Spatial Filtering for the Control of Smart Structures: An Introduction

Spatial Filtering for the Control of Smart Structures James E. Hubbard, Jr. Spatial Filtering for the Control of Sma...

Author: James E. Hubbard

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Spatial Filtering for the Control of Smart Structures

James E. Hubbard, Jr.

Spatial Filtering for the Control of Smart Structures An Introduction

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Prof. James E. Hubbard, Jr. University of Maryland 100 Exploration Way Hampton, VA 23666-6147 USA [email protected] [email protected]

ISBN 978-3-642-03803-7 e-ISBN 978-3-642-03804-4 DOI 10.1007/978-3-642-03804-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009938110 © Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To the Young Dragons – James, Drew and Jordan and The Fair Princess – Adrienne and The Ruler of the Kingdom– The Alpha and Omega

I understood that to grow a dream You need more than the one I was. You need the Believe of childhood The Do of Youth and the Think of Experience. - A Tale of Wonder, Wisdom and Wishes - Susan V. Bosak

An Historical Prologue and Preface

What follows is my personal perspective on early events that played a significant role in the formation of the field now known as Smart Structures. It is by no means meant to be all inclusive or definitive in any way, but merely an account of personal experiences that ultimately lead to the development of the material contained and presented herein. On March 23, 1983 then President Ronald Reagan announced his intentions to develop a new system to reduce the threat of nuclear attack and end the strategy of mutual deterrence in an address to the nation entitled, Address to the Nation on Defense and National Security. The system he proposed became known as “Star Wars,” after the popular movie, because it was meant to provide a protective shield over the nation from space. His speech mobilized the entire nation on a research and development path toward this end. Investigations were conducted into new areas such as space based radar, large aperture antennae and large flexible mirror concepts. These proposed systems represented an entirely new class of structures that proved to provide new challenges in materials, structures, control systems and modeling. For example antennae needed to monitor large areas of real estate in the continental United States required apertures on the order of 100 m. This coupled with the hefty cost of launch to space, on the order of $10,000 per pound, resulted in the design of light weight, highly flexible, lightly damped structures. Analysis of such structures revealed some never before seen characteristics such as very high modal densities, large numbers of paired modes due to the symmetries associated with the designs, lightly damped modes and concomitant large order models. It became clear that the research community and the academic community in particular needed to develop new tools and techniques to cope with the issues associated with these Large Space Structures (LSS). During this period Dr. Tony Amos, then a Program Manager with the Air Force Office of Scientific Research (AFOSR) began holding a series of invitation only workshops to discuss these systems, associated problems and potential solutions. The list of invitee’s included members of government, academia and the private sector who were all active in this area of research. Senior, mid-career and junior researchers from diverse fields that encompassed structural vibrations, active control, fluid dynamics, applied mathematics and more were in attendance. The group vii

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included a number of well-known, distinguished scholars such as Professor Leonard Meirovitch in the field of vibrations, Professor Holt Ashley the field of aeroelastic structures and unsteady aerodynamics, and Professor Michael Athans in the field of control theory. This was during a period in which multidisciplinary collaborations were rare and it quickly became apparent to the group that we lacked a common nomenclature for discussion. A simple example involved terminology to describe structural damping. Some spoke of loss factor, some of damping ratio and others of “Q” factor. This of course led to numerous philosophical debates and vociferous discussions on the fundamentals required for characterization and performance assessment for LSS. This group met regularly for several years and culminated with the formation of an annual meeting at Virginia Polytechnic and State University hosted by Professor Meirovitch entitled VPI & SU/AIAA Symposium on Dynamics and Control of Large Structures in Blacksburgh, Virginia. It became clear over time that LSS were lightly damped structures that could have demanding performance tolerances such as the shape or profiles required for antennae in space based radar applications. It also became clear that these systems could benefit from advanced active control techniques for damping augmentation and performance enhancement. In the ensuing years a number of test platforms were constructed to allow researchers to gain first hand experimental knowledge of this new class of structures. Prominent among these was the NASA Hoop Column Antenna; a Langley Research Center conceived design for increased sensitivity to ground or space-based signals. The antenna consisted of a deployable central column and a 15-m hoop, stiffened by cable into a structure with a high tolerance surface and offset feed location. The surface was been configured to have four offset parabolic apertures, each about 6 m in diameter, and made of gold plated molybdenum wire mesh. Vibration analysis of this structure yielded modal densities that had not routinely been encountered before revealing 70 significant modes over a bandwidth of just 4.1–6.2 Hz. Conventional control synthesis techniques often lead to large order state space models and controllers, ill-conditioned matrices that required inversion and sometimes needed to account for spatially distributed nonlinearities. Traditionally engineers modeled and designed such systems using linear, time invariant, lumped parameter methods and techniques for control. This yielded a system of low order ordinary differential equations (typically with constant coefficients) that could be readily analyzed using modern computational and linear algebra tools. When applied to large order systems new issues of model reduction, stability, physical realizability etc. began to surface. This suggested to me at the time that these structures represented true continuums that displayed both temporal and spatial dynamics and should be modeled and controlled using techniques appropriate for Distributed Parameter Systems (DPS). Distributed Parameter Systems may involve parameters that are time varying and distributed over certain spatial domains. The dynamic behavior of these systems is governed by partial differential equations, integral, or integro-differential, and occasionally by more general functional equations. Because of the fundamental nature of such systems (all physical systems have spatial extent), and the importance of applications areas, the study of DPS has had the attention of mathematicians and control theorist for many years. The dichotomy was that while much had been developed in


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this area by way of mathematics, very little if any control systems were built because the resulting designs could not be physically realized. It was at this time that I became aware of the important work that was being done by R.L. Forward and C.J. Swigert in so-called “electronic damping” during the late seventies and early eighties. These were two Air Force researchers that had been experimenting with using lead zirconium titanate (PZT) to damp optical structures with good results. Their approach of using such exotic materials as actuators for active vibration control was quite unconventional and novel. These materials were lightweight and used very little power and hence appeared to be ideal for application to large space structures. Unlike conventional actuation devices which applied control authority at a single point in space; it appeared that these materials represented actuators that applied control forces which were distributed over space and time and hence were characteristic of DPS. The merging of this class of actuators with DPS “plants” for control seemed a natural undertaking. A single actuator could cover an entire structure and provide a relatively low order controller that was physically realizable. In 1985 Thomas Bailey, a Masters Thesis student of mine at MIT, built and tested such a system. The structure consisted of a steel cantilevered beam with tip mass, covered by a thin sheet of polyvinylidene fluoride (PVDF) as an active damper for vibration control. Both the structure and the actuator were modeled as DPS and energy based control techniques (Lyapunaov’s 2nd method) were used to synthesize an “electronic damper”. The result was a demonstration of a truly adaptive structure which could significantly increase its damping when subjected to an outside disturbance. The experiment received much interest from the DPS community and Professor H.T. Banks of Brown University who was well versed in the issues of DPS control subsequently contacted me to discuss the implications of my experiment. This began an odyssey of lectures and seminars around the country, encouraged by Professor Banks, to initiate collaborations in this area. These included a visit to the Institute for Computer Applications in Science and Engineering (ICASE), the Air Force Rocket Propulsion Laboratory (RPL) and presentations to the International Federation of Automatic control (IFAC). These lectures and seminars help shaped my ideas and the material presented in this textbook. I am particularly grateful to Professors John Brown, Gary Rosen and Steve Gibson for the insights that they provided into the nature of continuum systems. As time progressed several notable experimental test platforms became available. Dr. Alok Das for example established the RPL Experiment, which was a flexible satellite test bed located in my laboratory at the Charles Stark Draper Laboratory. Dr. Jer-nan Juang established a cantilevered beam testbed at the NASA Langley Research Center and his work and contributions there were highly regarded. These platforms and others help move the development of hardware implementation rapidly along. Dr. Francis Moon and his then student C.K. Lee pioneered the application of PVDF sensors to both 1-D and 2-D structures. Dr. Edward Crawley and his students produced seminal papers on the use of piezo crystals for the control of LSS. Professor Andrew Von Flowtow developed a novel means of resistor shunting of piezoelectric crystals to produce an elegant solution to the active damping of flexible

x


beams. Professor Amr Baz pioneered the use of active materials to develop active constrained layer (ACLAD) damping solutions and Professor Alison Flatau broadened the application of electrostrictive materials to this class of problems. Professor Chris Fuller extended the application of active materials to control sound radiation from vibrating structures. Dr. Dan Inman developed techniques for sensing and actuation using a single transducer. There were many strong contributions by others to the state of application; too many to due justice here, but those listed had a personal and significant impact on my thinking and work presented here. In 2001 Drs. Alok Das and Ben K. Wada chronicled these contributions in an SPIE Milestone Series of Selected Papers on Smart Structures for Spacecraft. These papers form the basis for what has now become simply the field of Smart Structures. This was also a time of much activity in the development of Modern Robust Control Theory and major developments were taking place in the design, synthesis and realization of temporal filters for the control of LSS. Little work was being done however toward a structured design of the associated spatial filters needed for the control of such plants. Issues such as sensor and actuator placement were being addressed on an ad hoc basis, treated separately from overall control system synthesis. In actual practice a significant amount of information is needed to describe large scale systems. Traditional State Space approaches lead to the need for large numbers of sensors and actuators to identify and control such structures. The spatially distributed/continuum nature of vibratory structures makes it difficult to apply modern lumped parameter control philosophy and techniques. While there exists a substantial amount of technical literature on the control of DPS, there are still relatively few applications or practical implementations of the theory. Modal representations are commonly employed to succinctly approximate a structures behavior. This representation is of course complete when all terms of the expansion are included. Dr. Mark Balas in a series of seminal papers demonstrated the practical limitations of the then current techniques and the unique challenge that such structures posed to the controls engineer. Often for practical implementation when one must truncate the modal expansion, it is then difficult to determine the number of modes required to accurately model the structure, and to reconcile the location of sensors and actuators and to address overall system stability issues. Computational limitations can also necessitate the need for truncation and model order reduction. Reduced-order models have been shown to suffer from control and observation spillover effects. Control and observation spillover can cause closed loop instability for even a simple flexible beam problem that is otherwise open loop stable. The work presented in this textbook addresses the issue of the design and implementation of distributed parameter control schemes which exploit both spatially distributed sensing and actuation through the use of modern smart material technology. The merging of DPS with distributed parameter transducers leads to simple, realizable control system designs. It is hoped that this text will provide a significant reference for practicing professionals, students and researchers in the area of transducer design using smart materials for smart structures.


xi

Finally I would be remiss if I didn’t acknowledge the contributions of my many students over the years that have contributed to the development of the techniques presented in this book. Thomas Bailey developed the basic tenants for using spatially distributed actuation and energy based strategies for structural control. John Plump applied these techniques successfully to the RPL structure and later defined the concept of active constrained layer damping. Shawn Burke developed a unified approach to structural control using all of my previous students’ works and his extensive background in the field of acoustics and control. Jeannie Sullivan extended our knowledge of the use of spatially distributed active materials for control with applications to two dimensional structures. There are of course many more to numerous to list here including my most recent students in the University of Maryland’s Morpheus Laboratory who helped with the editing and problems sets given in this text. Much of the clerical, administrative and editorial work done here is due in large part to the dedication of Laurie Postlewait, Mollie Buechel and Carolyn Sager. Finally I dedicate this book in its entirety to my lifelong role models Lillie Echols and James Edward Hubbard Sr. My sincere appreciation also goes out to my mentors and technical advisors over the years from M.I.T., Drs. Stephen H. Crandall, Donald C. Fraser, Wesley L. Harris, Hank Paynter and Herbert H. Richardson. The graphic support of the National Institute of Aerospace and Mr. Rene H. Penzia is gratefully acknowledged.

Contents

1 Smart Structure Systems . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Smart Structure Architecture and Performance . . . . . . 1.3 Smart Material Transducer Considerations . . . . . . . . 1.4 Continuum Representation of Smart Structures . . . . . 1.5 Time Domain Representation of Smart Structure Models 1.6 Organization of the Book . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 5 9 15 18 19

2 Spatial Shading of Distributed Transducers . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spatial Shading of Distributed Transducers . . . . . . . . . . 2.2.1 Design by Example: A Center of Pressure Sensor . . 2.2.2 Approximating Shaded Apertures . . . . . . . . . . . 2.3 Analytical Modeling of Spatial Shading Functions for Distributed Transducers . . . . . . . . . . . . . . . . . . 2.3.1 A Compact Analytical Representation of Distributed Transducers . . . . . . . . . . . . . . . . 2.3.2 Two Dimensional Representation of Distributed Transducers with Nearly Arbitrary Spatial Shading . 2.4 Application to Two-Dimensional Shading Using Skew Angle 2.4.1 Applications Including Finite Skew Angle of Material Axes . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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25 25 26 26 29

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34

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35

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45 51

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55 60 61

3 Active Vibration Control with Spatially Shaded Distributed Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Control System Synthesis Based on the Lyapunov Direct Method . 3.3 Control System Synthesis for Beams . . . . . . . . . . . . . . . . 3.3.1 Collocated Distributed Transducers and Lyapunov Control 3.3.2 Performance Limitations of Control Designs with Shaded Distributions . . . . . . . . . . . . . . . . . . . .

69 69 70 71 74 76 xiii

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Contents

3.3.3 Performance Limitations of Uniformly Shaded Transducers 3.3.4 Performance Limitations of Linearly Shaded Transducers . 3.3.5 Design Guidelines on Spatial Shading for Vibration Control 3.4 Control System Synthesis for Plates . . . . . . . . . . . . . . . . 3.4.1 Performance Limitations of Uniformly Shaded Actuators for Plates . . . . . . . . . . . . . . . . . . . . . 3.4.2 Performance Limitations of Non-uniformly Shaded Actuators for Plates . . . . . . . . . . . . . . . . . . . . . 3.4.3 The Unique Compatibility of Distributed Transducers for Arbitrary Spatial Shadings . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Multi-Dimensional Transforms and MIMO Representations of Smart Structures . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Convolution and the Spatially Distributed Plant . . . . . . 4.2.1 Green’s Function Representations for Temporally Stationary Systems . . . . . . . . . . . . . . . . . 4.3 Multi-Input Multi-Output (MIMO) Representations of Smart Structures . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Performance Measures for Smart Structures with MIMO Representations . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Performance Metrics . . . . . . . . . . . . . . . . . . . . 5.3 Assessment of Performance Metrics Using Singular Values 5.3.1 Command Following . . . . . . . . . . . . . . . . 5.3.2 Disturbance Rejection . . . . . . . . . . . . . . . . 5.3.3 Sensor Noise . . . . . . . . . . . . . . . . . . . . 5.4 Metrics for Controllability and Observability . . . . . . . . 5.4.1 Controllability . . . . . . . . . . . . . . . . . . . . 5.4.2 Observability . . . . . . . . . . . . . . . . . . . . 5.5 Example: Active Damping of a Simply Supported Beam . 5.5.1 Spatially Uniform Actuator Distributions . . . . . . 5.5.2 Linear or “Ramp” Actuator Distributions . . . . . . 5.6 Metrics for Achieving Stability and Robustness for Control of Smart Structures . . . . . . . . . . . . . . . . . . . . . 5.6.1 Additive Error Uncertainty . . . . . . . . . . . . . 5.6.2 Multiplicative Error Uncertainty . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 80 81 84 86 92 94 95 95 96

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101 101 105

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107

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113 119

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121 121 121 124 124 127 128 128 129 130 130 131 133

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137 139 141 142 142 143

Contents

6 Shape Control: Distributed Transducer Design . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Shape Control and the Notion of Discrete Spatial Bandwidth . . . 6.2.1 Orthonormal Expansions and the Discrete Spatial Transform 6.2.2 Minimization of the Integrated Mean Square Profile Error . 6.3 Plant Representations in Terms of an Expansion Basis Set . . . . . 6.3.1 The Generic Green’s Function Representation . . . . . . . 6.3.2 The Symmetric Green’s Function Representation . . . . . 6.4 Input/Output Coupling and Transducer Shading . . . . . . . . . . 6.4.1 The Singular Value Decomposition and Performance Metrics for Shape Control . . . . . . . . . . 6.5 Spatially Distributed Sensors and Shape Estimation . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

145 145 146 147 149 151 151 153 155 156 162 166

7 Shape Control, Modal Representations and Truncated Plants . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Shape Error and Feed Forward Correction . . . . . . . . . . . . . 7.3 A Complete Dynamic Shape Control Case Study . . . . . . . . . 7.3.1 Case Background . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Airfoil Shapes and the Discrete Spectrum Parameterization 7.3.3 The Concept of Eigenfoils . . . . . . . . . . . . . . . . . 7.3.4 Morphing Airfoil Design Considerations . . . . . . . . . . 7.3.5 Actuator Placement and Input/Output Coupling . . . . . . 7.3.6 Morphing Airfoil Rib: Discrete Parameterization and the System Model . . . . . . . . . . . . . . . . . . . . 7.3.7 State Space Canonical Form . . . . . . . . . . . . . . . . 7.3.8 Morphing Airfoil Closed Loop Shape Controller Synthesis 7.3.9 Morphing Airfoil Closed Loop Shape Control Simulation . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 167 173 173 174 176 178 179 182 183 184 191 195 196

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203

Chapter 1

Smart Structure Systems

1.1 Introduction Today’s advanced structural systems are required to meet increasingly stringent requirements. Not only must they meet traditional load carrying objectives but they may also be required to be lightweight, low cost and incorporate embedded electronic materials and processors for control and health monitoring purposes. Such structures may be considered a system of systems that allow them to adapt to changing operating conditions and environments in order to achieve robust performance. Because of their adaptability these structural systems are often referred to as Smart Structures. The Sci-Tech [1] dictionary defines Smart Structures as structures that are capable of sensing and reacting to their environment in a predictable and desired manner, through the integration of various elements, such as sensors, actuators, power sources, signal processors, and communications networks. In addition to carrying mechanical loads, smart structures may alleviate vibration, reduce acoustic noise, monitor their own condition and environment, automatically perform precision alignments, or change their shape or mechanical properties on command [2]. Here we will be particularly focused on the design of sensors, actuators and control systems for Smart Structures. More specifically we consider primarily, but not exclusively, the class of sensors and actuators which can be either embedded within a structural material or adhered to its surface. We will also explore controller designs which permit the behavior of the structure to respond to external stimuli to meet specified performance metrics.

