Vibration Control of Active Structures (Solid Mechanics and Its Applications)

Vibration Control of Active Structures Edition

SOLID MECHANICS AND ITS APPLICATIONS Volume 96 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 a list of related mechanics titles, see final pages.

Vibration Control of Active Structures An Introduction Edition

by

ANDRÉ PREUMONT Université Libre de Bruxelles, Active Structures Laboratory, Brussels, Belgium

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” . . . le travail éloigne de nous trois grands maux: l'ennui, le vice et le besoin. ” Voltaire, Candide (XXX)

Contents xv

Preface to the second edition Preface to the first edition

xvii

1

Introduction 1.1 Active versus passive 1.2 Smart materials and structures 1.3 Control strategies 1.3.1 Feedback 1.3.2 Feedforward 1.4 The various steps of the design 1.5 Organization of the book 1.6 References 1.7 Problems

1 1 5 7 7 9 10 12 13 15

2

Some concepts of structural dynamics 2.1 Equation of motion of a discrete system 2.2 Vibration modes 2.3 Modal decomposition 2.3.1 Structure without rigid body modes 2.3.2 Structure with rigid body modes 2.3.3 Example 2.4 Transfer function of collocated systems 2.5 Continuous structures 2.6 Guyan reduction 2.7 References 2.8 Problems

17 17 18 20 20 22 25 26 30 31 33 34

3 Actuators, piezoelectric materials, and active structures 3.1 Introduction 3.2 Proof-mass actuator 3.3 Reaction wheels and gyrostabilizers vii

37 37 38 40

Vibration control of active structures

viii 3.4

3.5 3.6 3.7 3.8

3.9 3.10 3.11

Piezoelectric actuators 3.4.1 Constitutive equations 3.4.2 Linear actuator 3.4.3 Laminar actuator 3.4.4 Laminar sensor 3.4.5 Example: Tip displacement of a cantilever beam 3.4.6 Spatial modal filters Passive damping with piezoceramics Active cantilever beam Active truss Piezoelectric shell 3.8.1 Two-dimensional constitutive equations 3.8.2 Kirchhoff shell Finite element formulation References Problems

4 Collocated versus non-collocated control 4.1 Introduction 4.2 Pole-zero flipping 4.3 Collocated control 4.4 Non-collocated control 4.5 Notch filter 4.6 Pole-zero flipping in the structure 4.7 Effect on the Bode plots 4.8 Relation to the mode shapes 4.9 The role of damping 4.10 References 4.11 Problems 5 Active damping with collocated pairs 5.1 Introduction 5.2 Direct Velocity Feedback 5.2.1 Lead compensator 5.3 Acceleration feedback 5.3.1 Direct Velocity Feedback 5.3.2 Second order filter 5.3.3 SISO system with many modes 5.3.4 Multidimensional case 5.4 Positive Position Feedback 5.4.1 SISO system 5.4.2 Multidimensional case 5.5 Integral Force Feedback 5.5.1 Modal damping

40 41 46 48 51 52 53 54 55 58 63 64 65 68 70 73 75 75 76 78 80 80 82 84 84 87 88 88 91 91 93 94 96 97 97 98 100 101 101 102 103 104

CONTENTS

6

ix

5.6

Remarks 5.6.1 Controllability, observability 5.6.2 Actuator and sensor dynamics 5.7 References 5.8 Problems

107 107 107 109 110

Active vibration isolation 6.1 Introduction 6.2 Passive isolator 6.3 The “sky-hook” damper 6.4 Force feedback 6.5 Flexible clean body 6.5.1 Free-free beam with isolator 6.6 6 d.o.f. isolator 6.7 Decentralized control of the 6 d.o.f. isolator 6.7.1 Remarks 6.8 Pointing control 6.9 Vehicle suspension 6.10 References 6.11 Problems

113 113 114 116 118 119 121 124 127 128 129 131 133 134

7 State space approach 7.1 Introduction 7.2 State space description 7.2.1 Single degree of freedom oscillator 7.2.2 Flexible structure 7.2.3 Inverted pendulum 7.3 System transfer function 7.3.1 Poles and zeros 7.4 Pole placement by state feedback 7.4.1 Example: oscillator 7.5 Linear Quadratic Regulator 7.5.1 Symmetric root locus 7.5.2 Inverted pendulum 7.6 Observer design 7.7 Kalman Filter 7.7.1 Inverted pendulum 7.8 Reduced order observer 7.8.1 Oscillator 7.8.2 Inverted pendulum 7.9 Separation principle 7.10 Transfer function of the compensator 7.10.1 The two-mass problem 7.11 References

137 137 139 139 140 141 142 144 145 146 148 149 149 151 153 154 155 156 156 157 158 159 162

Vibration control of active structures

x

7.12 Problems

163

8 Analysis and synthesis in the frequency domain 8.1 Gain and phase margins 8.2 Nyquist criterion 8.2.1 Cauchy’s principle 8.2.2 Nyquist stability criterion 8.3 Nichols chart 8.4 Feedback specification for SISO systems 8.4.1 Sensitivity 8.4.2 Tracking error 8.4.3 Performance specification 8.4.4 Unstructured uncertainty 8.4.5 Robust performance and robust stability 8.5 Bode gain-phase relationships 8.6 The Bode Ideal Cutoff 8.7 Non-minimum phase systems 8.8 Usual compensators 8.8.1 System type 8.8.2 Lead compensator 8.8.3 PI compensator 8.8.4 Lag compensator 8.8.5 PID compensator 8.9 References 8.10 Problems

165 165 166 166 167 170 171 171 172 173 174 175 178 181 183 185 185 187 187 189 189 189 190

9 Optimal control 9.1 Introduction 9.2 Quadratic integral 9.3 Deterministic LQR 9.4 Stochastic response to a white noise 9.4.1 Remark 9.5 Stochastic LQR 9.6 Asymptotic behaviour of the closed-loop 9.7 Prescribed degree of stability 9.8 Gain and phase margins of the LQR 9.9 Full state observer 9.9.1 Covariance of the reconstruction error 9.10 Kalman-Bucy Filter (KBF) 9.11 Linear Quadratic Gaussian (LQG) 9.12 Duality 9.13 Spillover 9.13.1 Spillover reduction 9.14 Loop Transfer Recovery (LTR)

193 193 193 194 196 197 197 198 200 201 202 204 204 205 205 206 209 210

CONTENTS

xi

9.15 Integral control with state feedback 9.16 Frequency shaping 9.16.1 Frequency-shaped cost functionals 9.16.2 Noise model 9.17 References 9.18 Problems

211 212 212 215 216 217

10 Controllability and Observability 10.1 Introduction 10.1.1 Definitions 10.2 Controllability and observability matrices 10.3 Examples 10.3.1 A cart with two inverted pendulums 10.3.2 Double inverted pendulum 10.3.3 Two d.o.f. oscillator 10.4 State transformation 10.4.1 Control canonical form 10.4.2 Left and right eigenvectors 10.4.3 Diagonal form 10.5 PBH test 10.6 Residues 10.7 Example 10.8 Sensitivity 10.9 Controllability and observability Gramians 10.10 Relative controllability and observability 10.10.1 Internally balanced coordinates 10.11 Model reduction 10.11.1 Transfer equivalent realization 10.11.2 Internally balanced realization 10.11.3 Example 10.12 References 10.13 Problems

221 221 222 222 224 224 226 226 227 228 229 230 230 231 232 233 234 235 235 237 237 237 238 240 241

11 Stability 11.1 Introduction 11.1.1 Phase portrait 11.2 Linear systems 11.2.1 Routh-Hurwitz criterion 11.3 Liapunov’s direct method 11.3.1 Introductory example 11.3.2 Stability theorem 11.3.3 Asymptotic stability theorem 11.3.4 Lasalle’s theorem 11.3.5 Geometric interpretation

245 245 246 247 248 249 249 249 251 251 252

Vibration control of active structures

xii

11.4 11.5 11.6 11.7 11.8 11.9

11.3.6 Instability theorem Liapunov functions for linear systems Liapunov’s indirect method An application to controller design Energy absorbing controls References Problems

253 254 255 256 257 259 259

12 Semi-active control 12.1 Introduction 12.2 Magneto-rheological (MR) fluids 12.3 MR devices 12.4 Semi-active control 12.5 Open-loop control 12.6 Feedback control 12.6.1 Continuous control 12.6.2 On-off control 12.6.3 Force feedback 12.7 References 12.8 Problems

263 263 264 266 267 268 268 268 273 274 275 275

13 Applications 13.1 Digital implementation 13.1.1 Sampling, aliasing and prefiltering 13.1.2 Zero-order hold, computational delay 13.1.3 Quantization 13.1.4 Discretization of a continuous controller 13.2 Active damping of a truss structure 13.2.1 Actuator placement 13.2.2 Implementation, experimental results 13.3 Active damping generic interface 13.3.1 Active damping 13.3.2 Experiment 13.3.3 Pointing and position control 13.4 Active damping of a plate 13.4.1 Control design 13.5 Active damping of a stiff beam 13.5.1 System design 13.6 The HAC/LAC strategy 13.6.1 Wide-band position control 13.6.2 Compensator design 13.6.3 Results 13.7 Volume displacement sensors 13.7.1 QWSIS sensor

277 277 278 279 280 281 282 282 284 286 287 288 290 290 291 293 294 295 297 299 299 302 303

CONTENTS 13.7.2 Discrete array sensor 13.7.3 Spatial aliasing 13.7.4 Distributed sensor 13.8 References 13.9 Problems 14 Tendon Control of Cable Structures 14.1 Introduction 14.2 Tendon control of strings and cables 14.3 Active damping strategy 14.4 Basic Experiment 14.5 Approximate linear theory 14.6 Application to space structures 14.6.1 Guyed truss experiment 14.6.2 JPL-MPI testbed 14.6.3 Free floating truss experiment 14.6.4 Microvibrations 14.7 Application to cable-stayed bridges 14.7.1 Laboratory experiment 14.7.2 Control of parametric resonance 14.7.3 Large scale experiment 14.8 References

xiii 305 308 309 316 319

321 321 323 324 325 327 329 329 331 331 334 335 335 336 338 344

Bibliography

347

Index

361

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 cable-structures. 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 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 Frédéric Bossens for the cable-structures, Vincent Piefort for the modelling of piezoelectric structures, Pierre De Man and Arnaud François in vibroacoustics, Ahmed Abu Hanieh and 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 xv

Vibration control of active structures

xvi

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. André Preumont Brussels, November 2001.

Preface to the first edition

I was introduced to structural control by Raphaël 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 Liège. 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, 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. André Preumont Bruxelles, July 1996. xvii

Chapter 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. 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. Recently‚ it has been demonstrated that 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; the time slice required to perform the computations is so small that the atmospheric turbulence can be considered as frozen. Prototypes of such control systems have recently been tested successfully for ground-based astronomy. The foregoing example is not the only one where active structures have 1

2

Vibration control of active structures

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 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 La Silla, Chile, consists of having a relatively flexible primary mirror connected at the back to a set of a hundred or so actuators (e.g. Wilson et al.). As in the previous example, the control system uses an image analyser 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 and the optical mode amplitudes of the wave front changes can be determined experimentally with the image analyser:

J is a rectangular matrix‚ because the number of actuators is larger than the number of optical modes of interest. Once the modal errors have been

Introduction

3

evaluated‚ the correcting forces can be calculated from

where is the pseudo-inverse of the rectangular matrix J. This is the minimum norm solution to Equ.(l.l) (Problem P. 1.1). The second alternative‚ adopted on the Ten-Meter Telescope (TMT) of 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 three computer controlled degrees of freedom (2 pointing and 1 piston); the control system is used to achieve the optical quality of a monolithic mirror‚ to compensate for wind disturbances and minimize the impact of the telescope dynamics on the optical performance (Aubrun et al.). As a third example‚ also related to astronomy‚ consider the future interferometric missions planned by NASA or ESA (one such a mission‚ called ”Terrestrial Planet Finder” aims at detecting earth-sized planets outside the solar system; other missions include the mapping of the sky with an accuracy one order better than that achieved by Hypparcos). 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 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 cannot be achieved with a precision monolithic structure‚ but rather by active means as suggested in Fig.1.2. Let us first consider the supporting truss: given its size and environment‚ 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 (perhaps as far as the orbit of Jupiter) to ensure maximum sensitivity; this makes the weight issue particularly important. Another interesting subsystem necessary to achieve the stringent specifications is the six d.o.f. vibration isolator at the interface between the attitude

4

Vibration control of active structures

Introduction

5

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. The same general purpose vibration isolator may be used at the interface between the truss and the independent telescopes; in this case however‚ its vibration isolation capability is combined with the steering (pointing) 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 path length between every telescope and the beam combiner‚ so that these distances are kept identical to a fraction of the wavelength (e.g. ). 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. In recent years‚ there has been a growing interest in semi-active control‚ particularly for vehicle suspensions; this has been driven by the reduced cost as compared to active control‚ due mainly to the absence of a large power actuator. A semi-active device can be broadly defined as a passive device in which the properties (stiffness‚ damping‚ ...) can be varied in real time with a low power input. Although they behave in a strongly nonlinear way‚ semi-active devices are inherently passive and‚ unlike active devices‚ cannot destabilize the system; they are also less vulnerable to power failure. Semi-active suspension devices may be based on classical viscous dampers with a variable orifice‚ or on magneto-rheological (MR) fluids.

1.2

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 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

6

Vibration control of active structures

(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, SMAs 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 the NITINOL; SMAs are little used in vibration control, and will not be discussed in this book. 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 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 (PV DF or 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 The best known piezoceramic is the Lead Zirconate Titanate (PZT).

Introduction

7

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. Magneto-rheological (MR) fluids consists of viscous fluids containing micron-sized 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 behaviour under electrical field; they are called electro-rheological (ER) fluids; however‚ their performances (limited by the electric field breakdown) are significantly 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 field‚ one order of magnitude more than the piezoceramics. 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. We will see that piezopolymers can be shaped to react only to a limited set of vibration modes (modal filters).

1.3

Control strategies

There are two radically different approaches to disturbance rejection: feedback and feedforward. Although this text is entirely devoted to feedback 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.

1.3.1

Feedback

The principle of feedback is represented in Fig.1.4; the output is compared to the reference input ‚ and the error signal‚

of the system is passed

8

Vibration control of active structures

into a compensator and applied to the system The design problem consists of finding the appropriate compensator such that the closed-loop system is stable and behaves in the appropriate manner. 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

(Problem P.1.2)‚ this requires 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 (a position‚ or the pointing of an antenna) to a desired value in spite of external disturbances in some frequency range. From the previous formula and

we readily see that this requires large values of GH in the frequency range where is sought. implies that the closed-loop transfer function is close to 1, which means that the output tracks the input 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 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:

Introduction

9

The bandwidth of the control system is limited by the accuracy of the model; there is always some destabilization of the flexible modes outside (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 must always be two orders of magnitude larger than to preserve reasonably the behaviour of the continuous system. This puts some hardware restrictions on the bandwidth of the control system.

1.3.2

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 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

10

Vibration control of active structures

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 narrowband 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 ); 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.

1.4

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. 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. If an active

Introduction

11

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 behaviour 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 behaviour. At this point‚ if the 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

12

Vibration control of active structures

affect the behaviour 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.

1.5

Organization of the book

Structural control and the emerging area of smart structures belong to the general field of Mechatronics; they consist of a mixture of mechanical engineering‚

Introduction

13

structural mechanics‚ control engineering‚ material science and computer science. This book has been written for structural engineers willing to acquire some background in structural control; it has been assumed that the reader is familiar with structural dynamics and has some basic knowledge of linear system theory‚ including Laplace transforms‚ 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 is also assumed in signal processing. Chapter 2 recalls briefly some concepts of structural dynamics; chapter 3 considers the modelling of some active structures; chapters 4 to 9 are mostly devoted to the design of single input single output (SISO) compensators with an increasing degree of complexity‚ both in the frequency and in the time domain (state space); this part includes active damping with collocated actuator/sensor pairs (chap.5) and active isolation with decentralized controllers (chap.6). Chapter 10 is devoted to controllability and observability; chapter 11 discusses stability; chapter 12 discusses the semi-active control; chapter 13 describes some applications and discusses briefly some issues in transforming a continuous controller into a digital one. Chapter 14 is entirely devoted to the control of cable structures. Each chapter is supplemented by a set of problems; it is assumed that the reader is familiar with some computer aided control engineering software such as MATLAB or SIMULINK. Chapters 1 to 8‚ part of chapter 9 and some applications of chapter 13 could constitute a one semester graduate course in structural control.

1.6

References

J. N. A UBRUN ‚ K. R. L ORELL ‚ T. W. H AVAS & W. C. H ENNINGER ‚ Performance Analysis of the Segment Alignment Control System for the Ten-Meter Telescope‚ Automatica‚ vol. 24‚ no. 4‚ 437–453‚ 1988. R. H. CANNON‚ Dynamics of Physical Systems‚ McGraw-Hill‚ 1967. G. F. FRANKLIN‚ J. D. POWELL & A. EMANI-NAEMI‚ Feedback Control of Dynamic Systems‚ Addison-Wesley‚ 1986. C. R. FULLER‚ S. J. ELLIOTT & P. A. NELSON‚ Active Control of Vibration‚ Academic Press‚ 1996. M. V. GANDHI & B. S. THOMPSON‚ Smart Materials and Structures‚ Chapman & Hall‚ 1992. P. A. NELSON & S. J. ELLIOTT‚ Active Control of Sound‚ Academic Press‚ 1992. K. UCHINO‚ Ferroelectric Devices‚ Marcel Dekker‚ 2000.

14

Vibration control of active structures

R. N. WILSON‚ F. FRANZA & L. NOETHE‚ Active Optics. 1. A System for Optimizing the Optical Quality and Reducing the Costs of Large Telescopes‚ Journal of Modern Optics‚ vol. 34‚ no. 4‚ 485–509‚ 1987. General literature on control of flexible structures R. L. CLARK‚ W. R. SAUNDERS & G. P. GIBBS‚ Adaptive Structures‚ Dynamics and Control‚ Wiley‚ New York‚ NY‚ 1998. W. K. GAWRONSKI‚ Dynamics and Control of Structures - A Modal Approach‚ Springer‚ 1998. C. H. HANSEN & S. D. SNYDER‚ Active Control of Sound and Vibration‚ E&FN Spon‚ London‚ 1996. D. C. HYLAND‚ J. L. JUNKINS & R. W. LONGMAN‚ Active Control Technology for Large Space Structures‚ J. of Guidance‚ vol. 16‚ no. 5‚ 801–821‚ Sep.-Oct. 1993. D. J. INMAN‚ Vibration‚ with Control‚ Measurement‚ and Stability‚ PrenticeHall‚ 1989. H. JANOCHA‚ editor‚ Adaptronics and Smart Structures (Basics‚ Materials‚ Design and Applications)‚ Springer‚ 1999. S. M. JOHSI‚ Control of Large Flexible Space Structures‚ Lecture Notes in Control and Information Sciences‚ Springer-Verlag‚ vol. 131‚ 1989. J. L. JUNKINS‚ editor‚ Mechanics and Control of Large Flexible Structures‚ vol. 129‚ 1990. J. L. JUNKINS & Y. KIM‚ Introduction to Dynamics and Control of Flexible Structures‚ AIAA Education Series‚ 1993. L. MEIROVITCH‚ Dynamics and Control of Structures‚ Wiley‚ 1990. D. K. MIU‚ Mechatronics - Electromechanics and Contromechanics‚ SpringerVerlag‚ 1993. R. E. SKELTON‚ Dynamic System Control - Linear System Analysis and Synthesis‚ Wiley‚ 1988. D. W. SPARKS JR & J. N. JUANG‚ Survey of Experiments and Experimental Facilities for Control of Flexible Structures‚ AIAA J.of Guidance‚ vol. 15‚ no. 4‚ 801–816‚ July–Aug. 1992.

Introduction

1.7

Problems

P.1.1

Consider the underdeterminate system of equations

15

Show that the minimum norm solution‚ i.e. the solution of the minimization problem such that is

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 and the disturbance to the output are respectively

P. 1.3 Based on your own experience‚ describe one application in which you feel an active structure may outclass a passive one; suggest a configuration for the actuators and sensors.

Chapter 2

Some concepts of structural dynamics 2.1

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:

or, in matrix form,

The general form of the equation of motion governing the dynamic equilibrium between the external, elastic, inertia and damping forces acting on a non-gyroscopic, discrete, flexible structure with a finite number of degrees of freedom is where and are the vectors of generalized displacements (translations and rotations) and forces (point forces and torques) and M, K and C are respectively 17

Vibration control of active structures

18

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 P.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:

The coefficients

2.2

and

are selected to fit the structure under consideration.

Vibration modes

Consider the free response of a undamped (conservative) system of order is governed by If one tries a solution of the form eigenvalue problem

and

It

must satisfy the

Because M and K are symmetric, K is positive semi definite and M is positive definite, the eigenvalue must be real and non negative. is the natural frequency and is the corresponding mode shape; the number of modes is equal to the number of degrees of freedom, 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

Some concepts of structural dynamics

19

From Equ.(2.5), we see that if the structure is released from initial conditions and it oscillates at the frequency according to always keeping the shape of mode Left multiplying Equ.(2.5) by one gets the scalar equation

and, upon permuting

and

we get similarly,

If we substract these equations, taking into account that a scalar is equal to its transpose and that K and M are symmetric, we get

which shows that the mode shapes corresponding to distinct natural frequencies are orthogonal with respect to the mass matrix.

