root locus of the optimal control when the natural frequency has been reduced to ωn = 0.8.

9.11 Problems

9.11

213

Problems

P.9.1 Consider the s.d.o.f. oscillator described by Equ.(9.9). For non-zero initial conditions, sketch the free response in the phase plane (x1 , x2 ). Show that the image point rotates clockwise along a spiral trajectory. Relate the decay rate of the spiral to the damping ratio. P.9.2 Show that the inverted pendulum of Fig.9.2 is governed by Equ.(9.18) to (9.20), where u is the displacement of the support point. P.9.3 Consider an inverted pendulum similar to that of Fig.9.2, but mounted on a cart of mass M and controlled by an horizontal force u applied to the cart. If x stands for the horizontal displacement of the cart, show that, for small θ, the governing equations can be approximated by x¨ + lθ¨ = gθ (M + m)¨ x + mLθ¨ = u Write the equations in state variable form. P.9.4 For the inverted pendulum of Problem 9.3, assuming that the full state is available, ﬁnd a feedback control that balances the stick and keeps the cart stationary near x = 0. P.9.5 Consider the inverted pendulum of Fig.9.2. Using the same procedure as for the regulator in section 9.5.2, ﬁnd the analytical expression of the observer gains which locates the observer poles at (αωn , ξ). P.9.6 Consider an inverted pendulum with ω0 = 1 (Fig.9.2). (a) Compute the transfer function H(s) of the compensator such that the regulator and the observer poles are respectively at ωn = 2, ξ = 0.5 and αωn = 6, ξ = 0.5. (b) Draw a root locus (1 + gHG = 0) for this compensator. Observe that the closed-loop system is conditionally stable. Find the critical gains corresponding to the limits of stability. Sketch the Nyquist diagram for the nominal system.

10

Analysis and Synthesis in the Frequency Domain

10.1

Gain and Phase Margins

Consider the root locus plot of Fig.6.5; any point s in the locus is solution of the characteristic equation 1 + gG(s)H(s) = 0. Therefore, we have |gG(s)H(s)| = 1

φ(gGH) = −π

The locus crosses the imaginary axis at the point of neutral stability. Since s = jω, the following relations hold at the point of neutral stability: |gG(jω)H(jω)| = 1

(10.1)

φ(gGH) = −π Returning to the Bode plot of Fig.6.6, one sees that changing g amounts to moving the amplitude plot along the vertical axis; the point of neutral stability is obtained for g = 0.003 × 1.61 = 0.0048, when the amplitude curve is tangent to the 0 dB line, near the frequency ω = 1, above which the phase exceeds −1800. The system is stable if the algebraic value of the phase φ of the open-loop system is larger than −π for all frequencies where the amplitude |gGH| is larger than 1. The system is unstable if this condition is violated. A measure of the degree of stability is provided by the gain and phase margins: • The gain margin (GM) indicates the factor by which the gain must be increased to reach the neutral stability. • The phase margin (PM) is the amount by which the phase of the openloop transfer function exceeds −π when |gGH| = 1. The corresponding frequency is called the crossover frequency. The situation described above corresponds to a stable open-loop system G(s); the gain margin is a decreasing function of g, the system becomes unstable when the gain exceeds some critical value. There are more complex situations where the system is conditionally stable, when the gain belongs to some A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 215–246. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

216

10 Analysis and Synthesis in the Frequency Domain

interval g1 < g < g2 ; it becomes unstable when the gain decreases below the threshold g1 (see Problem 9.6). These cases can be handled with the Nyquist criterion.

10.2

Nyquist Criterion

10.2.1

Cauchy’s Principle

Consider the feedback system of Fig.10.1, where G(s) stands for the combined open-loop transfer function of the system and the compensator (shorthand for gGH in the previous section). The conformal mapping G(s) transforms the contour C1 in the s plane into a contour C2 in the G(s) plane. One of the properties of the conformal mapping is that two intersecting curves with an angle α in the s plane map into two intersecting curves with the same angle α in the G(s) plane. Assume G(s) is written in the form

m (s − zi ) G(s) = k ni=1 (10.2) (s − pi ) i=1 For any s on C1 , the phase angle of G(s) is given by φ(s) =

m

φi −

i=1

n

(10.3)

ψj

j=1

where φi and ψj are the phase angles of the vectors connecting respectively the zeros zi and the poles pi to s. From Fig.10.1, it can be seen that, if there

G

-

Im(s)

Im[G(s)] C1

G(s)

s

φi

Re(s)

C2

φ

ψi

Fig. 10.1 Nyquist contour in the s and G(s) planes.

Re[G(s)]

10.2 Nyquist Criterion

217

are neither poles nor zeros within the contour C1 , φ(s) does not change by 2π when s goes clockwise around C1 . The contour C2 will encircle the origin only if the contour C1 contains one or more singularities of the function G(s). Because the contour C1 is travelled clockwise, one pole of G(s) within C1 produces a phase change of 2π, that is one counterclockwise encirclement of the origin by C2 . Conversely, a zero produces a clockwise encirclement. Thus, the total number of clockwise encirclements of the origin by C2 is equal to the number of zeros in excess of poles of G(s), within the contour C1 . This is Cauchy’s principle.

10.2.2

Nyquist Stability Criterion

The foregoing idea provides a simple way of evaluating the number of singularities of the closed-loop system in the right half plane, from the poles and zeros of the open-loop transfer function G(s). The contour C1 is selected in such a way that it encircles the whole right half plane as indicated in Fig.10.2. If there are poles on the imaginary axis, indentations are made as indicated in the ﬁgure, to leave them outside C1 . The closed-loop transfer function is F (s) =

G(s) 1 + G(s)

(10.4)

and the closed-loop poles are solutions of the characteristic equation 1 + G(s) = 0 If G(s) =

(a)

n(s) d(s)

(b) C1 R=∞

R=∞

C1

ε

Fig. 10.2 Contour encircling the right half plane. (a) With no singularity on the imaginary axis. (b) With three poles on the imaginary axis.

218

10 Analysis and Synthesis in the Frequency Domain

1 + G(s) =

d(s) + n(s) d(s)

(10.5)

This equation shows that the poles of 1 + G(s) are the same as those of the open-loop transfer function G(s). Let P be the [known] number of unstable poles of the open-loop system. On the other hand, the zeros of 1 + G(s) are the poles of the closed-loop system; we want to evaluate their number Z within C1 . From Cauchy’s principle, if we consider the mapping 1 + G(s), the number N of clockwise encirclements of the origin by C2 when s goes clockwise along C1 is N =Z−P (10.6) Now, instead of considering the encirclements of the origin by 1 + G(s), it is completely equivalent to consider the encirclements of −1 by G(s). Thus, the number of unstable poles of the closed-loop system is given by Z =N +P

(10.7)

where N is the number of clockwise encirclements of −1 by G(s), when s follows the contour C1 , and P is the number of unstable poles of the openloop system. All physical systems without feedthrough are such that G(s) → 0 as s → ∞ (with feedthrough, G(s) goes to a constant value). As a result, only the part of the plot corresponding to the imaginary axis (−∞ < jω < ∞) must be considered. The polar plot for positive frequencies can be drawn from the Bode plots; that for negative frequencies is the mirror image with respect to the real axis, because G(−jω) = G∗ (jω). If there are poles of G(s) on the imaginary axis and if the indentations are made as indicated in Fig.10.2, the poles are outside the contour. According to Equ.(10.2) and (10.3), each pole contributes with an arc at inﬁnity and a rotation of −π, that is 1800 clockwise. Returning to the system of Fig.6.5 and 6.6, its Nyquist plot is shown in Fig.10.3: The contribution of the positive frequencies is plotted in full line, and the circle at inﬁnity corresponds to the indentation of the double pole at the origin (from ω = 0− to ω = 0+ ); the contribution of the negative frequencies (not shown) is the mirror image of that of the positive frequencies with respect to the real axis. For small gains (as shown in the ﬁgure), the number of encirclements of −1 is zero and the system is stable; for larger gains, there are two encirclements and therefore two unstable poles. This is readily conﬁrmed by the examination of Fig.6.5. Since the instability occurs when G(jω) encircles −1, the distance from G(jω) to −1 is a measure of the degree of stability of the system. The relative stability is measured by the gain and phase margins (Fig.10.4). As we shall see, there is a direct relationship between the phase margin and the sensitivity to parameter variations and the disturbance rejection near crossover.