1.2 Smart Structure Architecture and Performance In this chapter we paint the “Big Picture” in terms of the search for a robust methodology for the real time control of Smart Structures which will set the stage for the material to follow. It is evident from the above that Smart Structures are inherently self contained control systems. The spatially distributed nature of continuum systems such as vibrating structures makes it difficult to apply modern lumped J.E. Hubbard, Spatial Filtering for the Control of Smart Structures, C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-03804-4_1,

1

2

1 Smart Structure Systems

Error

Desired Performance

Control Effectors

Measured Output (Performance)

Vibrating Structure

Controller

Fig. 1.1 Smart structure component architecture

parameter control philosophy and methodologies. When applied these techniques require that a significant amount of precise information be used to describe the behavior of vibrating structures in order to achieve the requisite fidelity. A Smart Structure is basically a distributed parameter system. The purpose of this book is to present control schemes which allow for the practical implementation of distributed parameter control schemes which exploit both spatially distributed sensing and actuation. This powerful union of distributed parameter systems with distributed parameter transducers and related concepts yields simple, realizable control system designs. It should be noted however that the analytical developments presented here provide insight into designs which make use of both distributed and discrete transducers for structural control. Consider the representation in Fig. 1.1 of a Smart Structure depicted here using functional component block elements. Here we depict the Smart Structure System with actuators, sensors and a controller designed to modify the dynamic response of the structure to meet some desired performance in the presence of a changing environment and/or operating conditions. The desired performance may include vibration suppression, shape control, acoustic radiation mitigation and the like. The structure must also have the ability to accommodate uncertainties in the plant model, disturbance environment, transducer dynamics etc. The block diagram of Fig. 1.2 can be modified to more fully represent these goals according to the Multi-Input Multi Output (MIMO) system depicted in Fig. 1.2. d0

di Desired Behavior

r

+ –

e

K (Computer)

u u

actuators

n

Fig. 1.2 The smart structure as a MIMO system

P (Structure)

Actual Behavior

yp

y sensors

1.2

Smart Structure Architecture and Performance

3

Here;

P, represents the “Plant” or physical system to be controlled, i.e. a flexible wing, bridge, component, beam etc. ⎡ ⎤ yp1 ⎢ yp2 ⎥ ⎢ ⎥ yp , is the plant output in the MIMO representation, yp = ⎢ . ⎥ ⎣ .. ⎦ ⎡

ypm

⎤

up1 ⎢ up2 ⎥ ⎢ ⎥ up is the plant input vector, up = ⎢ . ⎥ and control command voltages to ⎣ .. ⎦ upm actuators on or embedded within the structure. K represents the controller or compensator gain matrix and our goal here in part is to learn how to design K to meet performance. ⎡ ⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ u, is the input or “control vector”, u = ⎢ . ⎥ generated by the controller ⎣ .. ⎦ um

⎡

r1 ⎢ r2 ⎢ partially in response to a reference command vector r = ⎢ . ⎣ ..

⎤ ⎥ ⎥ ⎥ which ⎦

rm in some way reflects the desired behavior of the structure. Finally⎡y is ⎤ the y1 ⎢ y2 ⎥ ⎢ ⎥ actual output or behavior of the structure as represented by y = ⎢ . ⎥ . ⎣ .. ⎦ ym As discussed earlier we seek to achieve a robust performance in the presence of noise, disturbances and model uncertainties. ⎡ ⎤This is reflected in Fig. 1.2 n1 ⎢ n2 ⎥ ⎢ ⎥ by the inclusion of sensor noise, n = ⎢ . ⎥ which is associated with ⎣ .. ⎦ nm the measurement of the system output and disturbances di , d0 . Also note the presence of an error signal e which is associated with the difference between the desired and actual performance of the structure. We can define more specifically a tracking error as the difference between the reference command vector and the actual output behavior vis, eT = r − y . Now we may be a bit more specific about the desired characteristics a Smart Structure in the sense that we must design the controller or compensator K on the basis

4


of some “nominal” plant model such that the closed loop system of Fig. 1.2 exhibits the following properties: (1) (2) (3) (4) (5)

Stability Small “tracking error” “Good” command following “Good” disturbance rejection “Good” sensor noise attenuation,

where obviously a rigorous definition of the subjective quality “Good” will need to be determined. In addition we would like the above criteria to hold even though we may have uncertainties present such as; (1) Actual plant characteristics P are not known but only a model P0 is available. (2) Uncertainty in the true representation of disturbances (3) Uncertainty in the true representation of noise. We may explore the implications of controller design and plant model by using the block diagram of Fig. 1.2 to obtain a relationship between the desired output and the parameters of interest e.g. y = f (r,di ,d0 ,n,P,K). Assuming that P and K are linear time invariant, from the block diagram we see that; y = yp + d0

(1.1)

y = Pup + d0

(1.2)

y = P(u + di ) + d0

(1.3)

y = KP(r − y − n) + Pdi + d0 .

(1.4)

Expanding (1.4) and grouping like terms yields; PKy + y = PKr + Pdi + d0 − PKn.

(1.5)

which in terms of the desired output y becomes; y = [I + PK]−1 PKr + [I + PK]−1 Pdi + [I + PK]−1 d0 − [I + PK]−1 PKn. (1.6) Equation (1.6) illustrates the importance of the plant model and proper choice of controller gains in mitigating the effects of disturbances, noise and performance. As the controller gains become “large” (1.6) approaches the desired result of y = r and the actual output tracks the desired input. In practice the magnitude of the gains becomes limited by issues such as stability, power requirements, control authority of the actuators available etc. Of course the controller gains are manifest in matrix

1.3

Smart Material Transducer Considerations

5

d0(x,t)

di(x,t)

r(x,t)

+ e(x,t) –

up(x,t) K(x,t)

P(x,t)

u(x,t)

yp(x,t)

y(x,t) sensors

actuators

n(x,t)

Fig. 1.3 The smart structure as a distributed parameter system

form and the subjective concept of large vs. small and good vs. bad must be rigorously quantified. We will address these performance metrics in subsequent chapters. In addition because Smart Structures are in fact distributed parameter systems, all parameters, gains and plant models are functions of space and time and we must develop an integrated synergistic methodology for the design of such systems (See Fig. 1.3). The choice of plant model and sensor/actuator selection is crucial to the achievement of performance and in the next sections we explore the implications on design.

1.3 Smart Material Transducer Considerations In this section we offer a brief overview of smart materials as a preface to the remainder of this text. While we focus on a few specific material types, the techniques presented here are generic in the sense that they may be readily applied to any active material capable of spatially distributed transduction. As we have seen the choice of sensors and actuators is tantamount to the successful design of a specific Smart Structure control architecture. With the recent development of material design techniques that allow new materials to be custom designed at the molecular and in some cases the atomic level, a new generation of sensing and actuating materials has emerged and there has been a proliferation of research activity in the area of active or so-called smart materials for use in smart structures [3–5]. Here we adapt the definition used by Leo [6] that smart materials are materials that exhibit coupling between multiple physical domains. According to [6] coupling occurs when a change in the state variable of one physical domain causes a change in the state variable of another or separate physical domain. That coupling typically takes the form of energy transduction and work production as illustrated in Fig. 1.4. These materials are usually fabricated such that their geometric and material properties permit energy inputs in the forms of electrical, mechanical, thermal etc. to be transformed into useful work output such as strain, motion, heat etc.

6

1 Smart Structure Systems Actuation Energy IN

Material Properties

Work OUT

Sensing Work IN

Material Properties

Energy OUT

Fig. 1.4 Smart material transduction for sensing and actuation

Table 1.1 illustrates the considerations and trade-offs that Smart Structures designers must take into consideration when considering the use of materials with different transduction properties. Modern material systems engineers are now designing smart material systems which exploit these coupling properties to achieve specific functionalities. The design of such systems is covered in considerable detail in [6]. These designs are truly a multi-disciplinary endeavor encompassing and integrating such diverse fields as nanotechnology, piezoelectrics, polymer chemistry, information technology, biometics, photonics and Electro-Rheological fluids just to name a few. From the standpoint of Smart Structures, this class of materials offers the promise of lightweight, low-cost, low power, high fidelity, embedded transducers for sensing and actuation. In addition they represent transducers which can feature innovative time dependent, programmable functionality and a responsiveness which may dynamically change depending on its environment and the role demanded by the Smart Structure system that it is a part of. The designer needs to select the material which is best suited to the application of interest and performance goals. Figure 1.5 shows how far smart materials have progressed in that now commercial grade materials are available in bulk quantities.

Table 1.1 Smart material transduction comparisons Transduction

Efficiency

Bandwidth

Power density

Piezoelectric Shape memory Electrorheological Electrostatic Thermomechanical Magnetostrictive Electromagnetic Diamagnetism .. .

High Low Medium Very high Very high Medium High High .. .

Fast Medium Medium Fast Medium Fast Fast Fast .. .

High Very high Medium Low Medium Very high High High .. .

1.3

Smart Material Transducer Considerations

7

Fig. 1.5 Production grade smart materials

In particular sheets of piezoelectric and self healing polymers, rolls of shape memory wires and high performance micro-fiber composites may be obtained from any number of commercial sources. Ultimately there must be parametric tradeoffs in material weight, complexity of implementation and integration, signal conditioning and processing requirements, bandwidth sensitivity, dynamic range and cost. In this section we address some practical considerations in the choice and selection of materials for sensing and actuation in Smart Structures. The design techniques presented in subsequent chapters are largely independent of the material specifics and more on the transduction functionality of Fig. 1.4. Table 1.2 contains some examples of several of the more popular materials currently in wide use for the design of Smart Structures. The question often arises however as to specific material choices for particular applications and we offer guidelines below in order to keep this in the proper context. Table 1.2 Smart material examples Materials

Application

Transduction property

Nickel-titanium Cadmium sulphide Terbium iron Barium titanate Germanium Electro-rheological fluids Copper oxide Quartz .. .

Actuator Sensor/actuator Sensor/actuator Sensor/actuator Sensor Actuator Sensor Sensor .. .

Shape-memory Piezoelectric Magnetostrictive Ferroelectric Photoconductive Viscoplastic Photoelectric Pyroelectric .. .

8


Piezoelectric materials have been in widespread use for some time in conventional accelerometers, solid state gyroscopes and the like. As actuators these materials produce a charge on their surfaces as a result of an applied electric field. Conversely when a strain field is applied to the material it produces a surface charge which may be collected using a conducting electrode to provide a signal which is proportional to the applied strain, i.e. sensing. Because the effect is electronic in nature the time response of such transducers can be very fast. The linear theory of such materials is well understood [7] but the effects of hysteresis, creep, charge leakage, depoling and constitutive non-linearity’s have yet to be thoroughly investigated in the Smart Structure application. For example if the electric field to which the material is exposed during operation is excessive, the material can saturate and lose its effectiveness. Shape Memory Alloys were discovered by the Naval Ordinance Laboratory and are nickel-titanium based materials. These materials have the unique property that an increase in temperature causes a phase change causing the material to go from a martensitic state to an austenitic state resulting in a change in its geometry. For a material such as nitinol this can yield remarkable properties such as an increase in the modulus of elasticity by a factor of 25 times before heating. The thermal nature of the transduction limits devices made from such materials to operation at slow to moderate bandwidths. Electro-rheological fluids consist of a colloidal suspension of fine dielectric particles in an insulating host or medium. Particle size typically ranges from 1 to 100 μm and in the presence of an applied electric field the rheological properties of the host fluid change. An applied electric field between 0 and 4 kV/mm can result in a change of several orders of magnitude in the shear modulus in an order of less than 1 ms. Table 1.3 illustrates the characteristics that might be used to select the proper material for an actuator application in a Smart Structure. If performance requires a DC component of control, for example in structural shape control, then clearly one would choose Nitinol over piezoelectric ceramics for actuation. On the other hand if an acoustic response is needed over the full range of human hearing then piezo-ceramics might be in order.

Table 1.3 Smart transducer trade-off comparisons Materials

Piezoelectric ceramics

Nitinol

Cost Linearity Response (Hertz) Sensitivity (microstrain) Maximum operating Temp. Maximum strain (microstrain) Embedability Technical maturity .. .

Moderate Good 10 Hz–20 kHz 0.001–0.01 200◦ C 550 Excellent Good .. .

Low Good DC –10 kHz 0.1–1.0 300◦ C 5000 Excellent Good .. .

1.4

Continuum Representation of Smart Structures

9

As sensors and actuators for Smart Structures these materials must be capable of sensing the mechanical motion of the structure to which it is attached, i.e. bending, twisting and dilatation or stretching with high fidelity over the bandwidth of operation. The fundamental elements of such systems include beams, plates, membranes and strings which are inherently distributed parameter systems exhibiting both spatial and temporal dynamics. Smart materials yield spatially distributed transducers that themselves have spatial and temporal dynamics which must be considered in the design of Smart Structures. These dynamics are convolved with those of the structure to yield the system input output characteristics which determine performance as space-time filters. Conventional point sensors/actuators have been the subject of investigations for decades and their application to lumped parameter systems is well understood. Smart materials and their use as spatially distributed transducers is a recent development and effective design strategies and tools need to be developed. Transducers made from these materials offer new degrees of design freedom in the spatial domain such as (1) Orientation of the skew angle between the structural principle axis and the active material axes for optimum performance (2) Shaping or spatially weighting the active field (electric, magnetic etc.) associated with the sensing and actuation functions to meet design goals (3) Varying the “polarization” profile of the active material within the given laminae or substrate structure (4) Optimal embedding/laminating techniques and profiles for performance. In this textbook we will specifically focus our attention on techniques which address the first three categories above. These techniques have been tried and proven in numerous applications by the author both academically and in commercial products. The analytical developments presented will provide insight into the design and utilization of both distributed and discrete transducers for structural measurement and control, i.e. Smart Structures.

1.4 Continuum Representation of Smart Structures In addition to the proper selection of sensors and actuators, one of the primary tasks of the Smart Structure systems designer is to determine a mathematical model of the Smart Structure i.e. the Plant model. The particular choice of the form and structure of the plant model can have profound effects on controller design and hence the systems ability to meet performance. We will now examine some of the implications of distributed parameter model representation. For the purpose of illustration consider the somewhat generic structure below consisting of a cantilever beam with a tip mass. We wish to control the lateral vibrations of this structure to some predetermined tolerance. The beam is laminated with a smart material actuator for the purpose vibration control. For the example here we

10


y PVDF

Mt : masstip

Beam

It : tip inertia

w(x,t) x Fig. 1.6 Example smart structure representation

choose a piezoelectric polymer film actuator in the form of Polyvinylidene Flouride or PVDF [8]. These polymer films can be readily bonded to lightweight structures undergoing large dynamic strains without significantly affecting their elastic performance. When PVDF film is polarized uniaxially, an electric field applied across its faces will result in a longitudinal strain in the x-direction shown in Fig. 1.6. It is important here to note that the strain occurs over the entire surface area of the film making it spatially distributed or a distributed parameter actuator. As we shall see in subsequent chapters this makes it possible to vary the applied control spatially as well as with time. When a voltage V(x,t) is applied to an unbonded piece of film it produces a strain, εp (x,t) in the film given by εp (x,t) = V(x,t) ·

d31 h2

(1.7)

where d31 is the appropriate static piezoelectric constant, h2 is the film thickness in the y-direction and both are assumed to be constant along the length of the film layer. The control film is bonded to and oriented along the beam so that a positive voltage will yield a positive strain, i.e. extension. When the PVDF film is bonded to the cantilever beam it introduces a “prestrain” in the film which is required to maintain its length along the beam during deflection. The composite structure now experiences a longitudinal strain which is given by

εl (x,t) = εp (x,t) ·

E2 h2 (E1 h1 + E2 h2 )

(1.8)

where E is the modulus of elasticity and h is the thickness of the layer with the subscripts 1 and 2 referring to the beam and film respectively. When a control voltage is applied to the film the resulting longitudinal strain along with the prestrain acts through a moment arm defined by the distance of midplane of the film layer to the neutral axis of the composite beam to produce a net spatially distributed torque on the structure T(x,t) = C · V(x,t)

(1.9)

1.4


11

where C is a constant that depends on the beam material properties and its geometry and allows the torque to be expressed per unit voltage. Equation (1.9) is thus the actuator transduction equation for the distributed parameter system. We now employ an analytical mechanics approach to formulate the equations of motion. This approach has the advantage of leading to a unifying formulation such as Hamilton’s principle. Hamilton principle can be stated as: An admissible motion of a system, i.e. a motion which satisfies force continuity, geometric compatibility and constitutive relationships, between specified states at time t1 and t2 is a natural motion if and only if the variational indicator vanishes for arbitrary admissible variations [9]. The Lagrangian for a system can be expressed in the form L = KE − PE

(1.10)

where KE and PE are the kinetic and potential or strain energy of the system respectively assuming the system is holonomic. Hamilton’s principle can be stated mathematically as

L dt =

δ T

(KE − PE) dt = 0.