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

where is the Kronecker delta index is the modal mass of mode Since the mode shapes can be scaled arbitrarily, it is usual to normalize them in such a way that If one defines the matrix of the mode shapes the orthogonality relationships read

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 satisfied. A rigid body mode is such that there is no strain energy associated with it It can be demonstrated that this implies that the rigid body modes can therefore be regarded as solutions of the eigenvalue problem (2.5) with

Vibration control of active structures

20

2.3 2.3.1

Modal decomposition Structure without rigid body modes

Let us perform a change of variables from physical coordinates coordinates according to where

to modal

is the vector of modal amplitudes. Substituting into Equ.(2.2), we get

Left multiplying by we obtain

and using the orthogonality relationships (2.8) and (2.9),

If the matrix is diagonal, the damping is said classical or normal. In this case, the modal fraction of critical damping (in short modal damping) is defined by One can readily check that the Rayleigh damping (2.3) complies with this condition and that the corresponding modal damping ratios are

The two free parameters and 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 with the notations

The following values of the modal damping ratio can be regarded as typical: satellites and space structures are generally very lightly damped 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 most dissipation takes place in the joints, and the damping increases with the strain level. For civil engineering applications, is typical and, when

Some concepts of structural dynamics

21

radiation damping through the ground is involved, it may reach 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 bandlimited excitation, its response is dominated by the normal 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. Now, let us consider the steady state response to a harmonic excitation The response will also be harmonic with the same frequency, substituting into Equ.(2.2), we find that the complex amplitudes of F and X are related by The matrix is the dynamic generalization of the flexibility matrix and is called the dynamic flexibility matrix. Equ.(2.16) can be obtained alternatively by Fourier transformation of Equ.(2.2). If one considers the same problem in modal coordinates, the modal amplitudes will also be harmonic, and, from Equ.(2.14), we obtain

From Equ.(2.10),

Comparing Equ.(2.16) and (2.18), we obtain the modal expansion of the dynamic flexibility matrix:

where the sum extends to all the modes. expresses the complex amplitude of the structural response of degree of freedom when the structure is

22

Vibration control of active structures

exposed to a steady state harmonic excitation at degree of freedom the structure has no rigid body modes, we can write Equ.(2.19) for obtain the modal expansion of the static flexibility matrix:

If we

For a limited frequency band if we select in such a way that the dynamic expansion can be split into the contribution of the low frequency modes which respond dynamically, and that of the high frequency modes which respond statically:

Using Equ.(2.20), we can transform the foregoing result in such a way that the high frequency modes do not appear explicitly in the expansion:

The static contribution of the high frequency modes to the flexibility matrix is often called the residual mode; we shall denote it R. It is independent of the frequency and introduces a feedthrough component in the transfer matrix: part of the output is proportional to the input. In terms of Laplace transform, the numerator and the denominator of some components of have the same power in (such a transfer function is said to be not strictly proper). We shall see later that truncating the modal expansion of a transfer function without introducing a residual mode can lead to substantial errors in the prediction of the open-loop zeros (e.g. see Fig.3.15) and, as a result, of the performance of a control system.

2.3.2

Structure with rigid body modes

The approximation (2.22) applies only at low frequency, If the structure has rigid body modes, the first sum can be split into rigid and flexible modes; however, the residual mode cannot be used any more, because no longer exists. This problem can be solved in the following way. The displacements are partitioned into their rigid and flexible contributions according to where and are the matrices whose columns are the rigid body modes and the flexible modes, respectively. Assuming no damping, to make things formally

Some concepts of structural dynamics

23

simpler, and taking into account that the rigid body modes satisfy we obtain the equation of motion

Left multiplying by and using the orthogonality relations (2.6) and (2.7), we see that the rigid body modes are governed by

or Substituting this result into Equ.(2.24), we get

or where we have defined the projection matrix

such that check that

is orthogonal to the rigid body modes. In fact, we can easily

24

Vibration control of active structures

P can therefore be regarded as a filter which leaves unchanged the flexible modes and destroys 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

Since the solution may contain an arbitrary contribution from the rigid body modes. On the other hand, 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.2.2). The modified system is statically determinate and its stiffness matrix can be inverted. If we denote by the flexibility matrix of the modified system, the general solution of (2.30) is

where is a vector of arbitrary constants. The contribution of the rigid body modes can be eliminated with the projection matrix P, leading to

is the pseudo-static flexibility matrix of the flexible modes. On the other hand, left multiplying Equ.(2.24) by we get

where the diagonal matrix is regular. It follows that the pseudo-static deflection can be written alternatively

Comparing with Equ.(2.31), we get

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:

Some concepts of structural dynamics

25

where the contribution from the residual mode is

Note that 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 but they differ only by a contribution of the rigid body modes, which is destroyed by the projection matrix P, leading to the same Let us illustrate the procedure by an example.

2.3.3 Example Consider the system of three identical masses of Fig.2.3. There is one rigid body mode and two flexible ones:

and From Equ.(2.27), the projection matrix is

Vibration control of active structures

26

or

We can readily check that

and the self-equilibrated loads associated with a force (Fig.2.3.a)

applied to mass 1 is

If we impose the statically determinate constraint on mass 1 (Fig.2.3.b), the resulting flexibility matrix is

leading to

The reader can easily check that other dummy constraints would lead to the same pseudo-static flexibility matrix (Problem P.2.3).

2.4

Transfer function of collocated systems

Consider a diagonal component of the dynamic flexibility matrix of a undamped system. It is the transfer function between the generalized force (point force or torque) and the response of the corresponding degree of freedom (respectively displacement or angle). If the force is generated by an actuator, and the displacement is measured by a sensor, the actuator and sensor are collocated. We shall see later that control systems using collocated actuator/sensor pairs usually enjoy special properties which are the consequence of alternating poles and zeros along the imaginary axis. To establish this property, consider the diagonal term of Equ.(2.34):

Some concepts of structural dynamics

27

is real since the system is undamped, and all the numerators in the modal expansion are positive. As a result, it is readily demonstrated that

The behaviour of is represented in Fig.2.5: The amplitude of the transfer function goes to at the resonance frequencies (corresponding to a pair of purely imaginary poles in the system). In every interval between consecutive resonance frequencies, there is a frequency where the amplitude of the frequency response function (FRF) vanishes; in structural dynamics, these frequencies are called anti-resonance frequencies; they correspond to purely imaginary transmission zeros in the open-loop transfer function It is worth mentioning that A harmonic excitation at an anti-resonance frequency produces no response at the degree of freedom where the excitation is applied; the structure behaves as if an additional restraint had been added. The antiresonance frequencies are in fact identical to the resonance frequencies of the system with the additional restraint (Fig.2.4), and the acting force is identical to the reaction force in the additional restraint during the free vibration of the modified system. In contrast to the resonance frequencies, the anti-resonance frequencies do depend on the actuator location. If the diagram of another diagonal component is examined, the frequencies where the plot goes to (resonances) are unchanged, but the frequencies where do change (as do the natural frequencies of the modified system when we change the location of the additional restraint). In all cases, however, there will be one and only one anti-resonance between two consecutive resonances.

28

Vibration control of active structures

Observing Fig.2.5 and Equ.(2.36), we see that neglecting the residual mode is equivalent to moving the downwards by Doing that, all the anti-resonance frequencies tend to be overestimated by an amount which depends on the slope of in the vicinity of the zeros. We shall see later that errors in the spacing between poles and zeros of the open-loop system can lead to substantial errors in the performance of the closed-loop system. For a system with non-collocated actuator and sensor, the numerators of the various terms in the modal expansion of become they can be either positive or negative, and the property (2.37) is lost. Interlacing poles and zeros are no longer guaranteed in this case. In system theory, it is customary to write the system transfer functions in the form

In the present case, the undamped collocated system has alternating imaginary poles and zeros at and (Fig.2.6.a). The transfer function reads

If some damping is added, the poles and zeros are slightly moved into the left half plane as indicated in Fig.2.6.b, without changing the dominant feature of

Some concepts of structural dynamics

29

interlacing. A collocated system always exhibits Bode and Nyquist plots similar to those represented in Fig.2.7. Each flexible mode introduces a circle in the Nyquist diagram; it is more or less centered on the imaginary axis which is intersected at and the radius is proportional to the inverse of the modal damping, In the Bode plots, a 180° phase lag occurs at every natural frequency, and is compensated by a 180° phase lead at every imaginary zero; the phase always oscillates between 0 and , as a result of the interlacing property of the poles and zeros.

30

2.5

Vibration control of active structures

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 This equation is based on the Euler-Bernouilli 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, is the mass per unit length and the distributed external load per unit length. If the beam is uniform, the free vibration is governed by

The boundary conditions depend on the support configuration: a simple support implies and (no displacement, no bending moment); for a clamped end, we have and (no displacement, no rotation); a free end corresponds to and (no bending moment, no shear), etc... A harmonic solution of the form can be obtained if and satisfy

with the appropriate boundary conditions. This equation defines a eigenvalue problem; the solution consists of the natural frequencies (infinite in number) and the corresponding mode shapes The eigenvalues are tabulated for various boundary conditions in textbooks on mechanical vibrations (e.g. Geradin & Rixen, p. 187). For the pinned-pinned case, the natural frequencies and mode shapes are

Just as for discrete systems, the mode shapes are orthogonal with respect to the mass and stiffness distribution:

Some concepts of structural dynamics

31

The generalized mass corresponding to Equ.(2.44) is As with discrete structures, the frequency response function between a point force actuator at and a displacement sensor at is

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 P.2.5).

2.6

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 (2.2) by condensing the degrees of freedom with little or no inertia and which are not excited by external forces. The degrees of freedom to be condensed, denoted in what follows, are often referred to as slaves; those kept in the reduced model are called masters and are denoted To begin with, consider the undamped forced vibration of a structure where the slaves are not excited and have no inertia; the governing equation is

or

According to the second equation, the slaves the masters :

are completely determined by

Substituting into Equ.(2.49), we find the reduced equation

which involves only Note that in this case, the reduced equation has been obtained without approximation.

32

Vibration control of active structures

The idea in this, so-called, Guyan reduction is to assume that the masterslave relationship (2.51) applies even if have some inertia (sub-matrix not identically 0) or applied forces. We assume the following transformation

The reduced mass and stiffness matrices are obtained by substituting the above transformation into the kinetic and strain energy:

with The second equation produces as in Equ.(2.52). If external loads are applied to , the reduced loads are obtained by equating the virtual work or

Finally, the reduced equation of motion reads

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 P.2.7). It can be demonstrated that the error in the mode shape associated with the Guyan reduction is a increasing function of the ratio

Some concepts of structural dynamics

33

where is the natural frequency of the mode and is the first natural frequency of the constrained system, where all the degrees of freedom (masters) have been blocked is the smallest solution of det . Therefore, should be kept far above the frequency band where the model is expected to be accurate. Considering the effect of additional constraints on can often be used as a guide in the selection of the degrees of freedom .

2.7

References

K. J. BATHE & E. L. WILSON, Numerical Methods in Finite Element Analysis, Prentice-Hall, 1976. R. E. D. BISHOP & D . C. JOHNSON, The Mechanics of Vibration, Cambridge University Press, 2nd ed., 1979. R. H. CANNON, Dynamics of Physical Systems, McGraw-Hill, 1967. R. W. CLOUGH & J. PENZIEN, Dynamics of Structures, McGraw-Hill, 1975. R. R. CRAIG JR., Structural Dynamics, Wiley, 1981. M. GERADIN & D. RIXEN, Mechanical Vibrations, Theory and Application to Structural Dynamics, Wiley, 1993. P. C. HUGHES, Attitude Dynamics of Three-Axis Stabilized Satellite with a Large Flexible Solar Array, J. Astronautical Sciences, vol. 20, 166–189, Nov.– Dec. 1972. P. C. H UGHES , Dynamics of Flexible Space Vehicles with Activ e Attitude Control, Celestial Mechanics Journal, vol. 9, 21–39, March 1974. T. J. R. HUGHES, The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, 1987b. D. J. INMAN, Vibration, with Control, Measurement, and Stability, PrenticeHall, 1989. L. MEIROVITCH, Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, 1980. V. J. MODI, Attitude Dynamics of Satellites with Flexible Appendages - A Brief Review, J. Spacecraft and Rockets, vol. 11, 743–751, Nov. 1974. O. C. ZIENKIEWICZ & R. L. TAYLOR, The Finite Element Method, McGrawHill, fourth ed., 1989, (2 vol.).

Vibration control of active structures

34

2.8

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 where 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 P.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 2.3.3. Show that changing the dummy constraint to mass 2 does not change the pseudo-static flexibility matrix P.2.4 Consider a simply supported beam with the following properties: EI = 10.266 It is excited by a point force at (a) Assuming that a displacement sensor is located at (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 high-frequency 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 created by a unit force applied at on a simply supported beam is

The symmetric operator is often called ”flexibility kernel” or Green’s function. (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 and (c) Do the same as (b) when the sensor location is Notice that the interlacing property of the poles and zeros no longer holds. P.2.5 Consider the modal expansion of the transfer function (2.47) and assume that the low frequency amplitude G(0) is available, either from static

Some concepts of structural dynamics calculations or from experiments at low frequency. approximated by the truncated expansion

35

Show that

can be

P.2.6 Show that the impulse response matrix of a damped structure with rigid body modes reads

where and 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 to starting from the clamped end. We perform various Guyan reductions in which we select according to: (a) all (12 degrees of freedom, no reduction); (b) all (6 d.o.f.); (c) all (6 d.o.f.); (d) (6 d.o.f.); (e) (3 d.o.f.); (f) (3 d.o.f.); For each case, compute the natural frequency of the first three modes and the first natural frequency 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 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

where is the torque applied to the main body, and are the modal masses and the natural frequencies of the constrained modes of the flexible appendages and are the modal participation factors of the flexible modes [i.e. 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). [Hint: Decompose the motion into the rigid body mode and the components of

36

Vibration control of active structures

the constrained flexible modes, express the kinetic energy and the strain energy, write the Lagrangian in the form

and write the Lagrange equations.]

Chapter 3

Actuators‚ piezoelectric materials‚ and active structures 3.1 Introduction The actuators play a critical role in active structures technology and‚ in many cases‚ they constitute the limiting factor. Two broad categories 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 the structure. After a brief review of the principle of the proof-mass actuator and the single axis gyrostabilizer‚ this chapter discusses the constitutive equations of piezoelectric materials and analyses the dynamics of various structures equipped with piezoelectric actuators and sensors‚ namely beams‚ trusses and shells. The duality between the sensor equation and the equivalent piezoelectric loads is emphasized. The danger of using the oversimplified beam theory for predicting the transmission zeros of slender structures with collocated actuator and sensor is pointed out. 37

Vibration control of active structures

38

3.2

Proof-mass actuator

A reaction mass is attached to a current-carrying coil moving in a magnetic field created by a permanent magnet; the moving part is attached to the chassis by a spring and damper (Fig.3.1). The current in the coil is governed by

where is the resistance and L the inductance of the coil, and is the back electromotive force (e.m.f.) due to the motion. In the Laplace domain,

where is the impedance of the coil. The equation of motion of the moving part is

where F is the electromagnetic force‚ proportional to the current in the coil

Note that the same constant velocity and the back e.m.f.;

appears in the relationship between the coil is called the voice coil constant. Combining the

Actuators‚ piezoelectric materials‚ and active structures

39

two foregoing equations‚ one gets the relationship between the displacement of the moving mass and the voltage applied

This equation shows that the back e.m.f. appears as an additional damping in the system. On the other hand‚ accelerating the mass with produces a reaction force on the supporting structure equal to

or‚ in Laplace terms‚

Combining with Equ.(3.6)‚ one gets the transfer function between the force applied to the structure and the voltage V:

If one neglects the inductance of the coil‚ function takes the form

is a constant and the transfer

where is the natural frequency associated with the spring-mass system and is the damping ratio (including contributions from the structure and from the back e.m.f.). The Bode plots (amplitude and phase) of the actuator are shown in Fig.3.1. Above some critical frequency the proof-mass actuator behaves like an ideal force generator (constant amplitude, no phase difference). The resonance frequency (and therefore the critical frequency ) can be adapted by changing the mass Because the reaction mass is attached to the structure by a spring, the proof-mass actuator has no authority over the rigid body modes (in contrast to reaction wheels). Operation at low frequency requires a large stroke, which is technically difficult. Medium to high frequency actuators (above 40 Hz) can be obtained with loudspeakers (slightly modified with an added mass ) or commercial electrodynamic shakers provided with an appropriate mass at the tip, to lower the natural frequency to the desired value. An application is discussed later. The proof-mass system can also behave as a passive absorber, if its natural frequency is tuned to one of the structural modes of the structure (e.g. Hagood & Crawley).

40

3.3

Vibration control of active structures

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.3.2. Rotating the gimbal about the axis with an angular velocity produces torques:

where is the angular momentum along the axis‚ and is the deviation of the rotor axis with respect to the vertical. The servo motor on the gimbal axis is velocity controlled. The angle 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).

3.4

Piezoelectric actuators

The piezoelectric effect was discovered by Pierre and Jacques Curie in 1880. The direct piezoelectric effect consists of the ability of certain crystalline materials (ceramics) to generate an electrical charge in proportion to an externally

Actuators‚ piezoelectric materials‚ and active structures

41

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 ceramic. 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 a built-in electric dipole‚ but the net electric dipole on a macroscopic scale is zero. When the crystal is cooled in the presence of an electric field‚ the dipoles tend to align‚ leading to an electric dipole on a macroscopic scale. After cooling and removal of the poling field‚ the dipoles cannot return to their original position‚ and the ceramic body becomes permanently piezoelectric‚ with the ability to convert mechanical energy to electrical energy and vice versa. The most popular piezoelectric materials are Lead-Zirconate- Titanate (PZT) which is a ceramic‚ and Polyvinylidene fluoride (PVDF) which is a polymer; their main properties are given in Table 3.1. 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. The direction of expansion with respect to the direction of the electrical field depends on the constants appearing in the constitutive equations. The material can be manufactured in such a way that one of the coefficients dominates the others. Piezoelectric actuators are becoming increasingly important in micropositioning technology. In the modelling of piezoelectric actuators for structural control applications‚ there is a great similarity between the inverse piezoelectric effect and thermal deformations. Table 3.1 gives the properties of typical piezoelectric materials.

3.4.1 Constitutive equations In this section we use the notations of the IEEE standard on piezoelectricity. In an unstressed one-dimensional dielectric medium, the electric displacement D (charge per unit area, expressed in is related to the electric field

E (V/m) by where is the dielectric constant of the material. Similarly, in a one-dimensional elastic body placed in a zero electric field, the strain S and the stress T are related by

where

is the compliance of the material (inverse of the Young’s modulus).

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Vibration control of active structures

For a piezoelectric material‚ the electrical and mechanical constitutive equations are coupled:

In Equ.(3.11), the piezoelectric constant relates the strain to the electric field E in the absence of mechanical stress and refers to the compliance when the electric field is constant. In Equ.(3.12), relates the electric displacement D to the stress under a zero electric field (short-circuited electrodes); is expressed in or Coulomb/Newton. is the dielectric constant under constant stress. The above equations can be transformed into

which are usually rewritten as

where

is the Young’s modulus under constant electric field (in is the constant relating the electric displacement to the strain, for short-circuited electrodes (in is called the coupling coefficient of the piezoelectric material. This name comes from the fact that, at frequencies far below the mechanical resonance frequency of the piezo, can be expressed as

A high value of is desirable for efficient transduction. From Equ.(3.14), one sees that the dielectric constant under constant strain is related to that under constant stress by Equation (3.13) is the starting point for the formulation of the equation of a laminar piezoelectric actuator, while Equ.(3.14) is that for a laminar sensor. In matrix form, Equ.(3.13) and (3.14) read

It is interesting to point out the analogy between Equ.(3.11) and the classical one-dimensional equation of thermoelasticity

Actuators‚ piezoelectric materials‚ and active structures

43

where is the temperature difference and is the thermal expansion coefficient. In Table 3.1‚ it is important to note that the piezoelectric coefficient [as appears in the actuator equation (3.13)] is 300 times larger for PZT than for PVDF; this indicates the superior actuation capability of piezoceramics as compared to piezopolymers. Figure 3.3 shows the two basic types of piezoelectric actuators (e.g. see Physik Instrumente catalogue): The stacked design (linear actuator) and the laminar design (spatially distributed actuator); they are briefly reviewed in the following sections. For multidimensional media‚ the constitutive equations keep more or less the same form as above‚ except that the quantities involved are tensors (e.g. see IEEE Std‚ Lerch or Piefort). If the direction of polarization coincides with direction 3‚ the constitutive equations can be written in matrix form:

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Vibration control of active structures

Actuators‚ piezoelectric materials‚ and active structures

45

Examining the actuator equation (3.17)‚ we note that when an electric field is applied parallel to the direction of polarization‚ an extension is observed along the same direction; its amplitude is governed by the piezoelectric coefficient Similarly‚ a shrinkage is observed along the directions 1 and 2 perpendicular to the electric field‚ the amplitude of which are controlled by and respectively. Piezoceramics have an isotropic behaviour in the plane‚ On the contrary‚ when the PVDF is polarized under stress‚ its piezoelectric properties are highly anisotropic‚ with Equation (3.17) also indicates that an electric field normal to the direction of polarization 3 produces a shear deformation controlled by the piezoelectric constant (the same occurs if a field is applied). An interesting feature of this type of actuation is that is the largest of all piezoelectric coefficients ( for PZT). This has led to the interesting torque actuator design of Fig.3.4 (Glazounov et al.).

Vibration control of active structures

46

3.4.2

Linear actuator

A linear piezo actuator consists of a stack of thin ceramic layers of thickness between 0.1 and 1 mm separated by electrodes (Fig.3.3). The polarization axis and the applied electric field are perpendicular to the layer and coincide with the direction of expansion of the actuator‚ so that the actuation capability is controlled by the coefficient. The required voltage for layers of 0.1 mm is 100 V (they are called low voltage piezo: LVPZ) while 1 mm layers require 1000 V (high voltage piezo: HVPZ). Both types have similar strain capabilities; the difference is that low voltage piezos have a higher electric capacitance and require a significantly larger current. LVPZ are characterized by low voltage and high current while HVPZ are high voltage and low current devices; this affects the amplifier design. The maximum expansion can be 0.1 to 0.13 %‚ which means that a piezo translator of 100 mm can have a maximum stroke of about The size of the disks can be selected to achieve the appropriate stiffness. For dynamic applications‚ a mechanical preload is introduced to compensate for the small extensional strength of the stack. The stacked design is often referred to as When no external load is applied‚ the change of length is related to the voltage applied‚ by the approximate relationship

where is the number of disks in the stack and V is the applied voltage. Two types of piezotranslators are shown in Fig.3.5. The first one follows exactly the principle of Fig.3.3.a. In the second design‚ the piezoelectric stack is mounted along the main axis of an elliptical structure which acts as a motion amplifier; the expansion of the actuator is in the direction perpendicular to that of the piezo stack; this design leads to more compact actuators.