10.2 Nyquist Criterion

219

Im Circle at infinity

ω=0

−

Re

-1

ω=0

+

Fig. 10.3 Nyquist plot for the non-collocated control of the two-mass problem.

Im 1

GM

Unit circle

G ( jω )

Re

-1 PM

ωc

Fig. 10.4 Deﬁnition of the gain margin GM and phase margin PM on the Nyquist plot.

Since the closed-loop transfer function F (s) is uniquely determined by G(s) [Equ.(10.4)], loci of constant magnitude |F (jω)| = M and of constant phase φ[F (jω)] = N can be drawn in the complex plane G(s); they happen to be circles. The M circles are shown in Fig.10.5; the larger magnitudes correspond to smaller circles near −1. Thus, the maximum ampliﬁcation Mp corresponds to the smallest circle, tangent to the Nyquist plot of G(s); it is reached for a frequency ωp close to the crossover frequency ωc (where the Nyquist plot crosses the unit circle). This is why there is a direct relationship between the maximum ampliﬁcation Mp and the phase margin P M . In most cases, P M > 600 prevents overshoot of F (Fig.10.6). The bandwidth ωb of the

220

10 Analysis and Synthesis in the Frequency Domain Im G ( jω )

Amplification

M <1

M >1 M = Mp

M =1

Re G ( jω ) −1

PM

ωc

G ( jω )

1 − 2

ωp Unit circle

PM = 60°

Fig. 10.5 Loci of constant magnitude of the closed-loop transfer function (M circles). The maximum ampliﬁcation Mp is associated with the smallest circle tangent to the Nyquist plot.

dB

wp

PM=22° PM=45°

wc

0 -3

PM=90°

log w

wb

-20 Fig. 10.6 Relation between the overshoot of the closed-loop transfer function and the phase margin. |F | is maximum at ωp ; the bandwidth is the frequency ωb where |F | = −3dB and |G| = 1 at the crossover frequency ωc . In most cases, ωb ∼ ωp ∼ ωc .

control system is deﬁned as the frequency corresponding to an attenuation of -3 dB in the closed-loop transfer function. Since ωb ∼ ωp ∼ ωc , the bandwidth can be approximated by the crossover frequency for design purposes.

10.3 Nichols Chart

10.3

221

Nichols Chart

The Nyquist plot is a convenient tool for evaluating the number of encirclements and the absolute stability of the system. However, because of the linear scale for the magnitude, the Nyquist plot is not always practical in the vicinity of -1. The Nichols chart plot is often more useful for evaluating the relative stability when the open-loop system is stable (Fig.10.7). It consists of a plot of the dB magnitude vs. phase angle of the open-loop transfer function G(jω). There is a one to one relationship between the Nichols chart and the Nyquist plot, but the former brings a considerable ampliﬁcation to the vicinity of -1, for easy evaluation of the gain and phase margins, and the logarithmic scale allows a much wider range of magnitude in the graph. Unlike the Nyquist plot, the Nichols chart plot can be obtained from the summation of the individual magnitude and phase angle contributions of the poles and zeros, and a change of gain moves the curve along the vertical axis. These two advantages are shared by the Bode plots, but the Nichols chart plot combines the gain and phase information into a single diagram. As for

Constant ma gnitude M 0.25 dB

30

|G|

0 dB

0.5 dB

dB 20

1 dB B

-1 d

3 dB

10

6 dB

B

-3 d

PM

0

-6 dB

GM

-10

-12 dB

-100

-150

-180

-200

-250

f (G) o

-50

-20 dB

-20

Fig. 10.7 Nichols chart: dB magnitude vs. phase angle of the open-loop transfer function G(jω) and loci of constant magnitude of the closed-loop transfer function (M -curves).

222

10 Analysis and Synthesis in the Frequency Domain

the Nyquist plot, in order to assist in the design, it is customary to draw the loci of constant amplitude M of the closed-loop system; they are no longer circles.

10.4

Feedback Speciﬁcation for SISO Systems

Consider the SISO feedback system of Fig.10.8, where r is the reference input, y the output, d the disturbance and n the sensor noise (unlike the disturbance, the sensor noise does not directly aﬀect the output).

d

e

r

G

y

-

n Fig. 10.8 Feedback System.

10.4.1

Sensitivity

In this section, we evaluate the sensitivity of the closed-loop transfer function to the variations of the open-loop transfer function; we assume that d = n = 0. The closed-loop transfer function is y G =F = r 1+G

e 1 = r 1+G

(10.8)

The sensitivity of F to parameter changes is related to that of the open-loop transfer function G by ∂F 1 ∂G = ∂p (1 + G)2 ∂p or δF = or

(10.9)

1 δG (1 + G)2

δF 1 δG = F 1+G G

(10.10)

S = (1+G)−1 is called the sensitivity function and 1+G the return diﬀerence. Equation (10.10) states that, in the frequency range where |1 + G| 1, the sensitivity of the closed-loop system to parameter changes is much smaller than that of the open-loop system. This is one of the objectives of feedback. On the other hand, if the phase margin is small, G(jω) goes very near −1 at crossover, and F becomes much more sensitive than G.

10.4 Feedback Speciﬁcation for SISO Systems

10.4.2

223

Tracking Error

Referring to Fig.10.8, the governing equations are e=r−y−n y = Ge + d = G(r − y − n) + d or y=

G 1 (r − n) + d 1+G 1+G

(10.11)

The tracking error is e∗ = r − y =

1 G (r − d) + n 1+G 1+G

(10.12)

From Equ.(10.11) and (10.12), we see that good tracking implies that • The sensitivity function S = (1 + G)−1 must be small in the frequency range where the command r and the disturbance d are large. • The closed-loop transfer function F = G(1 + G)−1 must be small in the frequency range where the sensor noise is large. Note, however, that S+F = 1

(10.13)

Therefore, S and F cannot be small simultaneously; this means that disturbance rejection and noise rejection cannot be achieved simultaneously.

10.4.3

Performance Speciﬁcation

In the previous section, G referred to the open-loop transfer function, including the system and the compensator. In this section, G will refer explicitly to the system and H to the compensator; the open-loop transfer function is GH (Fig.10.9). The general objective of feedback is to achieve sensitivity reduction, good tracking and disturbance rejection at low frequency (ω < ω0 ) with a sensor signal which has been contaminated with noise at high frequency (ω > ω1 ). From the foregoing section, this is translated into the following design constraints: |1 + GH(ω)| ≥ ps(ω) ω ≤ ω0 (10.14) where ps(ω) is a large positive function (performance speciﬁcation), deﬁned in the frequency range where good tracking and disturbance rejection is necessary [zero steady-state error requires that ps(0) = ∞], and |GH(ω)| 1

ω > ω1

(10.15)

224

10 Analysis and Synthesis in the Frequency Domain

GH

r

e

H

u

sensor noise

ps(ω)

y

G

-

1

ω0

ω1

steady state error, disturbance rejection Fig. 10.9 Feedback speciﬁcation.

where ω1 deﬁnes the frequency above which the sensor noise becomes significant. Of course, in addition to that, the closed-loop system must be stable. Thus minimum values for the gain and phase margins must also be speciﬁed: GM > μ

(10.16)

PM > φ

Equivalently, we may require that the GH√curve remains outside a particular M circle from the point −1+j0 (e.g. M = 2). We shall see later that stability places constraints on the slope of |GH| near crossover (in the frequency range near |GH| = 1).