(1.11)

T

The variational operator δ and ∂ ∂ ∂t, as well as δ and ∂ ∂x are commutative, and the integrations with respect to time and space are thus interchangeable. For lateral motions of the composite beam Smart Structure of Fig. 1.6 the kinetic energy is given by 2 2 2 ∂w 1 ∂w ∂ w 1 ρ·A dx+ + It · Mt · KE = 2 ∂t 2 ∂t ∂x∂t

x=l

(1.12) where ρ is the density, A is the cross-sectional area of the particular layer under consideration, l is the length of the beam, ρA is the total mass per unit length of the composite beam and is the domain of interest. The system potential energy is similarly 2 2 2 ∂ w 1 ∂ w −T · PE = EI · dx 2 ∂x2 ∂x2

(1.13)

where I is the area moment of the inertia of a layer about the neutral axis, EI is the total bending stiffness of the composite i.e. EI = E1 · h1 + E2 · h2 . The system is holonomic and the torque is included in the potential energy relation. From (1.11) to (1.13) Hamilton’s principle yields

12


δ

ρA ·

(KE − PE) = T

T

∂w ∂t

·δ

∂w ∂t

dx

2 2 ∂ w ∂w ∂ w ∂w + It · ·δ + Mt · ·δ ∂t ∂t ∂x∂t ∂x∂t x=l 2

2 2 ∂ w ∂ w ∂ w ·δ −T ·δ dx dt = 0. − EI · ∂x2 ∂x2 ∂x2

(1.14)

We can now integrate (1.14) by parts as necessary over the domains , 0 < x < l ˙ must vanish at and T = [t1 , t2 ]; and use the fact that the variations δw and δ (w) , t . Using the beam boundary conditions and requiring that δw the fixed times t 1 2 and δ ∂w ∂x vanish at x = 0 we may obtain the equations of motion. For the cantilever beam shown in Fig. 1.6 the geometric boundary conditions require that ∂w = 0 w= . ∂x x=0

(1.15)

The differential equation of motion becomes − EI ·

∂ 4w ∂x4

− ρA ·

∂ 2w ∂t2

+

∂ 2T =0 ∂x2

(1.16)

through the spatial domain , 0 < x < l. At the tip of the beam x = l we can write 2 ∂ w ∂ 3w − It · +T =0 − EI · ∂t2 ∂x ∂x2 3 2 ∂T ∂ w ∂ w − Mt · = 0. + EI · − ∂x ∂t2 ∂x3

(1.17) (1.18)

Combining (1.9), (1.16), (1.17), and (1.18) yields the governing equations for the Smart Structure of Fig. 1.6 as EI ·

∂ 4w ∂x4

+ ρA ·

∂ 2w ∂t2

=C·

∂ 2 V(x,t) ∂x2

(1.19)

with boundary conditions ∂w = 0 for x = 0, ∂x 2 3 ∂ w ∂ w EI · + I = C · V(x,t) for x = l, · t ∂x2 ∂t2 ∂x w=

(1.20) (1.21)

1.4


EI ·

∂ 3w ∂x3

− Mt ·

13

∂ 2w ∂t2

=C·

∂V . ∂x

(1.22)

These are the governing equations for a flexible distributed parameter system which has been laminated with a smart material for the purposes of control. Note that the control voltage acts generally on both the interior and boundary of the composite beam. This is a consequence of using a distributed parameter actuator that can provide both a distributed (system) control and a boundary control if the control voltage is varied spatially as well as temporally. This will require the design and synthesis of both spatial and temporal filters in order to achieve maximum performance. It will be shown in subsequent chapters that the union of distributed parameter systems as represented by flexible structures and distributed parameter transducers, such as those offered via the exploitation of modern smart materials, can lead to simple, robust, realizable smart structure control system design. At this stage of the design process the governing equation might typically be put into state space form to facilitate the use and exploitation of modern control system TM design tools and techniques such as MATLAB , LQG/LTR, H∞ and the like. The system as represented above may be put into this convenient canonical state space form x˙¯ = |A| x¯ + |B| u¯

(1.23)

where x¯ is the “state vector”, u¯ is the “input (or control) vector”, |A| is the “state matrix”, |B| is the “input matrix”. For a given governing equation, the state space equation will vary depending on the particular choice of state vector selected. Generally one state vector is chosen such that the system state equation and boundary conditions can be easily described. Also an accessible state vector is needed if full state feedback is desired. For the transverse vibrating beam illustrated here we choose X¯ (x,t) = [w (x,t) ,

∂w/ ∂t]T

(1.24)

where the superscript T denotes the transposed vector. The system state equation can be written as ⎤ ⎡ ⎡ ⎤ 0 1 0 ∂X ⎣ ⎦ · X + ⎣ C ⎦ ∇ 2 V (x,t) EI ∂4 = (1.25) − 0 · ∂t 4 ρA ρA ∂x and the boundary conditions are ⎡

1 0

⎤

⎥ ¯ ⎢ ⎦ · X(x,t) = 0 ⎣ ∂ 0 ∂x

for

x = 0,

(1.26)

14


⎡

⎤ Mt ∂ ·

⎢ EI ∂t ⎥ C 10 ∂V/∂x ⎢ ⎥ for x = l. ⎢ ⎥ X (x,t) = 01 V ⎣ EI 2 ⎦ ∂ I t · ∂ 2 /∂x2 EI ∂x∂t ∂ 3 /∂x3 −

(1.27)

These equations can be expressed more compactly by introducing the following differential operators ⎡

0

1

⎤

⎥ ⎢ ⎢ 4 ⎥ ⎥ ⎢ ∂ Ax = ⎢ EI ⎥ ⎣ − ρA · ∂x4 0 ⎦ ⎡

1 0

(1.28)

⎤

⎥ ⎢ ⎥ ⎢ A1 = ⎢ ∂ ⎥ 0⎦ ⎣ ∂x

(1.29)

⎡

A2

⎤

Mt ∂ · ∂ 3 ∂x3 − ⎢ EI ∂t ⎥ =⎣ ∂2 ⎦ I t ∂ 2 ∂x2 · EI ∂x∂t

(1.30)

along with the matrix operators ⎡ B=⎣

0

⎤ ⎦

C ρA

⎡ B2 =

1 0

⎤

C ⎣ ⎦. EI 0 1

(1.31)

Now the governing equations may be written as ∂ X¯ ¯ = |Ax | · X(x,t) + |B| · u (x,t) ∂t

(x,t) ∈ xT

for

(1.32)

with boundary conditions ¯ =0 A1 · X(x,t) ¯ = B2 · V¯ A2 · X(x,t)

for x = 0;

t ∈T

for x = l;

t ∈T

(1.33) (1.34)

where u (x,t) = ∂ 2 V(x,t)/ ∂x2 is the spatially distributed interior control input to the Smart Structure system, and V¯ is the boundary control input. It should be noted that V¯ is a vector consisting of two components, ν1 = ∂V(l,t)/ ∂t and ν2 = V(l,t)/ ∂t, and the scalar function V is the control voltage. Along with the initial conditions for displacement and velocity

1.5

Time Domain Representation of Smart Structure Models

¯ X(x,0) = X¯ 0 (x)

15

(1.35)

Eqs. (1.32), (1.33), (1.34), and (1.35) are the system governing equations or Plant P(x,t) model to be used in the architecture of Fig. 1.3. One can ascertain that the presence of the continuous spatial and temporal partial differential operators makes the problem of control system design a difficult one. Ideally one might employ one of several Galerkin methods for converting a continuous operator problem to a discrete problem [9] and then precede with standard time domain control synthesis techniques. A popular approach is to cast the governing equations in terms of natural structural modes. The subsequent modal representation of the Smart Structure is then truncated to some finite order that is deemed tractable for implementation. In the next section we present such an approach and discuss the implication for the design and control of Smart Structures.

1.5 Time Domain Representation of Smart Structure Models In this section we consider the class of flexible systems that includes interior and boundary control of vibrating strings, membranes, thin beams and thin plates by adopting the time domain approach and nomenclature used by Balas [10, 11]. We make use of a Newtonian based model derived using the principles of force and momentum. This class of systems can be described using the generalized wave equation; m(x)wtt (x,t) + 2ξ 1/2 wt (x,t) + w(x,t) = F(x,t)

(1.36)

where the subscript ( ◦ )tt denotes partial differentiation and the operator λ is a time invariant, symmetric, non-negative operator. The displacement of the structure is w(x,t), the mass density is given by m(x) and ξ (x) is a non-negative damping coefficient. An exogenous input to the system is denoted as F(x,t). The governing equation (1.36) along with its appropriate boundary conditions constitutes a SturmLiouville system which admits a solution in terms of a complete set of orthogonal functions ψk (x) which constitute the mode shapes of the structure along with the corresponding mode frequencies ωk . For Smart Structures the Input or exogenous force in (1.36) represents a degenerate spatially distributed control force that is modulated in time to achieve a desired performance. This class of actuators can be referred to as degenerate in the sense that the force distribution is separable in space and time. We assume for example that the control force distribution is provided by a set of M point force actuators as represented by F(x,t) =

M

bi (x)fi (t)

(1.37)

i=1

where bi (x) is the spatial weighting or spatial filter function and fi (t) the temporal filter function that is applied to the exogenous force input. For a set of point force

16


actuators the spatial filter function representing xi actuator locations can be written using the generalized Dirac delta function bi (x) = δ(x − xi ).

(1.38)

The output displacement and velocities may similarly be measured using P point sensors according to with

y(t) = Cw(x,t) + Dwt (x,t)

(1.39)

yj (t) = cj w(zj ,t) + dj wt (zj ,t)

(1.40)

where cj , dj are constants and zi the sensor locations. The solution to equation (1.36) can be expressed as an expansion in terms of a complete set of the eigenfunctions {ψi (x)} w(x,t) =

L

wk (t) ψk (x)

(1.41)

k=1

where the complete set constitutes L = ∞. The equation of motion can now be written as wtt (x,t) + 2ξ 1/2 wt (x,t) + w(x,t) = BL f (t)

(1.42)

with 1/2 being a diagonal matrix of eigenvalues and with actuator input distribution expanded as eigenfunction expansion, i.e. BL is an LxM matrix with entries bkl = ψk (zl ). At this point the Smart Structure designer must decide on a reasonable truncation of L which gives good model fidelity and allows for a practical implementation. This usually means selecting only the so-called “dominant modes” of the system. The paradox here is that the flexible systems which can benefit most from Smart Structures tend to be lightly damped and have a high number of dominant modes. Consider the NASA Langley Research Centers Hoop Column Antennae which had 70 dominant modes over a bandwidth of just 4.1–6.2 Hz [12]. Even for more benign systems the engineer must assume that L is sufficiently large to adequately describe the system. These large scale systems have both analytical challenges as well as practical limitations to implementation [13]. Consequently we must settle on a subset of controlled modes N < L and the displacement becomes portioned into a controlled and residual partition w(x,t) = wN (x,t) + wR (x,t)

(1.43)

and wN (x,t) =

N k=1

wk (t) ψk (x)

(1.44)

1.5

Time Domain Representation of Smart Structure Models

17

due to the practical limitations of this approach. The sensor equation becomes y(t) = CCN w(x,t) + DCN wt (x,t)

(1.45)

where CNT is an Nx P matrix with entries {ψk (zl )}, in other words the output is expressed as a weighted expansion in terms of a truncated set of the orthogonal eigenfunctions. The system as represented above may be put into convenient canonical state space form x˙¯ = |A| x¯ + |B| u¯

(1.46)

y¯ = |C| x¯ + |D| u¯

(1.47)

where x¯ is the “state vector”, y¯ is the “output vector”, u¯ is the “input (or control) vector”, |A| is the “state matrix”, |B| is the “input matrix”, |C| is the “output matrix”, and |D| is the “feedthrough (or feedforward) matrix”. For simplicity here, |D| is chosen to be the zero matrix, i.e. the system is chosen not to have direct feedthrough. For the example presented here we choose a state vector of controlled modesT according ˙ R and the to νN (t) = wTN w˙ TN and similarly for the residuals νR (t) = wTR w state equations become

with

ν˙¯ N (t) = |AN | ν¯ N (t) + |BN | f¯ (t)

(1.48)

ν˙¯ R (t) = |AR | ν¯ R (t) + |BR | f¯ (t)

(1.49)

y¯ (t) = |CN | ν¯ N (t) + |CR | ν¯ R (t)

(1.50)

0 IN |AN | = 1/2 , − N −2ξ N

0 |BN | = , BN |CN | = C CN D CN ,

0 IR |AR | = 1/2 − R −2ξ R

0 |BR | = , BR |CR | = C CR D CR .

(1.51)

(1.52) (1.53)

At this stage given the plant model along with pre-selected actuator and sensor distributions, one can begin the process of synthesizing a controller to meet performance using a number of well accepted time domain techniques. The state space representation of the plant model is a “time-domain approach” which provides a compact way of modeling and analyzing systems with multiple inputs and outputs. However the minimum number of state variables required to represent a given system, is usually equal to the order of the system s defining differential equation where according to (1.42) is dependant on the number of modes required for fidelity. For

18


flexible structures that are lightly damped, this can be significant. A model with a large number of degrees of freedom can cause numerical difficulties, uncertainties and high computational cost for even the simplest of distributed parameter systems [13]. For control the approach is often to develop a modal controller for the reduced subset of targeted modes, ignoring the residual modes. The effect of ignoring these modes is then analyzed for its impact on performance and stability. When the residual modes of the system are ignored in the control design process, the result can be an unstable interaction between the control system and the flexible modes of the structure. This interaction can manifest itself as an overflow or spillover onto the dynamics of the modes unaccounted for or “unseen” by the design. In addition, energy from these modes can in turn spillover into the measurements taken by the sensors resulting in inadequate performance of the smart structure system. In other words the sensor outputs become contaminated by the residual modes via the term |CR | vR (t) resulting in a feedback control which results in the excitation of these modes via |BR | f (t). These effects are classified as observation and control spillover respectively. In addition, it is difficult to reconcile the location of sensors and actuators on the structure [14] to mitigate these effects and physically realizable implementations of these control systems tend to suffer from poor stability and robustness characteristics. In general, active structural control techniques seek to cancel known plant dynamics and replace them with a set of desired dynamics. Unfortunately, a large amount of model uncertainty is present in structural systems [15]. There may be parametric uncertainty in the mass and stiffness properties of the structure and this will manifest itself as uncertainty in the natural frequencies of the structure [16]. Distributed parameter systems as represented by flexible structures can theoretically have an infinite number of modes and in practice can have a large number of modes present within their performance bandwidth. This results in parametric uncertainty in the model order if a controller requires truncation of modes [17]. In addition the disturbance environment is also often poorly known. The disturbances may be transient or continuous, either stochastic or deterministic. Finally it should be noted that the complications which arise from model truncations as outlined above apply equally well to finite element approximations [18].

1.6 Organization of the Book This book is written for advanced graduate students and practicing engineers who are interested in the practical design and control of distributed parameter systems. It is assumed that the reader has a basic knowledge of the fundamentals of vibration and control as taught in a typical undergraduate and first year graduate engineering curriculum. It is for example assumed that the reader has a thorough understanding of structural modes and modes shapes, linear algebra, ordinary differential equations as well as modern state space formulation and classical control. The methods and techniques presented in this book present an approach to the design and implementation of distributed parameter control schemes which allow

Problems

19

both spatially distributed sensing and actuation as a means of mitigating the problems outlined in this section. These methods do not require plant model truncations and offer the possibility of controlling all modes of the flexible smart structure using relatively simple compensator designs. In addition the analytical techniques presented provide insight into the use of both discrete and spatially distributed transducers for measurement and control. In Chap. 2 we discuss the design of spatially distributed transducers as continuum sensors and actuators. More specifically we describe the development and application of a method for modeling distributed transducers with arbitrary spatial distribution. This approach allows distributed transducer shape or spatial filters to be incorporated into the control design process for multi-dimensional structures as an additional design parameter. A compact representation is presented for the modeling of such devices and the method can be used for a wide variety of applications. The method itself is general and is thus applicable to many types of transducers, including piezoelectric, electrostrictive, and magnetostrictive devices. In Chap. 3 we combine the representations of Chap. 2 with spatial filtering techniques to yield a structured methodology for synthesizing stabilizing compensators for the vibration control of flexible Smart Structures in which distributed sensors and actuators are embedded into structural components to provide “built in” or “smart” active vibration control. We reveal the advantages of using fully distributed sensors and actuators in the design of Smart Structures. A non-modal energy based scheme for control synthesis is introduced and is based on Lyapunov’s Direct Method [19]. The technique is non-model based and hence does not suffer from the limitations discussed above related to model truncations. In Chap. 4 we introduce the notion of “spatial frequency as it relates to the modeling of distributed parameter systems. The modeling approach taken is based on the use of Green’s Functions and is therefore applicable to system representations using modal expansions and convolution kernels. In Chap. 5 we use fundamental concepts in modern multivariable control systems to construct performance measures over both time and space domains for distributed parameter systems. In Chaps. 6 and 7 the concepts previously introduced and discussed are leveraged and extended to the shape control problem and several examples are given including a morphing aircraft problem. The morphing aircraft problem serves to unite all of the concepts presented in this text in a synergistic manner which should provide concept cohesion for the reader. The reader should in general be familiar and comfortable with transform methods, orthogonality concepts and orthogonal functions as well as multivariable control concepts. Given this background or one similar the reader should have no difficulty following the materials presented herein.

Problems (1.1) Suppose you are designing a structure in the shape of a cantilever beam to hold a magnetic disk reader for a hard-drive, which is to be actuated on the order of 1 kHz. Discuss a smart material and a mechanism you could employ to accomplish this, as well as the trade-offs with the chosen smart material.

20


a. A cantilever beam is to be used to model the bending of an aircraft wing which has a half span of 1 m. Discuss the choice of a smart material and a mechanism which will actively curl the wing tips to achieve steady-state changes in the vehicle configuration. b. A major concern for passenger aircraft today is turbulence. In all instances an aircraft operator wishes to minimize the effects of turbulence or gusts on the aircraft in order to improve the ride quality for the passengers. This can be achieved with a method called gust alleviation, whereby small control surfaces on the aircraft are deployed to counteract the effects of turbulence. Suppose the requirements had been given to you to find a smart material actuator that could provide high bandwidth and moderate deflection. Use the guidelines from this chapter to select a material, or set of candidate materials, you suspect would be suitable for this application. (1.2) Based on your knowledge gained from this chapter, consider and discuss the best possible approaches to actively damping the system shown in Fig. 1.2a below using smart material actuators. a. Consider and address specifically: (i) system properties such as mass, geometry, string tension, (ii) frequency response, (iii) control authority, and (iv) actuator placement. b. Repeat the problem, but consider the smart material role as a sensor used as a feedback mechanism instead of as an actuator. Does your material selection change? Does the placement change? What other considerations are different for the sensor design versus the actuator design?

Fig. 1.2a Distributed parameter system

(1.3) It was discovered in the 1950s that using a simple cavity with a small orifice on one end and releasing acoustic waves on the opposite end would produce a jet issuing from the cavity through the orifice. Recommend the improvements and modification to this idea by incorporating smart materials as a novel part of this device. Name a few recommendations and discuss their advantages and disadvantages. (1.4) Applying certain assumptions, synthetic the jet actuator described in question 1.3 can be represented as a two degree of freedom model in terms of inertia, stiffness, damping, and forcing coefficients. A synthetic jet actuator is a coupled mechanical-Helmholtz resonator system and in this example has a

Problems

21 Table 1.4a Synthetic jet actuator design parameters Actuator parameters

Value

Orifice diameter (D) Orifice depth (L) Cavity diameter (Dc) Cavity height (H) Diaphragm displacement Diaphragm frequency (f)

1 mm 1.5 mm 32 mm 6 mm 0.065 mm 2200 Hz

built-in oscillating unimorph. Periodic motion of the diaphragm is coupled with the air oscillating in the actuator orifice. a. Identify the fundamental modes and the main parameters/coefficients that will drive the final frequency response of a synthetic jet actuator with parameters presented in Table 1.4a. b. Implement these coefficients in a set of two degree of freedom equations of motion (written in a matrix form) and derive the response of this particular system to harmonic excitation. c. Increase the diameter of the orifice by 2 and then 3 times and explain the changes in the frequency responses. What conclusions can you draw about potential optimization techniques for such a system? (1.5) Consider the system defined by the block diagram shown in Fig. 1.5a below. In this system depiction, the actuator and sensor dynamics are explicitly modeled separately from the plant dynamics, and the input disturbances affect the input to, and the output of, the plant. For this MIMO system, derive the following transfer functions: a. From control input to the measurement ym b. From the actuator output δ to plant output yp c. From the error signal e to the measurement ym (1.6) Active damping of a vibrating string: model basics. Consider the block diagram in Fig. 1.6a. a. Find the open loop (OL) transfer function (TF). b. Find the closed loop (CL) TF.