Actuators‚ piezoelectric materials‚ and active structures

47

The actual voltage/expansion diagram is nonlinear and exhibits a fairly high hysteresis (Fig.3.6). The maximum width of the hysteresis curve can reach as much as 15 % of the total expansion. For quasi-static applications‚ the hysteresis can be compensated in closedloop by measuring the actual expansion of the stack (for example by gluing a strain gage on the piezo stack). Closed-loop compensation of the hysteresis in dynamic applications such as active damping does not seem to be necessary‚ and may even be dangerous for the stability of the system. Many lightweight space structures consist of trusses. Active members are capable of both structural functions (load transmission) and simultaneously generating active control forces. The actuation system can be based on jackscrews (e.g. see Robertshaw et al.) or on piezotranslators (e.g. Fanson et al.). The former exhibit high rigidity‚ large amplitude‚ low bandwidth‚ low precision and low power efficiency; such actuators are used mainly for reconfiguration. By contrast‚ piezotranslators allow a stroke of up to a few hundreds microns‚ with high bandwidth and high precision (resolution in the nanometer range). Applications using linear piezoelectric actuators will be considered in subsequent chapters.

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48

3.4.3

Laminar actuator

In the laminar design‚ the piezoelectric layer is bonded to the structure (Fig.3.3.b). The geometrical arrangement is such that the useful direction of expansion is normal to that of the electric field. Thus‚ the activation capability is governed by the piezoelectric constant or more precisely by note that is only one half of The piezoelectric material may either be ceramics (PZT) or polymers (PVDF); typical thickness of PZT strips is their sizes are generally small and they are covered with silver electrodes. PVDF strips are flexible and may be much thinner and larger; however‚ the actuation capability of PVDF (measured by ) is 300 times smaller than that of PZT (see Table 3.1). PZT is always isotropic in the plane while PVDF may be manufactured both isotropic and anisotropic‚ even almost unidirectional It is the shape of the electrode which determines the active part of the material. Consider the beam of Fig.3.7 with a thin layer of piezoelectric material bonded to one side. We assume that the thickness of the piezoelectric strip is small compared to that of the beam‚ The piezoelectric strip is used as actuator by controlling the voltage V applied to the electrodes‚ creating a constant electric field V/ We assume that the electrodes have a variable width (this can be achieved either by etching the original surface electrode‚ or by cutting the piezoelectric layer). If one rewrites Equ.(3.13) with standard engineering notation‚ one gets‚ within the piezoelectric layer‚

Actuators‚ piezoelectric materials‚ and active structures

where denotes the Young’s modulus of the piezo‚ and following usual relation applies to the beam

49

The

where E is the Young’s modulus of the beam. The equilibrium equation of the beam is and‚ according to the Euler-Bernouilli assumption‚ the axial deformation and the curvature are related by where is the distance to the neutral axis. Introducing this into Equ.(3.20) and (3.21) and integrating over the cross-section‚ one gets

or

where we have assumed a constant moment arm across the thickness of the film. In this equation‚ the bending stiffness EI refers to the supporting structure and the piezoelectric film together‚ and is the current width of the electrode. Substituting into Equ.(3.22)‚ one gets

Comparing with Equ.(2.40)‚ we see that when the width of the electrodes varies along the beam‚ the piezoelectric actuator produces a distributed load proportional to the second derivative of

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Vibration control of active structures

Similarly‚ for an electrode of constant width‚ the distributed actuator is equivalent to adding concentrated moments at the boundaries of the actuator‚ as represented in Fig.3.8. From Equ.(3.24)‚ the piezoelectric moment is Equivalently‚ a sudden change in the first derivative produces a point force at the discontinuity (Fig.3.8):

at some location

This result is readily obtained by integrating the right hand side of Equ.(3.25) over a small interval about and taking the limit when the size of the interval goes to zero. Equations (3.25) to (3.27) give the equivalent loads associated with various shapes of the electrodes. We note that the actuation authority is strongly dependent on the electrode shape and that it is proportional to the piezoelectric constant as mentioned before‚ is substantially larger for PZT than for PVDF. Equations (3.25) to (3.27) are based on the assumption that the thickness of the piezo strip is much smaller than that of the beam more accurate calculations show that the distance appearing in these equations is actually the distance between the mid-plane of the beam and the mid-plane of the piezo strip.

Actuators‚ piezoelectric materials‚ and active structures

3.4.4

51

Laminar sensor

Consider a laminar sensor with short-circuited electrodes‚ so that a zero electric field is enforced (E = 0). According to Equ.(3.14)‚ the amount of charge per unit area is related to the strain by or‚ with structural engineerning notations‚

Combining with Equ.(3.23) and assuming that the strain is constant over the thickness of the strip (consequence of we get

The corresponding electric charge is equal to the integral of the electric displacement over the electrode area‚

If the polarization profile is uniform ( current amplifier (Fig.3.9.a) is

constant)‚ the output voltage of a

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Vibration control of active structures

is the constant of the amplifier. Thus‚ the output signal is proportional to the integral of the time derivative of the curvature weighted by the width of the electrode. If a charge amplifier is used (Fig.3.9.b)‚ the output voltage is proportional to the electric charge in the electrode:

If the width of the electrode‚

is constant‚

The sensor output is thus proportional to the difference of slopes at the extremities of the piezo strip. As for the actuator‚ we observe that the sensing capability is proportional to These results assume that a more accurate calculation shows that the distance appearing in Equ.(3.29) to (3.31) is actually the distance between the mid-plane of the beam and the mid-plane of the piezo.

3.4.5 Example: Tip displacement of a cantilever beam

Consider a cantilever beam clamped at and covered with a piezoelectric strip of triangular shape (Fig.3.10). The integral involved in Equ.(3.28) giving the electric charge accumulated on the triangular electrode can be integrated by parts twice according to

The integral in the right hand side vanishes since and taking into account that and we get

Actuators‚ piezoelectric materials‚ and active structures

53

Thus‚ the output signal is proportional to the tip displacement of the beam.

3.4.6 Spatial modal filters Consider the case of the actuator first. Let

be the modal expansion of the transverse displacements‚ where are the mode shapes and the modal amplitudes. Upon substituting this into the equation of motion (3.25) and using the orthogonality of the mode shapes‚ Equ.(2.45) and (2.46)‚ one gets the following equation which controls the amplitude of mode

This equation shows that the actuator can be designed in such a way that a specific mode is uncontrollable‚ by enforcing that the integral in the right hand side vanishes. Moreover‚ the electrode can be tailored in such a way that only a single mode is excited. This is achieved if the electrode profile is such that

Indeed‚ according to the first orthogonality relation (2.45)‚ this electrode profile can excite only mode without affecting the other modes. As far as the sensor is concerned‚ the output signal of the amplifier‚ Equ.(3.29)‚ can be decomposed into its modal components as

This equation shows that any specific mode can be made unobservable by choosing the electrode profile in such a way that the integral vanishes. Just as for the actuator‚ because of the second orthogonality relation (2.46)‚ the following electrode profile will produce a sensor which is sensitive only to mode In principle‚ if a good structural model is available for the set of modes that one wishes to control‚ it is possible to design a sensor which is sensitive only to these modes‚ and orthogonal to the residual modes. This offers an attractive possibility for spillover alleviation. Note that for a uniform beam‚ the mode shapes are such that as a result of (2.42). It follows that the modal actuator (3.34) and the modal sensor (3.36) have the same shape.

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Vibration control of active structures

Figure 3.11 illustrates the modal filters used for a uniform beam with various boundary conditions; the change of sign indicates a change in the polarity of the strip, which is equivalent to negative values of As an alternative, the part of the sensor with negative polarity can be bonded on the opposite side of the beam. The reader will notice that the electrode shape of the simply supported beam is the same as the mode shape, while for the cantilever beam, the electrode shape is that of the mode shape of a beam clamped at the opposite end.

3.5

Passive damping with piezoceramics

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55

A piezoelectric material works as an energy transformer, from mechanical energy to electrical and vice versa. A piezoelectric device subjected to mechanical strain can be regarded as a charge (or current) generator in parallel with a capacitance C (typical values for linear actuators range from to If the structure vibrates, the charge is generated at the frequency of vibration. One way to damp a structure equipped with a piezoelectric device is to connect the piezo to a resistor and an inductor as in Fig.3.12, to obtain a resonant LC circuit tuned on the natural frequency of the structure:

the problem is that‚ for practical values of and C‚ this leads to very large values of L‚ which would not be practical to realize by classical means. However‚ a tunable active inductor can be realized electronically with a gyrator circuit (Edberg et al.). Figure 3.12.b shows experimental results obtained with the truss of Fig.3.17 where one of the piezo translators is connected as indicated above. We see that the damping ratio depends very much on the tuning of to small errors can substantially reduce the efficiency of the system. We shall see later that active damping is often less sensitive to the parameters of the electromechanical system.

3.6

Active cantilever beam

Consider a cantilever beam provided with a pair of rectangular piezoelectric actuator and sensor as in Fig.3.13. The system is modelled by finite elements;

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Vibration control of active structures

the mesh is such that there is a node at both ends of the piezos (each node has two degrees of freedom‚ one translation and one rotation ). We seek the open-loop transfer function between the voltage applied to the actuator and the output voltage of the sensor. According to the foregoing sections‚ the rectangular actuator is equivalent to a pair of torques M with opposite signs and proportional to V:

The gain of the actuator can be computed from the actuator size and the material properties with Equ.(3.26). In the general form (2.2) of the equation of motion‚ the vector force is where the influence vector has the form the only nonzero components correspond to the rotational degrees of freedom of the nodes at and in the model. It follows that the modal excitations are

where is the relative angle (difference of slope) between the ends of the actuator, for mode . Similarly, the output of the charge amplifier, Equ.(3.31), is also proportional to the difference of the rotations at the ends of the sensor, Expanding in modal coordinates, we get

where are the modal components of the slope difference between the ends of the sensor; the gain depends on the sensor size‚ the material properties and the gain of the charge amplifier. Note that if the actuator and sensor are collocated in the sense of the Euler-Bernouilli beam theory‚ the slope differences appearing in the actuator and sensor equations are identical‚

Combining Equ(3.40)‚ (3.41) and the equations governing the structural dynamics‚ (2.19)‚ we obtain the frequency response function (FRF) between the voltage V applied to the actuator and the voltage at the output of the charge amplifier:

The discussion of section 2.3.1 regarding the truncation of the modal expansion applies fully (Problem P.3.4). Equ.(3.42) has the general form (2.36) with no contribution from the rigid body modes‚ because the piezoelectric loads

Actuators‚ piezoelectric materials‚ and active structures

57

are self equilibrating. This leads to alternating poles and zeros as in Fig.2.6.b. Figure 3.14 shows typical experimental results obtained with a system similar to that shown in Fig.3.13. Observe that the open-loop FRF does not exhibit any roll-off at high frequency. Note also that a small linear phase shift appears in the phase diagram‚ due to the fact that these results have been obtained digitally‚ at a sampling frequency of l0000Hz (the effect of sampling will be discussed briefly in chapter 13). Because of the high-pass behaviour of the charge amplifier‚ discrepancies can be observed also at very low frequency‚ typically below 0.1Hz to 1 H z ‚ depending on the charge amplifier characteristics. If a refined model is required‚ the dynamics of the charge amplifier can be represented by a second order high-pass filter

with the appropriate corner frequency When the modal expansion (3.42) is truncated after mode (quasi-static correction) reads

modes‚ the residual

Its influence on the open-loop transfer function may be quite significant‚ as illustrated in Fig.3.15 which displays the amplitude plots with and without residual mode (truncation after 3 modes). After correction‚ the truncated transfer function follows almost exactly the full expansion (not shown for clarity) for

Vibration control of active structures

58

Without correction, the truncated transfer function differs substantially from the full expansion, in particular near the zeros which are strongly dependent on the residual mode this is particularly important because the zeros play a key role in the performance of the control system. We also notice the presence of an extra zero in the corrected diagram; it is due to the feedthrough component which brings back the zero at infinity (due to roll-off) to a finite frequency. The feedthrough component will be reexamined in section 3.8 where the adequacy of the beam model for collocated systems will be questioned.

3.7

Active truss

Figure 3.16 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.3.17 shows an example of active truss equipped with two active struts (this truss was built in the late 80 ’s and uses out-dated high-voltage piezotranslators). When a voltage is applied to a unconstrained linear piezoelectric actuator‚ it produces an expansion which‚ if we neglect the hysteresis‚ is proportional

Actuators‚ piezoelectric materials‚ and active structures

59

Vibration control of active structures

60

to the applied voltage where is the gain of the actuator. If the actuator is placed in a truss‚ its effect on the structure can be represented by equivalent piezoelectric loads acting on the passive structure. As for thermal loads‚ the pair of self equilibrating piezoelectric loads applied axially to both ends of the active strut (Fig.3.16) has a magnitude equal to the product of the stiffness of the active strut‚ and the unconstrained piezoelectric expansion

Assuming no damping‚ the equation governing the motion of the structure excited by a single actuator is

where is the influence vector of the active strut in the global coordinate system. The non-zero components of are the direction cosines of the active bar (Problem P.3.6). As for the output signal of the force transducer‚ it is given by

where is the elastic extension of the active strut‚ equal to the difference between the total extension of the strut and its piezoelectric component The total extension is the projection of the displacements of the end nodes on the active strut‚ (Problem P.3.6). Introducing this into Equ.(3.47)‚ we get

Note that‚ because the sensor is located in the same strut as the actuator‚ the same influence vector appears in the sensor equation (3.48) and the equation of motion (3.46). If the force sensor is connected to a charge amplifier of gain the output voltage is given by

Note the presence of a feedthrough component from the piezoelectric extension Proceeding as in the previous section‚ we obtain the modal excitations

and the modal expansion of the output

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61

Combining the above equation with that of the dynamics in modal coordinates‚ we get the FRF between the applied voltage V and the output voltage of the charge amplifier:

where we have neglected the damping to simplify the equation. It can be written alternatively

where we have defined

We see that the numerator and the denominator of this expression represent respectively twice the strain energy in the active strut and twice the total strain energy when the structure vibrates according to mode is therefore called the modal fraction of strain energy in the active strut. From Equ.(3.52)‚ we see that determines the residue of mode that is the amplitude of the contribution of mode in the transfer function between the piezo actuator and the force sensor; it can therefore be regarded as a compound index of controllability and observability of mode is readily available from commercial finite element programs; it can be used to select the proper location of the active strut in the structure (Problem P.3.7): the best location is that with the highest for the modes that we wish to control (Preumont et al.). The FRF (3.52) has alternating poles and zeros (Fig.3.18). One can easily see from Equ.(3.52) and (3.48) that the FRF between the piezoelectric displacement and total displacement of the end nodes of the active strut‚ is

At

the total displacement

produced by the piezoelectric loads

is

where is the stiffness of the truss (with the active strut considered as passive) from the end point of the active strut. Writing Equ.(3.54) at and comparing with (3.55)‚ we obtain

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Vibration control of active structures

This ratio can be easily calculated from a static analysis (it is always smaller than 1). As in section 2.3.1‚ the modal expansion of can be truncated after modes provided that a quasi-static correction is applied:

and‚ taking into account Equ.(3.56)‚

This equation involves only the low frequency modes of the truss. It can be written alternatively

The foregoing results apply to an undamped truss with a single piezoelectric linear actuator collocated with a force sensor; the extension to the case with modal damping is straightforward and is left as an exercise (Problem P.3.8).

Actuators‚ piezoelectric materials‚ and active structures

3.8

63

Piezoelectric shell

The elementary case of a piezoelectric beam has been discussed in section 3.4.3‚ where we introduced the laminar actuators and sensors; analytical expressions for the equivalent piezoelectric loads of a laminar actuator and the output of a laminar sensor were developed with the Euler-Bernouilli beam theory. The open-loop transfer function of an active beam with collocated actuator and sensor was analysed in section 3.6 and the importance of the residual mode (high frequency dynamics) on the location of the open-loop transmission zeros was pointed out. In fact‚ in many situations involving nearly collocated actuators and sensors‚ the Euler-Bernouilli beam theory is inappropriate‚ because the transfer function becomes more sensitive to the detailed strain transmission path within the coupled system (structure-actuator-sensor) which requires more sophisticated modelling techniques accounting for membrane (in-plane) deformations as well as bending ones. This is illustrated by the following experiment. Consider the cantilever plate of Fig.3.19; the steel plate is 0.5 mm thick and 4 piezoceramic strips of thickness are bounded symmetrically as indicated in the figure. The size of the piezos is respectively 55 mm x 25 mm for and and 55 mm × 12.5 mm for and is used as actuator while the sensor is taken successively as and Since they cover the same extension along the beam‚ the various sensor locations cannot be distinguished in the sense of the Euler-Bernouilli beam theory (except for the sign or a constant factor‚ because and are on opposite sides and the size of is twice that of and ) 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.3.20. We see that the transmission zeros vary significantly from one configuration to the other (as we shall see later‚ this will dramatically change the performance

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Vibration control of active structures

of the closed-loop system). 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; these membrane strains are neglected in the classical beam theory but they can be handled by shell theory. This introductory text on active vibration control is not the place for extensive developments of the theory of piezoelectric shells which can be found in the literature (Lee); we will only write the actuator and sensor equations for piezoelectric shells in the piezoelectric axes and compare them with their equivalent forms for beams. Finally‚ we will write the general form of the dynamic equation of a piezoelectric structure.

3.8.1

Two-dimensional constitutive equations

In this section‚ we will denote vector quantities by {} in order to improve the understanding and to conform with the usual notations of the theory of laminate composites. Consider a two-dimensional piezoelectric lamina in a plane ( ); the poling direction is (normal to the lamina) and the electric field is also applied along In the piezoelectric principal axes‚ the constitutive equations read

where

Actuators‚ piezoelectric materials‚ and active structures

65

are the stress and strain vector‚ respectively‚ [C] is the matrix of elastic constant‚ E is the component of the electric field along D is the of the electric displacement and the dielectric constant.

3.8.2

Kirchhoff shell

Next‚ consider a piezoelectric lamina bonded to a shell structure (Fig.3.21). If the global axes coincide with the piezoelectric axes of the lamina‚ the constitutive equations can be integrated over the thickness of the shell in the form

where {N} is the vector of in-plane resultant forces and {M} the vector of bending moments;

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Vibration control of active structures

is the deformation vector of the mid-plane and tures:

is the vector of curva-

The matrices A‚B‚D are the classical stiffness matrices of shell theory (e.g. Agarwal & Broutman); is the thickness of the piezoelectric lamina and is the distance between its mid-plane and the mid-plane of the shell. If the piezoelectric lamina is connected to a charge amplifier as in Fig.3.9.b‚ the voltage between the electrodes is set to V = 0 and the sensor equation (3.64) can be integrated over the electrode to produce the sensor output

where the integral extends over the surface of the electrode (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 strain‚ while the second one is due to bending. If the piezoelectric properties are isotropic the surface integral can be further transformed into a contour integral using one of the Green integrals:

This integral extends over the contour of the electrode (Fig.3.22); 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. Similarly‚ for a piezoelectric actuator made of isotropic material‚ the equivalent piezoelectric loads consist of a in-plane force‚ normal to the contour of the electrode‚ and a constant moment‚ acting along the contour of the electrode (Fig.3.23): Comparing these results with the equivalent bending moment (3.26) of the beam theory‚ we see that the beam theory accounts only for the component of the bending moment normal to the beam axis and totally neglects the in-plane force (Fig.3.24). Conversely‚ the sensor signal given by the beam theory accounts only for the component of the rotation along the contour normal to the beam axis.

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67

68

3.9

Vibration control of active structures

Finite element formulation

The finite element formulation of a piezoelectric continuum can be derived from Hamilton’s principle in which the potential energy density and the virtual work are properly transformed to include the piezoelectric energy (Allik & Hughes). In addition to the classical displacements as in any F.E. formulation‚ the generalized variables include the electric potentials between the electrodes of the piezos. The resulting dynamic equations read

where M is the mass matrix, the stiffness matrix, the coupling matrix connecting the mechanical variables and the electrical variables the matrix of electrical capacitance; is the vector of external mechanical loads applied to the structure and q the electric charges brought to the electrodes. For voltage driven electrodes, the electric potential is controlled and Equ.(3.70) can be rewritten

where represents the equivalent piezoelectric loads. Once the mechanical displacements have been computed, the electric charges appearing on the electrodes can be computed from Equ.(3.71). Prom Equ.(3.72), we see that the dynamics of the system with short-circuited electrodes is the same as if there were no piezoelectric electromechanical coupling. Conversely, open electrodes correspond to a charge condition in this case we can substitute from Equ.(3.71) into Equ.(3.70)

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This equation shows that the piezoelectric coupling tends to increase the global stiffness of the system if the electrodes are left open (this is because the terms of are negative). The natural frequencies of the system with open electrodes are larger than those with short-circuited electrodes; this difference is usually small but it tends to increase as the amount of embedded piezoelectric material increases. Finally‚ if the electrodes are connected via a passive electrical network of impedance matrix Z‚ Equ.(3.70) and (3.71) must be complemented by the network equation (Fig.3.25). To conclude this section‚ let us reexamine the modeling of the cantilever beams of Fig.3.15 and 3.19. It turns out that‚ when the thickness of the beam is small and the actuator and sensor are collocated‚ the membrane strains contribute significantly to the transfer function. As a result‚ the discrepancy between the beam theory and the shell theory is large‚ particularly near the transmission zeros of the FRF which are strongly affected by the feedthrough from the strain actuator to the strain sensor (Fig.3.26).

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Figure 3.27 shows computed FRF based on the shell theory for the cantilever plate of Fig.3.19; the model is based on Mindlin-type finite elements (Piefort). The agreement with the experimental results of Fig.3.20 is qualitatively good, although the numerical results still tend to overestimate the spacing between the poles and the zeros. As we shall see in the chapters devoted to active damping, the closed-loop poles are located on branches going from the open-loop poles to the open-loop zeros; overestimating their spacing amounts to overestimating the performance of the control system.