10.4.4

Unstructured Uncertainty

Unstructured uncertainty When the uncertainty on the physical system is known to aﬀect some speciﬁc physical parameters like, for example, the natural frequencies and the damping, it is called structured. If little is known about the underlying physical mechanism, it is called unstructured. The unstructured uncertainty will be characterized by an upper bound of the norm of the diﬀerence between the transfer function of the actual system, G (ω) and its model G(ω). It can be considered either as additive: G (ω) = G(ω) + ΔG(ω),

|ΔG(ω)| < la (ω),

ω>0

(10.17)

where la(ω) is a positive function deﬁning the upper bound to the additive uncertainty (including the parameter changes and the neglected dynamics), or multiplicative: G (ω) = G(ω)[1 + L(ω)],

|L(ω)| < lm (ω),

ω>0

(10.18)

10.4 Feedback Speciﬁcation for SISO Systems

225

where lm (ω) is a positive function deﬁning an upper bound to the multiplicative uncertainty. lm (ω) is usually small ( 1) at low frequency where the model is accurate, and becomes large ( 1) at high frequency, due to the neglected dynamics of the system (Fig.10.10). Because lm (ω) deﬁnes only the magnitude of the uncertainty, it can be associated with any phase distribution; therefore, it clearly deﬁnes a worst case situation. Evaluating la (ω) or lm (ω) is not a simple task. For large ﬂexible structures, more modes are often available than it would be practical to include in a design model; these extra modes can be used to evaluate the associated uncertainty (Problem 10.4). l m (ω ) >> 1

0 dB

<< 1

Fig. 10.10 Feedback System.

10.4.5

Robust Performance and Robust Stability

In the face of uncertainties, the design objectives of section 10.4.3 must be fulﬁlled for the perturbed system. Referring to Fig.10.11, the stability of the system will be guaranteed in presence of uncertainty if the distance to instability, |1+GH| is always larger than the uncertainty ΔGH. The stability robustness condition is therefore |1 + GH| ≥ |H|la (ω)

(10.19)

This guarantees that the number of encirclements will not be altered by any additive uncertainty ΔG bounded by la (ω), for any phase distribution. Similarly, since la (ω) = lm (ω)|G(ω)|, condition (10.19) can be written in terms of the multiplicative uncertainty as

or

|1 + GH| ≥ |GH|lm (ω)

(10.20)

1 GH ≥| | lm (ω) 1 + GH

(10.21)

Normally, this condition will never be violated at low frequency, where the uncertainty is small and the loop gain is large. At high frequency, |

GH | ∼ |GH| 1 + GH

226

10 Analysis and Synthesis in the Frequency Domain

and Equ.(10.21) is reduced to |GH| <

1 , lm (ω)

wherever ω is such that

lm (ω) 1

(10.22)

This condition expresses the gain stability. It is not conservative if the phase is totally unknown, as in the case of pole-zero ﬂipping. Multiplication of (10.20) by |GH|−1 gives an alternate form of the stability condition: |1 + (GH)−1 | ≥ lm (ω)

(10.23)

perturbed system

Additive ΔG

−1

G

Multiplicative G

ΔGH

1 + GH

1+ L

GH

Uncertainty circle Radius : H la (ω ) = GH lm (ω ) nominal system

Fig. 10.11 Nyquist plot of the nominal and the perturbed systems.

Now, if we want to achieve the performances with the perturbed system, Equ(10.14) must be replaced by |1 + (1 + L)GH| ≥ ps(ω) This equation will be satisﬁed if |1 + GH|(1 − lm ) ≥ ps(ω) A suﬃcient condition is |GH| ≥

ps(ω) 1 − lm (ω)

(10.24)

The design tradeoﬀ for |GH| is explained in Fig.10.12. The shaded region at low frequency is excluded for robust performance, and that at high frequency for robust stability. Considering the vicinity of the crossover frequency (Fig.10.13), we can make the following observations:

10.4 Feedback Speciﬁcation for SISO Systems

227

Sensor noise

w1 Robust performance Tracking error & disturbance rejection Stability robustness sensor noise rejection Fig. 10.12 Design tradeoﬀ for |GH|.

Fig. 10.13 Nyquist plot near crossover. Relation between the phase margin and the return diﬀerence at crossover.

• At crossover, |GH| = 1, and the stability robustness condition (10.20) becomes |1 + GH| ≥ lm (ωc ) (10.25) This means that accepting a magnitude error lm (ωc ) = 1 at crossover requires a phase margin of 600 . From this, one can anticipate that the

228

10 Analysis and Synthesis in the Frequency Domain

bandwidth of the closed-loop system cannot be much larger than the frequency where lm = 1. • At crossover, the return diﬀerence is related to the phase margin by |1 + GH| = 2 sin(

PM ) 2

(10.26)

This establishes a direct connection between the phase margin and the sensitivity and the disturbance rejection near crossover. The following section addresses the conﬂict between the quality of the loop near crossover (good phase margin) and the attenuation rate of GH.

10.5

Bode Gain-Phase Relationships

Figure 10.12 suggests that good feedback design could be achieved by having a large gain at low frequency and a fast enough decay rate at high frequency. Unfortunately, things are more complicated, because the closed-loop stability of the nominal system requires that the phase remains larger than −π as long as the gain is larger than 1. It turns out that for stable, minimum phase systems (i.e. with neither poles nor zeros in the right half plane), the phase angle and the amplitude are uniquely related. This relationship is expressed by the Bode Integrals. In this section, we shall state the main results without proof; the interested reader can refer to the original work of Bode, or to Horowitz or Lurie. •

Integral # 1

Consider the unity feedback with the stable, minimum phase open-loop transfer function G(jω) = |G|ejφ . If the amplitude diagram has a constant slope corresponding to n poles in a log-log diagram [n × (−20 dB)/decade], the phase is φ = −n 900 (10.27) In the general case, the phase at a frequency ω0 is given as a weighted average of the gain slope at all frequencies, but with a stronger weight in the vicinity of ω0 : 1 ∞ d ln |G| φ(ω0 ) = W (u)du (10.28) π −∞ du where u = ln(ω/ω0 ) and the weighting function W (u) is deﬁned by W (u) = ln[coth(|u|/2)] W (u) is strongly peaked near u = 0 (Problem 10.5); its behavior is not too far from that of a Dirac impulse, W (u) 12 π 2 δ(u), so that φ(ω0 )

π d ln |G| ]u=0 2 du

10.5 Bode Gain-Phase Relationships

229

This relation is very approximate, unless the slope of ln |G| is nearly constant in the vicinity of ω = ω0 ; it applies almost exactly if the slope is constant over two decades. In this case, one readily sees that −20 dB/decade ⇒

d ln |G| = −1 ⇒ φ = −π/2 du

d ln |G| = −2 ⇒ φ = −π du The ﬁrst integral indicates that a large phase can only be achieved if the gain attenuates slowly. It follows that the roll-oﬀ rate in the region near crossover must not exceed −40 dB/decade and it must often be smaller than this, in order to keep some phase margin. −40 dB/decade ⇒

•

Integral # 2

Assume G(s) is stable and has an asymptotic roll-oﬀ corresponding to more than one pole (n > 1 at inﬁnity). According to section 10.4.2, the sensitivity function S = (1 + G)−1 represents the fraction of the command r, or of the disturbance d, which is transmitted into the tracking error. |S| must be small in the frequency range where r or d are large. However |S| cannot be small everywhere, because the second Bode integral states that ∞ ln |S| dω = 0 (10.29) 0

This relation states that if |S| < 1 in one frequency band, there must be |S| > 1 in another frequency band; sensitivity can only be traded from one frequency band to another, and good disturbance rejection in some frequency range can be achieved only at the expense of making things worse than without feedback outside that frequency range (Problem 10.9). If the open-loop system has unstable poles pi , the second integral becomes ∞ ln |S|dω = π Re(pi ) (10.30) 0

i

where the sum extends to the unstable poles. This shows that for a system unstable in open-loop, the situation is worse, because there is more sensitivity increase than decrease; fast unstable poles are more harmful than slow ones because they contribute more to the right hand side of Equ.(10.30). •

Integral # 3

The third integral states that some reshaping of |G| can be performed within the working band (normalized to ω = 1) without aﬀecting the phase outside the working band. This arises from:

230

10 Analysis and Synthesis in the Frequency Domain

ω=1

(ln |G| − ln |G|∞ )d arcsin ω = − ω=0

1

∞

φ √ dω ω2 − 1

(10.31)

If the open-loop transfer function is altered in the working band, in such a way that the integral on the left hand side is unchanged (ln |G|∞ is the same for both transfer functions since only the working band is altered), the weighted phase average is preserved outside the working band. This situation is illustrated in Fig.10.14. G dB

G2 G1

log ω ω=1

working band

φ°

φ1

φ2 Fig. 10.14 Eﬀect of reshaping the open-loop transfer function in the working band, which preserves the phase distribution outside the working band.