Fig. 1.5a MIMO system block diagram

22


Fig. 1.6a Vibrating string model block diagram

Fig. 1.6b Vibrating string lumped parameter model

c. Derive the OL TF for the approximation of a vibrating string as shown in Fig. 1.6b. How does the approximation differ from an actual vibrating string? What are the key assumptions? What conclusions can no longer be drawn due to the approximation? d. Plot the OL response for various values of R between −1 and 1. Let G (s) be the TF found in part c. Assign appropriate values for the other variables. e. Find the system closed loop response letting K (s) be a: (1) proportional, (2) integral, and (3) differential controller. Combine the control types to examine the system response (e.g. PI, PD). For all cases neglect the noisen . f. Let the noise be pink noise. Examine different frequency ranges for the pink noise and determine system sensitivity. (1.7) A structure is modeled as a cantilever beam with a tip mass, as shown in Fig. 1.7a. The tip mass has normalized mass Mt = 1.20, normalized inertia lt = 3.71x10−2 , and the ability to generate a transverse force T (t) . The governing equation describing the displacement of the beam is wtt + wxxxx = f (x,t) where w (x,t) is the transverse deformation, f (x,t) is the distributed forcing, and x ∈ [0,1] .

Problems

23

Fig. 1.7a Cantilever beam with tip mass

a. Determine the transcendental equation for this system and solve the resulting Eigenvalue problem for the first four modes. Provide the Eigenvalues, natural frequencies of vibration, and the modal shape equations for the first four modes, as well as plots of the mode shapes. b. Suppose an accelerometer is placed at the tip of the beam to measure the local acceleration A (1,t) = wtt (1,t) . Design a classical control law using the accelerometer measurements to generate the tip force T (t) in order to damp out vibrations. Simulate the effectiveness of this controller to disturbances and initial displacement profiles. Discuss the limitations of the controller and the implications of using discrete transducers to control a distributed structure. (1.8) Consider a pinned-pinned beam with a single discrete displacement sensor L L L along the beam length. For whose location can be prescribed as , , or 4 2 5 each of these possible sensor locations, derive an expression for the measured output y (t). Assume a 10-to-1 scaling between the measured output and the displacement for an infinite bandwidth amplifier. In addition, assume no direct feed-through to the output. (1.9) Fluid flows are spatially distributed systems best described by a set of partial differential equations known as Navier Stokes equations. Navier Stokes equations are a function of both spatial and temporal independent variables. In this problem we will consider a channel flow with the flow homogeneous in the x and z directions. The equations governing small, three-dimensional perturbations to the mean flow U are given by the linearized Navier Stokes (LNS) equations and continuity equations du dU dp 1 du +U + v=− + u dt dx dy dx Re dv dv dp 1 +U =− + v dt dx dy Re dw dw dp 1 +U =− + w dt dx dz Re du dv dw + + =0 dx dy dz where is the Laplacian, Re is the Reynolds number and ν is the kinematic viscosity. The state vector of the system is comprised by u, v, and w, which

24


are disturbance velocity components in the x, y, and z directions, as well as the pressure p. a. Show that it is possible to use a smaller number of fields to encode the state of the system, i.e. reduce the flow perturbation problem in {u,v,w,p} with second order partial derivatives to a problem {v,w} with fourth order partial derivatives (Hint: eliminate pressure from the equations and apply continuity). Place the reduced state governing equations in a state-space form. What do you notice about the coupling of these equations? b. Add a three-dimensional forcing term to the previous problem and re-write the governing equations in a state space form.

Chapter 2

Spatial Shading of Distributed Transducers

2.1 Introduction In general transducers may be parameterized by their placement, type (e.g. translational versus rotary), spatial distribution and spatial aperture. In this chapter a modeling method is presented for the design of one and two-dimensional spatially distributed strain induced and non-strain induced transducers with arbitrary spatial aperture and distribution. We make the distinction between discrete and spatially distributed transducers in that a discrete transducer provides transduction at a discrete point in or on a given system while distributed transducers can provide transduction at many points. This transduction occurs within the spatial aperture of the transducer and can cover large areas of the system. If the wavelength of the input to a sensor is large compared to its measurement aperture then the sensor and its concomitant measurement is said to be discrete. Consider the case illustrated in Fig. 2.1. The exogenous input to any given system may be characterized by its spectral profile, i.e. its amplitude and frequency content. Here we illustrate the frequency components in terms of wavelength. For example if the highest frequency of interest is 440 Hz and the wave speed in the structural material of interest is 350 m/s then the associated wavelength is given by λ = 350 440 = 0.8 m. If an accelerometer with a measurement aperture of d = 0.08 meters is attached to the structure then λ >> d and the measurement is discrete. In contrast using a spatially distributed smart material (such as piezo-crystals, films, fiber optics, shape memory alloys etc.) sensor to completely cover the structure can result in an aperture

1 Input Amplitude (force, strain acceleration)

A

B

C

Wavelength (meters, feet, etc.)

Fig. 2.1 Exogenous input amplitude and wavelength characterization

J.E. Hubbard, Spatial Filtering for the Control of Smart Structures, C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-03804-4_2,

25

26

2 Spatial Shading of Distributed Transducers

length many times the input wavelength, if d = 8 m for example then λ n ≡

0 xn dx = −∞

< x − a >n+1 , n+1

n ≥ 0.

(2.18)

In addition the functions < x−a >−1 and < x−a >−2 are exceptions (as denoted by writing the exponent below the bracket) and equal zero everywhere except when x = a where they are infinite, such that (2.17) and (2.18) are true, vis. x < x − a >−2 dx =< x − a >−1

(2.19)

< x − a >−1 dx =< x − a >0 .

(2.20)

−∞ x −∞

Note that the function < x − a >−1 is the unit impulse or unit discrete load function and < x − a >−2 is a unit concentrated moment or doublet function. Figure 2.8 is a graphical presentation of this family of singularity functions. The < x−a >−2 x

0 x=a < x−a >−1 0 < x−a >0 1 0

x=a

x=a

x

x

< x−a >1 0

Fig. 2.8 Family of singularity functions fn (x) =< x − a >n

< x−a >2 0

x x=a

x x=a

2.3

Analytical Modeling of Spatial Shading Functions for Distributed Transducers

37

first member of the family is a doublet and can be used to represent discrete loads such as point moments e.g. M0 < x−a >−2 . Similarly the second member shown is the Dirac delta function and can be used to represent discrete point loads e.g. F0 < x − a >−1 . Spatially distributed uniform loads can be represented via the function u0 < x − a >0 more commonly known as the step function and W0 < x − a >1 is a linearly increasing spatially distributed load or ramp function. Many practical beam loadings can be synthesized using a superposition of these basic singularity functions. The integration laws (2.18), (2.19), and (2.20) allow for the mathematical manipulation of the functions as needed. Recall for example the smart structure cantilever beam example presented in Chap. 1. The governing equation for this system was determined to be EI ·

∂ 4w ∂x4

+ ρA ·

∂ 2w ∂t2

=C·

∂ 2 V(x,t) ∂x2

(1.19)

where the control input was manifest as the Laplacian of the spatially distributed load provided by a piezofilm of uniform spatial distribution attached along the length of the structure (See Fig. 2.9). The spatially uniform control distribution can be defined in terms of the singularity functions u0 < x − a >0 as V(x,t) = (x) u(t) = Vmax < x >0 − < x − 1 >0 u(t)

(2.21)

in accordance with Eq. (2.16) for a beam of unity length where the transducer’s spatial shading (x) is shown below in Fig. 2.10. Note that, due to the distributed nature of the transducer, more than one singularity function is required. Because the transducer described above is an induced strain

y PVDF

Mt : tip mass

Beam

It : tip inertia

w(x,t) x Fig. 2.9 Example smart structure representation

1 Λ(x)

Fig. 2.10 Spatially uniform control distribution (shading)

0 0

1

38


device, its spatial input/output characteristics are determined by the Laplacian of the distribution defined in (2.21) as indicated by the governing equation (1.19) thus

xx (x,t) = Vmax < x >−2 − < x − 1 >−2

(2.22)

where xx (x) denotes the Laplacian of the transducer’s spatial shading function. A sketch of the effective loading interpretation of Eq. (2.22) is shown in Fig. 2.11 and consists of a pair of point moments located at the beams boundaries. If the film were used for sensing instead of actuation then the representation in Fig. 2.11 would indicate the measurement of angular displacement at the boundaries. The implications for structural control will be discussed in detail in subsequent chapters of this text. This result may be generalized to the broader class of Bernoulli-Euler structures with nearly arbitrary boundary conditions by considering a more generic representation of the smart structure as shown in Fig. 2.12, Here the structure is represented as a thin elastic structure with an active film layer adhered to one face to be used as an actuator. The active film layer is a smart material with the capability of responding to an applied electric field across its faces that is generated by the time varying voltage Vf (t) and which causes expansion, contraction and dilatation in its three principle dimensions. For small deflections the governing equation for this system is given by ˆ xˆ ∈ , D∇ˆ 4 wˆ + ρ wˆ ˆtˆt = m∇ˆ 2 V,

(2.23)

where w ˆ is the lateral deflection of the component, m is the films electro-mechanical coupling, D is the flexural rigidity of the composite structure, ρ is its area density, and Vˆ is the spatially distributed control voltage applied across the faces of the active film. Note that in general the spatial coordinate xˆ is restricted to xˆ = (x,y) at

< x >−2

Λxx(x)

Fig. 2.11 Loading interpretation for a uniformly shaded control distribution

< x–1 >−2

1

0

Vf (t)

y Active film x

Fig. 2.12 Active film/beam composite

Elastic component

x

2.3


39

most thus representing one and two dimensional structures and (•)t denotes partial differentiation. Both the lateral deflection and the applied control voltage are functions of space and time representing the smart structure as a distributed parameter system. Finally the system is completely described over the domain Ω and with its boundaries contained in G. At this stage it is important to point out several salient features. Equation (2.23) is a linear time invariant inhomogeneous equation whose input is given by the control distribution defined by m∇ˆ 2 Vˆ which is in fact a spatially distributed bending moment. The constant m is a constitutive constant which is a function of structural material and geometric properties and determines the magnitude of the applied bending moment per unit volt. If the material properties and/or the geometry of the composite structure change along its length, e.g. as a result of the application of transducer shading, then the control moment is a function of xˆ and hence represents a spatially shaded distributed moment. For a one dimensional structure such as the composite beam shown below in Fig. 2.13 the constant m can be determined to be m = −d31 (h1 + h2 )

E1 h1 E2 B 2 (E1 h1 + E2 h2 )

(2.24)

where h is thickness, B is width, E is the elastic modulus, (•)1 and (•)2 denote beam and film parameters respectively and d31 is a constitutive constant which relates the applied electric field and induced strain. Similarly if the control voltage is spatially varying the result is again a spatially shaded distributed moment. As presented earlier, the distributed voltage may be realized by shaping the electrode applied to the active film thus shading or spatially weighting the applied control moment. Equation (2.23) can be further generalized by considering the following nondimensionalizations,

y

h1

Active film layer

h2

Beam

x Fig. 2.13 Basic composite distributed transducer/beam configuration

40


x = Lxˆ , t=

ˆt L2

D ρ,

w=

wˆ L,

V=

ˆ VD , mL2

(2.25)

which permits (2.23) to be conveniently rewritten as ∇ 4 w + wtt = ∇ 2 V.

(2.26)

This governing equation combined with the appropriate boundary conditions may now be used to investigate the design and corresponding behavior of smart structures with distributed transducers. Note that the input V(x,t) appears in terms of its Laplacian as a result of the structures moment curvature relationship. The resulting analysis will be valid for any coordinate system in which ∇ 4 and ∇ 2 are defined. The concomitant boundary conditions are assumed to be homogeneous with respect to both discrete and distributed system elements and with respect to the control input as well. (The control actuator spatial distribution can be considered as extending to, but never reaching the boundaries in the mathematical sense.) The boundary conditions may now be generalized to ∂ ∂n

w (,t) =

w (,t) = 0,

w (,t) = ∇ 2 w (,t) = 0, ∇ 2 w (,t) =

∂ ∂n

2 ∇ w(,t) = 0,

∂ ∂n w (,t)

∂ ∂n

2 ∇ w(,t) = 0.

=

(2.27)

The boundary conditions of (2.27) describe clamped, pinned, free and sliding boundaries respectively. The Macauley notation of singularity functions introduced earlier can now be used to investigate the behavior of a broad class of these Bernoulli-Euler type structures with nearly arbitrary boundary conditions involving both uniform and non-uniform spatial shadings. As a final example consider the linear shading of a one-dimensional transducer defined over the domain ∈ [0,a] as represented in Fig. 2.14. This distribution can be compactly represented as

(x) = b x 0 −

b 1 b x + x − a 1 a a

(2.28)

whose shading characteristics decrease from a maximum b, to zero over the aperture [0,a]. Again note that more than one singularity function is needed for the description required by (2.28). The spatially distributed control input can be defined as

2.3


41

y

Fig. 2.14 One-dimensional linearly shaded distribution

b

Λ(x) 0

x a

0

b b V(x,t) = (x) u(t) = b x 0 − x 1 + x − a 1 u(t) a a

(2.29)

and in accordance with (2.26)

b b ∇ V = b x −2 + x −1 + x − a −1 u(t). a a 2

(2.30)

The result consists of a doublet function and a delta function at x = 0 and a single delta function at x = a as shown in Fig. 2.15. If this transducer were used as an actuator, this shading would result in the production of a moment and two forces which satisfies force and moment equilibrium. Conversely if the transducer were a sensor, this shading would correspond to angular and lateral displacement. Note that the shading function can in general be represented as a superposition of singularity functions:

(x) =

n

ci x − di n

(2.31)

i=1

y b

b a

Λxx(x) b 0

x a

b a Fig. 2.15 Laplacian of the linearly shaded transducer distribution

42


where the constant ci represents the amplitudes of the component singularity functions and di defines the length scale of the aperture of interest. It was illustrated earlier that singularity functions can be used to describe both discrete and spatially distributed transducers. The distinction between the two now becomes evident in that a transducer is determined to be “distributed” if its spatial shading function

(x) is given as a superposition of at least two singularity functions corresponding to a single, irreducible device. The distinctive qualifier of “a single, irreducible device” is meant to distinguish between transducers and transducer arrays. For real physical smart structures, one-dimensional shading is actually an approximation for the continuous variation of the conversion properties of the transducer over the aperture such as shown in Fig. 2.14. This approximation, however, becomes invalid if the beam behaves like a plate. For example if there are modes of vibration along the transverse direction of the structure, then the problem is no longer one-dimensional and the shaped transducer approximation to onedimensional shading will not be valid. In this case, the distributed transducer does not behave as a beam but instead is two-dimensional and can no longer be modeled using one-dimensional singularity functions. It is possible to model a two-dimensional transducer using two-dimensional McCauley functions. Consider the application of a spatially uniform bi-axial layer of active film to a rectangular plate as depicted in Fig. 2.16. Assuming that the film is an induced strain device, the non-dimensional governing equation for such a system is given by Eq. (2.26). Assuming a degenerate control distribution which is separable in space and time, a spatially uniform control distribution as illustrated in Fig. 2.17 can be defined here as 0 u(t), (2.32) V(x,t) = Vmax (x) u(t) = Vmax x 0 − x − lx 0 y 0 − y − ly where lx and ly are the non-dimensional lengths of the sides parallel to the x and y axes respectively. The control input to the smart structure manifest itself as the Laplacian of the control distribution and is given by

Fig. 2.16 Rectangular plate structure with bi-axial film applied

2.3


43

Fig. 2.17 Spatially uniform 2-D distribution

∇2 V =

Vmax

0 0 x −2 − x − lx −2 y − y − ly

+ x 0 − x − lx 0 y −2 − y − ly −2 u(t).

(2.33)

As an actuator this distribution is seen to exert distributed angular moments along the boundaries x = 0 and y = 0 where the net loading provides for static equilibrium due to the films self reacting nature. In order to determine this distributions effectiveness as a controller one must formally consider the appropriate boundary conditions for the smart structure application of interest. It is apparent for example that spatially uniform shading would not be effective in controlling a plate with clamped boundaries on all four sides but may provide control for pinned or free boundaries. In the next chapter a formal methodology for vibration controller synthesis will be presented based on the shading analysis introduced here. An extension of the one dimensional linear shading of Fig. 2.14 to two dimensions can be introduced as a distribution that is a product of “ramp” functions vis.

44


x x 0 0 x − 1 − x − lx

(x) = 1 − lx lx

0 x y 0 y − 1 − y − ly . 1− ly ly

(2.34)

This double ramp or “snowplow” shading is illustrated in Fig. 2.18 below. The control loading exerted by this shading and given by its Laplacian has components

xx (x) = x −2 −

yy (x) =

1−

x lx

1 lx

! 0 x −1 − x − lx −1 1 − lyy y 0 − 1 − lyy y − ly ,

x 0 − 1 − lxx x − lx 0 y −2 −

1 ly

! y −1 − y − ly −1 . (2.35)

It is observed that this shading distribution will exert spatially distributed moments along the sides x = 0 and y = 0, in addition to point loads along the boundaries. Thus far the techniques and examples presented have involved shading and generalized functions defined in orthogonal coordinates, and thus only rectangular shaped transducers have been examined. In the following subsection, a more formal approach to the use of singularity functions to describe spatial shading is presented which enables one to use multidimensional distributions to model nearly arbitrary transducer spatial weightings.

Fig. 2.18 Snowplow 2D shading function

2.3


45

2.3.2 Two Dimensional Representation of Distributed Transducers with Nearly Arbitrary Spatial Shading In order to adequately describe nearly arbitrary shadings, singularity functions must form a complete set. In this context an orthonormal system of functions may be considered complete in the class of square-integrable functions called L2 or in some subset M of this class, if Parseval’s equation holds for any shading function (x) of L2 or of M respectively. In other words the integral of the square of the absolute value of the functions L2 over the interval [a,b] must be finite. The sense of Parseval’s theorem is such that if a specified orthonormal set of functions is complete in M then any function of M can be represented with arbitrary precision using linear combinations of the functions in the set. More specifically, a shading function (x) can thus be approximated if it is constrained to be contained in the set of squareintegrable shadings i.e. in a system of L2 functions. In addition to the above the following definitions are necessary [25]; Definition A system of L2 functions (not necessarily orthogonal) in the range [a,b] shall be considered closed in all of L2 or in the subset M if the only functions of M orthogonal to all functions of the given system are those which vanish everywhere in [a,b]. This definition essentially means that if a system of functions is closed then there are no missing functions in the set. The only function that is orthogonal to members contained in the set is the trivial function (x) ≡ 0, which is of no practical interest. Now having a definition of a closed set the following can be shown; Theorem Any orthonormal system of functions which is shown to be “closed” in a given subset M of L2 is also “complete” there. Using this theorem the completeness of singularity functions may be demonstrated by simply proving that they form a closed set. 2.3.2.1 The Completeness of Singularity Functions To prove that singularity function representations of smart material spatial shading functions are complete it is merely necessary to show that the system of polynomials that they form is complete on the finite interval [a,b]. More specifically that the system 1, x, x2 , x3 , . . .