3.10

References

B. D. AGARWAL & L. J. BROUTMAN‚ Analysis and Performance of Fiber Composites‚ Wiley‚ second ed.‚ 1990. H. ALLIK & T. R. J. HUGHES‚ Finite Element Method for Piezoelectric Vibration‚ Int. J. for Numerical Methods in Engineering‚ vol. 2‚ 151–157‚ 1970. E. H. ANDERSON‚ D. M. MOORE‚ J. L. FANSON & M. A. EALEY‚ Development of an Active Member Using Piezoelectric and Electrostrictive Actuation for Control of Precision Structures‚ SDM Conference‚ AIAA paper 90-1085CP‚ 1990. S. E. BURKE & J. E. HUBBARD‚ Active Vibration Control of a Simply Supported Beam Using Spatially Distributed Actuator‚ IEEE Control Systems Magazine‚ pp. 25–30‚ Aug. 1987.

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G. S. CHEN‚ B. J. LURIE & B. K. WADA‚ Experimental Studies of Adaptive Structures for Precision Performance‚ SDM Conference‚ AIAA paper 89-1327CP‚ 1989. E. F. CRAWLEY & J. DE LUIS‚ Use of Piezoelectric Actuators as Elements of Intelligent Structures‚ AIAA Journal‚ vol. 25‚ no. 10‚ 1373–1385‚ 1987. E. F. CRAWLEY & K. B. LAZARUS‚ Induced Strain Actuation of Isotropic and Anisotropic Plates‚ AIAA Journal‚ vol. 29‚ no. 6‚ 944–951‚ 1991. E. K. DIMITRIADIS‚ C. R. FULLER & C. A. ROGERS‚ Piezoelectric Actuators for Distributed Vibration Excitation of Thin Plates‚ Trans. ASME‚ J. of Vibration and Acoustics‚ vol. 113‚ 100–107‚ Jan. 1991. D. L. EDBERG‚ A. S. Bicos & J. S. FECHTER‚ On Piezoelectric Energy Conversion for Electronic Passive Damping Enhancement‚ Proceedings of Damping’91‚ San Diego‚ 1991. J. L. FANSON‚ G. H. BLACKWOOD & C. C. CHEN‚ Active Member Control of Precision Structures‚ SDM Conference‚ AIAA paper 89-1329-CP‚ 1989. R. L. FORWARD‚ Electronic Damping of Orthogonal Bending Modes in a Cylindrical Mast Experiment‚ AIAA Journal of Spacecraft‚ vol. 18‚ no. 1‚ 11–17‚ Jan.-Feb. 1981. A. E. GLAZOUNOV‚ Q. M. ZHANG & C. KIM‚ A New Torsional Actuator Based on Shear Piezoelectric Response‚ SPIE Conference on Smart Structures and Materials‚ vol. 3324‚ 82–91‚ 1998a‚ san Diego‚ CA. A. E. GLAZOUNOV‚ Q. M. ZHANG & C. KIM‚ Piezoelectric Actuator Generating Torsional Displacement from the D15 Shear Strain‚ Appl. Phys. Lett. 72‚ 2526‚ 1998b. N. W. HAGOOD & E. F. CRAWLEY‚ Experimental Investigation of Passive Enhancement of Damping for Space Structures.‚ AIAA Journal of Guidance‚ vol. 14‚ no. 6‚ 1100–1109‚ Nov.–Dec. 1991. W. L. HALLAUER JR‚ Recent Literature on Experimental Structural Dynamics and Control Research‚ in J. L. JUNKINS‚ editor‚ Mechanics and Control of Large Flexible Structures‚ chap. 18‚ AIAA Progress in Astronautics and Aeronautics‚ Vol. 129‚ 1990. S. HANAGUD‚ M. W. OBAL & A. J. CALISE‚ Optimal Vibration Control by the Use of Piezoceramic Sensors and Actuators‚ AIAA Journal of Guidance‚ vol. 15‚ no. 5‚ 1199–1206‚ Sep.–Oct. 1992. W.-S. HWANG & H. C. PARK‚ Finite Element Modeling of Piezoelectric Sensors and Actuators‚ AIAA Journal‚ vol. 31‚ no. 5‚ 930–937‚ May 1993.

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IEEE Std‚ Standard on Piezoelectricity‚ ANSI/IEEE Std 176-1987‚ Jan. 1987. A. D. JACOT & D. J. LISKA‚ Control Moment Gyros in Attitude Control‚ AIAA J. of Spacecraft and Rockets‚ vol. 3‚ no. 9‚ 1313–1320‚ Sep. 1966. C.-K. LEE‚ Theory of Laminated Piezoelectric Plates for the Design of Distributed Sensors/Actuators- Part I : Governing Equations and Reciprocal Relationships‚ J. Acoust.-Soc. Am.‚ vol. 87‚ no. 3‚ 1144–1158‚ March 1990. C.-K. LEE‚ W.-W. CHIANG & T. C. O’SULLIVAN‚ Piezoelectric Modal Sensor/Actuator Pairs for Critical Active Damping Vibration Control‚ J. Acoust.Soc. Am.‚ vol. 90‚ no. 1‚ 374–384‚ July 1991. C.-K. LEE & F. C. MOON‚ Modal Sensors/Actuators‚ Trans. ASME‚ J. of Applied Mechanics‚ vol. 57‚ 434–441‚ June 1990. R. LERCH‚ Simulation of Piezoelectric Devices by Two- and Three-Dimensional Finite Elements‚ IEEE Transactions on Ultrasonics‚ Ferroelectrics‚ and Frequency Control‚ vol. 37‚ no. 3‚ May 1990. N. LOIX & A. PREUMONT‚ Remarks on the Modelling of Active Structures with Colocated Piezoelectric Actuators and Sensors‚ ASME Vibration Conference‚ Boston‚ vol. 84-3‚ 335–339‚ Sep. 1995. L. D. PETERSON‚ J. J. ALLEN‚ J. P. LAUFFER & A. K. MILLER‚ An Experimental and Analytical Synthesis of Controlled Structure Design‚ SDM Conference‚ AIAA paper 89-1170-CP‚ 1989. Philips Application Book on Piezoelectric Ceramics‚ (J. Van Randeraat & R. E. Setterington‚ Edts)‚ Mullard Limited‚ London‚ 1974. Physik Instrumente catalogue‚ Products for Micropositioning (PIGmbH‚ FRG). V. PIEFORT‚ Finite Element Modeling of Piezoelectric Active Structures‚ Ph.D. thesis‚ Université Libre de Bruxelles‚ Active Structures Laboratory‚ 2001. A. PREUMONT‚ J. P. DUFOUR & C. MALEKIAN‚ Active Damping by a Local Force Feedback with Piezoelectric Actuators‚ AIAA J. of Guidance‚ vol. 15‚ no. 2‚ 390–395‚ March–April 1992. H. H. ROBERTSHAW‚ R. H. WYNN‚ S. L. HENDRICKS & W. W. CLARK‚ Dynamics and Control of a Spatial Active Truss Actuator‚ SDM Conference‚ AIAA paper 89-1328-CP‚ 1989. K. U CHINO ‚ Ferroelectric Devices‚ Marcel Dekker‚ 2000. J. R. VlNSON‚ The Behavior of Shells Composed of Isotropic and Composite Materials‚ Kluwer Academic Publishers‚ 1993.

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D. C. ZIMMERMAN‚ G. C. HORNER & D. J. INMAN‚ Microprocessor Controlled Force Actuator‚ AIAA Journal of Guidance‚ vol. 11‚ no. 3‚ 230–236‚ May– June 1988.

3.11

Problems

P.3.1 Consider a beam covered with a piezo patch and an electrode of the shape shown in Fig.3.8.b. Sketch and calculate the equivalent loading given by this electrode configuration. (Note: just like thermal loads‚ piezoelectric loads are internal loads which must be self-equilibrated.) P.3.2 Consider a simply supported beam with collocated piezoelectric rectangular actuator and sensor extending longitudinally from to Using the results of section 2.5‚ show that the expression appearing at the numerator of Equ.(3.42) can be written analytically

P.3.3 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.3.4 Consider the active cantilever beam of transfer function (3.42). Assuming that G(0) is available from static calculations or from an experiment at low frequency‚ show that the truncated modal expansion including a quasi-static correction can be written

P.3.5 Consider a beam covered with a piezoelectric actuator of triangular shape (the axis of symmetry of the triangle coincides with that of the beam). Show that the equivalent piezoelectric loads consist of two point forces of equal magnitude and opposite signs at each end, and a torque at the larger end. Check that these forces and moment are self equilibrating. P.3.6 Consider the active truss of Fig.3.16. (a) For each active strut, write the influence vector b of the piezoelectric loads in global coordinates. (b) Assuming small displacements, check that the projection of the differential displacements of the end nodes of the active struts are given by where b is the result of (a). P.3.7 Consider a nine bay planar truss similar to that of Fig.3.16. 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.

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(a) Free-free boundary conditions. (b) Clamped-free boundary conditions. [Hint: Use the modal fraction of strain energy as index in the selection of the active strut location.] P.3.8 Show that with modal damping Equ.(3.59) is transformed into

Chapter 4

Collocated versus non-collocated control 4.1

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.4.1.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

75

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76

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 poles and zeros along the imaginary axis is reversed as in Fig.4.1.b. This situation is called a pole-zero flipping. It is responsible for a phase uncertainty of 360° and the only protection against instability is provided by the damping (systems which are prone to such a huge phase uncertainty can only be gain-stabilized).

4.2

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 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 of the closed-loop characteristic equation when the real parameter goes from zero to infinity. If the open-loop transfer function is written

the locus goes from the poles (for to the zeros (as open-loop system, and any point P on the locus is such that

of the

Collocated versus non-collocated control

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where are the phase angles of the vectors joining the zeros to P and are the phase angles of the vector joining the poles to P (see Fig.4.10). There are branches of the locus going asymptotically to infinity as 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 must remain constant after flipping. As a result, a nice stabilizing loop before flipping is converted into a destabilizing one after flipping (Fig.4.2). 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 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. 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.

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4.3

Collocated control

Consider the two-mass problem of Fig.4.3. The system has a rigid body mode along the axis; it is controlled by a force f applied to the main body M. A flexible appendage is connected to the main body by a spring and a damper In this section, 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 and 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 P.2.8. The system equations are :

With the notations

the transfer functions between the input force f and

and

are respectively :

Approximation (4.8) recognizes the fact that, for low damping the far away zero will not influence the closed-loop response. There are two poles near the imaginary axis. In which refers to the collocated sensor, there are two zeros also near the imaginary axis, at 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 is small, the polynomials in the numerator and denominator are almost equal, and there is a pole-zero cancellation.

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Let us consider a lead compensator

It includes one pole and one zero located on the negative real axis; the pole is to the left of the zero. Figure 4.4 shows a typical root locus plot for the collocated case when M = 1, and The parameters of the compensator are T = 10 and Since there are two more poles than zeros the root locus has two asymptotes at ±90°. 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 The lead compensator always increases the damping of the flexible mode (see Problem P.4.5). 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 in this case), so that the angles of the asymptotes remain ±90° 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.

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4.4

Non-collocated control

Figure 4.5 shows the root locus plot for the lead compensator applied to the non-collocated open-loop system characterized by the transfer function Equ.(4.7), with the following numerical data: M = 1, 0.02. The excess number of poles is in this case so that, for large gains, the flexible modes are heading towards the asymptotes at ±60°, in the right half plane. For a gain the closed-loop poles are located at and (these locations are not shown in Fig.4.5 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.4.6; the phase and gain margins are indicated. One observes that even with this small bandwidth (crossover frequency the gain margin is extremely small. A slightly lower value of the damping ratio would make the closed-loop system unstable (Problem P.4.1).

4.5

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:

The zeros of the notch filter, at are selected right below the flexible poles. The double pole at aims at keeping the compensator proper (i.e. the

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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.4.7.a for and This compensator allows a larger bandwidth than the lead compensator alone (Problem P.4.2). 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.4.7.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, is reduced from to the other data are identical to that of Fig.4.5), 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 to with the zeros at 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 P.4.3).

82

4.6

Vibration control of active structures

Pole-zero flipping in the structure

In the previous section, a pole-zero flipping was created by lowering the natural frequency of the flexible mode from above to below the imaginary zero of the notch filter. This led to a sharp alteration of the root locus plot near the flexible poles. This was used as a warning of the danger of using notch filters for systems with large uncertainty. In this section, we use a simple example to demonstrate the occurrence of a pole-zero flipping in a structure for small variations of the parameters. Consider once again the three mass system of Fig.2.1; if there is no damping, the equation of motion is

Transforming into Laplace form (assuming zero initial conditions) one gets

With the same notation as in the previous example, the transfer functions between the applied force f and the displacement of the various masses are respectively

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83

The poles of the system are located on the imaginary axis at

has two pairs of zeros, independent of the mass ratio, located at and As usual, they are identical to the natural frequencies of the constrained modes, when is blocked. For the poles and the zeros of cancel each other; for they alternate on the imaginary axis, as we expect for a collocated system. on the other hand, has one pair of zeros at and has no zero at all. The evolution of the poles with the mass ratio is shown in Fig.4.8. The zeros are also shown in the figure; they do not depend on One observes that at there is a pole-zero flipping in In the vicinity of can be written

The pole-zero pattern of the open-loop system is shown in Fig.4.9 for values of close to 1.

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4.7

Effect on the Bode plots

From Equ.(4.1),

the phase of

for a specific value

is given by

where is the phase angle of the vector joining the zero to and is the phase angle of the vector joining the pole to (Fig.4.10). Accordingly, an imaginary zero at produces a phase lead of 180° for and an imaginary pole produces similarly a phase lag of 180°. Therefore, a pole-zero flipping near the imaginary axis produces a phase uncertainty of 360° in the frequency range between the pole and the zero (Problem P.4.4). 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 in that frequency range.

4.8

Relation to the mode shapes

Because of their dramatic influence on the phase of the open-loop transfer function, it is essential for the designer to be able to predict all the imaginary zeros within or near the bandwidth of the control system, for a given actuator and sensor location. This fact is not given proper importance by structural engineers involved in identification; in fact, it is still a tradition, in the modal analysis and model up-dating community, to be more concerned with the poles (resonance

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85

frequencies) than with the zeros (anti-resonance). It turns out that the zeros are essentially related to the mode shapes of the structure, which unfortunately often tend to be more sensitive to the parameter uncertainty than the natural frequencies. Consider the modal expansion of the undamped transfer function

where and are the modal amplitudes at the actuator and sensor locations, normalized such that An interesting result, due to Martin, states that if two neighbouring modes are such that their residues and have the same sign, there is always an imaginary zero between the two poles (Problem P.4.7). Based on this result, one can anticipate that the pole-zero flipping is associated with a change of sign, that is to a nodal point in the mode shapes. This is illustrated in Fig.4.11, which shows the evolution of the zeros of a simply supported beam with a point force actuator located at when the displacement sensor is moved to the right One sees that, as the sensor moves away from the actuator, the zeros migrate along the imaginary axis. When the sensor reaches which is a nodal point of mode 5, the fourth imaginary zero becomes identical to for Similarly, the third zero crosses at which is a node of mode 4. The same phenomenon occurs for other along the beam. Note that the same observation can be made for the three mass system of section 4.6: It is easy to show that, for the nodal point of mode 1 is located between the mass points 1 and 2; for it coincides with 2, and it moves between 2 and 3 for Extensive studies have been conducted on simple structures, to understand

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how the zeros migrate when the sensor moves away from the actuator (Miu, Spector & Flashner). The following observations have been made: 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 (Problem P.4.8). 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.4.12). Systems with right half plane zeros are called non-minimum phase. Thus, non-collocated control systems are always non-minimum 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, as we shall see in chapter 8. In addition to purely imaginary and purely real zeros, some conservative systems do possess complex zeros, symmetrically located with respect to

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the imaginary axis; this was shown numerically on a multiply connected spring-mass system and confirmed experimentally on a beam clamped at one end for a special actuator and sensor configuration (Loix et al.) Besides, simulation results suggest that the presence of complex zeros increases the sensitivity of the closed-loop system with respect to the parameter variations.

4.9

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 180°. According to the stability criterion, the amplitude of the open-loop transfer function must satisfy whenever the phase lag exceeds 180°. Since the amplitude of in the roll-off 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 P.4.1). 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 et al. or Ikegami & Johnson). 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.

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4.10

References

R. H. CANNON & D. E. ROSENTHAL, Experiment in Control of Flexible Structures with Noncolocated Sensors and Actuators, AIAA Journal of Guidance, vol. 7, no. 5, 546–553, Sep.–Oct. 1984. G. F. FRANKLIN, J. D. POWELL & A. EMANI-NAEMI, Feedback Control of Dynamic Systems, Addison-Wesley, 1986. W. B. GEVARTER, Basic Relations for Control of Flexible Vehicles, AIAA Journal, vol. 8, no. 4, 666–672, April 1970. P. C. HUGHES & T. M. ABDEL-RAHMAN, Stability of Proportional Plus Derivative Plus Integral Control of Flexible Spacecraft, AIAA J. Guidance and Control, vol. 2, no. 6, 499–503, Nov.–Dec. 1979. R. IKEGAMI & D. W. JOHNSON, The Design of Viscoelastic Passive Damping Treatments for Satellite Equipment Support Structures, Proceedings of DAMPING’86, AFWAL-TR-86-3059, 1986. C. D. JOHNSON, D. A. KIENHOLZ & L. C. ROGERS, Finite Element Prediction of Damping in Beams with Constrained Viscoelastic Layers, Shock and Vibration Bulletin, , no. 51, 78–81, May 1981. N. LOIX, J. KOZANEK & E. FOLTETE, On the Complex Zeros of Non-Colocated Systems, Journal of Structural Control, vol. 3, no. 1–2, June 1996. G. D. MARTIN, On the Control of Flexible Mechanical Systems, Ph.D. thesis, Stanford University, 1978. D. K. MIU, Physical Interpretation of Transfer Function Zeros for Simple Control Systems with Mechanical Flexibilities, ASME J. Dynamic Systems Measurement and Control, vol. 113, 419–424, Sep. 1991. V. A. SPECTOR & H. FLASHNER, Sensitivity of Structural Models for Noncollocated Control Systems, ASME J. Dynamic Systems Measurement and Control, vol. 111, no. 4, 646–655, Dec. 1989. V. A. SPECTOR & H. FLASHNER, Modeling and Design Implications of Noncollocated Control in Flexible Systems, ASME J. Dynamic Systems Measurement and Control, vol. 112, 186–193, June 1990.

4.11

Problems

P.4.1 Consider the lead compensator for the non-collocated control of the two-mass system (section 4.4).

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(a) Determine the value of the damping ratio which would reduce the gain margin to zero. (b) What would be the gain margin if instead of P.4.2 Consider the lead compensator plus notch filter (4.10) for the noncollocated control of the two-mass system (section 4.5). Draw the corresponding Bode plots. Select a reasonable value of the gain and compare the bandwidth, the gain and phase margins with those of the lead compensator of Fig.4.6. P.4.3 (a) Repeat the previous problem when the frequency of the appendage is lower than that of the notch filter 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 Comment on the importance of tuning the notch filter. P.4.4 Consider the transfer function (4.14) of the three mass system with (a) Compare the Bode plots for and Comment on the phase diagram and its influence on the robustness. (b) For both values of plot the root locus for the lead compensator (4.9) with T = 10 and P.4.5 Consider the PD regulator

applied to the open-loop structure

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

the

Conclude on the stability condition (Gevarter). [Hint: Use a perturbation method, in the vicinity of and write the closed-loop characteristic equation.] P.4.6 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.4.7 Consider an undamped structure. Show that if two neighbouring modes are such that their residues and have the same sign, there is always an imaginary zero between the two poles (Martin).

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P.4.8 Consider a system of identical masses M simply connected with springs of stiffness k; assume that a point force is applied on mass and a displacement sensor is connected to mass Show that the zeros of the transfer function are the resonance frequencies of the two substructures (from 1 to and from ), formed by constraining the masses and (Miu). P.4.9 Consider the non-collocated control of the two-mass problem [the system transfer function is given by Equ.(4.7)] with M = 1. For various values of the mass ratio 0.01, 0.001, assuming a lead compensator (4.9), draw a diagram of the bandwidth of the control system, as a function of the damping ratio for the limit of stability (GM = 0).

Chapter 5

Active damping with collocated pairs 5.1

Introduction

The role of damping in the gain stabilization of a control system in the roll-off region has been pointed out in the foregoing chapter. The damping also reduces the settling time of the transient response to impulsive loads. Indeed, since the modal expansion of the impulse response matrix corresponding to (2.19) is

and are a Fourier transform pair, see problem P.2.6], one readily sees that the time constant (the memory) of each modal contribution is proportional to If, for example, and which are common values for spacecrafts, the time to reduce the impulse response by a factor of 10 is longer than comparable to that of one orbit revolution of the spacecraft. If one wants, for example, to maintain a micro-gravity environment or the pointing of a telescope, in spite of the transient loads from the thrusters or the human activity, one easily appreciates the need for damping augmentation. Similarly, the damping reduces the amplitude of the FRF in the vicinity of the resonances and, as a result, the steady state response to wideband disturbances [the variance of the stationary modal response to a white noise excitation is proportional to In this chapter, we concentrate on active means for damping augmentation, in short active damping. We will focuse on control schemes with guaranteed stability (at least to a large extent). As already stressed earlier, the use of collocated actuator and sensor (i.e. physically located at the same place and energetically conjugated, such as 91

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force and displacement or velocity, or torque and angle) leads to an alternating pole/zero pattern, on the imaginary axis if the structure is undamped, or slightly in the left half plane if the structure is lightly damped. Thanks to this property, a number of active damping schemes with guaranteed stability have been developed and successfully tested with various types of actuators and sensors. They can be implemented in a decentralized manner, with each actuator interacting only with the collocated sensor. In this case, the control system consists of independent SISO loops, whose stability can be readily established from the root locus of where is the structure transfer function between the actuator and the collocated sensor, is the active damping compensator, and is the scalar gain. For practical implementation purposes, however, one should be careful that has enough roll-off at high frequency, to accommodate the actuator and power amplifier dynamics and the inevitable phase lag due to sampling in digital controllers. This implies that some roll-off should appear in if there is a feedthrough component in (which often occurs in collocated systems). Nearly collocated systems may sometimes be preferable to strictly collocated ones, to reduce the feedthrough component in while preserving the interlacing property of the poles and zeros in the frequency band where the active damping is significant. It is also important to note that guaranteed stability does not imply guaranteed performance of the control system. Good performance requires the proper sizing and location of the actuator and sensor, to achieve good controllability and observability. This will be reflected by well separated poles and zeros of the open-loop system, leading to wide loops in the root locus plot. The crucial role played by the actuators in the design of the control system was stressed in earlier chapters. The sensors are also technologically critical, for the following reasons: (i) Often, physical problems have natural outputs, because not all the variables (stress, acceleration, velocity,...) are within the range of normally available sensors. For example, vibration amplitudes below one micron may be difficult to measure, while the corresponding accelerations may be easily measured if the structure is stiff, (ii) Apparently similar control problems may involve displacements, velocities and accelerations varying by several order of magnitudes, which may preclude some measurement systems, (iii) The presence of the sensor may significantly alter the dynamic properties of the system, (iv) Even if, in principle, displacements, velocities and accelerations can be computed from one another, this is rarely the case in practice, for noise reasons; besides, only relative displacement and velocity measurements are possible. It is therefore important that the compensator design be consistent with the actual hardware used in the implementation. In this chapter, we describe stable, local controllers, for various types of

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collocated actuator/sensor pairs. They are summarized in Table 5.1.