•

Integral # 4

This integral says that the greater the phase lag, the larger will be the feedback in the working band: ∞ φ dω = π{ln |G|∞ − ln |G|0 } (10.32) −∞ ω In particular, if two loop shapes G1 and G2 have the same high frequency behavior, but G2 has a greater phase lag than G1 , G2 has a larger magnitude in the working band than G1 ∞ φ1 − φ2 G2 dω = π ln | |0 (10.33) ω G1 −∞ The third and fourth integrals tell us that, in order to achieve a large gain within the working band, the phase lag must be as large as possible outside

10.6 The Bode Ideal Cutoﬀ

231

the working band. The stability limitations on the phase lag are reﬂected in feedback limitations in the working band. The next section illustrates how these phase-magnitude relationships can be translated into design.

10.6

The Bode Ideal Cutoﬀ

The ﬁrst Bode integral tells us how the phase at one frequency is aﬀected by the gain slope in the vicinity (about one decade up and down). It also says that local phase increase can be achieved by lowering the gain slope (this situation is illustrated at high frequency in Fig.10.14). Figure 10.15 shows a Nichols chart of the desired behaviour: One wants to keep the feedback |G|0 constant and as large as possible within the working band (ω < 1 in reduced frequency), then reduce it while keeping gain and phase margins of x dB and yπ (shaded rectangle in Fig.10.15). It is therefore logical to keep the phase constant and compatible with the phase margin:

w=1

^

Fig. 10.15 Nichols chart of the two segment problem.

φˆ = (y − 1)π

(10.34)

The compensator design is thus reduced to that of enforcing an open-loop transfer function (system plus compensator) G(jω) to map the two segments of Fig.10.15; the result is shown in Fig.10.16: The open-loop transfer function with a constant gain in the working band and a constant phase lag φˆ outside the working band is

232

10 Analysis and Synthesis in the Frequency Domain

G(jω) =

√

|G|0

(10.35)

ˆ 2φ

[ 1 − ω 2 + jω] π

This transfer function has an ideal behavior at low frequency. However, the roll-oﬀ at high frequency is directly related to the phase margin. Improved high frequency noise attenuation requires higher roll-oﬀ. Because of the ﬁrst integral, we know that if we simply add one segment with a higher slope to the amplitude diagram of Fig.10.16, it will be reﬂected by an additional phase lag near crossover, which is incompatible with the stability margin. The cure to this is to have ﬁrst a ﬂat segment x dB below 0 dB, for approximately one or two octaves, followed by a segment with a higher slope, taking care of the sensor noise attenuation. The ﬂat segment at −x dB provides the extra phase lead near crossover which compensates for the extra phase lag associated with the higher rolloﬀ at high frequency. The ﬁnal design is sketched in Fig.10.17. The reader can compare it to the design tradeoﬀ of Fig.10.12. Once the loop transfer function GH has been obtained, that of the compensator can be deduced and approximated with a ﬁnite number of poles and zeros.

30 0

G (dB)

-30 -50 -90 -120 -150 0.01

0.1

1

10

0 -20 -40

f°

-60 -80

PM = 90°

-100

PM = 60°

-120 -140 0.01

0.1

1

10

Fig. 10.16 Bode plots of the two segment problem, for two values of the phase margin.

10.7 Non-minimum Phase Systems

233

-40(1-y) dB/ decade wc

w=1

x dB

working band

1-2 octaves

-20 n dB/ decade

^

Fig. 10.17 The Bode Ideal Cutoﬀ.

10.7

Non-minimum Phase Systems

In the previous sections, we assumed that G(s) has no singularity in the right half plane; such systems are called minimum phase (this section will explain why). The eﬀect of right half plane poles was brieﬂy examined in Equ.(10.30). In this section, we consider the eﬀect of right half plane zeros. To begin with, assume that G(s) has a single right half plane zero at s = a. We can write G(s) = G0 (s).A(s)

(10.36)

where A(s) is the all-pass function A(s) = −

s−a s+a

(10.37)

and G0 (s) is the minimum phase transfer function obtained by reﬂecting the right half plane zero into the left half plane [G0 (s) no longer has a singularity in the right half plane]. The Nyquist diagram of A(s) is represented in Fig.10.18; A(jω) follows the unit circle clockwise, from φ = 0 at ω = 0 to φ = −1800 at ω = ∞. A(s) takes its name from the fact that |A(ω)| = 1 for all ω; G(s) and G0 (s) have the same magnitude for all ω and the same phase at ω = 0, but for ω > 0, their phase diﬀer by that of A(ω); at ω = a/2, Δφ = −530 and at ω = a, Δφ = −900 . We have seen in section 6.7 that ﬂexible structures with non-collocated actuators and sensors do have non-minimum phase zeros. If they lie well outside the bandwidth of the system (ωc a), they do no harm, because

234

10 Analysis and Synthesis in the Frequency Domain

Im(s)

Im[A(s)]

Re(s) -a

Re[A(s)]

a

Fig. 10.18 Nyquist diagram of the all-pass function A(s) = − s−a . s+a

the corresponding all-pass function brings only very little phase lag near crossover. On the contrary, if a non-minimum phase zero lies at a distance comparable to the bandwidth, ωc ∼ a and φ[A(a)] = −π/2; this means that in the design, the phase angle of G0 (ω) cannot exceed −π/2 (without any phase margin) and, consequently, its falling rate cannot exceed −20 dB/decade instead of −40 dB/decade for a minimum phase system. The situation can easily be generalized to an arbitrary number of right half plane zeros. If G(s) has k right half plane zeros at ai , it can be written in the form (10.36) where G0 (s) is minimum phase and A(s) is the all-pass function A(s) = (−1)k

k ) s − ai i=1

s + ai

(10.38)

A typical pole-zero pattern of A(s) is shown in Fig.10.19. It is easy to see that |A(ω)| = 1 for all ω and that the phase angle decreases by kπ when ω goes from 0 to ∞,or 2kπ when ω goes from −∞ to ∞. This extra clockwise rotation about the origin, associated with the all-pass function, can change the number of encirclements of −1, especially if some zeros are close to the bandwidth; it is therefore decisive in stability considerations. The phase lag associated with the non-minimum phase transfer function G(jω) is always greater than that of G0 (jω), by that of A(jω). This is why G0 (jω) is called minimum phase, because no other stable function with the same amplitude characteristics can have any less phase lag. The design of a compensator for a non-minimum phase system can be done by considering the minimum phase system G0 (s) and modifying the design speciﬁcations to account for the extra phase lag generated by the all-pass function. As we have seen, this puts stronger restrictions on the attenuation rate near crossover to avoid instability (Problem 10.7).

10.8 Usual Compensators

235

Im(s)

Re(s)

Fig. 10.19 Pole-zero pattern of a typical all-pass function.

A pure time delay T has a transfer function D(s) = e−sT

(10.39)

Just as the all-pass function, it has a constant magnitude and a phase lag φ = ωT which increases linearly with the frequency. A pure delay can always be approximated as closely as needed by an all-pass network (Pad´e approximants), so that the above conclusions also apply (Problem 10.8).

10.8

Usual Compensators

Amongst the desired features of the closed-loop system are the stability properties, expressed by the gain and phase margins, the transient response characteristics, the bandwidth, the high frequency attenuation for sensor noise rejection and the ability to maintain the output at a desired value with minimum static error. We have seen in the previous sections that these requirements are interdependent and often conﬂicting. The Ideal Bode Cutoﬀ deﬁnes a reasonable compromise. The ease with which it can be implemented, however, depends very much on the system properties. This section brieﬂy reviews the most popular PD, Lead, Lag, PI, PID compensators. Before that, we deﬁne the system type which controls the steady state error and tracking ability.