(2.36)

is closed and complete in the class of square-integral functions L2 . Using the definitions of the previous section, if a function g(x) is orthogonal to all of the functions contained in (2.36) then it must be orthogonal to every linear combination of such functions, that is to say to each and every polynomial. Consider the special case of the polynomial of degree 2n in (2.37)

46


Pn (x) = 1 +

ε2 − (x − x0 )2

n

(b − a)2

,

(2.37)

where ε and x0 are constants that shall be defined subsequently. We will now show that the orthogonality condition given by b In ≡

Pn (x) (x) dx = 0

(2.38)

a

can only be satisfied if (x) = 0 and thus the set is closed and the functions are thus complete in the sense described earlier. The following proof parallels the development found in [25]. Let (x) represent a continuous function on the finite interval [a,b] and thus there must be at least one point x0 in this interval such that (x0 ) = 0. Also assume that (x) is positive definite so that (x0 ) = β > 0 in the defined interval. The positive constant ε can always be chosen to be small enough so that (x0 − ε, x0 + ε) is completely contained in the finite interval [a,b] and 1 β > 0; 2

(x) >

x0 − ε ≤ x ≤ x0 + ε

(2.39)

as stated. Also 0> 1 ;

ω ∈ r,

k ∈ Kr .

(5.27)

The requirement that the loop transfer function be large for good command following is visualized in Fig. 5.2 for a continuous spatial transform. The range of desired spatial and temporal response is shown as a rectangular box wavenumber/frequency space. To achieve “good” performance the surface represented by the minimum singular values of T (k,ω) must not penetrate this box. The higher the loop gain in this region, the smaller the tracking error. We can expand T (k,ω) include sensor and actuator space/time dynamics in accordance with the discussion and nomenclature of Chap. 4 as T (k,ω) = h (k,ω) p (k,ω) q (k,ω) K (k,ω) .

(5.28)

5.3

Assessment of Performance Metrics Using Singular Values

σ[T (k,ω)]

127

Freq.

ωr Wavenumber

Singular Value Surface

kr

Fig. 5.2 A visualization of k, ω tracking performance specification

With the careful choice of the spatially distributed compensator K (k,ω) and sensor and actuator filter functions p (k,ω) and q (k,ω), T (k,ω) can be made “large” to meet the requirement for good command following. It is clear now that sensor and actuator “placement” influences the control spatial bandwidth response and (5.27) in combination with (5.28) provide a means of selecting sensor/actuators distributions and candidate compensators in order to achieve the desired temporal and spatial command following for smart structures as distributed parameter systems.

5.3.2 Disturbance Rejection The performance metric for rejection of disturbances and providing insensitivity to plant modeling errors can be assessed in a similar manner as that above. The tracking error in the absence of noise or command inputs is given as e0 (k,ω) = −S (k,ω) d (k,ω) .

(5.29)

Again we can invoke Cauchy-Schwartz to manipulate (5.29) and obtain e0 (k,ω)2 ≤ S (k,ω)2 d (k,ω)2 .

(5.30)

e0 (k,ω)2 ≤ σmax [S (k,ω)] d (k,ω)2 .

(5.31)

and thus

This reveals that the requirement for rejecting disturbances and providing insensitivity to modeling errors is σmax [S (k,ω)] . > 1 ; ω ∈ d,

k ∈ Kd .

(5.33)

Because the singular value can be thought of as a scalar “gain control”, (5.33) requires high loop gain in both frequency and wavenumber bands d and Kd .

5.3.3 Sensor Noise We now consider the issue of sensor noise and its effect on the tracking error. The tracking error in the absence of disturbances and command inputs can be written as e0 (k,ω) = C (k,ω) n (k,ω) .

(5.34)

This can be manipulated as before to yield e0 (k,ω)2 ≤ σmax C (k,ω)2 n (k,ω)2 .

(5.35)

and hence the insensitivity to noise requires that σmax [C (k,ω)] . 0 and exploit the fact that an unconstrained minimization must be smaller than, or at most as large as a constrained minimization then (I + Tnom )z2 (I + Tnom )x2 ≤ min z =0 z2 x2 ; X∈Null

min

(5.67)

since as shown x is not arbitrary but restricted to lie in the null space of (I + Tnom ). This constraint can be expressed in terms of the singular value as σmin

(I + Tnom )z2 (I + Tnom )x2 ≤ min . z2 X∈Null x2

(5.68)

Similarly we note that a constrained minimization is always bounded by an unconstrained maximization and thus Eadd x2 Eadd z2 ≤min . z =0 X∈Null x2 z2 min

(5.69)

We may now write that (I + T) is singular if σmin [I + Tnom (k,ω)] ≤ σmax [Eadd (k,ω)]

(5.70)

and conclude that a condition for the robustness of stability is that (I + T) is nonsingular and (5.60) is BIBO stable with respect to the additive modeling error matrix Eadd if σmax [Eadd (k,ω)] < σmin [I + Tnom (k,ω)], for all (k,ω) .

(5.71)

Note that (5.71) is conservative in the sense that the condition is sufficient but not necessary.

5.6

Metrics for Achieving Stability and Robustness for Control of Smart Structures

141

5.6.2 Multiplicative Error Uncertainty Finally we consider the case when the difference between the nominal and true plant is represented by the multiplicative error matrix Emult at the plants output vis, T(k,ω) = [I + Emult (k,ω)]Tnom (k,ω) = Tnom (k,ω) + Emult (k,ω)Tnom (k,ω).

(5.72)

Again we know that if (I + T) is singular then (I + Tnom + Tnom Emult ) will be singular as well. There exist a vector quantity x with x2 > 0 in the null space of (I + Tnom + Tnom Emult ) such that (I + Tnom + Emult Tnom ) x = 0

(5.73)

(I + T−1 nom + Emult )Tnom x = 0.

(5.74)

which can be re-written as

If the forward loop is stable then (I + T−1 nom + Emult )x = 0,

(5.75)

(I + T−1 nom ) x = −Emult x.

(5.76)

and

Proceeding in the same manner as before we obtain the result that if (I + T) is singular then σmin [I + T−1 nom (k,ω)] ≤ σmax [Emult (k,ω)]

(5.77)

σmax [Emult (k,ω) < σmin [I + T−1 nom (k,ω)], for all (k,ω)

(5.78)

so that

is a sufficient (but not necessary) condition for (I + T) to be non-singular thus insuring that the closed loop system is stable with respect to the multiplicative modeling error. Equation (5.78) can be written in terms of its singular values as σmax [C(k,ω)]
L, must be minimized.

6.4.1 The Singular Value Decomposition and Performance Metrics for Shape Control The singular value as introduced in Chap. 5 is useful in the quantification of performance metrics for shape control. Consider the submatrix M (α) and its singular value decomposition M(α) = U

VH .

(6.36)

where U and V are, respectively, left and right singular vectors for the corresponding singular values of M (α) contained in . The right singular vector V spans the control signal input space of M (α) while the left singular values span the output space. If the left singular values of M (α)are equal to the unit vectors ei , i.e. U = I, the output space decouples and the system will exactly achieve the required spatial bandwidth of basis shapes independently and/or in linear combination. If we consider the shape basis as representing “directions” in the output space ψi (x), then each shape will be equally attainable as the ratio of maximum and minimum singular values of M (α) i.e. the condition number approaches 1 as these denote the system gains in each direction. In addition the system will be well controlled with respect to ψi (x) when the minimum singular value σmin (M) is large. As noted earlier, the choice of actuator types and spatial distributions fixes M (α) and thus the requirements stated here can be used to screen candidate actuator distributions. In practice the screening process can become tedious for systems with even a modest number of inputs and outputs. One means of mitigating this is via the use of “input/output coupling operators [56]. Here the technique involves expressing the singular value decomposition of M (α) in terms of an input/output factorization, M(α) = EF.

(6.37)

and the corresponding spatially band-limited form of the input/output relation (6.31) then becomes

6.4

Input/Output Coupling and Transducer Shading

157

y = EFu,

(6.38)

y ≡ [c1 (ω)...cN (ω)]T ,

(6.39)

u ≡ [u1 (ω)...uN (ω)]T,

(6.40)

where

and the requirement L = N is assumed to be met. We define an intermediate variable z, z = Fu,

(6.41)

y = Ez.

(6.42)

and

We now choose E and F such that (6.63) can be satisfied using the singular value decomposition (6.62), 1

F=

2

VH ,

(6.43)

1

E=U

2

.

(6.44)

The matrix F transforms the input u to the gain space z of the singular values, while the matrix E transforms from gain space to the requisite output shape space, y. The ability of each actuator to drive the system to any given shape or a linear combination of shapes can be evaluated by letting u=

N

ηi ei ,

(6.45)

i=1i

with {ei } being the standard unit vectors. Then, z=

N i=1

ηi Fei =

N

ηi Fi

(6.46)

i=1

setting ηi = 1without loss of generality, each i-th column of F is the contribution of the i- th actuator, from the input (control) space to the gain space. In addition, zi 2 = Fi 2

(6.47)

158

6 Shape Control: Distributed Transducer Design

can be thought of as a measure the i-th actuator’s contribution to controlling all N shapes. Example: Shape Control of a Simply Supported Beam (Actuator Design) We now demonstrate the use of the techniques presented using “design by analysis”. Consider the problem of controlling the shape of a non-dimensional simply supported beam. We choose the desired shape basis functions to be the eigenfunctions of the system and use the first 4 sinusoids, ψi (x) = sin (iπ x); i = 1..4, 0 < x < 1.

(6.48)

The beam will be driven by abutting distributed piezoelectric film actuators. Four actuators are required to meet the bandwidth specification. Each actuator has been arbitrarily chosen to have a rectangular aperture, and is assumed for simplicity to have infinite temporal bandwidth, hence qj (x,x) = δ (x − cj ) − δ (x − cj − j ),

(6.49)

where 0 < c j < 1, and j is the aperture width. The elements of the transformation matrix M (α)can be written in the following generic Green’s function form αij (ω) =

∞

fjk (ω)hik (ω).

(6.33)

k=1

Because we have chosen the basis shape functions as the eigenfunctions of the system we have

ϕk (ξ ) qj (ξ ,ω)dξ , λk (ω)

(6.25)

ψi (x)ϕk (x)dx = 1.

(6.26)

fjk (ω) = D

with hik (ω) = D

This Sturm-Louisville system admits an Eigen system and can be represented using a bilinear expansion as follows, fjk (ω) = D

ϕk (x) qj (x)dx. λk (ω)

(6.50)

Using (6.48) and (6.49) we can write (6.50) as fjk (ω) =

1 λk

D

3

4 sin (iπ x) δ x − cj − δ x − cj − j dx

(6.51)

6.4


1 fjk (ω) = λk

159

1 x − cj dx − sin (iπ x) sin (iπ x) δ x − cj − j dx . λk D D ( ( )* + )* + δ

(1)

(2)

(6.52) The integrals (1) and (2) can be evaluated using the following general mathematical identities: f (a) δ (a + b) dx = f (−b) (6.53) f (a) δ (a) dx = −f (a) δ (a) dx = −f (0) . (6.54) Using these we can extrapolate the following

f (a) δ (a + b) dx = f (−b)

(6.55)

and we may now evaluate integral (1) using these by choosing a = x, f (a) = sin (iπ x), and b = −cj . Integral (1) in Eq. (6.52) now becomes

sin (iπ x) δ x − cj dx = −iπ cos iπ cj .

(6.56)

Similarly we can evaluate integral (2) by changing the parameter b = −cj − j and then (6.57) sin (iπ x) δ x − cj − j dx = −iπ cos iπ cj + j . Now combining we may use these results to rewrite (6.52) as fjk (ω) =

1 λk

⎤

⎡

⎥ ⎢ (−iπ ) ⎣cos iπ cj − cos iπ cj + j ⎦ . ( )* +

(6.58)

(3)

Expression (3) in Eq. (6.58) can be evaluated using the trigonometric identity cos iπ cj − cos iπ cj + j = cos

1 2 iπ

2cj + j − 12 iπ j − cos 12 iπ 2cj + j + 12 iπ j .

Equation (6.59) can be further simplified by letting a = b = 12 iπ j then using the fact that,

1 2 iπ

(6.59)

2cj + j and

160


cos (a − b) − cos (a + b) = 2 sin (a) sin (b) .

(6.60)

Expression (3) in Eq. (6.58) now is rewritten as

2 sin

1 1 iπ 2cj + j sin iπ j . 2 2

(6.61)

Combining the results for expression (1), (2), and (3) Eq. (6.58) becomes fjk (ω) =

1 1 1 2 sin iπ 2cj + j sin iπ j . λk 2 2

(6.62)

The eigenvalues be expanded as λk (ω) = −iπ ω2 + (iπ)2

(6.63)

and we can then write the following fjk (ω) =

2 sin

1 2 iπ

2cj + j sin 12 iπ j ω2 + (iπ )2

.

(6.64)

The elements of the transformation matrix M (α) become αij (ω) =

2 sin

1 2 iπ

2cj + j

sin

ω2 + (iπ )2

1 2 iπ

j

.

(6.65)

We examine the elements of the transformation matrix M (α) (at zero frequency) in order to assess actuator distributions which: 1. Minimize the condition number of M (α) 2. Realize a balanced actuator participation in performing the control task 3. Decouple the output space. [Note: The shape control task could have been posed at any frequency ω > 0 equally as well. However, in the present formulation the spatial compensation (e.g. actuator distribution) is designed first. After the actuator distribution is set, suitable MIMO control methods can be employed to develop a temporal compensator to achieve dynamic performance.] By way of example we choose four candidate distributions for assessment designated I through IV and shown in Fig. 6.6. The corresponding input coupling operators are shown in Fig. 6.7. For actuator distribution I the condition number was determined to be 22.5818 with a minimum singular value of 0.0061. Distribution I

6.4


161

I.

II.

III.

IV. Fig. 6.6 Active film actuator distributions

0.4

I

0.2 0

1

2

3

4

0.4

I

0.2

zi

0 2

1

2

3

4

0.4

II

0.2 0

1

2

3

4

0.4

I

0.2 0

1

2

3 Actuator Index

Fig. 6.7 Input coupling operators for actuator distributions I–IV

4

162


has a relatively large condition number and the corresponding input coupling operator plot in Fig. 6.7 shows that actuators 2 and 3 are doing most of the work. Conversely, this implies that the control voltages sent to actuators 1 and 4 will be larger than those to actuators 2 and 3 for a given output. Distribution II is observed to be asymmetric about the beams mid-span, and yields an even larger condition number of 25.5867 and a minimum singular value of 0.0052. Each actuator contributes to the shape control task in proportion to its size, as shown in Fig. 6.7. This suggests that a symmetric distribution (note that the sinusoids possess symmetry about mid-span) having actuators with equal-length apertures may be best. The symmetric distribution III, shown in Fig. 6.6, has a significantly lower condition number at 4.3296 with minimum singular value at 0.0253; subject to the constraint that the actuators have abutting rectangular apertures. Its input coupling operators, Fig. 6.7, show a more balanced participation of the actuators over the entire bandwidth of the shape control task. In addition, it was found that the matrix of left singular vectors U associated with distribution III is equal to the identity matrix; the output is decoupled by this spatial compensation. Since distribution III did not have equivalued input coupling operators, distribution IV was synthesized. The input coupling operators associated with distribution IV are all equal, hence all actuators participate equally over the entire bandwidth of the shape control task. It has a larger condition number of 5.2511 with minimum singular value of 0.0194 and its left singular vector matrix is not decoupled, as was the case for distribution III. Thus, input coupling operators cannot be used alone in screening actuator distributions for shape control. Rather, they are used as figures of merit in the analysis, with minimum condition number and a decoupled output space having priority in the final design selection. In the system example discussed here they reflect the tradeoffs in actuator participation over the entire spatial bandwidth due to the beam’s strain/curvature relation and boundary conditions.

6.5 Spatially Distributed Sensors and Shape Estimation The methods presented here for the choice of spatial distribution of distributed actuators may be applied to sensors as well. One key difference however is that not only must the spatial distribution of sensor sites be chosen but the sensors must also be used to estimate the shape (profile) output. Given a specified number and type of sensors, the shape expansion technique approach used earlier can be used to estimate the Fourier coefficients as decomposed on an orthonormal basis of shape functions. The coefficients may of course then be used to calculate the band limited reconstruction of the shape. Consistent with our earlier discussions on the modeling of sensors in Sect. 3.3.1, the output of the l-th sensor in a sensing array can be written as pl (x,ω)L[y(x,ω)]dx. (6.66) sl (ω) = D

6.5

Spatially Distributed Sensors and Shape Estimation

163

where pl (x,ω) is a sensor filter function, or spatio-temporal filter and L[ · ] is a linear spatial operator that reflects the operation of sensor. We may now expand the sensor output in terms of a band limited orthogonal basis of shape functions as before i.e., sl (ω) =

pl (x,ω)L[ D

N

cl (ω)ψl (x)]dx.

(6.67)

pl (x,ω)L[ψi (x)]dx.

(6.68)

i=1

Since L is a linear spatial operator we write sl (ω) =

N

ci (ω) D

i=1

=

N

cl (ω)βil (ω),

i=1

for βil (ω) ≡

pl (x,ω)L[ψi (x)]dx.

(6.69)

D

For P sensors, one can construct a matrix input/output relation or transformation matrix between the shape coefficients ci (ω) and the sensor outputs sl (ω) ⎤ ⎡ ⎤⎡ ⎤ β11 (ω) β21 (ω) · · · βN1 (ω) c1 (ω) s1 (ω) ⎢ s2 (ω) ⎥ ⎢ β12 (ω) β22 (ω) · · · βN2 (ω) ⎥ ⎢ c2 (ω) ⎥ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎢ .. ⎥ = ⎢ .. .. .. ⎥ ⎢ .. ⎥ , ⎣ . ⎦ ⎣ . . . ⎦⎣ . ⎦ sp (ω) β1P (ω) β2P (ω) · · · βNP (ω) cN (ω) ⎡

(6.70)

or more compactly s = M(β)c.

(6.71)

We can now estimate the shape function expansion coefficients given the measurement vectors, and reconstruct the shape using, c = [M(β)]♦ s,

(6.72)

where ( · )♦ denotes the pseudo inverse and requires that the matrix M(β) have full rank. Note that there must be at least as many sensors as shape functions i.e. P ≥ N, and more if one wishes to over-determine the problem to prevent aliasing from shapes ψi with index i > N which are outside of the bandwidth of interest. Note that the judicious choice of the number of sensors P must be made by the smart structure designer. The choice is driven by many factors such as anti-aliasing constraints [57, 58], inherent structural filtering characteristics and sensor aperture weighting.

164


The singular value decomposition of M(β)provides a mechanism for the assessment of the requirements for shape control and is M(β) = U

VH ,

(6.73)

where U is the matrix of left singular vectors which span the sensor signal space, is the matrix of singular values, and V is the matrix of right singular vectors which span the shape space. All “directions” (e.g. shapes) in ψi space will be equally measurable if the ratio of the maximum and minimum singular values of M is as near to 1 as possible, since these singular values denote gains in the various signal/shape directions; this ratio is the matrix condition number. The shapes will also be “best” estimated if σmin (M) is large, implying more inherent gain in the sensing system since M(β) is fixed by the sensor types and locations/apertures, these requirements become criteria for screening candidate sensor distributions for shape estimation and reconstruction.