5.2

Direct Velocity Feedback

Let be a set of velocity measurements, and a set of collocated control forces, and are energetically conjugated (in the sense that their dot product has the dimension of power). Dissipation can be achieved by proportional (negative) feedback on The governing equations are

Equation (5.3) is the equation of motion including the perturbation (dimension and the control force (dimension acting through the influence matrix

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B (Problem P.5.1). The structural damping has been omitted for simplicity. Equation (5.4) is the output equation relating the velocity output to the time derivative of the degrees of freedom (d.o.f) of the structure. The fact that the same matrix B appears in Equ.(5.3) and (5.4) is a consequence of collocation. The control law (5.5) is chosen in such a way that G is positive definite. This guarantees that the power is dissipated, because If G is diagonal, the controller is decentralized because every actuator interacts only with its collocated sensor. Substituting Equ.(5.4) and (5.5) into (5.3), one gets

Therefore, the control forces appear as a viscous damping (electrodynamic damping). The damping matrix is positive semi definite, because the actuators and sensors are collocated. The velocity distributions which belong to the null space of (such that remain undamped; such distributions have vibration nodes at the sensor locations.

5.2.1

Lead compensator

Let us examine the SISO case a little closer. In this case, the matrix B degenerates into a control influence vector From Equ.(2.19), we easily see that, if there is no damping, the open-loop transfer function between the control force and the collocated displacement is

where the sum extends to all the modes. We know that the corresponding poles and zeros alternate on the imaginary axis. It is not desirable to implement the derivative compensator as in Equ.(5.5), because it tends to amplify the noise at high frequency; it is preferable to add

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a low-pass filter, leading to:

A pole has been added at some distance along the negative real axis. This compensator behaves like a differentiator at low frequency The block diagram of the control system is shown in Fig.5.1; a displacement sensor is now assumed and the structural damping is again omitted for simplicity. Typical root locus plots are shown in Fig.5.2 for two values of the low-pass filter corner frequency The closed-loop pole trajectories go from the open-loop poles to the open-loop zeros following branches which are entirely contained in the left half plane. Since there are two more poles than zeros, there are two asymptotes at ±90°. The system is always stable, and this property is not sensitive to parameter variations, because the alternating pole-zero pattern is preserved under parameter variations. At this point, it is appropriate to explain the physical meaning of the fact that the closed-loop poles are located near the zeros of the open-loop system for large values of the gain We know that the imaginary zeros are in fact antiresonance frequencies (i.e. frequencies at which a harmonic excitation produces no displacement where the force is applied); they are identical to the natural frequencies of the modified system in which a support has been added instead of the actuator (Fig.2.4). In fact, as increases from zero, the compensator brings more and more damping into the structure; for larger the closed-loop

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poles travel on the loops and go to the open-loop zeros, the damping decreases and finally vanishes when Then, the controller excites the structure at the anti-resonance frequencies and, as a result, the point of application of the control force remains fixed, corresponding to a nodal point being enforced by the control system in the closed-loop modes: the control system acts as a support (Fig.5.3). The very lightly damped closed-loop modes, which are not excitable from the input, are likely to be excited by any perturbation which does not enter at the input. Other implementations of the Direct Velocity Feedback and the Lead compensator are possible (Problem P.5.2). The root locus shows that large damping can be achieved only if the loops going from the poles to the zeros are wide in the left half plane, which occurs only if the poles and zeros are well separated.

5.3

Acceleration feedback

This section addresses the case in which the output of the system is the acceleration and the control input is a force. Acceleration is often easier to measure than displacement or velocity, particularly when the structure is stiff.

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Direct Velocity Feedback

The easiest way to use the acceleration is to integrate it to obtain the velocity. Thus, Direct Velocity Feedback is obtained with the compensator

The stability properties of this compensator have been examined in the foregoing section. In practice, however, piezoelectric accelerometers use charge amplifiers which behave as high-pass filters [Equ.(3.43)]. This does not significantly affect the results if the corner frequency of the charge amplifier is well below the vibration modes of the structure (Problem P.5.4). Next, we present a compensator involving a second order filter which also enjoys guaranteed stability and exhibits a larger roll-off at high frequency. First, we consider the case of a single degree of freedom oscillator; then, we extend the results to SISO systems with many modes, and to MIMO systems.

5.3.2

Second order filter

The basic idea is to pass the acceleration signal through a second order filter with substantial damping, and generate a force feedback proportional to the output of the filter. For a single degree of freedom oscillator, the equations of the system and the compensator are respectively Structure : Compensator :

The control force per unit mass, is obtained as the product of a scalar gain has the dimension of natural frequency squared) by the output of a second order filter with properties excited by the structure acceleration as input. Although the compensator equation is solved by a computer and does not correspond to a physical system, can be interpreted as the relative displacement of a single d.o.f oscillator with properties seismically excited by the acceleration The block diagram of the control system is shown in Fig.5.4, where represents the accelerometer dynamics (assumed to be perfect). The closed-loop characteristic equation, reads

Using the Routh-Hurwitz criterion (see chapter 11), one can check that the combined system is always stable for (Problem P.5.6). Typical root locus plots are shown in Fig.5.5.

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SISO system with many modes

As we can see from Fig.5.4, the compensator is

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Its effect on a SISO system with many modes can be investigated by substituting Equ.(5.7) in the block diagram of Fig.5.4; the block diagram is that of Fig.5.6; the corresponding root locus is shown in Fig.5.7 for two values of in both cases, is used; in Fig.5.7.a, the frequency of the filter, is taken close to mode 2 while in Fig.5.7.b, it is selected close to mode 1. Comparing the figures, one sees that all the modes are positively damped, but the mode with the natural frequency close to is more heavily damped. Thus, the performance of the compensator relies on the tuning of the filter on the mode that we wish to damp (this aspect may become problematic if the system is subject to changes in the parameters). The maximum achievable damping ratio increases with a value of between 0.5 and 0.7 is recommended. For closely spaced modes, stability is still guaranteed, but a large damping ratio cannot be achieved simultaneously for the two modes; besides, small variations of the filter frequency may significantly change the root locus and the modal damping (Preumont et al.). If several modes must be damped, several compensators may be used in parallel:

where the are tuned on the targeted modes (Fig.5.8). The gains can be determined with some experimentation (in the sequential selection of the gains it is generally easier to start with the filter of higher frequency, because the roll-off of the second order filter reduces the influence of the filter on the frequencies above

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As compared to the Direct Velocity Feedback (5.9), the new compensator has a larger roll-off at high frequency instead of which may decrease the risk of destabilizing high frequency modes. The need for tuning the filter frequency may be a drawback if the natural frequency is not known accurately, or is subject to changes. In the foregoing discussion, we have assumed that the sensors and the actuators are perfect. If the actuator dynamics is taken into account, unconditional stability can no longer be guaranteed. If, for example, a proof-mass actuator is used (with in Equ.(3.8), so that it behaves like a perfect force generator in the frequency range of interest), it can be shown that the closed-loop system becomes unstable for large gains (Problem P.5.5). The effect of the charge amplifier dynamics is left as an exercise (Problem P.5.4).

5.3.4

Multidimensional case

Next, we consider the more general case of a MIMO system with collocated actuator/sensor pairs and an array of second order filters. The structure and controller equations are Structure: Sensors: Controller:

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with the notations and The structural damping has been omitted for simplicity. As before, B is the control influence matrix, indicating the way the control acts on the structure, is the set of accelerations sensed by the collocated sensors. E is a rectangular matrix, which allows us to use more filters than actuators (so that modes can be damped with actuators). The controller equation consists of independent second order filters fed by linear combinations of the accelerations. The control is constructed by multiplying the controller internal variables by the diagonal positive gain matrix G and then by It can be proved that this system is unconditionally asymptotically stable providing the proof is based on Liapunov’s direct method (Sim & Lee).

5.4

Positive Position Feedback

The Positive Position Feedback (PPF) (Goh & Caughey) is appropriate for a structure equipped with strain actuators and sensors; the objective is, once again, to use a second order filter to improve the roll-off of the control system, allowing high frequency gain stabilization.

5.4.1

SISO system

The block diagram is represented in Fig.5.9. As compared to Fig.5.6, the output is now proportional to the displacements (e.g. strain sensor) and a minus sign appears in the controller block (together with the minus sign in the feedback loop, this produces a positive feedback.). Figure 5.10.a and b show the root locus when the controller is tuned on mode 1 and mode 2, respectively. We see that the tuning property of the controller is very similar to that of Fig.5.7 and, even in presence of a feedthrough component, the open-loop transfer function has a roll-off of However, there is a stability limit which is

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reached when the open-loop static gain is equal to 1 (Problem P.5.7). Note that this condition is independent of the damping.

5.4.2

Multidimensional case

The extension to SISO systems with many modes is parallel to that of the acceleration feedback. The modal selectivity is also achieved by tuning the filter frequency on the targeted mode. Several modes can be targeted simultaneously with the form (5.15) (with a negative sign in this case, to produce a positive feedback). As for the acceleration feedback, in the design of the compensator involving several modes, it is generally more convenient to start with the component of larger frequency. The MIMO implementation of the PPF is also very similar to that of the acceleration feedback: Structure: Sensors: Controller:

A necessary and sufficient condition for asymptotic stability can be established using Liapunov’s direct method (see Fanson & Caughey). An application of the

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PPF to the active damping of a plate equipped with piezoelectric actuators and sensors will be examined in chapter 13.

5.5

Integral Force Feedback

Consider the active truss of section 3.7. Each active strut consists of a piezoelectric linear actuator collocated with a force transducer. In this section, we consider the decentralized active damping with local controllers connecting each actuator to its collocated force sensor. The open-loop transfer function between the voltage applied to the piezo and the sensor signal is given by Equ.(3.52) or (3.59). A typical Bode plot of is represented in Fig.3.18; once again, there are alternating poles and zeros along the imaginary axis; there is a feedthrough component and, according to the foregoing discussion, some roll-off is necessary in the active damping compensator D(s) to achieve stability. It is readily established from the root-locus (Fig.5.11) that the positive Integral Force Feedback (IFF):

is unconditionally stable for all values of The negative sign in Equ.(5.24) is combined with the negative sign in the feedback loop (Fig.5.12) to produce a positive feedback. In practice, it is not advisable to implement plain integral control, because it would lead to saturation. A forgetting factor can be introduced by slightly moving the pole of the compensator from the origin to the negative real axis, leading to

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This does not affect the general shape of the root locus and prevents saturation. Note that piezoelectric force sensors have a built-in high-pass filter.

5.5.1

Modal damping

The behaviour of an active truss equipped with a single active strut has been analysed in section 3.7. Combining with the control law (5.24), we find the equations governing the control system: Structure: Sensor: Controller:

where is the unconstrained piezoelectric displacement (proportional to the voltage applied to the piezo, except for the hysteresis), is the relative displacement of the end nodes of the active element, is the force in the active strut and its stiffness. Note that the stiffness matrix K includes the active strut. is the equivalent piezoelectric load. Combining (5.27) and (5.28), we get

Taking the Laplace transform of (5.26) and substituting δ we get the closed-loop characteristic equation

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From this equation, we can deduce the open-loop transmission zeros, which coincide with the asymptotic values of the closed-loop poles as Taking the limit, we get which states that the zeros (i.e. the anti-resonance frequencies) coincide with the poles (resonance frequencies) of the structure where the active strut has been removed (corresponding to the stiffness matrix To evaluate the modal damping, Equ.(5.30) must be transformed in modal coordinates with the change of variables Assuming that the mode shapes have been normalized according to and taking into account that we have

The matrix is, in general, fully populated; if we assume that it is diagonally dominant, and if we neglect the off-diagonal terms, it can be rewritten

where

is the fraction of modal strain energy in the active member when the structure vibrates according to mode was defined by Equ.(3.53)]. Substituting Equ.(5.33) into (5.32), we find a set of decoupled equations

For small we can assume a solution of the form into (5.35) and neglecting the terms of the second order in P.5.9)

substituting we get (Problem

This interesting result tells us that, for small gains, the active damping ratio in a given mode is proportional to the fraction of modal strain energy in the active element. This result is very useful for the design of active trusses; the active struts should be located in order to maximize the fraction of modal strain energy in the active members for the critical vibration modes. is readily available from F.E. softwares. Notice that the damping ratio decreases with the frequency of the mode, as we would expect from the integral shape of the controller. A root locus plot of the closed-loop poles can be obtained as follows. If we denote

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we can write Equ.(5.35) as

or

This equation shows that, for each mode, the closed-loop poles follow the root locus corresponding to the open-loop transfer function

From the comparison with Equ.(5.31), which shows the asymptotic behaviour of the closed-loop system as the zeros at are readily identified as the natural frequencies of the structure when the active member has been removed; the poles are those of the open-loop structure and the integral controller. The root locus is shown in Fig.5.13. It can be shown that the maximum modal damping is given by

and it is achieved for (Problem P.5.10). The preceding result has been established for a single active member; if there are several active members operating with the same control law and the same gain this result can be generalized under similar assumptions. It can be shown that each closed-loop pole follows a root locus governed by Equ.(5.38)

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where the pole is the natural frequency of the open-loop structure and the zero is the natural frequency of the structure where the active members have been removed. The proof is left as an exercise (Problem P.5.11).

5.6 5.6.1

Remarks Controllability, observability

For all the control laws discussed in this chapter, the guaranteed stability is a consequence of the interlacing of the poles and zeros of the structure along the imaginary axis. This property is guaranteed for collocated systems, for any location of the actuator-sensor pair in the structure. However, if the pair is located where the controllability and observability (they always go together for collocated pairs) are weak, the corresponding loops in the root locus are very small and remain close to the imaginary axis; this severely restricts the achievable active damping. It is important to stress that good performance requires good controllability (Problem P.5.12).

5.6.2

Actuator and sensor dynamics

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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 [except the PPF)]. In practice, however, the open-loop transfer function (5.2) becomes where includes the sensor, actuator and the digital controller dynamics. In chapter 3, we saw that the low frequency behaviour of the proof-mass actuator (3.8) and the charge amplifier (3.43) can both be approximated by a second order high-pass filter. The high frequency behaviour of the actuators and sensors can often be represented by a second order low-pass filter

It is easy to see that the two extra poles in bring the asymptotes inside the right half plane and substantially alter the root locus for Figure 5.14 shows the effect of the second order low-pass filter (5.42) on the root locus plot of the Direct Velocity Feedback (5.8) (we have assumed and The active damping is no longer unconditionally stable and always has some destabilizing influence on the high frequency modes. Fortunately, the modes of the structure are not exactly on the imaginary axis, because of the structural damping (Fig.5.15); this allows us to operate the controller with small gains. The control system becomes insensitive to the actuator dynamics if

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is far beyond the cross-over frequency of (5.2). The effect of the low frequency dynamics of a proof-mass actuator on the acceleration feedback is examined in Problem P.5.5

5.7

References

J. N. AUBRUN, Theory of the Control of Structures by Low-Authority Controllers, AIAA J. of Guidance, vol. 3, no. 5, 444–451, Sep.–Oct. 1980. M. J. BALAS, Direct Velocity Feedback Control of Large Space Structures, AIAA J. of Guidance, vol. 2, no. 3, 252-253, 1979. A. BAZ, S. POH & J. FEDOR, Independent Modal Space Control with Positive Position Feedback, Trans. ASME, J. of Dynamic Systems, Measurement, vol. 114, no. 1, 96–103, March 1992. R. J. BENHABIB, R. P. IWENS & R. L. JACKSON, Stability of Large Space Structure Control Systems Using Positivity Concepts, AIAA J. of Guidance and Control, vol. 4, no. 5, 487–494, Sep.–Oct. 1981. J. L. FANSON & T. K. CAUGHEY, Positive Position Feedback Control for Large Space Structures, AIAA Journal, vol. 28, no. 4, 717–724, April 1990. R. L. FORWARD, Electronic Damping of Orthogonal Bending Modes in a Cylindrical Mast Experiment, AIAA Journal of Spacecraft, vol. 18, no. 1, 11–17, Jan.–Feb. 1981. W. B. GEVARTER, Basic Relations for Control of Flexible Vehicles, AIAA Journal, vol. 8, no. 4, 666–672, April 1970. C. G OH & T. K. C A U G H E Y , On the Stability Problem Caused by Finite Actuator Dynamics in the Control of Large Space Structures, Int. J. of Control, vol. 41, no. 3, 787–802, 1985. A. PREUMONT, J. P. DUFOUR & C. MALEKIAN, Active Damping by a Local Force Feedback with Piezoelectric Actuators, AIAA J. of Guidance, vol. 15, no. 2, 390–395, March–April 1992. A. PREUMONT, N. Loix, D. MALAISE & O. LECRENIER, Active Damping of Optical Test Benches with Acceleration Feedback, Machine Vibration, vol. 2, 119–124, 1993. D. SCHAECHTER, Optimal Local Control of Flexible Structures, AIAA J. of Guidance, vol. 4, no. 1, 22–26, 1981. E. S IM & S. W. L EE , Active Vibration Control of Flexible Structures with Acceleration or Combined Feedback, AIAA J. of Guidance, vol. 16, no. 2, 413–415, 1993.

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5.8

Problems

P.5.1 Consider a finite element representation of a, cantilever beam with 5 elements of equal length. Write the influence matrix B [Equ.(5.3)] for (a) a point force actuator at the tip (b) a torque actuator at the tip Check that the sensor output matrix [Equ.(5.4)] for collocated displacement and angular sensor is the transpose P.5.2 Compare the following implementations of the Lead and the Direct Velocity Feedback compensators:

Discuss the conditions under which these compensators would be applicable for active damping. P.5.3 Assuming that the modes are well separated, show that for small gain the Direct Velocity Feedback produces a closed-loop damping ratio

where the mode shapes are normalized in such a way that (Aubrun). [Hint: Use a perturbation method, in the vicinity of and write the closed-loop characteristic equation]. P.5.4 Consider a vibrating structure with a point force actuator collocated with an accelerometer. For the two compensators discussed in connection with the acceleration feedback, analyse the effect of the charge amplifier dynamics [Equ.(3.43)] on the root locus. Show that the stability is no longer guaranteed, but if the sensor dynamics does not affect the control system for small gains. P.5.5 Consider a s.d.o.f. oscillator controlled with a proof-mass actuator; analyse the effect of the actuator dynamics on the active damping by acceleration feedback with a second order filter. Show that if the natural frequency of the actuator is much lower than that of the filter the actuator dynamics has little influence on the root locus for small gains, but that the system becomes unstable for large gains.

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P.5.6 Using the Routh-Hurwitz criterion, show that the acceleration feedback based on the second order filter is always stable for [Start from the closed loop characteristic equation (5.13)] P.5.7 Consider the second order Positive Position Feedback applied to a s.d.o.f. oscillator; using the Routh-Hurwitz criterion, show that the stability limit is reached when the open-loop static gain is equal to 1. P.5.8 Show that for a collocated SISO system without feedthrough, the Positive Position Feedback can be implemented with the following first order controller

Show that for a s.d.o.f. oscillator, the stability condition is (Baz et al.). P.5.9 Consider a truss with an active strut consisting of a linear piezo actuator collocated with a force sensor. Show that, for small gain the Integral Force Feedback produces a closed-loop damping ratio

(Preumont et al.). [Hint: Start from the closed-loop characteristic equation (5.35) and assume P.5.10 Show that the maximum damping achievable in an active truss with the integral force feedback is

and that it is achieved for P.5.11 Consider a truss structure with several identical active members controlled with the same control law (IFF) and the same gain. Making the proper assumptions, show that each closed-loop pole follows a root locus defined by Equ.(5.39), where the natural frequency is that of the open-loop structure and is that of the structure where the active members have been removed. P.5.12 Compare the transfer functions of Fig.3.26 from the point of view of control authority. Comment on the consequences of the error in predicting the zeros.

Chapter 6

Active vibration isolation 6.1

Introduction

There are two broad classes of problems in which vibration isolation is necessary: 1. Operating equipments generate oscillatory forces which can propagate in the supporting structure (Fig.6.1.a).This situation corresponds to that of an engine in a car, or an attitude control reaction wheel assembly in a spacecraft. 2. Sensitive equipments may be supported by a structure which vibrates appreciably (Fig.6.l.b). This situation corresponds to, for example, a telescope in a spacecraft, 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. As already mentioned, this book is focused on the feedback strategies for active isolation; they apply to both deterministic and random disturbances. For deterministic sources such as a rotating unbalance, feedforward control can also be very effective (see Fuller et al., chap. 7). In this chapter, we first consider the single-axis isolation problem; the acceleration feedback (also called “sky-hook” damper) and the force feedback are compared. Next, we consider the general purpose 6-axis isolator based on a Stewart platform. The decentralized integral force feedback of a Stewart platform is examined. The joint pointing/isolation control is also briefly examined. Finally, we consider the application of the sky-hook damper to an active vehicle suspension. 113

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6.2

Passive isolator

First, let us consider the situation of an operating equipment generating a disturbance force (Fig.6.l.a). If the support is fixed, the governing equations are as follows:

In the Laplace domain,

where

and stand for the Laplace transform of respectively and and with the usual notations and The transmissibility of the support is defined in this case as the transfer function between the disturbance force applied to the mass and the force transmitted to the support structure; combining Equ.(6.3) and (6.4), we get

Next, consider the second situation illustrated in Fig.6.1.b; the disturbance is the displacement of the supporting structure (that we call “dirty body”) and the system output is the displacement of the sensitive equipment (referred to as “clean body”). Following a procedure similar to that leading to Equ.(6.5),

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it is easily established (Problem P.6.1) that the transmissibility of this isolation system, defined in this case as the transfer function between the displacement of the dirty body and that of the clean body, is given by

which is identical to Equ.(6.5); the two isolation problems can therefore be treated in parallel. The amplitude of the corresponding FRF (for is represented in Fig.6.2 for various values of the damping ratio We observe that All the curves are larger than 1 for and become smaller than 1 for Thus the critical frequency separates the domains of amplification and attenuation of the isolator. When the high frequency decay rate is that is -40 dB/decade, while very large amplitudes occur near the corner frequency (the natural frequency of the spring-mass system).