10.8.1

System Type

The system type measures the ability of the control system to track polynomials. Consider a unity feedback (Fig.10.8) with the open-loop transfer function

y(s) Kn (1 − s/zi) = G(s) = n

(10.40) e(s) s (1 − s/pi )

236

10 Analysis and Synthesis in the Frequency Domain

The transfer function between the reference input r and the error e is

e(s) 1 sn (1 − s/pi )

= = n

(10.41) r(s) 1+G s (1 − s/pi ) + Kn (1 − s/zi ) According to the ﬁnal value theorem, the steady state error, e∞ , is given by e∞ = lim s e(s) = lim {

sn+1 r(s)} + Kn

s→0 sn

s→0

(10.42)

The number n of integrators in the open-loop transfer function is called the system type. If n = 0, the system is type 0: a constant actuating signal e maintains a constant output y. e∞ = lim { s→0

s r(s)} 1 + K0

If r(t) is a step function, r(s) = r0 /s and e∞ =

r0 , 1 + K0

y∞ =

K0 r0 1 + K0

(10.43)

The output follows a step input with a steady state error proportional to (1 + K0 )−1 . If there is one integrator in G(s), n = 1 and the system is type 1: a constant actuating signal e maintains a constant rate of change of the output. A step input is tracked with zero steady state error and a ramp input, r = ct, is tracked with a constant error e∞ = lim { s→0

s2 c c }= s + K1 s2 K1

(10.44)

Similarly, a system with two integrators in the open-loop transfer function is type 2: a constant actuating signal e maintains a constant second derivative of the output (structures with rigid body modes belong to this category); step and ramp inputs are tracked without steady state error and a parabolic input, r = at2 , produces a constant error e∞ = lim { s→0

s3 2a 2a }= 2 3 s + K2 s K2

(10.45)

Kn is called static error coeﬃcient of the system of type n. From Equ.(10.40), Kn = lim sn G(s) s→0

(10.46)

If a polynomial input tk is applied to a type n system, there is a zero steady state error if k < n. If k = n, there is a constant steady state error deﬁned

10.8 Usual Compensators

237

by the static error coeﬃcient Kn ; if k > n, the reference signal cannot be tracked. From the foregoing discussion, we note that long term errors associated with persistent disturbances can be zero only if the poles of the disturbance (one pole at s = 0 for a step, one double pole at s = 0 for a ramp, etc...) are included among the poles of the open-loop transfer function GH(s); this is called the internal model principle.

10.8.2

Lead Compensator

The aim of the cascade compensator, Fig.10.20, is to alter the open-loop transfer function from G(s) to G(s)H(s), in order to improve the characteristics of the closed-loop system. The simplest one is the Proportional plus Derivative (PD): H(s) = g(T s + 1) (10.47) This compensator increases the phase above the breakpoint 1/T . Therefore, selecting 1/T slightly below the crossover frequency of G(ω)H(ω) produces an increase in the phase margin. The major drawback of the PD compensator is that the compensation increases with frequency and the high frequency attenuation rate is reduced, which is undesirable if the sensor noise is signiﬁcant. The Lead compensator eliminates this drawback by adding a pole at higher frequency Ts + 1 H(s) = g α<1 (10.48) αT s + 1 The phase lead is still signiﬁcant but the high frequency ampliﬁcation is considerably reduced, Fig.10.21. The maximum phase lead occurs at ω=√

1 αT

(10.49)

Normally, in order to maximize the beneﬁt of the phase lead, α and T are chosen in such a way that the crossover frequency ωc lies between the zero and the pole of the compensator.

r

e

Compensator

System

H(s)

G(s)

-

Fig. 10.20 Cascade compensator.

y

238

10 Analysis and Synthesis in the Frequency Domain

Lead : H (s) = g

Ts 1 (a < 1) aTs + 1

PI : H ( s) =

g (s + 1 T ) I s

H

H

g

g 1

1

T

1

aT

wc

wc

90°

TI

0°

f

f

0°

-90° 1 aT

Lag : H ( s) = g H

g s

Ts 1 (a > 1) aTs + 1

PID : H (s) = ( s + 1 T ) (TD s + 1) I

H

g

1

aT

1

T

wc

1

TI

1

TD

wc

90°

0°

f

f

0°

-90°

-90°

Fig. 10.21 Frequency response of the most usual compensators. The range of ωc indicates the normal location of the crossover frequency of the uncompensated system.

10.8.3

PI Compensator

The aim of the Proportional plus Integral (PI) compensator is to increase the gain at low frequency to eliminate the static error. The compensator is H(s) =

g 1 (s + ) s TI

(10.50)

The system type is increased by 1. The ampliﬁcation at low frequency is obtained at the expense of a phase lag of −900 below the breakpoint 1/TI .

10.9 Multivariable Systems

239

In order to minimize the phase lag at crossover, the breakpoint must be substantially below the crossover frequency (ωc TI 1).

10.8.4

Lag Compensator

The analytical form of the Lag compensator is the same as that of the Lead compensator, except that the pole and zero are placed in reverse order H(s) = g

Ts + 1 αT s + 1

α>1

(10.51)

Normally, the pole and zero are placed at low frequency, to minimize the phase lag at crossover. The lag compensator can be used 1. to reduce the gain at high frequency without aﬀecting that at low frequency, in order to reduce the crossover frequency and increase the phase margin; 2. to increase the gain at low frequency without aﬀecting the high frequency behavior, in order to improve the steady state error characteristics.

10.8.5

PID Compensator

The Proportional plus Integral plus Derivative (PID) compensator consists of cascaded PI and PD compensators H(s) =

g 1 (s + )(TD s + 1) s TI

(10.52)

The integral part aims at reducing the static error, while the PD part increases the phase near crossover. Reasonable breakpoint frequencies are 20 1 ∼ < ωc TI TD

(10.53)

If the increasing behavior at high frequency is undesirable, the PD part can be replaced by a Lead compensator which achieves some phase lead without changing the asymptotic decay rate of G(ω)H(ω).

10.9

Multivariable Systems

So far, this chapter has been devoted to SISO systems; the analysis and design of multi-input multi-output (MIMO) systems is much more complicated and a comprehensive treatment of the subject is outside the scope of this introductory text (e.g. see Maciejowski). However, without entering too much into the details, it is interesting to point out some of the salient features of the theory which are particularly relevant in structural control, and their relation to the theory of SISO systems.

240

10 Analysis and Synthesis in the Frequency Domain

(a)

(b)

s(GH)

d r

e

u

H

lower bound for s(GH)

y

G

-

upper bound for s(GH)

n

0dB

w0

sensor noise rejection

w1

steady state error, disturbance rejection

Fig. 10.22 Feedback speciﬁcation of a MIMO system. (a) Block diagram. (b) Singular values of the open-loop transfer matrix GH.

10.9.1

Performance Speciﬁcation

Referring to Fig.10.22.a, the governing equations of the closed-loop system are y = GHe + d = GH(r − y − n) + d y = (I + GH)−1 GH(r − n) + (I + GH)−1 d

(10.54)

which generalizes (10.11). Introducing the sensitivity matrix : S = (I + GH)−1

(10.55)

and the closed-loop transfer matrix : F = (I + GH)−1 GH

(10.56)

y = Sd + F (r − n)

(10.57)

S+F =I

(10.58)

Equ.(10.54) reads It is easy to check that which means that the disturbance d and the noise n cannot be rejected simultaneously in the same frequency band. According to Equ.(10.57), the disturbance rejection is controlled by the size of the sensitivity matrix, measured by its singular values σ(S).1 If the maximum and minimum singular values, denoted respectively by σ ¯ (S) and σ(S) are close to each other, the situation is very similar to that of a SISO system; otherwise, what matters is the maximum singular value, σ ¯ (S), which limits the disturbance rejection capability of the system: good disturbance rejection requires that σ ¯ (S) be small within the bandwidth. 1

About Singular Value Decomposition (SVD), see (Strang).

10.9 Multivariable Systems

241

Similarly, good tracking requires that all the singular values of F be close to 1, σ ¯ (F ) 1, σ(F ) 1 in the bandwidth. Finally, noise rejection requires that the maximum singular value σ ¯ (F ) be small at high frequency. These conditions can be translated into conditions on the singular values of the open-loop transfer matrix, σ(GH), Fig.10.22.b.

10.9.2

Small Gain Theorem

The small gain theorem plays an important role in the analysis of robust stability of MIMO systems. Referring to Fig.10.23, if G1 (s) and G2 (s) are stable systems, a suﬃcient condition for closed-loop stability is the small gain condition: σ ¯ [G1 (jω)G2 (jω)] < 1 ∀ω>0 (10.59) An alternative, more conservative condition is ∀ω>0

σ ¯ [G1 (jω)]¯ σ [G2 (jω)] < 1

+

G 1(s) G2 (s)

e2

u2

+

e1

+

+

u1

(10.60)

Fig. 10.23 Small gain theorem.