Example: Shape Control of a Simply Supported Beam (Sensor Design) The application of the shape reconstruction performance measures will now be demonstrated by an example problem of “design by analysis”. Consider the shape measurement of a pinned–pinned beam of length 40 in., where the desired shape functions are the first four sinusoids; ψi (x) = sin (

iπ x ); 40

i = 1 . . . 4,

0 < x < 40.

(6.74)

The output will be sensed by ten point displacement transducers having infinite temporal bandwidth (with respect to the control task) and therefore, pl (x) = δ(x − xl ), 0 < x,xl < 1,

(6.75)

and the transducer spatial operator is L = 1.

(6.76)

The elements of M(β) are then βil = sin (

iπ xl ). 40

(6.77)

For the case presented P =10, and N = 4, P > N and the output is oversampled. The matrix M(β) can then be calculated, and selected sensor positions studied to determine placements that minimize its condition number.

6.5

Spatially Distributed Sensors and Shape Estimation

σmin = 1.5767

165

conditon = 1.6027

p(x)

0

1

Fig. 6.8 Displacement sensor distribution I

σmin = 2.3292

conditon = 1.0890

p(x)

0

1

Fig. 6.9 Displacement sensor distribution II

σmin = 2.3452

conditon = 1.0000

p(x)

0

1

Fig. 6.10 Displacement sensor distribution III

Three candidate sensor distributions are sketched in Figs. 6.8, 6.9, and 6.10, and are hereafter designated distributions I, II, and III. The condition number and minimum singular value for each distribution appears in the corresponding figure. Distribution I was randomly chosen, and displays the largest condition number (∼1.6) of the three distributions. The fact that the condition number is close to 1 reflects the oversampling, even for this distribution. Distribution II is symmetric (as are the shape functions), and has a lower condition number as well as a larger minimum singular value. This makes it “better” than distribution I. Distribution III is equi-spaced and achieves an ideal condition number of 1. In addition, its minimum singular value is the largest of all the distributions. Consequently, it is the best distribution of ten point displacement transducers to estimate the first four sinusoidal shapes. This result does not preclude the use of other sensors for the shape reconstruction task but is an example of how the formalism can be applied to screen candidate sensor distributions. The matrix M(β) can be calculated and analyzed for various types of sensors, not only to screen placements for each type of transducer, but to compare various types of sensors as well using the performance measures developed here.

166


6.6 Summary In this chapter we have developed and presented techniques for the design and assessment of spatially distributed sensors and actuators for shape control. These are essentially performance measures which should be viewed as extensions to those developed earlier in Chap. 5. The shape control problem was cast in the context of shape representations in terms of orthogonal, band limited basis functions. A set of performance measures was derived for shape estimation/reconstruction that are consistent with the discrete spatial transform analysis derived for actuator placement. The sensor placement problem is the dual of the actuator placement problem. Conditions for sampling and optimum placement were derived that are useful not only within the context of shape control, but for shape estimation in general. Example problems for actuator and sensor placement were presented.

Problems (6.1) Visit the UIUC airfoil database at the website below, or search the web for a coordinate data file for the Eppler E422 high lift airfoil. Calculate a Fourier sine series expansion of the upper airfoil surface shape using 3 expansion elements and plot it against the desired shape. What is the mean squared error for this expansion? Determine the minimum number of expansion terms required to reduce this error by one order of magnitude. Repeat for the lower surface shape. http://www.ae.uiuc.edu/m-selig/ads/coord_database.html#N (6.2) Consider the simply supported beam shape control example presented in the chapter. Suppose each of the spatially distributed actuators on the beam were triangular pennant shapes, i.e. qj (x) = δ x − cj + δ x − cj − δ x − cj − j

a) Derive the expression for the elements of the transformation matrix M (α) . b) Compute the input/output coupling operators and discuss performance in terms of the system singular values and condition number. c) Discuss how each of the spatially distributed actuators might be collocated with compatible sensors.

Chapter 7

Shape Control, Modal Representations and Truncated Plants

7.1 Introduction In theory smart structures typically have dynamics that we may wish to control which occur in both space and time. As we have seen these systems often involve plant representations in the form of partial differential equations which are essentially infinite dimensional, distributed parameter systems. The idea of using a model which contains an infinite number of modes has been suggested as inappropriate for real physical systems. Real system representations are often truncations of the true plant’s characteristics using finite element or a modal analysis which admits eigenfunctions. When eigenfunctions are used, they represent a complete basis set with respect to the static and dynamic response of linear self-adjoint distributed parameter systems. When the eigenfunction expansion is truncated for practical reasons, the predicted response in the spatial domain will be in error. Paradoxically, using a model which contains an inordinate amount of modal dynamics, presents the design engineer with unwanted computational problems, particularly if the required control temporal bandwidth requirement is “low”. For shape control, these issues can be addressed by appending a quasi-static correction term to the truncated plant model.

7.2 Shape Error and Feed Forward Correction The static system response, or static influence function, may be expressed in terms of the plants static Green’s function as ys (x,0) =

hs (x,ξ ) u(ξ ,0)dξ ,

(7.1)

D

where hs (x,ξ ) is the static Green’s function. The system input can be expressed as a superposition of inputs, as per Eq. (6.17), so that Eq. (7.1) becomes J.E. Hubbard, Spatial Filtering for the Control of Smart Structures, C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-03804-4_7,

167

168

7 Shape Control, Modal Representations and Truncated Plants

N ys (x,0) = [ hs (x,ξ )qj (ξ ,0)dξ ]uj (0).

(7.2)

D

j=1

As we saw earlier in Chap. 6, the dynamic response of a self-adjoint distributed parameter system can be represented using a bilinear expansion in its eigenfunctions and if the eigenfunction expansion is truncated, the modally band-limited response is given by yd (x,ω) ∼ =

K

ϕk (x) D

k=1

ϕk (ξ ) u(ξ ,ω)dξ . λk (ω)

(7.3)

Because we choose to express the input as a superposition of inputs, the dynamic system response now becomes yd (x,ω) ∼ =

N K ϕk (ξ ) qj (ξ ,ω)dξ ]uj (ω) [ ϕk (x) D λk (ω) j=1 k=1 N K

[

=

j=1

k=1

(7.4)

ϕk (x) f jk (ω)]uj (ω), λk (ω)

where f

jk (ω)

≡

ϕk (ξ )qj (ξ ,ω)dξ .

(7.5)

D

We now note that at zero frequency, the static and dynamic representations of the output y must be equal and because of the modal truncation of the dynamic Green’s function, the responses predicted by (7.2) and (7.4) differ by the amount ys (x,0) − yd (x,0) =

N K ϕk (x) f jk (0)]uj (0) [ hs (x,ξ ) qj (ξ ,0)dξ − λk (0) D j=1

k=1

(7.6)

N dˆ j (x,0)uj (0), = j=1

for dˆ j (x,0) ≡

hs (x,ξ )qj (ξ ,0)dξ − D

K ϕk (x) k=1

λk (0)

f jk (0).

(7.7)

Therefore the true static response, as well as the dynamic response for the case when ω N. The correction vanishes because the eigenfunctions defining the shape control task ϕk , k = 1,....N, are orthogonal to those for k = N + 1,....,∞ that lie beyond the required discrete spatial bandwidth.

Example 1: Shape Control of a Non-dimensional String (Feed Forward Correction) As a simple example of the feed forward correction strategy, consider a string of unit length with fixed ends. The non-dimensional governing equation is ∂ 2y ∂ 2y − 2 = u(x,t), ∂t2 ∂x

0 < x < 1.

(7.20)

. The Fourier transform of the governing equation (7.20) in time yields the corresponding dynamic Green’s function and has the bilinear expansion h(x,ξ ,ω) =

∞ 2 sin (kπ x) sin (kπ ξ ) . π2 (kπ )2 + ω2

(7.21)

k=1

Statically, the string is described by the governing equation ∂ 2y = u (x) , ∂x2

(7.22)

which has a static Green’s function of the form & hs (x,ξ ) =

(1 − ξ )x for x ≤ ξ , (1 − x)ξ for ξ ≤ x.

' .

(7.23)

If the string is driven by a point load of unit magnitude located at ξ = 0.75, the corresponding static displacement of the string, using the static Green’s function (7.23), is plotted in Fig. 7.1. Since the string has no flexural rigidity, the slope at the excitation point is discontinuous, and the displacement is a linear function of position. The static response of the string to the same excitation, using the dynamic Green’s function (7.21), is plotted in Fig. 7.2 for 4, 6, 8 and 10 respectively in the bilinear expansion. The error in the predicted response is due to the truncation. Now, if a quasi-static correction is appended to the truncated dynamic Green’s function, the static response prediction becomes exact, as shown in Fig. 7.3. The feed through correction will be valid for frequencies below the resonance of the last mode retained in the truncated plant representation.

172


Fig. 7.1 String static Green’s function response at ξ = 0.75

Fig. 7.2 Band-limited approximation of string static response

7.3

A Complete Dynamic Shape Control Case Study

173

Fig. 7.3 String static response with feed through correction

7.3 A Complete Dynamic Shape Control Case Study In this section we formulate, pose and present in it’s entirety a case study of smart structure design and shape control in the form of a morphing airfoil. In presenting the study we will exploit the spatial filter design techniques developed and covered throughout this text. In addition we will employ well known and accepted temporal filter design techniques readily implemented using modern software tools such as Matlab(TM) . The study is meant to stand alone and thus by presenting a complete case in its entirety, it is hoped that the reader will obtain a global view of the techniques presented earlier and a deeper appreciation of their power and utility. The case is also meant to be a succinct review of the spatial filter design and at the risk of redundancy we will refresh the reader’s memory when necessary with any assumptions, constraints and background needed to complete the design.

7.3.1 Case Background Unmanned Aerial Vehicles (UAVs) have been receiving unprecedented support in recent years due in large part to their increasing utility as intelligence, reconnaissance and surveillance platforms. As their missions become more complex, vehicle endurance, range, maneuverability and expense become key drivers in their design

174


and configuration. Morphing aircraft technology has drawn widespread interest as a promising technology that may allow the design of multifunction multi-role vehicles. For this class of vehicles airfoil design will have a major impact on performance. Low camber airfoils yields higher L/D at high speeds while high camber yields higher L/D at low speeds. In order to effectively broaden their mission profile UAV’s will need to morph from one airfoil shape to another so as to seamlessly transition from soar to strike, observe to maneuver and the like. Vehicle designers will need to create the most effective actuator/structural designs to morph airfoils. These designs must be capable of morphing with respect to some nominal baseline shape and use minimal energy during the process. The use of smart materials and structures has been recognized as an efficient means of designing vehicles which can be reconfigured in response to changing conditions. In limited cases adaptive structures incorporated into modern vehicle technology have resulted in improved maneuverability, increased redundancy/survivability and reduced weight [59, 60]. Aircraft wing and airfoil morphing offer unique challenges with regard to increased control system design complexity, actuator/sensor transducer requirements and morphing structural materials. Traditional approaches to shape control use discrete transducers and lumped parameter modeling techniques. For real-time shape control these techniques can only address performance measures in a limited fashion, if at all. Distributed Parameter Control (DPS), i.e. the control of systems described by space and (usually) time, can more effectively address the spatial performance requirements dictated by the shape control problem. In addition to traditional temporal bandwidth requirements, shape control requires both a prescribed spatial bandwidth and a set of shapes that characterize the control task, e.g. airfoil shapes for efficient flow control. Distributed Parameter System control techniques and spatially distributed transducers are well suited to the design and implementation of dynamic shape control in modern systems. In this example case study modern robust multi-variable control techniques are extended and applied to the problem of dynamic airfoil morphing. Spatially distributed transducers and distributed parameter system design and analysis techniques are applied to seamlessly integrate MIMO system design methodologies with dynamic shape control synthesis.

7.3.2 Airfoil Shapes and the Discrete Spectrum Parameterization The machinery of modern control analysis and synthesis, while explicitly incorporating temporal frequency domain information for the synthesis of temporal servos, does nothing to address spatial performance and the synthesis of spatial servos. The distributed nature of the system to be controlled is either neglected or treated using ad hoc methods. Performance requirements for the shape control problem must be

7.3


175

posed in spatial terms. Using MIMO techniques one can construct an input/output relation representing a DPS in a temporal and spatial frequency domain. Given the success and availability of temporal frequency domain tools in classical and robust, multi-variable lumped parameter systems control theory, the introduction of spatial frequency transforms to distributed parameter systems and control follow naturally. An extensive treatment of spatial filtering concepts in distributed parameter control has been developed in previous chapters. Those techniques are applied here to the airfoil morphing control problem. The details of modern temporal filter design techniques are beyond the scope of this text, however key results are highlighted herein where necessary. Morphing from one shape to another requires that the morphing system use minimal energy during the process. Posing the control problem in terms shapes that are easily and readily attainable by the system can achieve this goal. The control problem for achieving the desired airfoil shape can readily be posed in terms of controlling a set of orthogonal basis shapes. Any desired shape would then be represented by y(x) = c1 ψ1 (x) + c2 ψ2 (x) + . . . ,

(7.24)

where {ψi (x)} are the orthogonal shape functions that represent the required component profiles, and thus the shape control task. Consider a shape y(x,ω)defined on x ∈ D. One possible requirement for the representation per Eq. (7.24) would be y(x,ω) −

lim

n→∞

n

ci (ω)ψi (x) = 0,

(7.25)

i=1

e.g. that the expansion should precisely equal the shape at every point in the domain D. In actual practice, such a point-by-point convergence is generally difficult to achieve. A more realistic requirement that can be satisfied under fairly general conditions is that the mean square error in representing the shape be minimized over the domain, D, vis. y(x,ω) −

lim

n→∞

n

2 ci (ω)ψi (x)

dx = 0.

(7.26)

i=1

D

The shape problem now reduces to one of finding the coefficients ci that minimize this error. A necessary and sufficient condition for the shape y(x,ω) to be approximated in the mean by an expansion in the orthonormal shape functions {ψi (x)} is that Parseval’s equation be satisfied i.e., ∞ i=1

c2i (ω)

=

y2 (x,ω)dx D

(7.27)

176


where ci (ω) are Fourier coefficients. An orthonormal system of functions {ψi (x)} can now be referred to as complete with respect to the shape y(x,ω) if the corresponding Fourier coefficients satisfy (7.27). It is important to note however, that even if the system is not complete, choosing the expansion coefficients ci (ω) to equal the Fourier coefficients of y(x,ω) with respect to {ψi (x)} leads to a representation minimizing the mean square error over D [9]. The shape control problem is now parameterized and becomes a matter of deriving an input/output representation of the distributed plant in terms of the expansion (7.24).

7.3.3 The Concept of Eigenfoils Many modern airfoils can be adequately described using the parameterization described above, i.e. a limited set of orthogonal basis or eigenfunctions and for the sake of this example case we shall refer to these as eigenfoils. Consider for example the MH 20 pylon racer high performance airfoil shown below in Fig. 7.4. The airfoil shape shown can be represented in decomposition on a basis of functions with a discrete spatial transform in a set of orthogonal functions, e.g. a Fourier series. The upper portion of the airfoil shape for example can be decomposed as y(x) = 0.0613 sin (π x) + 0.0203 sin (2π x) + 0.0048 sin (3π x) + 0.0054 sin (4π x); (7.28) which has a mean square fitting error of 0.0102 or roughly 1% and is shown graphically in Fig. 7.5 where Y is the airfoil thickness, X is the chordwise dimension and C is the nominal chord length.

M2HO

Designed for pylon racing model aircraft (FAI class F3D)

Characteristics • Thickness: 9.02% • Low moment coefficient of cm c/4. • Less drag than Eppler 221. • Can be used at Reynoldsnumbers of 350’000 and above Fig. 7.4 MH 20 pylon racer airfoil geometry

7.3


177

M2HO Airfoil Upper Profile 0.2

Y/C

0.1 0 –0.1 –0.2 0

0.2

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Coefficient Magnitude

X/C 0.08 0.06 0.04 0.02 0

1

2 3 Coefficient Number

4

Fig. 7.5 Fourier series approximation of the MH20 airfoil

The discrete spectrum indicated by (7.28) and plotted in Fig. 7.5 indicates that to a reasonable approximation the airfoil shape has a finite discrete spatial bandwidth. An interesting and somewhat intuitive interpretation of (7.28) is that the airfoil shape may be thought of as a superposition of four distinct functions or basis shapes. Hence if a control system is designed such that it can achieve each basis shape independently, then it can also synthesize the entire family of shapes represented by linear combinations of the basis. For the morphing airfoil problem, this family of shapes should by necessity span the performance envelope of interest. Airfoils which can be adequately described using a limited set of orthogonal basis or eigenfunctions are therefore amenable and predisposed to achieving the entire family of shapes represented by linear combinations of those basis. This property can be used to guide the design of smart structures which require morphing or dynamic shape control. Structural systems which are themselves self-adjoint distributed parameter systems (e.g. Sturm-Louisville systems) readily admit orthogonal eigenfunctions and have preferred shapes governed by these bases. These so-called eigenfoils allow for seamless integration of dynamic shape control into the structural design of the overall system.

178


7.3.4 Morphing Airfoil Design Considerations In this section a “proof of concept” design is undertaken and the application of the aforementioned techniques will be demonstrated using “design by analysis”. Consider the morphing airfoil system shown below in Fig. 7.6. The system consists of upper and lower flexible airfoil ribs joined together at their boundaries via pinned–pinned boundary conditions. Each rib or beam can be modeled as a Bernoulli-Euler beam with in-plane tension and structural damping. The displacement of each beam is therefore assumed to be expandable in a set of sinusoidal shape functions. For the pinned-pinned beam, these functions are orthogonal and therefore possess the property of spatial independence. A detail of one possible implementation is illustrated in Fig. 7.7. Our goal is to be able to command the shape of both the upper and lower airfoil ribs so as to assume the shape of the HF20 pylon racing airfoil shown in Fig. 7.4 and the family of shapes represented by any linear combination of its basis. The desired shape functions will be the first four sinusoids as given in Eq. (7.28). Tables 7.1 and 7.2 are tables of the beam and actuator properties respectively used in the plant model for the shape control task. We consider the shape control of only the upper rib modeled as a non-dimensional pinned-pinned beam assuming that the lower rib could be controlled simultaneously in a similar manner. For the purpose of control we will use spatially distributed actuators in the form of NASA Macro Fiber Composite (MFC) actuators [61]. These actuators have good control authority, stability and robustness properties and because they have spatial

Flexible Beam #1

pinned

Flexible Beam #2

pinned

Fig. 7.6 Morphing airfoil configuration

Micro-Fiber Composite Bending Actuators

Rubber Trailing Edge

Cantilevered Wing Rib

Rub Rubber ber Leading Lea Edge ding Edg

Simply Supported Boundary

Fig. 7.7 Morphing airfoil detail

7.3


179

Table 7.1 Airfoil rib material properties Parameter

Value

Material Length Thickness Density Modulus

904 Stainless steel 14.56 in. 0.015 in. 0.2815 in./lb3 3.047 × 107 psi

Table 7.2 Macro-fiber composite actuator material properties Parameter

Value

Length Width Thickness

3.4 in./active segment 1.1 in. 0.01181 in. (0.01281 film + bond) 0.065 in./lb3 4.4 ×106 psi 8.267 × 10−9 in./volt 0.12 in.