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The damping reduces the amplitude at resonance, but also tends to reduce the effectiveness at high frequency; the high frequency decay rate becomes

The design of a passive isolator involves a trade-off between the resonance amplification and the high frequency attenuation; the ideal isolator should have a frequency dependent damping, with high damping below the critical frequency to reduce the amplification peak, and low damping above to improve the decay rate. The objective in designing an active isolation system will be to add a force actuator working in parallel with the spring and dashpot (Fig.6.1.c), that will operate in such a way that there is no amplification below and the decay rate is -40dB/decade at high frequency, as represented in dotted line in Fig.6.2.

6.3

The “sky-hook” damper

Consider the single axis isolator of Fig.6.3. It consists of a soft spring acting in parallel with a force actuator generating a control force the objective is to isolate the clean body from the disturbance applied to the dirty body. With the usual Laplace notations, the equations governing the system are as follows: Dirty body:

Clean body:

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Spring:

In the sky-hook isolator, a feedback link is established between the absolute acceleration of the clean body, and the actuator force with an integral controller (Fig.6.4.a), leading to a

We see that the active control force is proportional to the clean body absolute velocity; this is why this control is called “sky-hook” damper (Fig.6.4.b).

Introducing Equ.(6.10) into (6.8) and using Equ.(6.9), we get

or

Thus, the transmissibility has a corner frequency at its high frequency decay rate is that is -40 dB/decade, and the control gain can be chosen in such a way that the isolator is critically damped the corresponding value of the gain is In this way, we achieve a low-pass filter without overshoot with a roll-off of -40 dB/decade. This transmissibility follows exactly the objective represented in Fig.6.2. Combining Equ.(6.7) to (6.9), the open-loop transfer function of the soft isolator of Fig.6.3 can be written

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The open-loop poles are the natural frequencies of the system without control. The rigid body modes do not appear in the transfer function (6.13) because they are not controllable from The root locus of the closed-loop poles as the gain of the controller increases is shown in Fig.6.5.

In developing the sky-hook damper, we have assumed that the passive part involves only a spring. If we include also a dash-pot of constant Equ.(6.9) must be replaced by

and Equ.(6.11) becomes

which shows that the high frequency roll-off is reduced to -20dB/decade. Thus, better performances will be achieved without passive damping (Problem P.6.2).

6.4

Force feedback

If we look at Equ.(6.8), we see that the acceleration of the clean body is proportional to the total force transmitted by the interface, As a result, the sky-hook damper can be obtained alternatively with the control configuration of Fig.6.6, where a force sensor has been substituted to the acceleration sensor. Equations (6.7) to (6.9) still hold and we must add the sensor equation

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It follows that the open-loop transfer function between the actuator force input and the force sensor output F is

It is identical to Equ.(6.13), except for the constant M relating the total force F to the clean body acceleration. As a result, the root locus of Fig.6.5 also applies to the integral force feedback damper which is totally equivalent to the sky-hook damper. Taking into account the constant M relating the clean body acceleration and the total transmitted force, the transmissibility of the active isolator now reads

which is totally equivalent to Equ.(6.12)(except for the slightly different meaning of The control strategies based on acceleration feedback and on force feedback appear as totally equivalent for the isolation of rigid bodies. However, the force feedback has two advantages which may be significant in some circumstances. The first one is related to sensitivity: force sensors with a sensitivity of are commonplace and commercially available; if we consider a clean body with a mass of 1000 kg (e.g. a telescope), the corresponding acceleration is Accelerometers with such a sensitivity are more difficult to find; for example, the most sensitive accelerometer available in the Bruel & Kjaer catalogue is (model 8318). The second advantage is stability when the clean body is flexible; this is discussed below.

6.5

Flexible clean body

When the clean body is flexible, the behaviour of the acceleration feedback and the force feedback are no longer the same, due to different poles/zeros config-

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urations of the two control strategies. In fact, different sensor configurations correspond to different locations of the zeros in the s-plane. To analyse this situation, consider the clean body with a flexible appendage as represented in Fig.6.7; the nominal numerical values used in the calculations are the mass of the flexible appendage is taken as a parameter to analyse the interaction between the flexible appendage and the isolation system. When 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 6.8 shows the root locus plots for 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 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. In contrast,

when

is large, the root locus plots are reorganized as shown in Fig.6.9 for For force feedback (Fig.6.9.a), the poles and zeros still alternate on

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the imaginary axis, leading to a stable root locus; this property is lost for the acceleration feedback (Fig.6.9.b), leading to an unstable loop for the lower mode. In practice, however, this loop is moved slightly to the left by the structural damping, and the control system can still operate for small gains (conditionally stable).

6.5.1

Free-free beam with isolator

To analyse a little further the situation when the clean body is flexible, consider the vertical isolation of a free-free continuous beam from the disturbance of a dirty body of mass (Fig.6.l0.a). This situation is representative of a large space structure with its attitude control system. Note that the rigid body modes are uncontrollable from the internal force In the numerical example described below, the length of the beam is the mass per unit length is the stiffness of the isolator is and the mass of the dirty body is the stiffness EI of the beam is taken as a parameter. Let be the natural frequencies of the flexible modes of the free-free beam alone (Fig.6.10.b) and be the transmission zeros corresponding to a force excitation and a collocated displacement sensor (or equivalently acceleration). According to section 2.4, are the natural frequencies of the system with an additional restraint at the connecting degree of freedom of the isolator. Because of the collocation (Fig.2.5), the poles and zeros are alternating on the imaginary axis, so that and satisfy

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

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(Fig.6.10.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 of the free-free beam. It can be shown that the residues in the modal expansion of the FRF between the force actuator and the force sensor F are all positive (Problem P.6.4). It follows that the poles and zeros alternate and the following relation holds:

This condition guarantees a fixed, interlacing, pole/zero pattern of the open-

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loop system and the stability of the closed-loop system when a force feedback is used. With an acceleration feedback (sky-hook damper, Fig.6.10.c) the poles are still while the zeros, obtained by enforcing a zero acceleration at the connecting d.o.f. are as for the free-free beam of Fig.6.10.b. This actuator/sensor configuration is no longer collocated, so that no condition similar to (6.19) or (6.20) holds between and When the beam is stiff, the interlacing property is satisfied and the stability is guaranteed, but as the beam becomes more flexible, the values of and decrease at different rates and a pole/zero flipping occurs when they both become equal to the natural frequency of the isolator (Fig.6.11). As a result, the system is no longer unconditionally stable when the flexibility is such that and above. As a conclusion to this section, it seems that the sky-hook damper imple-

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mentation (acceleration feedback) is preferable when the clean-body 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.6.8). On the contrary, the force feedback implementation looks preferable when the clean body is very flexible (e.g. space structure), to benefit from the guaranteed stability.

6.6

6 d.o.f. isolator

In the foregoing sections, we have analysed a single axis active isolator which combines a -40 dB/decade attenuation rate in the roll-off region with no overshoot at the corner frequency. To fully isolate two rigid bodies with respect to each other (e.g. an engine from the body of a car, or a telescope from the carrier spacecraft) we would need six such isolators judiciously placed, that could be controlled either in a centralized or (more likely) in a decentralized manner. For a number of space applications, generic multi-purpose 6 d.o.f. isolators have been developed with a standard Stewart platform architecture (Fig.6.12). These general purpose isolators can be used to isolate noisy components such as a reaction wheel assembly (RWA) from the rest of the spacecraft (the wheels apply low frequency torques to control the attitude of the spacecraft, but in doing so they also generate undesirable high frequency forces and torques shaking

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the base of the structure; this situation is depicted in Fig.6.1.a). Alternatively, they can also be used to isolate quiet subsystems such as a telescope or other sensitive payloads from the carrier spacecraft (Fig.6.1.b). The isolation objectives are illustrated in Fig.6.13; the ideal 6 d.o.f. isolation platform should transmit the attitude control torque within the bandwidth of the attitude control system and filter all the high frequency disturbances above

The Stewart platform of Fig.6.12 uses 6 identical active struts arranged in a mutually orthogonal configuration connecting the corners of a cube as indicated in Fig.6.14. The triangular base plate connects the nodes 1,3,5 and the payload plate connect the nodes 2,4,6. This cubic architecture is interesting because it provides a uniform control capability in all directions, a uniform stiffness in all directions, and it minimizes the cross-coupling amongst actuators (being orthogonal to each other). It also tends to minimize the spread of modal frequencies. Figure 6.14 depicts the geometry of the hexapod and the numbering system for the nodes and the struts; the basic frame has its origin at node 0; the reference (or payload) frame has its origin at the geometrical center of the hexapod, noted as node 8, and is perpendicular to the payload plate; the orientation of and is shown in Fig.6.14. If we neglect the flexibility of the struts and the bending stiffness of the flexible joints connecting the struts to the base and payload plates, the dynamic equations of motion can be obtained from rigid body dynamics. If the base is fixed and the payload is an axisymmetrical rigid body of mass with the principal axes of inertia aligned with principal moment

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of inertia and with the center of mass located at an offset distance from the geometrical center, along the vertical axis the dynamic equation (for small rotations) of the isolator is

where is the vector describing the small displacements and rotations in the payload frame is the vector of active control forces in strut 1 to 6 respectively, M and K are the mass and stiffness matrices of the isolator, given by

where is the stiffness of one strut. B is the influence matrix of the active control force:

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(B is the transposed of the Jacobian between the small displacements and rotations and the length increments of the legs; L is the length of the struts). The proof of Equ. (6.21) to (6.24) is left as an exercise (Problem P.6.5). The natural frequencies of the isolator and its payload can be obtained by solving the eigenvalue problem on the left side of Equ. (6.21); the or “bounce” mode and the or “torsional” mode are decoupled, with natural frequencies given by

where and is the radius of gyration normalized to the strut length. For most cases, and The remaining four modes are lateral bending coupled with shear; their natural frequencies occur in two identical pairs, solutions of the characteristic equation

where is the radius of gyration normalized to the strut length and is the center of mass offset normalized to the strut length. Note that if the center of mass is at the geometric center and if and the hexapod will have 6 identical natural frequencies, all equal to

6.7

Decentralized control of the 6 d.o.f. isolator

Let us assume that each leg is equipped with a force sensor as in Fig.6.6; since each leg consists of a passive spring of stiffness in parallel with a force actuator, the output equation reads

where is the vector containing the 6 force sensor outputs, is the vector of leg extensions from the equilibrium position, is the strut stiffness and is the vector of active control forces. Taking into account

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the relationship between the leg extension and the payload frame displacement, we have

Once again, we note that the same matrix B appears in Equ.(6.21) and (6.28) because the sensors and actuators are collocated. Using a decentralized integral force feedback with constant gain we find the controller equation

Combining Equ.(6.21), (6.28) and (6.29), we obtain the closed-loop equation

and, taking into account that

If we transform into modal coordinates, and take into account the orthogonality relationships (2.8) and (2.9), the characteristic equation is reduced to a set of uncoupled equations

Thus, every mode follows the characteristic equation

or

The corresponding root locus is shown in Fig.6.15. It is identical to Fig.6.5 for a single-axis isolator; however, unless the 6 natural frequencies are identical, a given value of the gain 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.

6.7.1

Remarks

(i) The foregoing model is based on the assumption that there is no structural damping and that the only contribution to the stiffness matrix is the axial

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stiffness of the struts, so that In practice, however, the spherical joints at the connection between the legs and the base plates of the Stewart plateform are replaced by flexible connections with high longitudinal stiffness and low bending stiffness (such flexible tips can be seen in Fig.3.5), in such a way that the stiffness matrix has an additional contribution, from the elastic joints. Thus, and the closed loop equation becomes

According to this equation, the transmission zeros, which are the asymptotic solutions of Equ.(6.34) as are no longer at the origin but are solutions of the eigenvalue problem

This shift of the zeros may have a substantial influence on the practical performances of actual Stewart platforms. (ii) The foregoing model assumes that the legs of the Stewart platform have no mass. In fact, the magnetic circuit of the voice coil actuator is fairly heavy and the local dynamics of the legs may interfere with that of the Stewart platform, resulting in a reduced attenuation of the isolation system.

6.8

Pointing control

At this stage, it is worth pointing out that, in addition to providing vibration isolation, a Stewart platform may also be used for pointing of, for example, a

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telescope towards a target star (the reference pointing accuracy of the Hubble Space Telescope is but future missions are likely to require better accuracy by at least one order of magnitude). If are the desired pointing angles, the decoupling matrix D relating to the corresponding leg extensions in the nominal position is obtained from the appropriate columns of the Jacobian

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Figure 6.16 shows a possible control strategy. Note that the six-leg Stewart platform has some built-in redundancy which can be exploited to account for actuator failure. Figure 6.17 shows the block diagram of the combined control strategy for pointing and isolation.

6.9

Vehicle suspension

Figure 6.18 shows a quarter-car model of a vehicle. Although this 2 d.o.f. model is too simple for performing a comprehensive analysis of the ride motion, it is sufficient to gain some insight in the behaviour of passive and active suspensions in terms of vibration isolation (represented by the body acceleration suspension travel and road holding (represented by the tyre deflexion Typical numerical values used in the simulation reported later are also given in the figure (taken from Chalasani, 1984). The stiffness corresponds to the tyre; the suspension consists of a passive part and an active one, assumed to be a perfect force actuator acting as a sky-hook damper in this case (the active control force is applied on both sides of the active device, to the body and to the wheel of the vehicle). Figure 6.19 shows the FRF from the roadway vertical velocity to the car body acceleration for the passive suspension alone; several values of the damping coefficient c are considered. The first peak corresponds to the body resonance (also called sprung mass resonance) and the second one to the wheel resonance (unsprung mass resonance). The passive damping cannot control the body resonance without reducing the isolation at higher frequency. Next, a sky-hook damper is added. Figure 6.20 shows the corresponding FRF from to for various values of the control gain. Note that

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133

the body resonance can be damped without reducing the isolation at higher frequency but the peak in the FRF corresponding to the wheel resonance cannot be changed by the active control. Figure 6.21 compares the amplitude of the FRF and for two values of the gain. This figure shows that the absolute velocity of the body rolls-off much faster (i.e. has much lower frequency components) than the relative velocity This point will be further discussed when we examine the semi-active suspension devices which try to emulate the sky-hook damper by acting on the flow parameters of the damper acting on the relative velocity.

6.10

References

R. M. C HALASANI , Ride Performance Potential of Active Suspension Systems, Part1: Simplified Analysis Based on a Quarter-Car Model, ASME Symposium on Simulation and Control of Ground vehicles and Transportation systems, Anaheim, CA, Dec. 1984. C. R. F ULLER , S. J. E LLIOTT & P. A. N ELSON , Active Control of Vibration, Academic Press, 1996.

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Z. J. G ENG & L. S. H AYNES , Six Degree-of-Freedom Active Vibration Control Using the Stewart Platforms, IEEE Transactions on Control Systems Technology, vol. 2, no. 1, 45–53, March 1994. C. E. KAPLOW & J. R. VELMAN, Active Local Vibration Isolation Applied to a Flexible Space Telescope, AIAA J. Guidance and Control, vol. 3, no. 3, 227–233, May–June 1980. D. K ARNOPP , Design Principles for Vibration Control Systems Using SemiActive Dampers, Trans. ASME Journal of Dynamic Systems, Measurement and Control, vol. 112, 448–455, Sep. 1990. D. C. KARNOPP & A. K. TRIKHA, Comparative Study of Optimization Techniques for Shock and Vibration Isolation, Trans. ASME, Journal of Engineering for Industry, series B, vol. 91, no. 4, 1128–1132, 1969. R. A. LASKIN & S. W. SIRLIN, Future Payload Isolation and Pointing System Technology, AIAA J. Guidance and Control, vol. 9, no. 4, 469–477, July–Aug. 1986. J. E. Mc INROY, G. W. NEAT & J. F. O’BRIEN, A Robotic Approach to Fault-Tolerant, Precision Pointing, IEEE Robotics & Automation Magazine, pp. 24–31, Dec. 1999. J. SPANOS, Z. RAHMAN & G. BLACKWOOD, A Soft 6-Axis Active Vibration Isolator, Proceedings of the American Control Conference, Seattle, WA, pp. 412–416, June 1995. D. STEWART, A Platform with Six Degrees of Freedom, Proc. Instn. Mech. Engrs, vol. 180, no. 15, 371–386, 1965–66. D. THAYER, J. VAGNERS, A. VON FLOTOW, C. HARDHAM & K. SCRIBNER, Six-Axis Vibration Isolation System Using Soft Actuators and Multiple Sensors, Proc. of Annual American Astronautical Society Rocky Mountain Guidance and Control Conference (AAS-98-064), pp. 497–506, 1998.

6.11 P.6.1

Problems

Consider the passive isolator of Fig.6.1.b. Find the transmissibility of the isolation system. P.6.2 Analyse the effect of the passive damping on the transmissibility of the sky-hook damper. Is it beneficial or detrimental?

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P.6.3 Write the differential equations governing the system of Fig.6.7 in state variable form. Using the following values of the parameters: Write the open-loop frequency response for the acceleration feedback and force feedback configurations and draw the corresponding poles/zeros pattern. In both cases draw the root locus for an integral controller. Do the same for investigate the effect of structural damping. P.6.4 Consider the modal expansion of the open-loop FRF of the system of Fig.6.10.d. Show that the residues are all positive and that this results in alternating poles and zeros. P.6.5 Consider the cubic Stewart platform of Fig.6.14: (1) Using energetic considerations, show that the mass matrix M, the stiffness matrix K, and the control force influence matrix B are given by Equ.(6.22) to (6.24). (2) Show that the natural frequencies are given by Equ.(6.25) and (6.26). [Hint: write the Jacobian relating the leg extension to the motion of the payload frame The stiffness matrix is given by where is the stiffness of one strut.] P.6.6 Consider a Stewart platform of cubic architecture supporting an axisymetric payload such that (1) Select the spring stiffness in order to achieve a corner frequency of about 10Hz. (2) Calculate the corresponding natural frequencies. (3) Select the controller gain to achieve a reasonable damping ratio for the 6 modes. P.6.7 Consider the active isolation problem described in the figure below, where is assumed to be a white-noise process. Show that the optimal control minimizing the objective function

is

where the coefficients and depend on the weighting coefficients and This is the “sky-hook” damper (Karnopp & Trikha, 1969). (The solution of this problem requires a prior reading of Chapters 7 and 9). P.6.8 (1) Develop a state space model of the vehicle suspension of Fig.6.18 (use and as state variables). (2) Plot the amplitude of the FRF of the passive suspension for various

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values of the damping coefficient Compare with Fig.6.19. (3) Consider the active suspension with a sky-hook damper control law. Plot the amplitude of the FRF for the nominal parameters and various values of the gain Compare the frequency content of the absolute velocity of the body with that of the relative velocity P.6.9 Consider the fully active suspension of Fig.6.18 Using the LQR theory, design a state feedback which minimizes the performance index

The terms involved in the state penalty of this performance index represent respectively the vibration isolation (comfort), the suspension travel and the road holding (unsprung mass velocity). The fourth one is the control force. Try various values for the weighting coefficients and Compare the fullstate feedback to the sky-hook damper (Chalasani, 1984). (The solution of this problem requires a prior reading of Chapters 7 and 9).

Chapter 7

State space approach 7.1

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 9. In the state variable form, a linear system is described by a set of first order linear differential equations

with the following notations: state vector, input vector, output vector, = system noise, = measurement noise, 137

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A = system matrix, B = input matrix, C = output matrix, D = feedthrough matrix, E = system noise input matrix. For SISO systems, both and are scalar functions. Equation (7.1) 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 emphasized in chapter 5. The system noise 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 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.

State space approach

7.2

139

State space description

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.

7.2.1

Single degree of freedom oscillator

First, consider the familiar second order differential equation of a single degree of freedom oscillator:

This second order equation implies that there will be two state variables; they can be selected as

With this choice, Equ.(7.3) can be rewritten as a set of two first order equations

or

This equation explicitly shows the system and input matrices A and B. If one measures the displacement, the output matrix is C = (1 0) while if one uses a velocity sensor, 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.(7.5), one gets

The output and feedthrough matrices are respectively

and

The choice (7.4) of the state variables is not unique and it may be convenient to make another choice :

The advantages are that both state variables express a velocity, and that the free response trajectories in the phase plane are slowly decaying spirals (Problem P.7.1). With this choice, Equ.(7.3) can be rewritten

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This form is sometimes preferred to (7.6) because the system matrix is dimensionally homogeneous.