10.9.3

Stability Robustness Tests

From the small gain theorem, it is possible to derive suﬃcient conditions for stability which have useful applications: Additive uncertainty Consider the perturbed unit feedback system with additive uncertainty of Fig.10.24.a. If one breaks the loop at z and v, one ﬁnds that z = −(I + G)−1 v and the block diagram can be recast into that of Fig.10.24.b. According to the small gain theorem, a suﬃcient condition for stability is that σ ¯ (ΔG). σ ¯ [(I + G)−1 ] < 1 ω>0 (10.61) or σ ¯ (ΔG).

1 <1 σ(I + G)

242

10 Analysis and Synthesis in the Frequency Domain

which is equivalent to σ ¯ (ΔG) < σ(I + G)

(a)

ω>0

(b)

DG z

DG

v

+ -

(10.62)

+

G

z

v - (I + G) -1

+

Fig. 10.24 (a) Additive uncertainty. (b) Seen from the break points z and v.

Multiplicative uncertainty Consider the perturbed unit feedback system with multiplicative uncertainty of Fig.10.25.a., or equivalently Fig.10.25.b. Upon breaking the loop at z and v, one ﬁnds z = −(I + G)−1 Gv and the block diagram can be redrawn as in Fig.10.25.c. According to the small gain theorem, a suﬃcient condition for stability is σ ¯ (L). σ ¯ [(I + G)−1 G] < 1 ω>0 (10.63) or σ ¯ (L).

1 σ[G−1 (I

+ G)]

<1

which is equivalent to σ ¯ (L) < σ(I + G−1 )

ω>0

(b)

(a) + -

+

I+L

G

(c)

(10.64)

L

z

+ +

-

v G

L

z

v -1

-(I + G) G Fig. 10.25 (a) and (b) Multiplicative uncertainty. (c) Seen from the break points z and v.

10.9 Multivariable Systems

10.9.4

243

Residual Dynamics

A frequent form of uncertainty encountered in structural control is the residual dynamics: the control model includes only the quasi-static response and the primary modes, G0 (s), and the residual (high frequency) modes are considered as uncertainty, GR (s), Fig.10.26. Because many controlled structures are lightly damped, the residual modes may be destabilized by the controller H(s); this phenomenon is known as spillover ; it will be studied in detail in the next chapter. In this section, we use the foregoing inequalities to establish a lower bound to the stability margin. The system of Fig.10.26 may be recast in the standard form of additive uncertainty by taking G = G0 H and ΔG = GR H. It follows that the stability robustness becomes: σ ¯ (GR H) < σ(I + G0 H)

ω>0

(10.65)

Alternatively, the system may also be recast in the standard form for multiplicative uncertainty by taking G = HG0 and L = G−1 0 GR . Accordingly, the stability robustness test (10.64) becomes −1 σ ¯ (G−1 ] 0 GR ) < σ[I + (HG0 )

ω>0

(10.66)

Note that these tests being suﬃcient conditions, they are both conservative, with diﬀerent degrees of conservatism. Typical representations of these tests are shown in Fig.10.27. In the inequality (10.66), the left hand side does not involve the controller at all; it measures the relative size of the contribution of the residual dynamics in the global response as a function of ω. The peaks in the uncertainty curves are at the frequencies of the residual modes and their amplitude is controlled by the damping. The minimum distance between the two curves may be regarded as a lower bound to the gain margin. Residual

(a)

(b) +

GR + -

+ Primary

DG

-

G

+

G = GO H

+

DG = GR H

+

GO H Control

(c) + -

I +L

G

G = HGO -1

L = GO GR

Fig. 10.26 (a) System with residual dynamics. (b) Additive uncertainty. (c) Multiplicative uncertainty.

244

10 Analysis and Synthesis in the Frequency Domain

(a) GM

10

A

s [ I + (HG0) -1] 1

0.1 structural uncertainty

0.01 (b)

-1 s [ G0 GR ]

f1

10

s [ I + (G0H) ]

1

GM A

0.1

s [ GR H ]

uncertainty 0.01

f1

0.01

0.1

1 [Hz]

10

Fig. 10.27 System with residual dynamics. (a) Robustness test based on multiplicative uncertainty. (b) Robustness test based on additive uncertainty. The minimum distance between the two curves may be regarded as a lower bound to the gain margin.

10.10

Problems

P.10.1 Consider an inverted pendulum with ω0 = 1 (Fig.9.2). (a) Compute the transfer function H(s) of the state feedback compensator such that the regulator and the observer poles are respectively at ωn = 2, ξ = 0.5 and αωn = 6, ξ = 0.5.

10.10 Problems

245

(b) Draw a root locus and the Nyquist diagram for this compensator. Using the Nyquist criterion, show that the closed-loop system is conditionally stable. Find the critical gains corresponding to the limits of stability. P.10.2 Draw the Nichols chart plot for the two-mass problem of section 6.3.2. P.10.3 Consider a direct velocity feedback applied to a structure with three ﬂexible modes at ω1 = 1 rad/s, ω2 = 3 rad/s and ω3 = 5 rad/s; the modal damping is uniform, ξi = 0.02. Assume that the actuator and sensor are noncollocated and that the uncertainty on the mode shapes is responsible for some uncertainty on the ﬁrst pair of zeros which vary from z1 = ±j2.8 rad/s to z1 = ±j3.2 rad/s. The second pair of zeros are ﬁxed to z2 = ±j4 rad/s. Plot the open-loop transfer function on a Nichols chart, for various locations of the uncertain zeros. Comment on the eﬀect of the zero on the stability margins. P.10.4 Consider a simply supported beam excited by a point force at xa = 0.2 l and provided with a displacement sensor at xs = 0.3 l. The nominal system is such that l = 1m, m = 1kg/m, EI = 10.266 10−3 N m2 and the modal damping is uniform ξi = 0.01. Compute an estimate of the transfer function G (ω). Next, consider the following modelling errors: (a) Modal truncation: the modal expansion (2.58) is truncated after 4, 6 and 10 modes. (b) Quasi-static correction: same as (a), but start from the result of Problem 2.5. (c) Parametric uncertainty: the model is based on EI = 11. 10−3N m2 and ξi = 0.02. (d) Errors in the mode shapes are simulated by using φ(xs ) = φ(0.25 l) in the model. For each situation, compute the transfer function of the model, G(ω), the additive and the multiplicative uncertainty, respectively ΔG(ω) and L(ω) (gain and phase). Comment on the adequacy of the unstructured uncertainty to represent the various situations. P.10.5 Plot the weighting function W (u) appearing in the ﬁrst Bode integral (10.28). P.10.6 Consider a SISO system with the following performance speciﬁcations: |G| > 30 dB |G| < −30 dB

ω < 1 rad/s ω > 10 rad/s

Using the ﬁrst Bode integral, discuss the feasibility of the design. P.10.7 Consider a non-minimum phase system with a real zero at a = 5ωc . Following the Bode Ideal Cutoﬀ, draw the magnitude and phase diagrams

246

10 Analysis and Synthesis in the Frequency Domain

for the corresponding minimum phase system G0 (s) which produce a gain margin of x dB and a phase margin of yπ for G(s). Do the same for a = 2ωc . P.10.8 Show that at low frequency, the nonrational function e−s can be approximated by the following all-pass functions of increasing orders: A1 (s) = A2 (s) = A3 (s) =

1 − s/2 1 + s/2

1 − s/2 + s2 /12 1 + s/2 + s2 /12

1 − s/2 + s2 /10 − s3 /120 1 + s/2 + s2 /10 + s3 /120

Ai (s) are called the Pad´e approximants of e−s (T s may be substituted for s to allow for any desired delay). For each case, draw the pole-zero pattern and the Bode plots. Compare the phase diagrams and comment on the domain of validity of the various approximations. [Hint: Expand the exponential and the all-pass functions into a McLauren series and match the coeﬃcients.] P.10.9 Consider a stable, minimum phase system G(s) with a high frequency attenuation rate larger than −20 dB/decade. Using a Nyquist diagram and geometrical arguments, show that there is always a limit frequency above which there is an ampliﬁcation of the disturbances.

11

Optimal Control

11.1

Introduction

The optimal control approach for SISO systems has already been introduced in chapter 9. In this chapter, we extend it to Multi-Input Multi-Output (MIMO) systems. The general results are stated without demonstration and the discussion is focused on the aspects which are important for the control of lightly damped ﬂexible structures.