Density Modulus Static piezo-electric constant,d33 Interstitial segments

extent are well suited to the shape control task. The MFC has proven to be particularly useful in both rotary-wing and fixed-wing aeronautical applications. It has the desirable features of high strain energy density, directional actuation, conformability and durability.

7.3.5 Actuator Placement and Input/Output Coupling In order to evaluate and decide on the proper actuator placement we apply the techniques described in Chap. 6 and repeated in condensed form here for clarity. In the shape control problem the goal is to drive the output of the distributed system (i.e. the upper airfoil rib) to each of the orthonormal shape functions {ψi (x)} up to a discrete band limit i = L, both independently and in combination, i.e.

ydesired (x,ω) =

L

βi (ω)ψi (x).

(7.29)

i=1

Equation (7.29) can be restated in matrix form in terms of the plant and actuator characteristics by executing a “harmonic balance” in these expansion functions,

180


⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ α11 1 0 0 ⎢0⎥ ⎢1⎥ ⎢ 0 ⎥ ⎢ α21 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ β1 ⎢ . ⎥ + β2 ⎢ . ⎥ + . . . + βL ⎢ ⎢.⎥=⎢ . ⎢0⎥ ⎢0⎥ ⎢ 1 ⎥ ⎢ αL1 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ .. .. .. .. . . . .

⎤ α12 · · · α1 N ⎡ ⎤ u1 α22 · · · α2 N ⎥ ⎥ ⎥ .. .. ⎥ ⎢ ⎢ u2 ⎥ . . ⎥ . ⎢ . ⎥⎣ . ⎥ . ⎦ αL2 · · · αLN ⎥ ⎦ un .. .. . .

(7.30)

The left hand side of (7.30) is spanned by a set of orthonormal vectors βi which represent the coefficients of the desired output shape expansion and the actuator distributions on the airfoil rib fix the matrix of coefficients αij as well as the spatial and temporal response characteristics of the morphing structure. The partitioned submatrix shown in (7.30) must be invertible and have full rank in order to allow any given actuator distribution to achieve a given output. Therefore, the number of actuators N must be equal to or greater than the number of shapes L of the commanded output as expanded in the shape function basis {ψi (x)}, i = 1,. . .,L. In addition, the residual components of the output at higher spatial frequencies (i > L) must be minimized. If we denote the submatrix in (7.30) as M (α), it has been shown that it’s rank and condition number at zero frequency determine the degree of participation of each actuator via input coupling operators and the degree of decoupling that occurs in the output shape space. The input coupling operators may be computed by considering the airfoil rib control problem as control of a non-dimensional pinned–pinned beam. The desired shape functions are given in Eq. (7.28) as four sinusoids. The rib will be driven to the desired shape by abutting four MFC actuators along the beam as shown in Fig. 7.8. As previously discussed four actuators are required in order to meet the bandwidth requirements of (7.28). Each actuator has the characteristics given in Table 7.2, a rectangular aperture and is assumed for the sake of simplicity to have infinite temporal bandwidth. The exogenous command signal can be represented as degenerate, i.e. separable in space and time,

u(x,ω) =

4

qj (x,ω)uj (ω),

(7.31)

j=1

0

1

3.4” 0.12”

3.4” 0.24”

Fig. 7.8 MFC actuator distribution

3.4” 0.24”

3.4” 0.24”

0.12”

7.3


181

where we assume a uniform spatial distribution over the aperture of the actuators vis., qj (x) = δ (x − cj ) − δ (x − cj − j )

(7.32)

and where 0 < cj < 1, and j is the aperture width. The elements of the M (α) matrix given in (7.30) may be given in “generic” Green’s function form as, αij (ω) =

ψi (x)h(x,ξ ,ω)qj (ξ ,ω)dξ dx.

(7.33)

D

. For the pinned–pinned beam, the elements of the M(α) matrix are then αij (ω) =

2 sin

iπ 2

(2cj + j ) sin iπ2 j ω2 + (iπ )2

.

(7.34)

The input coupling operators for the distribution shown in Fig. 7.8 are given in Fig. 7.9 below. The plot is annotated with the associated condition number and minimum singular value of M(α). The plot shows a fairly balanced participation of the actuators over the entire bandwidth of shapes as reflected by the relatively low condition

Fig. 7.9 Input coupling operators

182


number (σmin ≈ 4). This distribution minimizes the condition number subject to the constraint that the MFC actuators have rectangular apertures with a separation dictated by the manufactured interstitial spacing. The left singular values of M(α) are equal to the identity matrix hence the output is decoupled by this spatial compensation. It must be noted that the input coupling operators must not be used alone to screen candidate distributions for dynamic shape control. They are instead intended for use as figures of merit in the analysis driven by the goal of achieving minimum condition number and a decoupled output space. Having set the actuator distribution and concomitant spatial performance metrics, MIMO control methods may now be employed to develop a temporal compensator to achieve the desired performance.

7.3.6 Morphing Airfoil Rib: Discrete Parameterization and the System Model For control system design purposes the “Plant”, which consists of the upper airfoil rib, is modeled as a Bernoulli-Euler beam with in-plane tension and internal structural damping. The model presented here follows closely the procedures outlined in Chap. 6. The governing equation for the airfoil rib including the control inputs from the four MFC spatially distributed actuators is, ∂ 2 Vi (x,t) ∂ 2 y(x,t) ∂ 2 y(x,t) ∂ 3 y(x,t) ∂ 4 y(x,t) + ρA = m − T − λ ,0 < x < L, ∂x4 ∂x2 ∂x2 ∂t ∂ 2t ∂x2 i=1 (7.35) where 4

EI

EI is the rib/MFC composite structures flexural rigidity T is the in-plane tension λ is the structural damping coefficient ρ is the rib/MFC composite’s mass per unit length A is the cross-sectional area L is the length of the rib m is the MFC actuator gain constant. The displacement of the airfoil rib is assumed to be expandable in a set of sinusoidal shape functions, y(x,t) =

∞ n=1

sin

nπ x yn (t), L

with the Fourier coefficients being defined by

(7.36)

7.3


yn (t) =

2 L

L

y(x,t) sin 0

183

nπ x dx. L

(7.37)

The input can be represented as a separable product of space and time for these so-called degenerate actuators, V(x,t) =

4

i (x)ui (t).

(7.38)

i=1

Assuming a uniform spatial distribution over the actuators aperture yields the result that

i (x) = h(x − ci ) − h [x − (ci + i )] ,

(7.39)

where ci is the start of the aperture, and i is its width. Similarly

i (x) = δ (x − ci ) − δ [x − (ci + i )].

(7.40)

and if the control distribution is expanded in the requisite shape functions the right hand side of (7.35) takes the form nπ x ∂ 2 V(x,t) ui (t). = q sin in L ∂x2 4

∞

(7.41)

i=1 n=1

The Fourier coefficients qin of the MFC actuator distributions i (x) are described by 4 2 L3 [x − (c + )] sin nπ x dx δ (x − c ) − δ qin = i i i L 0 L 4nπ nπ nπ i = − 2 sin (2ci + i ) sin 2L 2L L

(7.42)

The plant described in Eq. (7.35), thru the discrete parameterization presented, may now be represented in traditional MIMO state space canonical form.

7.3.7 State Space Canonical Form Defining the time derivative of the n-th Fourier coefficient yields νn (t) ≡ y˙ n (t). The n-th shape of the governing equation may then be described as

(7.43)

184


4 EI nπ 4 T nπ 2 λn nπ 2 m yn + + νn + ν˙ n = qin ui, ρA L ρA L ρA L ρA

(7.44)

i=1

In state space form this becomes

y˙ n ν˙ n

⎡

⎤

0 1 yn 4 2 2 ⎣ ⎦ EI λ nπ nπ nπ T = n νn − − − ρA L ρA ⎡L ⎤ ρA L u 1

⎥ m 0 0 0 0 ⎢ ⎢ u2 ⎥ + ⎣ u3 ⎦ ρA q1n q2n q3n q4n u4

(7.45)

where the state vector is defined as T Xp ≡ y1 ν1 · · · y4 ν4 ,

(7.46)

T Up ≡ u1 · · · u4 .

(7.47)

and the control vector as

The first four output shapes are now the system outputs and thus the output state vector can be defined as T y ≡ y1 · · · y4 .

(7.48)

Now the governing equation for the smart structure consisting of the airfoil rib and MFC actuators can be written in state matrix form as ˙ p = Ap Xp + Bp Up X

(7.49)

y = Cp Xp .

(7.50)

In this form we have achieved the parameterization that has the plant inputs as control voltages and the outputs are Fourier coefficients.

7.3.8 Morphing Airfoil Closed Loop Shape Controller Synthesis A temporal compensator for the morphing shape control task is designed using the Loop Transfer Recovery (LTR) method of modern robust control. LTR is a structured method of compensator synthesis. Modern synthesis tools can be found in the robust control toolbox offered by MatLab. The design methodology is well known and well documented. The multivariable design is implemented using a Linear Quadratic Gaussian (LQG) and a Loop Transfer Recovery (LTR) design

7.3


185

Table 7.3 Shape controller synthesis process 1. Determine scaled state space matrices 2. Determine system natural modes – eigenvalues and eigenvectors 3. Modal analysis-excitation of individual modes 4. Transmission zeros and directions 5. Controllability rank test 6. Observability rank test 7. Frequency response-scaled singular values 8. Singular value decomposition (SVD) analysis at DC 9. Step responses 10. LQG/LTR design at the plant output (i) Form design plant and augment with integrators at input (ii) Design plant SVD analysis (iii) Design target loop using kalman filter theory – Design target loop: matching at all frequencies – Adjust target loop bandwidth – Target singular values: loop, sensitivity, complementary sensitivity (iv) Recover target loop via cheap control problem 11. Compensator analysis: poles, zeros, singular values 12. Open loop analysis: poles, zeros, singular values 13. Closed loop analysis: poles, zeros, sensitivity and complimentary sensitivity

which incorporates integral control and singular value matching at low and high frequencies. Because the shape expansion parameterization produces state space models (for self-adjoint plants), other multi-variable control design methods such as H∞ , μ-synthesis etc. could also be used. The steps used in the control synthesis process are outlined in Table 7.3. A nominal plant as described by Eqs. (7.49) and (7.50) using the parameters of Tables 7.1 and 7.2 is assumed. A scaling of 10 • mils as outputs and V/100 as inputs was used for the compensator design and simulation. The airfoil rib was given a nominal tension of 4.9 lbs. The modal parameters are given in Table 7.4. The uncompensated forward loop transfer matrix singular values for the nominal plant model are shown in Fig. 7.10. The plant has some directionality as seen and indicated by a static condition number of 1.4135. The reason for this directionality is reflected in the earlier figures of coupling coefficient (See Fig. 7.9.) by the asymmetry in actuator coupling due to the constrained interstitial spacing between

Table 7.4 Beam/MFC dynamic parameters Mode

Frequency

λn

1 2 3 4

34.576 71.317 112.18 158.78

0.00184 0.00191 0.00167 0.00456

186


Fig. 7.10 Uncompensated nominal plant model SVD response

actuators. Because the singular value plot is not highly directional, compensator design and implementation should be simplified. The system eigenvectors were calculated and used to set the initial conditions needed to excite each mode. The response characteristics for the first four modes are plotted in Figs 7.11 and 7.12. An input to the open loop plant was formulated using the filtered step response profile shown in Fig. 7.13. The specifications for the control system design are as follows: • The nominal closed loop system is stable • The nominal closed loop system exhibits zero steady state error • The open loop singular values are matched at ALL frequencies and lie above 20 db for all frequencies below 1 Hz. The rib influence functions are determined by commanding a single actuator to drive the rib to its output profile. This was done for a command voltage of 100 V to each of the four MFC actuators separately. The MFC actuator drive voltages used here are conservative as these actuators have a drive range of –500 to 1500 V The purpose of this example case is to demonstrate the efficacy of the shape parameterization methodology and hence modest drive voltages are used. The normalized Fourier coefficient estimates 100yi /Vmax for the influence function

7.3


Fig. 7.11 Modes 1 and 2 response of airfoil rib

Fig. 7.12 Modes 3 and 4 response of airfoil rib

187

188


Fig. 7.13 Input command profile

Fig. 7.14 Influence function due to actuator 1

7.3




189

190



measurements as a function of command voltage are plotted in Figs. 7.14, 7.15, 7.16, and 7.17. Also shown is the resulting shape of the airfoil rib. The coupling between shapes is apparent in the coefficient plots and is consistent with the directionality noted earlier in the plant SVD analysis and the actuator coupling analysis. (See Figs. 7.6 and 7.7). The design parameters for the temporal compensator are listed in Table 7.5. The nominal plant of Fig. 7.10 was augmented with additional integrators in order to provide sufficient loop gain at low frequencies and in order to satisfy the tracking requirement for all shapes. This augmentation also enhances disturbance rejection at low frequencies, e.g. zero steady state error. Since the uncompensated plant has eight states and four inputs, the integrator-augmented design plant model has 12 states. The target loop was constructed from the nominal plant. The singular values are plotted in Fig. 7.18 for the target and recovered loops. The parameter ρ is the socalled “cheap control” recovery parameter and can be used to adjust the bandwidth of the recovered loop. The smaller the parameter the better the recovery, small ρ Table 7.5 LQG/LTR compensator design parameters μ

ρ

ωc (Hz)(Hz)

Order

0.025

0.1

1

12

7.3


191

Fig. 7.18 Forward loop singular value response for the target and recovered loop

implies large bandwidth and vice versa for large values of ρ. In practice ρ is adjusted to recover the target up to one decade above the crossover frequency of 1 Hz. The forward loop frequency response of Fig. 7.18 is non-directional up to the recovered bandwidth limit. This will lead to a more decoupled shape control response in the closed loop system as well as a more balanced participation of the MFC actuators. The closed-loop singular value response for the target and recovered loop is shown in Fig. 7.19. The parameter μ sets the measurement noise intensity used to adjust the Kalman Filter in the control synthesis process and ωc defines the control bandwidth.

7.3.9 Morphing Airfoil Closed Loop Shape Control Simulation The compensator design of Fig. 7.19 was tested by observing the sequenced step response of each of the four basis sinusoid shapes as expressed in Eq. (7.28). Each shape was commanded individually by inputting a step input of 100 V as individual Fourier coefficients. The results are presented in Figs. 7.20, 7.21, 7.22, and 7.23 show that the plant compensator design has decoupled the shapes during the airfoil ribs transient response. The Fourier Coefficients track the input step profile well. (Recall Fig. 7.5.)

192


Fig. 7.19 Closed-loop singular value responses for the target and recovered loop

Fig. 7.20 Basis shape 1 step response

7.3




193

194



Fig. 7.24 Basis shape superposition for HF20 airfoil

7.4

Summary

195

For the conservative design implemented here the specifications outlined previously are easily met. Because the controller decouples the basis shapes, its output can consist of any linear combination of the basis set of shapes. Recalling the airfoil decomposition illustrated in Fig. 7.5, the input to the controller is made to correspond to the Fourier components of the decomposed HF20 upper airfoil section. The result is shown in Fig. 7.24 as the reproduction of the HF20 profile. The profile achieved is accurate to within 1% rms and serves to validate the parameterization. The applied voltages were small, and hence the concomitant amplitudes are too small for practical application to real airfoil systems. The operating voltage range of the MFC actuator is listed as –500 to 1500 V. Any practical application would involve optimizing the rib structure subject to and constrained by the aerodynamic loadings and MFC control authority.

7.4 Summary A space/time transform parameterization was used to model the upper profile section of a morphing airfoil. The parameterization allows for the quantification of morphing spatial performance in terms of a spatial bandwidth. The approach implicitly assumes that the morphing control task can be described by a band-limited set of orthogonal shapes. The Loop Transfer recovery method of control synthesis was used to construct a robust compensator for closed loop dynamic shape control of a flexible airfoil rib. The space/time parameterization resulted in a spatially decoupled system with Fourier coefficients as inputs and orthogonal basis shapes as outputs. The classes of airfoil profiles which are amenable to the parameterization are called Eigenfoils. The MH 20 pylon racer airfoil was chosen as the reference Eigenfoil and this shape was achieved under command control to with in 1% accuracy. The “proof-of-concept” simulation presented demonstrates the efficacy of a dynamically commanded airfoil profile using this parameterization. The approach presented herein represents an effective actuator/structural design synthesis to morph airfoils. Eigenfoil morphing from one shape to another allows the system to use minimal energy during the process. In any practical application of the techniques presented here the airfoil should be morphed from a baseline shape and the airfoil rib structure must be optimized to yield the required displacements given the aerodynamic loads and actuator control authority. While the MFC actuators may lack the control authority to achieve this in large scale systems, there is a number of actuator options which when properly configured can be used to meet the desired objectives. This study therefore concludes that space/time parameterization when coupled with Eigenfoil representations of modern airfoil surfaces, can achieve efficient aircraft structure morphing.

196


Problems (7.1) Using the string vibration control problem in Chap. 1 (Problem 1.6), draw a new block diagram for a feedforward control implementation, and compare it with the diagram provided from the problem in Chap. 1. Discuss the differences in feedforward control versus feedback control. Give an example of an application in which feedforward control of a string might be useful. Discuss how you might go about implementing feedforward control of a string using smart materials. (7.2) For the airfoil shape control example of Sect. 7.3, calculate and plot the input coupling operators for the actuator spatial distributions below in Fig. 7.2a through d. Assume a temporal frequency equal to the first natural frequency of the beam model. What are the condition number and minimum singular value in each case? Use the same beam/actuator/model specifications provided in Sect. 7.3 as needed. (7.3) From Problem 7.2.a, calculate the Fourier coefficients for the given MFC actuator spatial distribution. Calculate the system state-space matrices for this configuration. Plot the time response of the system to a sinusoidal excitation of each of the 4 modes independently. Use the same beam/actuator/model specifications provided in Sect. 7.3 as needed.

Fig. 7.2a–d Actuator spatial distributions

Problems

197

(7.4) Repeat the airfoil morphing problem presented using four equally sized actuators. (7.5) Consider the airfoil morphing problem presented in Sect. 7.3.8. Compare the tracking response of the presented LQG/LTR controller with both an analogous LQR controller and a classical PD controller using noisy measurements. (7.6) Suppose the material properties of the airfoil morphing system are not well known and are to be identified while performing shape control. Equation (7.45) can be rewritten as

y˙ n v˙ n

=

0 0 θ1n θ2n

yn + θ2n vn

0 0 0 0 q1n q1n q1n q1n

⎡

⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ ⎣ u3 ⎦ u4

which is linear in the unknown parameters θin . Synthesize an adaptive control law to simultaneously track shape commands and estimate the unknown parameters θin .