7.2.2

Flexible structure

Next, consider the multi degree of freedom vibrating system governed by the set of second order differential equations

where is the input force influence matrix, indicating the way the input forces act on the structure. Equation (7.10) is expressed in physical coordinates. If one defines the state vector as it can be rewritten in state space form as

The foregoing state variable form is acceptable if M is invertible. Calculating is straightforward for a lumped mass system. However, Equ.(7.11) 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.(7.10) 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 (7.10) can be transformed into modal coordinates following the procedure described in chapter 2. If we perform the change of variables the governing equations in modal coordinates read

with the usual notations damping) and

(modal mass), (modal frequencies). In this equation,

(modal

where is the matrix of the mode shapes and is the input force influence matrix. The columns of contain the modal amplitudes at the actuators location, so that the vector represents the work of the input forces on the various mode shapes. Here, exactly as for the single degree of freedom oscillator, one can define the state variable as either

or

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141

In the latter case, the state space equation reads

This form is similar to Equ.(7.9), except that diagonal matrices are substituted for scalar quantities. Note that, in most cases, the size of the state vector in Equ(7.14) (which is twice the number of modes included in the model) is considerably smaller than in Equ.(7.11). The output equations for a set of sensors distributed according to the influence matrix are as follows displacement sensors

velocity sensors accelerometers

where the columns of 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 : and

7.2.3

Inverted pendulum

As another example of state space description, consider the inverted pendulum with a point mass at a distance from the support, as represented in Fig.7.2.a. The horizontal displacement of the support O is the input of the system. The differential equation governing the motion is

This equation also applies to more complicated situations where the pendulum consists of an arbitrary rigid body (Fig.7.2.b) if is taken as the equivalent length of the pendulum, defined as

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 is the moment of

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inertia about O). This can be readily established using the Lagrange or the Newton/Euler equations (Problem P.7.2). Equation (7.18) can be linearized about as When the displacement of the hinge is the input, it is convenient to make a change of variable and introduce the absolute position of the point mass, rather than the angle With this transformation, Equ.(7.20) becomes

where is the natural frequency of the pendulum. Note that, in general, the coordinate is different from that of the center of mass C in Fig.7.2.b. Using the state variables we find the state space equation

If the output of the system is the angle of the pendulum, the output and feedthrough matrices are

7.3

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.(7.1) and (7.2), because they have nothing to do with the present discussion.

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143

Laplace transforming the state equation

gives and, assuming zero initial conditions,

and, from the output equation (7.2), we have

or with In the scalar (SISO) case, 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.(7.6), in which the output is the displacement We readily obtain

This result could have been obtained more easily by Laplace transforming Equ.(7.3). Similarly, applying Equ.(7.28) to the inverted pendulum, using (7.23) and (7.24), we find

Once again, this result is straightforward from Equ.(7.20). 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

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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:

Using this theorem with

instead of A, and – B instead of B, we get

Comparing with Equ.(7.28), we see that the transfer function of a SISO system can be rewritten

Now, we demonstrate that, as in the classical pole-zero representation of transfer functions, the denominator and the numerator of Equ.(7.32) explicitly show the poles and zeros of the system.

7.3.1

Poles and zeros

The poles are the values such that the free response of the system from some initial state has the form Substituting this in the free response equation, one gets

or 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 which is the denominator of (7.32). The zeros of the system are the values of such that an input

applied from appropriate initial conditions The state vector has the form if the following condition is fulfilled:

produces a zero output,

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145

that is, if Under this condition, the output is

Therefore, having

requires that

The two conditions (7.37) and (7.39), can be cast in compact form as

The values of for which this system of homogeneous equations has a non-trivial solution are the zeros of the system. They are solution of

which is the numerator of (7.32). From Equ.(7.40), 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 other initial conditions would produce a transient output that would gradually disappear if the system is asymptotically stable.

7.4

Pole placement by state feedback

The idea in the state space approach is to synthesize a full state feedback

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 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.7.3), and leave the state reconstruction until later. A system of the order involves state variables. Therefore there are feedback gains that can be adjusted independently. Since there are poles in the system, either real or complex conjugate, they can be assigned arbitrarily by proper choice of the 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

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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 (7.42) into the system equation (7.1) yields

The closed-loop system matrix is A – BG. Its eigenvalues are the closed-loop poles; they determine the natural behaviour of the closed-loop system and are solutions of the characteristic equation

The state feedback design consists of selecting the gains so that the roots of (7.44) are at desirable locations. If the system is controllable, arbitrary pole locations (either real or complex conjugate pairs) can be enforced by choosing the in such a way that Equ.(7.44) is identical to

The gain values

7.4.1

are achieved by matching the coefficients of (7.44) and (7.45).

Example: oscillator

As an example, consider the single degree of freedom oscillator of Equ.(7.6)

The poles of the open-loop system are represented in Fig.7.4.a. We want to design a state feedback

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147

such that the poles of the closed-loop system are moved to the locations indicated in Fig.7.4.b. From Equ.(7.43), the closed-loop system matrix, A – BG, reads

and the characteristic equation

The desired characteristic equation is

Comparing Equ.(7.48) and (7.49), one sees that the desired behaviour will be achieved if the gains are selected according to

These equations indicate that the controller will act as a spring (of stiffness , and damper (of damping constant . This is a general observation about the state feedback which consists of a generalization of the PD control.

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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 (see section 10.4.1). 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: 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 10). 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 closed-loop poles is discussed in the next section.

7.5

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 analyse this problem for SISO systems; a more general formulation will be examined in chapter 9. We seek a state feedback (7.42) minimizing the performance index

where

is the controlled variable, defined by

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 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, , may or may not be the actual output of the system. If this is the case, H = C.

State space approach

7.5.1

149

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

where

is the open-loop transfer function between the input and the controlled variable. Equation (7.54) defines a root locus problem in terms of the scalar parameter the weighting factor of the control cost in the performance index. Note that and affect Equ.(7.54) 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.(7.53). Draw the symmetric root locus in terms of the parameter only the part of the locus in the left half plane.

and consider

Select a value of 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 9). Check with simulations that the corresponding control effort is compatible with the actuator. If not, return to the previous step.

7.5.2 Inverted pendulum To illustrate the procedure, consider the inverted pendulum described by the state space equation (7.23). If we adopt the absolute displacement as the controlled variable, H = (1 0), and the transfer function (7.55) reads

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The symmetric root locus is represented in Fig.7.5. The closed-loop poles consist of a complex conjugate pair on the left branch of the locus, Note that the transfer function used in the regulator design is different from the open-loop transfer function (7.30) between the input and the output variable (the latter is the tilt angle rather than the absolute position Using instead of in the performance index will not only keep the pendulum vertical, but will also prevent it from drifting away from its initial position. Once appropriate pole locations have been selected, the control gains G can be calculated as indicated in the previous section, by matching the closed-loop characteristic equation (7.44) to the desired one (7.45):

Equating the coefficients of the various powers in s provides

The open-loop system has a stable pole at and an unstable one at The cheapest optimal state feedback which stabilizes the system, obtained for on the root locus, simply relocates the unstable pole at (thus, the closed-loop system has two poles at The corresponding characteristic equation (7.59) is

State space approach

151

and the optimal gains are

7.6

Observer design

The state feedback (7.42) 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 In this way, the state feedback can be implemented on the reconstructed state, rather than on the actual state 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 (7.1) and (7.2).

The following form is assumed for the full state observer (also called Luenberger observer).

The first part of Equ.(7.64) simulates the system and the second contribution (innovation) uses the information contained in the sensor signal; 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, converges to zero. Combining Equ.(7.62) to (7.64), we find the error equation with the initial condition 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 observer poles can be assigned arbitrarily in the complex plane by choosing the values of the independent gains (observability will be discussed in chapter 10). The procedure for pole assignment is identical to that used for the regulator design : If a set of poles has been selected, the gains can be computed by matching the coefficients of the characteristic equation

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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 (7.62), (7.63) and the observer equation (7.64). This amounts to assuming perfect modelling. In practice, however, the actual system will be substituted for Equ.(7.62) and (7.63) while the observer equation (7.64) will be integrated numerically on 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.(7.64) must be taken as instead of this leads to the same error equation (7.65). If noise terms are included as in Equ.(7.1) and (7.2), the error equation becomes

The plant noise appears as an excitation; so does the measurement noise 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.(7.65), 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. The observer equation (7.64) is based partly on the system model, and partly on the measurement error, 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 , 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.

State space approach

7.7

153

Kalman Filter

The Kalman Bucy Filter (KBF) will be studied in detail in chapter 9. 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

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 where

is the transfer function between the plant noise

and the output

and the parameter is the ratio between the intensity of the plant noise and that of the measurement noise (noisy measurements correspond to small ). 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 When proper pole locations have been selected on the left half part of the locus, the gains can be computed by matching the coefficients of the characteristic equation (7.66) to that of the desired one (7.67). If none of the values of provides a desirable set of poles, another E matrix can be selected and the procedure repeated. In the LQR approach, the regulator design involves the symmetric root locus of the open-loop transfer function between the input and the controlled variable while in the KBF approach, the observer design is based on the transfer function between the system noise and the output .

If one assumes that the plant noise 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

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

7.7.1

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.(7.23) and (7.24), the transfer function between the noise an the output Equ.(7.74), is

Thus, although the output variable is different from the controlled variable is proportional to used in the regulator design [Equ.(7.56)]; this is because there is no feedthrough component in as there is in [Equ.(7.30)]. As a result, the root locus used in the regulator design can also be used for the observer design (Fig.7.6). 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 to where according to the rule of thumb mentioned before (Problem P.7.5). Finally, assume that the noise enters the system according to From Equ.(7.23) and (7.24), the transfer function between the noise and the

State space approach

155

output is readily obtained as

The corresponding symmetric root locus is shown in Fig.7.7, assuming One notices that one of the poles goes to the zero at as 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.

7.8

Reduced order observer

With the full state observer, the complete state vector is reconstructed from the output variable . 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).

156

7.8.1

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Oscillator

Consider the single degree of freedom oscillator governed by Equ.(7.6). The output measurement is the displacement The velocity equation is

We seek a first order observer governed by the following equations

where is the estimated velocity and and are free parameters; they are selected in such a way that the error equation governing the behaviour of is asymptotically stable, with an appropriate decay rate

Equations (7.78)-(7.80) give the error equation

It can be made identical to Equ.(7.81) if the coefficients are selected according to

From Equ.(7.81),

7.8.2

is the eigenvalue of the observer; it can be chosen arbitrarily.

Inverted pendulum

The same procedure can be applied to the inverted pendulum described by Equ.(7.23) and (7.24). The displacement is obtained from the output measurement and the input as

The velocity equation is

The velocity observer has again the form

State space approach

and the error equation

157

is readily written as

It becomes identical to (7.81) if the coefficients satisfy

The reduced order observer is therefore

where the only remaining parameter is the eigenvalue of the observer, which can be selected arbitrarily. Note that appears also as the feedthrough component of the measured variable in Equ.(7.93), leading to a direct effect of the measurement noise on the reconstructed velocity (amplified by 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.

7.9

Separation principle

Figure 7.8 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 closedloop equations are :

The complete system has state variables. If one uses the reconstruction error, as state variable instead of the reconstructed state vector one can write the closed-loop system equation as

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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.

7.10

Transfer function of the compensator

The transfer function of the compensator can be obtained from Equ.(7.94). It reads

The poles of the compensator are solutions of the characteristic equation

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 7.9 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

State space approach

159

the closed-loop properties; the compensator is always of the same order as the system.

7.10.1

The two-mass problem

Let us illustrate this point on the two-mass problem of Fig.4.3. Defining the state vector we can write Equ.(4.3) and (4.4) as

For a non-collocated displacement sensor,

The open-loop transfer function is given by Equ.(4.7). If the open-loop system has a pair of poles at open-loop transfer function is

where the compact notation

M = 1 and (flexible mode) the

is used for

Using the output 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 It is shown in Fig.7.10. Once the regulator and observer poles have been selected on the left side of the locus, the corresponding gains can be calculated by matching the coefficients of the characteristic equations (7.44) and (7.66) to the desired ones; for the pole locations indicated in the figure, we get

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The resulting compensator is

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 to produce a notch filter. With this nominal compensator, it is possible to draw a conventional root locus plot for (Fig.7.11), which describes the evolution of the poles of the closed-loop system when the scalar gain varies from 0 to The pole locations on this root locus for are, of course, the same as those selected on the symmetric root locus (Fig.7.10). To assess the robustness of the control system, Fig.7.12 shows the root locus plot when the natural frequency is changed from to 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 In that case, a pole-zero flipping occurs between the pole of the flexible mode and the zero of the notch filter (Fig.7.13). As we already stressed in chapter 4, the pole-zero flipping changes the departure angles from the poles and the arrival angles at the zeros by 180°, transforming the stabilizing loop of

State space approach

161

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Fig.7.11 into a destabilizing one as in Fig.7.13. This lack of robustness is typical of state feedback and notch filters.

7.11

References

A. E. BRYSON JR., Some Connections Between Modern and Classical Control Concepts, ASME, Journal of Dynamic Systems, Measurement, and Control, vol. 101, 91–98, June 1979. R. H. CANNON & D. E. ROSENTHAL, Experiment in Control of Flexible Structures with Noncolocated Sensors and Actuators, AIAA Journal of Guidance, vol. 7, no. 5, 546–553, Sep.–Oct. 1984. G. F. FRANKLIN, J. D. POWELL & A. EMANI-NAEMI, Feedback Control of Dynamic Systems, Addison-Wesley, 1986. T. KAILATH, Linear Systems, Prentice-Hall, 1980. H. KWAKERNAAK & R. SIVAN, Linear Optimal Control Systems, Wiley, 1972. D. LUENBERGER, An Introduction to Observers, IEEE Trans. Autom. Control, AC-16, pp. 596–603, Dec. 1971. D. LUENBERGER, Introduction to Dynamic Systems, Wiley, 1979.

State space approach

7.12

163

Problems

P.7.1 Consider the oscillator described by Equ.(7.9). For non-zero initial conditions, sketch the free response in the phase plane Show that the image point rotates clockwise along a spiral trajectory. Relate the decay rate of the spiral to the damping ratio. P.7.2 Show that the inverted pendulum of Fig.7.2 is governed by Equ.(7.18) to (7.20), where is the displacement of the support point. P.7.3 Consider an inverted pendulum similar to that of Fig.7.2, but mounted on a cart of mass M and controlled by an horizontal force applied to the cart. If stands for the horizontal displacement of the cart, show that, for small the governing equations can be approximated by

Write the equations in state variable form. P.7.4 For the inverted pendulum of Problem P.7.3, assuming that the full state is available, find a feedback control that balances the stick and keeps the cart stationary near P.7.5 Consider the inverted pendulum of Fig.7.2. Using the same procedure as for the regulator in section 7.5.2, find the analytical expression of the observer gains which locates the observer poles at P.7.6 Consider an inverted pendulum with (Fig.7.2). (a) Compute the transfer function of the compensator such that the regulator and the observer poles are respectively at and (b) Draw a root locus 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.

Chapter 8

Analysis and synthesis in the frequency domain 8.1

Gain and phase margins

Consider the root locus plot of Fig.4.5. Any point in the locus is solution of the characteristic equation Therefore, we have

The locus crosses the imaginary axis at the point of neutral stability. Since the following relations hold at the point of neutral stability:

Returning to the Bode plot of Fig.4.6, one sees that changing amounts to moving the amplitude plot along the vertical axis; the point of neutral stability is obtained for when the amplitude curve is tangent to the 0 dB line, near the frequency above which the phase exceeds –180°. The system is stable if the algebraic value of the phase of the openloop system is larger than for all frequencies where the amplitude 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 open loop transfer function exceeds when The corresponding frequency is called the crossover frequency. 165

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The situation described above corresponds to a stable open-loop system the gain and phase margins are decreasing functions of 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 interval it becomes unstable when the gain decreases below the threshold (see Problem P.7.6). These cases can be handled with the Nyquist criterion.

8.2

Nyquist criterion

8.2.1

Cauchy’s principle

Consider the feedback system of Fig.8.1, where stands for the combined open-loop transfer function of the system and the compensator (shorthand for in the previous section). The conformal mapping transforms the contour in the plane into a contour in the plane. One of the properties of the conformal mapping is that two intersecting curves with an angle in the plane map into two intersecting curves with the same angle in the plane. Assume is written in the form

Analysis and synthesis in the frequency domain

For any

on

167

the phase angle of G(s) is given by

where and are the phase angles of the vectors connecting respectively the zeros and the poles to From Fig.8.1, it can be seen that, if there are neither poles nor zeros within the contour does not change by when goes clockwise around The contour will encircle the origin only if the contour contains one or more singularities of the function Because the contour is travelled clockwise, one pole of within produces a phase change of that is one counterclockwise encirclement of the origin by Conversely, a zero produces a clockwise encirclement. Thus, the total number of clockwise encirclements of the origin by is equal to the number of zeros in excess of poles of within the contour This is Cauchy 's principle.

8.2.2

Nyquist stability criterion

The foregoing idea provides a simple way to evaluate 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 The contour is selected in such a way that it encircles the whole right half plane as indicated in Fig.8.2. If there are poles on the imaginary axis, indentations are made as indicated on the figure, to leave them outside The closed-loop transfer function is

and the closed-loop poles are solutions of the characteristic equation

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If

This equation shows that the poles of are the same as those of the open-loop transfer function Let P be the [known] number of unstable poles of the open-loop system. On the other hand, the zeros of are the poles of the closed-loop system; we want to evaluate their number within From Cauchy’s principle, if we consider the mapping the number of clockwise encirclements of the origin by when goes clockwise along

is Now, instead of considering the encirclements of the origin by completely equivalent to consider the encirclements of –1 by number of unstable poles of the closed-loop system is given by

it is Thus, the

where N is the number of clockwise encirclements of –1 by when s follows the contour and P is the number of unstable poles of the open-loop system. All physical systems without feedthrough are such that as (with feedthrough, goes to a constant value). As a result, only the part of the plot corresponding to the imaginary axis must be

Analysis and synthesis in the frequency domain

169

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 If there are poles of on the imaginary axis and if the indentations are made as indicated in Fig.8.2, the poles are outside the contour. According to Equ.(8.2) and (8.3), each pole contributes with an arc at infinity and a rotation of that is 180º clockwise. Returning to the system of Fig.4.5 and 4.6, its Nyquist plot is shown in Fig.8.3: 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 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.4.5. Since the instability occurs when encircles –1, the distance from 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.8.4). 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. Since the closed-loop transfer function is uniquely determined by [Equ.(8.4)], loci of constant magnitude and of constant phase can be drawn in the complex plane they happen to be circles. The M circles are shown in Fig.8.5; the larger magnitudes correspond to smaller circles near – 1. Thus, the maximum amplification corresponds to the smallest circle, tangent to the Nyquist plot of it is reached for a

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frequency close to the crossover frequency (where the Nyquist plot crosses the unit circle). This is why there is a direct relationship between the maximum amplification and the phase margin PM. In most cases, prevents overshoot of F (Fig.8.6). The of the control system is defined as the frequency corresponding to an attenuation of -3 dB in the closed-loop transfer function. Since the bandwidth can be taken equal to the crossover frequency for design purposes.

8.3

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.8.7). It consists of a plot of the vs. of the open-loop transfer function 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

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171

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 the Nyquist plot, in order to assist in the design, it is customary to draw the loci of constant amplitude M and constant phase N of the closed-loop system (Fig.8.7); they are no longer circles.

8.4

Feedback specification for SISO systems

Consider the SISO feedback system of Fig.8.8, where is the reference input, the output, the disturbance and the sensor noise (unlike the disturbance, the sensor noise does not directly affect the output).

8.4.1

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 The closed-loop transfer function is

Its sensitivity to parameter changes is related to that of the open-loop transfer function G by

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or or

is called the sensitivity function and 1 + G the return difference. Equation (8.10) states that, in the frequency range where 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, goes very near –1 at crossover, and F becomes much more sensitive than G.

8.4.2

Tracking error

Referring to Fig.8.8, the governing equations are

Analysis and synthesis in the frequency domain

173

or

The tracking error is

From Equ.(8.11) and (8.12), we see that good tracking implies that The sensitivity function range where the command

must be small in the frequency and the disturbance are large.

The closed-loop transfer function frequency range where the sensor noise is large.

must be small in the

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

8.4.3

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.8.9). The general objective of feedback is to achieve sensitivity reduction, good tracking and disturbance rejection at low frequency with a sensor signal which has been contaminated with noise at high frequency From the foregoing section, this is translated into the following design constraints:

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where 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 and

where 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:

Equivalently, we may require that the GH curve remains outside a particular M circle from the point We shall see later that stability places constraints on the slope of near crossover (in the frequency range near

8.4.4

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, and its model It can be considered either as additive:

where is a positive function defining the upper bound to the additive uncertainty (including the parameter changes and the neglected dynamics), or

Analysis and synthesis in the frequency domain

175

multiplicative:

where is a positive function defining an upper bound to the multiplicative uncertainty. is usually small at low frequency where the model is accurate, and becomes large at high frequency, due to the neglected dynamics of the system (Fig.8.10). Because defines only the magnitude of the uncertainty, it can be associated with any phase distribution; therefore, it clearly defines a worst case situation. Evaluating or 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 P.8.4).

8.4.5

Robust performance and robust stability

The design objectives in the face of uncertainties are that the conditions enumerated in section 8.4.3 must be fulfilled for the perturbed system. Referring to Fig.8.11, the stability of the system will be guaranteed in presence of uncertainty if the distance to instability, is always larger than the uncertainty The stability robustness condition is therefore

This guarantees that the number of encirclements will not be altered by any additive uncertainty bounded by for any phase distribution. Similarly, since condition (8.19) can be written in terms of the multiplicative uncertainty as

or

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

and Equ.(8.21) is reduced to

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 (8.20) by gives an alternate form of the stability condition:

Now, if we want to achieve the performances with the perturbed system, Equ(8.14) must be replaced by

This equation will be satisfied if

A sufficient condition is

The design tradeoff for is explained in Fig.8.12. The shaded region at low frequency is excluded for robust performance, and that at high frequency for robust stability.

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177

Considering the vicinity of the crossover frequency (Fig.8.13), we can make the following observations: At crossover, becomes

and the stability robustness condition (8.20)

This means that accepting a magnitude error at crossover requires a phase margin of 60º. From this, one can anticipate that the bandwidth of the closed-loop system cannot be much larger than the frequency where At crossover, the return difference is related to the phase margin by

This establishes a direct connection between the phase margin and the sensitivity and the disturbance rejection near crossover. We shall now address the conflict between the quality of the loop near crossover (good phase margin) and the attenuation rate of GH.