11.2

Quadratic Integral

Consider the free response of an asymptotically stable linear system x˙ = Ax

(11.1)

The quadratic expression xT Qx is often used to measure the distance from the equilibrium, x = 0. Asymptotic stability implies that x → 0 as t → ∞, and that the quadratic integral ∞ J= xT Qx dt (11.2) 0

converges, for any semi positive deﬁnite weighting matrix Q. Its value depends on the initial state x0 alone. To compute J, consider the decay rate of the positive quadratic form V (t) = xT P x, where P is the positive deﬁnite solution of the matrix Lyapunov equation P A + AT P + Q = 0 −V˙ (t) = −

d T (x P x) = −xT P x˙ − x˙ T P x dt

A. Preumont: Vibration Control of Active Structures, SMIA 179, pp. 247–274. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

(11.3)

248

11 Optimal Control

Substituting x˙ = Ax and using Equ(11.3), one gets −

d T (x P x) = −xT (P A + AT P )x = xT Qx dt

(11.4)

Since Q is semi positive deﬁnite, this result proves that V (t) is a monotonically decaying function [it is indeed a Lyapunov function for the system (11.1)]. Integrating Equ.(11.4) from 0 to ∞ provides the desired result: ∞ J= xT Qx dt = xT0 P x0 (11.5) 0

where P is solution of the Lyapunov equation (11.3).

11.3

Deterministic LQR

We now formulate the steady state form of the Linear Quadratic Regulator (LQR) problem for MIMO systems. This is the simplest and the most frequently used formulation; it can be extended to a cost functional with ﬁnite horizon, and to time varying systems. Consider the system x˙ = Ax + Bu (11.6) where the system matrix A is not necessarily stable, but it is assumed that the pair (A, B) is controllable (controllability and observability will be discussed in detail in chapter 12). We seek a linear state feedback with constant gain u = −Gx such that the following quadratic cost functional is minimized ∞ J= (xT Qx + uT Ru)dt

(11.7)

(11.8)

0

where Q is semi positive deﬁnite (Q ≥ 0) and R is strictly positive deﬁnite (R > 0). J has two contributions, one from the states x and one from the control u. The fact that R is strictly positive deﬁnite expresses that any control has a cost, while Q ≥ 0 implies that some of the states may be irrelevant for the problem at hand. If a set of controlled variables is deﬁned, z = Hx a quadratic functional may be deﬁned as ∞ ∞ T T J= (z z + u Ru)dt = (xT H T Hx + uT Ru)dt 0

0

(11.9)

(11.10)

11.3 Deterministic LQR

249

This situation is equivalent to the previous one with Q = H T H. In a similar way, one often uses the form R = S T S where S is a scaling matrix and the scalar is a design parameter. It can be shown (see e.g. Kwakernaak & Sivan) that the solution of the problem (11.6) to (11.8) is G = R−1 B T P

(11.11)

where P is the symmetric positive deﬁnite solution of the algebraic Riccati equation (11.12) P A + AT P + Q − P BR−1 B T P = 0 The Riccati equation is nonlinear in P ; the existence and uniqueness of the solution is guaranteed if (A, B) is a controllable pair (stabilizable is in fact suﬃcient) and (A, Q1/2 ) is observable (by Q1/2 , we mean a matrix H such that H T H = Q). Under these conditions, the closed-loop system x˙ = (A − BG)x

(11.13)

is asymptotically stable. From Equ.(11.5), the minimum value of the cost functional (11.8) is given by ∞ xT (Q + GT RG) x dt = xT0 P ∗ x0 (11.14) J= 0

∗

where P is the solution of the Lyapunov equation of the closed-loop system P ∗ (A − BG) + (A − BG)T P ∗ + Q + GT RG = 0

(11.15)

Substituting G = R−1 B T P , we get P ∗ A − P ∗ BR−1 B T P + AT P ∗ − P BR−1 B T P ∗ + Q + P BR−1 B T P = 0 This equation is identically satisﬁed by P ∗ = P , where P is solution of Equ.(11.12). Thus, the minimum value of the cost functional is related to the solution P of the Riccati equation by ∞ J = min (xT Qx + uT Ru)dt = xT0 P x0 (11.16) 0

Many techniques are available for solving the Riccati equation; they will not be discussed in this text because they are automated in most control design softwares. In principle, the LQR approach allows the design of multivariable state feedbacks which are asymptotically stable. A major drawback is, of course, that it assumes the knowledge of the full state vector x. Since the latter is, in general, not directly available, it has to be reconstructed.

250

11 Optimal Control

The poles of the closed-loop system depend on the matrices Q and R. Multiplying both Q and R by a scalar coeﬃcient leads to the same gain matrix G and the same closed-loop poles. In structural control applications, if the controlled variables are not clearly identiﬁed, it may be sensible to choose Q in such a way that xT Qx represents the total (kinetic plus strain) energy in the system. Usually R will be chosen R = R1 , where R1 is a constant positive deﬁnite matrix and is an adjustable parameter; its value is selected to achieve reasonably fast closed-loop poles without excessive values of the control eﬀort.

11.4

Stochastic Response to a White Noise

A white noise is a mathematical idealization of a stationary random process for which there is a total lack of correlation between the values of the process at diﬀerent times. If w is a vector white noise process of zero mean E[w(t)] = 0, the covariance matrix reads E[w(t1 )wT (t2 )] = Rw (t1 − t2 ) = W1 δ(t1 − t2 ) = W1 δ(τ )

(11.17)

where δ(τ ) is the Dirac function, τ = t1 − t2 is the delay separating the two times t1 and t2 , and W1 is symmetric semi positive deﬁnite (W1 ≥ 0). W1 is called the covariance intensity matrix, but this is often abbreviated to covariance matrix. If the components of the random vector are independent, W1 is diagonal. The corresponding power spectral density matrix is ∞ 1 1 ΦW (ω) = Rw (τ )e−jωτ dτ = W1 (11.18) 2π −∞ 2π This result shows that the power in the signal is uniformly distributed over the frequency. Although a white noise is not physically realizable [because its variance would not be ﬁnite, see the Dirac function in Equ.(11.17)], it is a convenient approximation which is appropriate whenever the correlation time of the signal is small compared to the time constant of the system to which it is applied. In what follows, it is assumed that all processes are Gaussian, so that they are entirely characterized by their second order statistics (i.e. the covariance matrix or the power spectral density matrix). Consider the stationary random response of a linear time invariant system to a white noise excitation w of covariance intensity matrix W1 . x˙ = Ax + Dw

(11.19)

11.5 Stochastic LQR

251

It can be shown that the covariance matrix of the steady state response, X = E[x(t)xT (t)]

(11.20)

satisﬁes the Lyapunov equation AX + XAT + DW1 D T = 0

(11.21)

(see e.g. Bryson & Ho). This equation is the dual (adjoint) of Equ.(11.3); it expresses the equilibrium between the damping forces in the system and the random disturbance acting on it. It has a unique positive deﬁnite solution if the system is stable. Now, consider the quadratic performance index J = E[xT Qx]. Since aT b = tr(abT ) we can write J = E[xT Qx] = E{tr[QxxT ]} = tr[QE{xxT }]} = tr[QX]

(11.22)

where we have used the fact that the expected value applies only to random quantities. The covariance matrix X is the solution of Equ.(11.21). This result can be written alternatively J = tr[P DW1 DT ]

(11.23)

where P is solution of Equ.(11.3); the proof is left as an exercise (Problem 11.1).

11.4.1

Remark

The notation used in this section is the most commonly used in the literature; however, for the reader who is not trained in stochastic processes, it deserves some clariﬁcation: The deﬁnition (11.17) of the covariance matrix involves two separate times t1 and t2 ; for a stationary process, the covariance matrix is a function of the diﬀerence (delay) τ = t1 − t2 . On the contrary, the covariance matrix involved in Equ.(11.20) and (11.21) is not a function of τ ; X is the particular value of the covariance function for τ = 0; it is the generalization of the mean square value for a vector process.