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Index

The letters ‘f’ and ‘t’ following the locators refer to figures and tables respectively A Accelerometers, 8, 25 Active film layer, 38, 39f, 84 Active vibration control system synthesis, spatially shaded distributed transducers, 69–99 for beams, 71–74 collocated distributed transducers and Lyapunov control, 74–76 control designs with shaded distributions, limitations, 76 linearly shaded transducers, limitations, 80–81 spatial shading design, 81–83 uniformly shaded transducers, limitations, 76–80 Lyapunov direct method, 70–71 for plates, 84–86 distributed transducers for arbitrary spatial shadings, 94–95 non-uniformly shaded actuators, limitations, 92–93 uniformly shaded actuators, limitations, 86–91 structural components, 69 Actuator(s), 1–5, 9, 15–16, 18–20, 81, 112–114, 129, 152, 162 coupling analysis, 190 degenerate, 183 distribution, 74–76, 94–95, 112, 129–130, 131–137, 142, 144, 153, 156, 160, 162, 170, 180, 182–183 force, 75, 117 MFC, 178–182, 184, 186 piezo, 75 piezoelectric film, 158

smart film, 81–82 uniformly shaded, 86–91 See also Sensor(s) Aircraft technology, 174 Airfoil design considerations, 178–179 airfoil rib material properties, 179t macro-fiber composite actuator material, 179t morphing airfoil configuration, 178f Analytical modeling, distributed transducers, 34 compact analytical representation, 35–45 composite beam, 38f, 39 degenerate transducers, 35 singularity functions, 35 two dimensional representation, 45 completeness of singularity functions, 45–47 distributional chain rule to multi-dimensions extension, 50–51 distributions using singularity functions, 47–50 Aperture shading, 30–31 See also Shaded apertures Arbitrary spatial shading, 45 completeness of singularity functions, 45–47 orthogonality condition, 46 distributional chain rule to multidimensions extension, 50–51 distributions using singularity functions, 47–50 2D projection, 47f rectangular shading, 49f, 54 spatial gain, 48, 49f

203

204 B Basis set of plant representations, 101, 151–155, 167, 195 expansion basis set of “shapes,” 152f generic Green’s function, 151–153 symmetric Green’s function, 153–155 Bernoulli-Euler beam, 66, 74, 80, 97, 111, 130, 178, 182 pinned–pinned, 130 Bernoulli-Euler structures, 38, 40, 94 BIBO, see Bounded Input-Bounded Output (BIBO) Bilinear expansion, 112, 153, 155, 158, 171 in eigenfunctions, 168 Bilinear Hilbert-Schmidt expansion, see Bilinear expansion Bounded Input-Bounded Output (BIBO), 137, 139–140 C Cantilever beam, 9, 10f, 12, 37, 71, 73 Cauchy-Schwartz inequality, 125 Center of force (COF), 26, 29 Center of pressure (COP), 26–33, 49 integral components, 30 sensor/measurement, 26 Clamped–clamped beam, 77, 83 effective loading, 87f Classical “boxcar” aperture, definition, 102 Closed loop shape controller synthesis, airfoil, 184–191 basis shape step response, 192f–194f Beam/MFC dynamic parameters, 185t closed-loop singular value responses, 192f function due to actuator, 188–189f input command profile, 188f loop singular value response, 192f LQG/LTR compensator, 190t response of airfoil rib, 187f shape controller synthesis process, 185t uncompensated nominal plant model SVD response, 186f Closed loop system, nominal, 138, 141, 191 properties, 2f, 4 Closed loop transfer matrix, 122, 123 Closed set, 45 COF, see Center of force (COF) Compact analytical representation, arbitrary spatial shading composite beam, 38f, 39 degenerate transducers, 35 singularity functions, 35

Index Constant gain controller, 73–74 drawback, 74 Control design methods, 69, 138, 185 H∝ , 13, 101, 116, 185 LQG/LTR, 13, 116, 185t, 190t MATLAB, 13, 173, 184 Controllability/observability, MIMO representations, 76, 128–130, 142 controllability, 129–130 observability, 130 state controllable, 129 state observable, 129 state observable (linear system), 129 Controller gains, 4 magnitude limiting issues, 4 Control-Lyapunov function, 70–72 Control spillover, 18 Control system synthesis, 70–95 based on Lyapunov direct method, 70–71 control-Lyapunov function, 71 power flow out of system, 71 for beams, 71–74 constant gain controller, 73–74 time optimal control law, 73 uniformly shaded active film actuator, 71 for plates, 84–86 non-uniformly/uniformly shaded actuators, limitations, 86–95 Control theory, 138 COP, see Center of pressure (COP) D Degenerate transducers, 35, 113 Dirac delta function, 16, 37 Discrete transducer, 2, 9, 25, 69, 76, 145, 174 Distributed parameter actuator, see Spatially distributed actuator Distributed parameter control, 2, 18, 69, 73, 105, 114, 142–143, 145, 174–175 Distributed parameter system actuator, 10, 13 exogenous inputs, 106 transducers, 2, 13 See also Smart structure systems Distributed sensors, 19, 26, 29, 32, 74, 94, 121, 129, 162 See also Sensor(s) Distributed transducers, spatial shading/shaping of, 25–67 advantages, 26

Index analytical modeling, 34 compact analytical representation, 35–45 two dimensional representation, 45 approximating shaded apertures, 29–34 active film layer, 38 composite shaped apertures, 32f subaperture, 31f, 32 uniform electroded/shaded aperture, 32f, 38f for arbitrary spatial shadings, 94–95 Green’s function, second form, 94 COP sensor, 26–29 aeronautics, 26 piezo-electric polymer film, PVDF, 26–27 shading function, 29 voltage constants/“g” coefficients, 27f and Lyapunov control, 74–76 parameters, 25 two-dimensional shading application using skew angle, 51–55 finite skew angle of material axes, applications, 55–59 Distributional chain rule, 50–51, 53 Dominant modes, 16 Double ramp shading, 44, 93 Doublet functions, distributed, 36, 41, 53–56, 59 Dynamic shape control, 145, 173–195 actuator placement and input/output coupling, 179–182 input coupling operators, 181f MFC actuator distribution, 180f E Eigenfoils, 176–177, 195 aircraft (FAI class F3D), 176f Fourier series approximation of the MH20 airfoil, 177f MH20 pylon racer airfoil geometry, 176f Electro-mechanical coupling, 28, 38, 84 Electro-rheological fluids, 6–8, 7t size range, 8 Electrostrictive materials, 19, 34–35 Eppler E422 high lift airfoil, 166 Equations of motion, 11–12 F Feedback control law, 18, 70, 101, 121 Fiber optic materials, 25, 35 Fourier coefficients, 150, 191

205 Fourier transform, 102 integral space/time transform, 117, 131 Free–free Bernoulli-Euler beam, 111 G Galerkin methods, 15 Gaussian curvature, 84 Generic Green’s function, 151–153 linear time invariant spatial distributed plant, 151 Green’s function, 94, 105–113 second form, 94 for stationary systems, 107–113 analogy, anti-causal, 108 damped convolution form, 110 green’s function for a bernoulli-euler beam, 111 green’s function for a string, 109–110 right convolution form, 108 right/left convolution form, 108–109 square-integrable (L2 space), 107 symmetric form, 111–113 See also Impulse response function H Hamilton’s principle, 11 Hermitian matrix, 125 Holonomic system, 11 H∞ optimal control theory, 101 Hydrophones (sensing apertures), 101 I Impulse response function, 105 Induced strain device, 28, 37–38, 50 Isoplanatic convolution, 108 K Kronecker delta function, 150 L Lagrangian expression, 11 Laplace transform, see Fourier transform Linearized Navier Stokes (LNS), 143 Linearly shaded aperture, 30, 31f, 41f Linearly shaded transducers, limitations, 80–81 with even and odd symmetries, 83f Laplacian of, 40f Linear or “ramp” actuator distributions, 133–137 closed loop response function, 137f loop transfer function, 134f ramp actuator augmented plant response, 134f sensitivity response function, 136f

206 spatial filter, 135 temporal filter, 135 Linear Quadratic Gaussian (LQG), 116, 184 Linear time invariant spatial distributed plant, 151 L2 /L2 functions, 45 LNS, see Linearized Navier Stokes (LNS) Longitudinal strain, 10, 84 Loop transfer function, 135 Loop transfer recovery (LTR), 106, 184 LQG, see Linear Quadratic Gaussian (LQG) LTR, see Loop transfer recovery (LTR) Lyapunov damping controller, 133 Lyapunov direct method, 70 control-Lyapunov function, 71 power flow out of system, 71 Lyapunov method (vibration control), 101 M Macro fiber composite (MFC) actuators, 178, 195 See also Actuator(s) Magnetostrictive materials, 19, 34–35, 75 Material axes, 9, 27, 51–52 skew angle shadings application, 55–59 Matlab(TM) , software tools, 173 McCauley notation, 40, 42 Mean square profile error definition, 150 recapitulations/assumptions, 149 MFC, see Macro fiber composite (MFC); Macro fiber composite (MFC) actuators MH 20 pylon racer airfoil, 195 MIMO, see Multi-input multi output (MIMO) system Modal transition matrix, 170 Mode shapes/frequencies, 15, 78, 90, 103, 153–154, 185t Mode targeting, 26 Multi-dimensional transforms/MIMO, 101 “ boxcar” aperture, 102f convolution/spatially distributed plant, 105–113 composition integral, 106 Green’s function forstationary systems, 107–113 time shifted response function, 106 weighted sum of impulse response functions, 106 LTIS to an impulse, 106f MIMO, 113–119 S-C-S-C plate’s, wavenumber acceptance of, 104f

Index structural “morphing,” spatial frequency, 101 temporal resonant frequency, 103 temporal/spatial coincidence, 103 wavenumber/frequency transform of a non-dimensional, 104f wavenumber response for boxcar aperture, 103f Multi-input multi output (MIMO) system, 2, 76, 105, 113–118, 121, 139, 146 augmented plant response matrix, 115 infinite string on an elastic foundation, 117f MIMO distributed parameter system, 115f modern control system design/analysis, 116 string on elastic foundation, 117–118 Multivariable/multidimensional distributions, 34, 44 N NASA Langley Research Centers Hoop Column Antennae, 16 Natural structural modes, 15 Naval Ordinance Laboratory, 8 See also Shape memory alloys Newtonian based model, 15 Nitinol, 8 Non-dimensional governing equation, 171 Non-induced strain transducers, 49 Non-strain inducing transducers, 25, 34, 49 Non-uniformly shaded actuators, limitations, 92–93 Nyquist stability criteria, 139 O One dimensional transducer, 34, 40, 41f P Parametric tradeoffs, comparisons, 7, 8t Parseval’s equation, 45, 151, 175 Photolithography techniques, 31, 33 Piezoelectric coefficients, 28 constant, 10, 27 element, 27, 31 material axes, 27 film actuators, 158 materials, 8, 31, 34, 51, 54 polymer film, 10, 28, 131 PVDF, 26–27, 28 transduction, 27 Pinned–pinned Bernoulli-Euler beam, 78, 130, 164, 181 non-dimensional, 178, 180

Index Plant model, 2, 4, 5, 71 nominal, 4, 139, 185, 186f state space representation, 17 structure of, 9 truncated, 19, 69, 71, 167 Plants static Green’s function, 167 Point by point convergence, 148 Point force actuator, 15–16, 75, 80–83 Polarization, 9, 27, 51, 54 Polyvinylidene fluoride (PVDF), 10, 26, 27t, 37f, 51–52 distributed torque, 10 prestrain, 10 strain/longitudinal strain, 10 Prestrain, PVDF, 10 PVDF, See Polyvinylidene fluoride (PVDF) R Ramp function, 37, 43, 81, 92, 93 Rectangular box wavenumber/frequency space, 126 Rib influence functions, 186 RMS, see Root mean square (RMS) Root mean square (RMS), 143 Rotary-wing/fixed-wing aeronautical applications, 179 S Self-adjoint matrix, see Hermitian matrix Self-adjoint partial differential operators, 111 Sensitivity transfer matrix, 122 Sensor(s) aperture, morphing, 29–30, 163 collecting electrode, 31 equation, 17, 75 functions, 6 selection, 5, 9 shape estimation, spatially distributed, 162–165 Shaded apertures, 29–34 Shaded distribution, control designs limitations, 76 advantages, 76 Shading function, 29, 34–51 Shading/input/output coupling transducer, 155–162 Shape control, distributed transducer design of, 145 basis set of plant representations, 151–155 expansion basis set of “shapes,” 152f generic Green’s function, 151–153 symmetric Green’s function, 153–155 discrete spatial bandwidth/shape control, notion of, 146–151

207 discrete spatial ranform of the shape profile, 147f discrete spatial transform of the upper profile, 149f mean square profile error, 150 MH20 airfoil, 148f orthonormal expansions/discrete spatial transform, 147–149 shape profile, 146f non-zero set point, distributed plants, 146 shading/input/output coupling transducer, 155–162 singular value decomposition/ performance metrics, 156–162 spatially distributed sensors/shape estimation, 162–165 displacement sensor distribution I/II/III, 165f sensor design, example, 164–165 Shape control, modal representations/truncated plants, 167 actuator spatial distributions, 196f dynamic shape control, 173–195 actuator placement and input/output coupling, 179–182 airfoil design considerations, 178–179 airfoil rib, 182–183 airfoil shapes/discrete spectrum parameterization, 174–176 background, case, 173–174 closed loop shape controller synthesis, airfoil, 184–191 closed loop shape control simulation, airfoil, 191–195 eigenfoils, concept of, 176–177 space canonical form, 183–184 shape error/feed forward correction, 167–173 Shape error/feed forward correction, 167–173 shape control of a non-dimensional string, 171–173 string static Green’s function response, 172f string static response with feed through correction, 173f Shape memory alloys, 8, 25, 75 property, 8 Single Input Single Output (SISO), 130 Singularity functions, 35–37, 40–42, 44–45, 47, 69, 76, 79 Macauly notation, 36

208 Singular value decomposition/performance metrics, 156–162 active film actuator distributions, 161f actuator design, example, 158–162 input coupling operators for actuator distributions, 161f Singular value decomposition (SVD), 125 Singular values, performance metrics using, 124–128 command following, 124–127 tracking performance specification, 127f disturbance rejection, 127–128 scalar “gain control,” 128 sensor noise, 128 SISO, see Single Input Single Output (SISO) Skew angle, 51, 52f 2-D shading application, 51–55 material axes, 55–59 nonzero, 59 zero, 54, 57 Smart material systems comparisons, 6t coupling properties, 6 examples, 7t transduction, 5, 6f transducers, 5, 26, 75, 94 Smart structures with MIMO representations, 121 achieving stability/robustness for control of smart structures, 137–141 additive error uncertainty, 139–140 multiplicative error uncertainty, 141 active damping of a simply supported beam, 130–137 linear or “ramp” actuator distributions, 133–137 spatially uniform actuator distributions, 131–133 uncompensated response for a damped supported beam, 132f controllability/observability, 128–130 controllability, metrics, 129–130 observability, 130 state controllable, 129 state observable (linear system), 129 metrics performance, 121–124 architecture of smart structure, 121f good command following, 123 good disturbance rejection, 123 good tracking, output error, 123 immunity to noise, 123–124

Index output error, definition, 122 sensitivity transfer matrix, definition, 122 singular values, performance metrics using, 124–128 command following, 124–127 disturbance rejection, 127–128 sensor noise, 128 Smart structure systems, 1–24 advantages/definition, 1 architecture and performance, 1–5 closed loop system, properties, 2f, 4 magnitude limiting issues, 4 MIMO system, 2 performance, 2 plant model, 4 vibrating structures, 1, 2f continuum representation, 9–15 governing equations, 12–13 smart material transducer considerations, 5–9 commercial/production grade materials, 6–7, 7t comparisons, 6t electro-rheological fluids, 8 examples, 7t fundamental elements, 9 parametric tradeoffs, comparisons, 7, 8t piezoelectric materials, 8 sensing and actuation functions, 6 shape memory alloys, 8 spatial domain, 9 time domain representation, 15–18 exogenous input, 15 mode shapes/frequencies, 15 Newtonian based model, 15 wave equation, 15 Snowplow shading, 44f See also Double ramp shading Solid state gyroscopes, 8 Space/time characteristic equation, 105 Space-time filters, 9, 103 Spanwise-homogenous/exogenous pressure distribution, 30, 31 Spatial domain, 9, 12, 107, 123, 167 design freedom, 9 Spatially coincident aperture, 30, 33f Spatially distributed actuator, 10, 13 Laplacian of, 37–38 Spatially uniform control distribution, 37, 37f–38f, 40–41, 42 2-D distribution, 43f

Index Laplacian of, 42 rectangular plate structure with bi-axial film, 42f Spatial parameter transducers, 9 Spatial shading for vibration control, 81–83 clamped–clamped beam, 83f observations, 81 symmetric modes, 82f vibration damping, 83 Spatial weighting, 15 Square-integrable functions, 45 See also L2 /L2 functions State matrix, 13, 184 State space equation, 13 State space form, 13, 17–18, 184 Strain energy, 11, 82, 84, 179 Strain inducing transducers, 25, 34, 50 piezo actuators, 75 Sturm-Liouville systems, 15, 111, 158 eigenvalue problem, 112 Subaperture, 32 SVD, see Singular value decomposition (SVD) Symmetric Green’s function, 153–155 System potential energy, see Strain energy System state equation, 13 T Temporal dynamics, smart materials, 9 Time-domain approach, 17 Time optimal control law, 73 characteristics, 73 disadvantage, 73 Tip mass, 9, 10t, 37f, 71

209 Tracking error, 3, 4, 76, 124, 126–128 Transducer spatial filter (characteristics), 102 Transform methods, spatially distributed systems, 101 Two-dimensional shading application using skew angle, 51–55 finite skew angle of material axes, applications, 55–59 piezoelectric equivalent loading, 59f forces or linear displacement, 53 moments or angular displacements, 53 stress/charge constant, 54 triangular shaded distribution, 52f, 53 Two-dimensional transducer, 34, 42, 49 U UAVs, see Unmanned Aerial Vehicles (UAVs) Uniformly shaded actuators for plates, limitations, 86–95 2-D distribution, 88f mixed order modes, 91 strictly odd/even modes, 91 transducers, limitations, 76–80 Fourier sine series, 79 pinned-pinned beam, 78 sensor equation, 78 singularity functions, 76 Unmanned Aerial Vehicles (UAVs), 173 V Vibration damping, 71, 77, 80, 83, 94, 135 W Wave equation, 15

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