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Bode gain-phase relationships

8.5

Figure 8.12 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, this is not as simple as that, because the closed-loop stability of the nominal system requires that the phase remains larger than 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 If the amplitude diagram has a constant slope corresponding to poles in a log-log diagram the phase is In the general case, the phase at a frequency is given as a weighted average of the gain slope at all frequencies, but with a stronger weight in the vicinity of

where

and the weighting function

is defined by

Analysis and synthesis in the frequency domain

is strongly peaked near far from that of a Dirac impulse,

179

(Problem P.8.5); its behaviour is not too so that

This relation is very approximate, unless the slope of is nearly constant in the vicinity of it applies almost exactly if the slope is constant over two decades. In this case, one readily sees that

The first integral indicates that a large phase can only be achieved if the gain attenuates slowly. It follows that the rolloff rate in the region near crossover must not exceed and it must often be smaller than this, in order to keep some phase margin.

Integral # 2 Assume one pole

is stable and has an asymptotic rolloff corresponding to more than at infinity). According to section 8.4.2, the sensitivity function represents the fraction of the command or of the disturbance which is transmitted into the tracking error. must be small in the frequency range where or are large. However cannot be small everywhere, because the second Bode integral states that

This relation states that there is as much sensitivity increase as there is decrease 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 (see also Problem P.8.9). If the open-loop system has unstable poles the second integral becomes

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.(8.30).

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Integral # 3 The third integral states that some reshaping of can be performed within the working band (normalized to without affecting the phase outside the working band. This arises from:

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 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.8.14.

Integral # 4 This integral says that the greater the phase lag, the larger will be the feedback in the working band:

In particular, if two loop shapes and have the same high frequency behaviour, but has a greater phase lag than has a larger magnitude

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in the working band than

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 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.

8.6

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.8.14). Figure 8.15 shows a Nichols chart of the desired behaviour: One wants to keep the feedback constant and as large as possible within the working band in reduced frequency), then reduce it while keeping gain and phase margins of dB and (shaded rectangle in Fig.8.15). It is therefore logical to keep the phase constant and compatible with the phase margin:

The compensator design is thus reduced to that of enforcing an open-loop transfer function (system plus compensator) to map the two segments of

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Fig.8.15. The result is shown in Fig.8.16: The open-loop transfer function with a constant gain in the working band and a constant phase lag outside the working band is

This transfer function has an ideal behaviour at low frequency. However, the rolloff at high frequency is directly related to the phase margin. Improved high frequency noise attenuation requires higher rolloff. Because of the first integral, we know that if we simply add one segment with a higher slope to the amplitude diagram of Fig.8.16, it will be reflected by an additional phase lag near crossover, which is incompatible with the stability margin. The cure to that is to have first a flat segment 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 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.8.17. The reader can compare it to the design tradeoff of Fig.8.12. 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.

Analysis and synthesis in the frequency domain

8.7

183

Non-minimum phase systems

In the previous sections, we assumed that had 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.(8.30). In this section, we consider the effect of right half plane zeros. To begin with, assume that has a single right half plane zero at We can write

where

is the all-pass function

and is the minimum phase transfer function obtained by reflecting the right half plane zero into the left half plane no longer has a singularity in the right half plane]. The Nyquist diagram of is represented in Fig.8.18; follows the unit circle clockwise, from at to at takes its name from the fact that for all and have the same magnitude for all and the same phase at but for their phase differ by that of at and at We have seen in section 4.8 that flexible structures with non-collocated actuators and sensors do have non-minimum phase zeros. If they lie well outside

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the bandwidth of the system they do no harm, because the corresponding all-pass function brings only very little phase lag near crossover. On the contrary, if the non-minimum phase zero lies at a distance comparable to the bandwidth, and this means that in the design, the phase angle of cannot exceed (without any phase margin) and, consequently, its falling rate cannot exceed instead of for a minimum phase system. The situation can easily be generalized to an arbitrary number of right half plane zeros. If has right half plane zeros at it can be written in the form (8.36) where is minimum phase and is the all-pass function

A typical pole-zero pattern of is shown in Fig.8.19. It is easy to see that for all and that the phase angle decreases by when goes from 0 to or when 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 is always greater than that of by that of This is why 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 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 P.8.7).

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A pure time delay T has a transfer function

Just as the all-pass function, it has a constant magnitude and a phase lag which increases linearly with the frequency. A pure delay can always be approximated as closely as needed by an all-pass network approximants), so that the above conclusions also apply (Problem P.8.8).

8.8

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 abilty 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 Lead, Lag, PI, PID compensators. Before that, we define the system type which controls the steady state error and tracking ability.

8.8.1

System type

The system type measures the ability of the control system to track polynomials. Consider a unity feedback (Fig.8.8) with the open-loop transfer function

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The transfer function between the reference input

and the error

According to the final value theorem, the steady state error,

is

is given by

The number of integrators in the open-loop transfer function is called the system type. If the system is type 0: a constant actuating signal maintains a constant output

If

is a step function,

and

The output follows a step input with a steady state error proportional to (1 + If there is one integrator in and the system is type 1: a constant actuating signal maintains a constant rate of change of the output. A step input is tracked with zero steady state error and a ramp input, is tracked with a constant error

Similarly, a system with two integrators in the open-loop transfer function is type a constant actuating signal 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, produces a constant error

is called static error coefficient of the system of type

From Equ.(8.40),

If a polynomial input is applied to a type system, there is a zero steady state error if If there is a constant steady state error defined by the static error coefficient if the reference signal cannot be tracked. From the foregoing discussion, we note that long term errors associated with persistant disturbances can be zero only if the poles of the disturbance (one pole at for a step, one double pole at for a ramp, etc...) are included among the poles of the open-loop transfer function this is called the internal model principle.

Analysis and synthesis in the frequency domain

8.8.2

187

Lead compensator

The aim of the cascade compensator (Fig.8.20) is to alter the open-loop transfer function from to in order to improve the characteristics of the closed-loop system. The simplest one is the Proportional plus Derivative (PD):

This compensator increases the phase above the breakpoint 1/T. Therefore, selecting 1/T slightly below the crossover frequency of 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

The phase lead is still significant but the high frequency amplification is considerably reduced (Fig.8.21). The maximum phase lead occurs at

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

8.8.3

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

The system type is increased by 1. The amplification at low frequency is obtained at the expense of a phase lag of –90º below the breakpoint In

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189

order to minimize the phase lag at crossover, the breakpoint must be substantially below the crossover frequency

8.8.4

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

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 behaviour, in order to improve the steady state error characteristics.

8.8.5

PID compensator

The Proportional plus Integral plus Derivative (PID) compensator consists of cascaded PI and PD compensators (Fig.8.21)

The integral part aims at reducing the static error, while the PD part increases the phase near crossover. Reasonable breakpoint frequencies are

If the increasing behaviour 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

8.9

References

H. W. BODE, Relations Between Attenuation and Phase in Feedback Amplifier Design, Bell System Technical Journal, July 1940. H. W. BODE, Network Analysis and Feedback Amplifier Design, Van Nostrand, N-Y, 1945.

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J. J. D’Azzo & C. H. H OUPIS, Feedback Control System Analysis McGraw-Hill, second ed., 1966.

Synthesis,

J. J. DISTEFANO, A. R. STUBBERUD & I. J. WILLIAMS, Feedback and Control Systems, Shaum’s Outline Series, McGraw-Hill, 1967. J. C. DOYLE & G. STEIN, Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis, IEEE Transaction on Automatic Control, vol. AC-26, no. 1, 4–16, Feb. 1981. G. F. FRANKLIN, J. D. POWELL & A. EMANI-NAEMI, Feedback Control of Dynamic Systems, Addison-Wesley, 1986. I. M. HOROWITZ, Synthesis of Feedback Systems, Academic Press, 1963. G. J. KISSEL, The Bode Integrals and Wave Front Tilt Control, AIAA Guidance, Navigation & Control Conference, Portland Oregon, 1990. B. J. LURIE & P. J. ENRIGHT, Classical Feedback Control (with MATLAB), Marcel Dekker Inc., 2000. J. M. MACIEJOWSKI, Multivariable Feedback Design, Addison-Wesley, 1989. R. E. SKELTON, Model Error Concepts in Control Design, Int. J. of Control, vol. 49, no. 5, 1725–1753, 1989.

8.10

Problems

P.8.1 Consider an inverted pendulum with (Fig. 7.2). (a) Compute the transfer function of the state feedback compensator such that the regulator and the observer poles are respectively at and (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.8.2 Draw the Nichols chart plot for the two-mass problem of section 4.4. P.8.3 Consider a direct velocity feedback applied to a structure with three flexible modes at and the modal damping is uniform, 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 to The second pair of zeros are fixed to 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.

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191

P.8.4 Consider a simply supported beam excited by a point force at and provided with a displacement sensor at The nominal system is such that and the modal damping is uniform Compute an estimate of the transfer function Next, consider the following modelling errors: (a) Modal truncation: the modal expansion (2.47) is truncated after 4, 6 and 10 modes. (b) Quasi-static correction: same as (a), but start from the result of problem P.2.5. (c)Parametric uncertainty: the model is based on and (d)Errors in the mode shapes are simulated by using in the model. For each situation, compute the transfer function of the model, the additive and the multiplicative uncertainty, respectively and (gain and phase). Comment on the adequacy of the unstructured uncertainty to represent the various situations. P.8.5 Plot the weighting function appearing in the first Bode integral (8.28). P.8.6 Consider a SISO system with the following performance specifications:

Using the first Bode integral, discuss the feasibility of the design. P.8.7 Consider a non-minimum phase system with a real zero at Following the Bode Ideal Cutoff, draw the magnitude and phase diagrams for the corresponding minimum phase system which produce a gain margin of dB and a phase margin of for Do the same for P.8.8 Show that at low frequency, the nonrational function can be approximated by the following all-pass functions of increasing orders:

are called the Padé approximants of (Ts 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.]

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P.8.9 Consider a stable, minimum phase system with a high frequency attenuation rate larger than Using a Nyquist diagram and geometrical arguments, show that there is always a limit frequency above which there is an amplification of the disturbances.

Chapter 9

Optimal control 9.1

Introduction

The optimal control approach for SISO systems has already been introduced in chapter 7. 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.

9.2

Quadratic integral

Consider the free response of an asymptotically stable linear system

The quadratic expression is often used to measure the distance from the equilibrium, Asymptotic stability implies that as and that the quadratic integral

converges, for any semi positive definite weighting matrix Q. Its value depends on the initial state alone. To compute J, consider the decay rate of the positive quadratic form where P is the positive definite solution of the matrix Liapunov equation

193

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Substituting

and using Equ(9.3), one gets

Since Q is semi positive definite, this result proves that V(t) is a monotonically decaying function [it is indeed a Liapunov function for the system (9.1)]. Integrating Equ.(9.4) from 0 to provides the desired result:

where P is solution of the Liapunov equation (9.3).

9.3

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 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 10). We seek a linear state feedback with constant gain

such that the following quadratic cost functional is minimized

where

is semi positive definite and R is strictly positive definite J has two contributions, one from the states and one from the control The fact that R is strictly positive definite expresses the fact that any control has a cost, while implies that some of the states may be irrelevant for the problem at hand. If a set of controlled variables is defined,

a quadratic functional may be defined as

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195

This situation is equivalent to the previous one with In a similar way, one often uses the form 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 (9.6) to (9.8) is where P is the symmetric positive definite solution of the algebraic Riccati equation 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 is observable (by we mean a matrix H such that Under these conditions, the closed-loop system

is asymptotically stable. From Equ.(9.5), the minimum value of the cost functional is given by

where

is the solution of the Liapunov equation of the closed-loop system

Substituting

we get

This equation is identically satisfied by where P is solution of Equ.(9.12). Thus, the minimum value of the cost functional is related to the solution P of the Riccati equation by

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 Since the latter is, in general, not directly available, it has to be reconstructed. 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

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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 represents the total (kinetic plus strain) energy in the system. Usually will be chosen where 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.

9.4

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 is a vector white noise process of zero mean, we have

and the covariance matrix reads

where is symmetric semi positive definite and is called the covariance intensity matrix, but this is often abbreviated to covariance matrix. If the components of the random vector are independent, is diagonal. The corresponding power spectral density matrix is

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.(9.17)], 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. Consider the stationary random response of a linear time invariant system to a white noise excitation of covariance intensity matrix

It can be shown that the covariance matrix of the steady state response,

satisfies the Liapunov equation

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197

(see e.g. Bryson & Ho). This equation is the dual (adjoint) of Equ.(9.3); 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 Since

we can write

where we have used the fact that the expected value applies only to random quantities. The covariance matrix X is the solution of Equ.(9.21). This result can be written alternatively

where P is solution of Equ.(9.3); the proof is left as an exercise (Problem P.9.1).

9.4.1 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 (9.17) of the covariance matrix involves two separate times and for a stationary process, the covariance matrix is a function of the difference (delay) On the contrary, the covariance matrix involved in Equ.(9.20) and (9.21) is not a function of X is the particular value of the covariance function for it is the vector generalization of the mean square value.

9.5 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

Find the control function of all information from the past, that minimizes the performance index

The solution of the problem is a constant gain linear state feedback

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where P is solution of the Riccati equation

We note that the solution does not depend on the noise intensity matrix 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

and the performance index (9.25) can be rewritten

From Equ.(9.23), where

is solution of the Liapunov equation of the closed-loop system

This equation is identical to (9.15). Substituting the gain matrix from (9.27), we can readily establish that it is identically satisfied by the solution P of the Riccati equation (9.28). Thus, and the minimum value of the performance index is

This result is the dual of (9.16) for the deterministic case.

9.6

Asymptotic behaviour of the closed-loop

In the particular case where there is a single controlled variable and a single input, the cost functional (9.10) is reduced to (7.52):

We know from section 7.5 that the closed-loop poles are the stable roots of the characteristic equation

where

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199

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

Equ.(9.34) becomes

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 of or their mirror image in the left half plane 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. As the penalty on the control cost decreases, branches go from the poles to the left half plane zeros if the system is minimum phase, or to their mirror image, if the zeros are located in the right half plane. The remaining branches go to infinity. Their asymptotic behaviour can be evaluated by ignoring all but the highest power in in the characteristic equation (9.34). From (9.36), we get

or

The stable solutions lie on a circle of radius in a configuration known as a Butterworth pattern of order (Fig.9.1). 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

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200

remain slow, they will not be noticeable on the controlled variable, because the open-loop zeros are also zeros of the closed-loop system [see Equ.(8.4)]. Extensions to MIMO systems with square open-loop transfer matrix can be found in Kwakernaak & Sivan.

9.7

Prescribed degree of stability

In the previous section, we have seen that as 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 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 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 (Fig.9.2). The idea is to use a modified cost functional

where and

is a positive constant. If one defines the modified variables the system equation in terms of and becomes

and the performance index is

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201

Equations (9.39) and (9.40) define a classical LQR problem in terms of the solution is a constant gain feedback

and

with where

is solution of the Riccati equation

The closed-loop system

is asymptotically stable. If one back substitutes

and

in Equ.(9.41), one gets

Thus, the optimal control is a constant gain linear feedback. Besides, since and is asymptotically stable, we know that as approaches infinity, at least as fast as The prescribed degree of stability has been achieved by solving a classical LQR problem for the modified system with the system matrix instead of A.

9.8

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 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. Figure 9.3.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.9.3.b; the corresponding open-loop transfer function is

It is possible to draw a Nyquist plot of and this allows us to define the gain and phase margins as in the previous chapter.

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The interesting result is that it can be demonstrated (see Anderson & Moore, p.68) that the LQR satisfies the following inequality.

This implies that the Nyquist plot of always remains outside a disk of unit radius, centered on –1 (Fig.9.4). Simple geometric considerations show that the LQR has a phase margin PM > 60°, and a infinite gain margin. The first result comes from the observation that any open-loop transfer function with a phase margin less than 60° 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.9.4.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

PM > 60°

For the situation depicted in Fig.9.4.a, the phase is close to 90° above crossover; as a result, the gain roll-off rate at high frequency is at most –20dB/decade.

9.9

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 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.

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Consider the system

where and are uncorrelated white noise processes with zero mean and covariance intensity matrices V and W, respectively. From chapter 7, we know that without noise, a full state observer of the form

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, and the error equation is the following

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It shows that both the system noise and the measurement noise 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.

9.9.1

Covariance of the reconstruction error

Comparing Equ.(9.50) with Equ.(9.19) to (9.21), we see that the steady state reconstruction error has zero mean, and a covariance matrix given by the Liapunov equation:

where we have used the fact that

and

are uncorrelated. It can be rewritten

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.

9.10

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 where is a vector of arbitrary coefficients. There is a universal choice of K which makes J minimum for all

where P is the covariance matrix of the optimal observer, solution of the Riccati equation This solution has been obtained as a parametric optimization problem for the assumed structure of the observer given by Equ.(9.49), but it is in fact optimal. The minimum variance observer is also called steady state Kalman-Bucy filter (KBF).

Optimal control

9.11

205

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

where

and are uncorrelated white noise processes with intensity matrices and V > 0. Find the control such that the cost functional

is minimized. The solution of this problem is a linear, constant gain feedback

where G is the solution of the LQR problem and is the reconstructed state obtained from the Kalman-Bucy filter. Combining Equ.(9.56), (9.59) and (9.50), we obtain the closed-loop equation

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 (9.49). The compensator equation has exactly the same form as that of Fig.7.9, 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.

9.12

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 9.1. 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 that minimizes the performance index

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for the system The solution is with where P is solution of

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

9.13

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 behaviour 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

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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 7.2.2). In what follows, the subscript refers to the controlled modes, which are included in the control model, and the subscript 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.9.5); 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

A perfect knowledge of the controlled modes is assumed. The full state observer is

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and the state feedback The interaction between the control system and the residual modes can be analysed by considering the composite system formed by the state variables where The governing equation is

This equation is the starting point for the analysis of the spillover (Fig.9.5). The key terms are and They arise from the sensor output being contaminated by the residual modes via the term (observation spillover), and the feedback control exciting the residual modes via the term (control spillover). Equ.(9.66) shows that if either or the eigenvalues of the system remain decoupled, that is identical to those of the regulator the observer and the residual modes They are typically located in the complex plane as indicated in Fig.9.6. 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 and 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 and Since the stability margin of the residual modes is small, even a small

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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 P.9.6).

9.13.1

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, and the contamination of the sensor output by the residual modes (observation spillover, 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:

where the weighting matrix W allows us to penalize some specific modes. This amounts to using the modified control weighting matrix

This control weighting matrix penalizes the excitation 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

The extra contribution to the covariance intensity matrix indicates to the filter that the measurement noise has the spatial shape of the residual modes is the covariance matrix of the observation spillover if 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 pinpoints 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.

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9.14

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Loop Transfer Recovery (LTR)

In section 9.8, we have seen that the LQR has guaranteed stability margins of GM = 1/2 to and PM > 60° 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 is the nominal plant noise intensity matrix, the KBF is designed with the following plant noise intensity matrix

where 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 (9.70), 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 it can be proved (Doyle & Stein) that, for square, minimum phase open-loop systems the loop transfer function from the control signal to the compensator output (loop broken at the input of the plant, as indicated in Fig.9.7) tends to that of the LQR:

As a result, the LQG/LTR recovers asymptotically the margins of the LQR as

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Note that The loop breaking point at the input of the plant, as indicated by × in Fig.9.7, is a reasonable one, because this is typically one of the locations where the uncertainty enters the system. 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 60° (it may reach 180° 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 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 [ in Equ.(7.72)] (Problem P.9.7).

9.15

Integral control with state feedback

Consider a linear time invariant system subject to a constant disturbance

If we use a state feedback to stabilize the system, there will always be a non-zero steady state error in the output Increasing the gain G would 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.(9.72) by leading to the augmented state vector With the state feedback

the closed-loop equation reads

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If G and

are chosen in such a way that they stabilize the system, we have

which means that the steady state error will be zero edge of the disturbance

9.16

without knowl-

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.

9.16.1

Frequency-shaped cost functionals

According to Parseval’s theorem, the cost functional of the LQR, Equ.(9.10), can be written in the frequency domain as

where and are the Fourier transforms of and and indicate the complex conjugate transposed (Hermitian). Equ.(9.78) shows clearly that the weighing matrices and S do not depend on meaning that all the frequency components are treated equally. Next, assume that we select frequency dependent weighing matrices

Clearly, if the shaping objectives are to produce a P+I type of controller and to increase the roll-off, we must select to put more weight on low frequency, to achieve some integral action, and to put more weight on high frequency,

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to attenuate the high frequency contribution of the control. Examples of such functions in the scalar case are

where the corner frequencies and and the exponent are selected in the appropriate manner. Typical penalty functions are represented in Fig.9.8. Likewise, a narrow-band disturbance can be handled by including a lightly damped oscillator at the appropriate frequency in Equation (9.78) can be rewritten

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 and with and being scalar functions. If we introduce the modified controlled variable

and control we get the frequency independent cost functional

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or, in the time domain,

This cost functional refers to the augmented system of Fig.9.9, including input filters on all input channels and output filters on all controlled variables. If a state space realization of these filters is available (Problem P.9.9), the complete system is governed by the following equations: Structure

Output filter [state space realization of

Input filter [state space realization of

These equations can be combined together as

with the augmented state vector

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and the notations

The state feedback is obtained by solving the LQR problem for the augmented system with the quadratic performance index (9.85). Notice that, since the input and output filter equations are solved in the computer, the states and are known; only the states of the structure must be reconstructed with an observer. The overall architecture of the controller in shown in Fig.9.10. It can be shown that the poles of the output filter (eigenvalues of ) appear unchanged in the compensator (Problem P.9.11); this property can be used to introduce a large gain over a narrow frequency range, by introducing a lightly damped pole in (Problem P.9.10).

9.16.2 Noise model As an alternative to the frequency-shaped cost functionals, loop shaping can be achieved by assuming that the plant noise has an appropriate power spectral density, instead of being a white noise. Thus, we assume that is the output of a filter excited by a white noise at the input. If the system is governed by

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and the plant noise is modelled according to

where is stable and is a white noise (Problem P.9.12). The two sets of equations can be coupled together to form the augmented system

or, with

and the appropriate definitions of

and

Since and 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

with the appropriate matrices and V. In Equ.(9.101), note that the filter dynamics is not controllable from the plant input, but this is not a problem provided that 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 and the input and output matrices and However, in contrast to the previous section, the poles of do not appear unchanged in the compensator (Problem P.9.13) and this technique may lead to difficulties for the rejection of narrow-band perturbations (Problem P.9.14).

9.17

References

B. D. O. ANDERSON & J. B. MOORE, Linear Optimal