11.5

Stochastic LQR

The linear quadratic regulator can be formulated in a stochastic environment as follows. Consider the linear time invariant system subjected to a white noise excitation of intensity W1 :

252

11 Optimal Control

x˙ = Ax + Bu + w1 ,

E[w1 w1T ] = W1

(11.24)

Find the control u, function of all information from the past, that minimizes the performance index J = E[xT Qx + uT Ru]

(11.25)

The solution of the problem is a constant gain linear state feedback u = −Gx

(11.26)

G = R−1 B T P

(11.27)

where P is solution of the Riccati equation P A + AT P + Q − P BR−1 B T P = 0

(11.28)

We note that the solution does not depend on the noise intensity matrix W1 and that it is identical to that of the deterministic LQR using the same weighting matrices Q and R. The closed-loop system behaves according to x˙ = (A − BG)x + w1

(11.29)

and the performance index (11.25) can be rewritten J = E[xT (Q + GT RG)x] From Equ.(11.23),

(11.30)

J = tr[P ∗ W1 ]

where P ∗ is solution of the Lyapunov equation of the closed-loop system P ∗ (A − BG) + (A − BG)T P ∗ + Q + GT RG = 0

(11.31)

This equation is identical to (11.15). Substituting the gain matrix from (11.27), we can readily establish that it is identically satisﬁed by the solution P of the Riccati equation (11.28). Thus, P ∗ = P , and the minimum value of the performance index is J = min E[xT Qx + uT Ru] = tr[P W1 ]

(11.32)

This result is the dual of (11.16) for the deterministic case.

11.6

Asymptotic Behavior of the Closed-Loop

In the particular case where there is a single controlled variable z and a single input u, the cost functional (11.10) is reduced to (9.52):

11.6 Asymptotic Behavior of the Closed-Loop

J=

∞

(z 2 + u2 )dt

253

(11.33)

0

We know from section 9.5 that the closed-loop poles are the stable roots of the characteristic equation

where

1 1 + G0 (s)G0 (−s) = 0

(11.34)

G0 (s) = H(sI − A)−1 B

(11.35)

is the transfer function between the input u and the controlled variable z = Hx. Only the part in the left half plane of the symmetric root locus is of interest. If

m (s − zi ) G0 (s) = k ni=1 (s − pi ) i=1 Equ.(11.34) becomes n )

(s − pi )(−s − pi ) +

i=1

m k2 ) (s − zi )(−s − zi ) = 0 i=1

(11.36)

For large values of , that is when there is a heavy penalty on the control cost, the closed-loop poles are identical to the stable poles pi of G0 (s), or their mirror image in the left half plane (−pi ) if they are unstable. Thus, the cheapest stabilizing control in terms of control amplitude simply relocates the unstable poles at their mirror image in the left half plane. n=1

n=2

n=4

n=3

n=5

45° 45°

60°

22°30'

36°

Fig. 11.1 Butterworth patterns of increasing order n.

As the penalty on the control cost decreases, m branches go from the poles to the left half plane zeros zi , if the system is minimum phase, or to their mirror image, if the zeros are located in the right half plane. The remaining n − m branches go to inﬁnity. Their asymptotic behavior can be evaluated by ignoring all but the highest power in s in the characteristic equation (11.34). From (11.36), we get

254

11 Optimal Control

1+

k 2 sm (−s)m =0 sn (−s)n

or s2(n−m) =

k2 (−1)n−m+1

(11.37)

The n − m stable solutions lie on a circle of radius (k 2 /)1/2(n−m) in a conﬁguration known as a Butterworth pattern of order n − m (Fig.11.1). For small values of , the fact that the system is very fast is reﬂected by the large distance of the faraway poles to the origin. Some of the poles, however, do not move away but are stuck at the open-loop zeros. Although they remain slow, they will not be noticeable on the controlled variable, because the openloop zeros are also zeros of the closed-loop system. Extensions to MIMO systems with square open-loop transfer matrix can be found in Kwakernaak & Sivan.

11.7

Prescribed Degree of Stability

In the previous section, we have seen that as → 0, some of the closed-loop poles go to inﬁnity according to Butterworth patterns, while the others go to the open-loop zeros. The latter depend on the choice of the matrix H deﬁning the controlled variables z. In practice, a single controlled variable cannot always be clearly identiﬁed, and the choice of H is often more or less arbitrary: The designer picks up a H matrix and ﬁnds a root locus, on which reasonable pole locations can be selected. If one fails to ﬁnd fast enough poles with acceptable control amplitudes, the process is repeated with another H matrix. It may require several iterations to avoid nearby poles. Anderson and Moore’s α−shif t procedure allows one to design a regulator with a prescribed degree of stability. The method guarantees that the closed-loop eigenvalues (of A − BG) lie on the left of a vertical line at −α (Fig.11.2).

−α

Fig. 11.2 Prescribed degree of stability.

11.8 Gain and Phase Margins of the LQR

The idea is to use a modiﬁed cost functional ∞ J= e2αt (xT Qx + uT Ru)dt

255

(11.38)

0

where α is a positive constant. If one deﬁnes the modiﬁed variables x ¯ = eαt x αt and u¯ = e u, the system equation in terms of x ¯ and u ¯ becomes x ¯˙ = (A + αI)¯ x + Bu ¯ and the performance index is J=

∞

(¯ xT Q¯ x+u ¯T R¯ u)dt

(11.39)

(11.40)

0

Equations (11.39) and (11.40) deﬁne a classical LQR problem in terms of x¯ and u¯; the solution is a constant gain feedback

with

u ¯ = −G¯ x

(11.41)

G = R−1 B T P¯

(11.42)

where P¯ is solution of the Riccati equation P¯ (A + αI) + (AT + αI)P¯ + Q − P¯ BR−1 B T P¯ = 0

(11.43)

The closed-loop system x x ¯˙ = (A + αI − BR−1 B T P¯ )¯ is asymptotically stable. If one back substitutes x ¯ and u ¯ in Equ.(11.41), one gets u = −Gx (11.44) Thus, the optimal control is a constant gain linear feedback. Besides, since ¯(t), and x¯(t) is asymptotically stable, we know that as t apx(t) = e−αt x proaches inﬁnity, x(t) → 0 at least as fast as e−αt . The prescribed degree of stability has been achieved by solving a classical LQR problem for the modiﬁed system with the system matrix A + αI instead of A.

11.8

Gain and Phase Margins of the LQR

The LQR has been developed as the solution of an optimization problem. The constant gain feedback assumes the perfect knowledge of the state. We momentarily stay with this assumption and postpone the discussion of the state reconstruction until the next section. In the previous chapter, we emphasized the role played by the gain and phase margins in the good performance of

256

11 Optimal Control

a control system and in the way the performance will be degraded if some change occurs in the open-loop system. A good phase margin improves the disturbance rejection near crossover, and eliminates the oscillations in the closed-loop system; it also protects against instability if some delay is introduced in the loop. This section establishes guaranteed margins for the LQR.

u -

x& = A x + B u

x

C

(sI − A )−1 B

-

G

G (a)

(b)

Fig. 11.3 (a) Classical representation of the state feedback. (b) Unity feedback representation.

Figure 11.3.a shows the classical representation of the state feedback; it is not directly suitable for deﬁning the gain and phase margins. However, the feedback loop can also be represented with a unity feedback as in Fig.11.3.b; the corresponding open-loop transfer function is G0 (s) = G(sI − A)−1 B

(11.45)

It is possible to draw a Nyquist plot of G0 (jω), and this allows us to deﬁne the gain and phase margins as in the previous chapter. The interesting result is that it can be demonstrated (see Anderson & Moore, p.68) that the LQR satisﬁes the following inequality. |1 + G(jω − A)−1 B| ≥ 1

(11.46)

This implies that the Nyquist plot of G0 (jω) always remains outside a disk of unit radius, centered on −1 (Fig.11.4). Simple geometric considerations show that the LQR has a phase margin P M > 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

282

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)

284

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)

286

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)

288

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).

290

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.

292

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.

338

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

340

14 Applications

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∗ ).

342

14 Applications

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).

344

14 Applications

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.

346

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,

348

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.

350

14 Applications

(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.

352

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).

354

14 Applications

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.

356

14 Applications

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

358

14 Applications

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

360

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)

362

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.

364

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

366

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)

368

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.

370

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).

372

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.

374

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).

376

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.

378

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).

380

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%.

382

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

386

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

388

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