Spatial Filtering for the Control of Smart Structures
James E. Hubbard, Jr.
Spatial Filtering for the Control of Smart Structures An Introduction
123
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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An Historical Prologue and Preface
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
An Historical Prologue and Preface
ix
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
An Historical Prologue and Preface
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.
An Historical Prologue and Preface
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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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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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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
1 Smart Structure Systems
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
1 Smart Structure Systems
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
1 Smart Structure Systems
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
Continuum Representation of Smart Structures
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
1 Smart Structure Systems
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
Continuum Representation of Smart Structures
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
1 Smart Structure Systems
⎡
⎤ 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
1 Smart Structure Systems
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
1 Smart Structure Systems
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
1 Smart Structure Systems
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
1 Smart Structure Systems
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
1 Smart Structure Systems
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 << d thus constituting a distributed measurement. A similar argument can of course be posed for actuation. In this chapter we explore the concept of spatial weighting and aperture shading of smart material transducers in order to achieve a desired performance measure. We begin by using a design by example to introduce the salient features of the approach. The design exploits the ability to spatially weight or shade the performance of distributed sensors to meet specified performance goals.
2.2 Spatial Shading of Distributed Transducers Spatial weighting, also referred to as spatial shading or shaping of distributed transducers has been explored by a number of researchers and used for mode targeting, control loop shaping and all mode sensing and control of beams and structures composed of beam elements [1–6]. One of the primary advantages that distributed transducers offer the smart structure designer is the ability to vary or spatially weight gains over their extended aperture. Through the application of this transducer shading, the spatial filter characteristics of the transducers and concomitant structural control system can be altered to achieve desired performance goals [7].
2.2.1 Design by Example: A Center of Pressure Sensor A design by example case study is now presented in an effort to provide the reader with an intuitive insight into the process of spatial shading and its advantages. In subsequent sections we take a more formal approach and develop in detail modeling techniques and methodologies for one and two-dimensional shadings. We wish to design a dynamic center of pressure sensor using smart materials. Real-time center of pressure measurement is an enabling technology with broad applications in the automotive and aerospace industries. In the development of aeronautics for example the center of pressure of an airfoil was considered an important parameter in the quest to have a heavier than air machine achieve stable fight at equilibrium conditions. The prediction of center of pressure was once believed to be a necessity in the design of aircraft with proper equilibrium characteristics. The real time measurement of center of pressure however using conventional methods and transducers has proved historically to be very difficult. Consider the measurement of center of force on a flat plate. “center of force” implies the centroid of force acting over a surface along a specified sensing aperture. For applications requiring “good” dynamic range above 10 Hz, “good” embedability and low cost, a piezo-electric polymer film such as polyvinylidene fluoride (PVDF) would be an excellent choice. This material is commercially available and robust over a broad range of environmental operating conditions. During the manufacture process after poling, the film becomes “active” and develops a surface charge
2.2
Spatial Shading of Distributed Transducers
27
Table 2.1 Typical properties of uni-axially poled PVDF Parameter
Units
Value
3 10
Density Permittivity d31 g31 k31
ε ε0
kg m3
1.78 12 23 216 12
−12 m V 10−3 Vm 10 N Percent at 1 kHz
proportional to a change in mechanical loading. The film can be produced with a single axis of polarization in the “1” direction (un-axial) or with a double axis of polarization in the “1” and “2” directions (bi-axial). Sensing transducers made from this film do not perform well under static conditions due to the rapid decay of the induced charge as a result of the materials inherent leakage. After a load is applied, the induced charge can be collected for measurement by depositing a surface electrode, usually a nickel or aluminum alloy, to encourage surface mobility. PVDF is commercially available in large sheets with the typical properties indicated in Table 2.1. We now consider the case of PVDF adhered to the rigid surface of a flat plate and acted upon by a spatially distributed exogenous force f (x, t) applied over the aperture area of the sensor to produce a resultant applied pressure (See Fig. 2.2). To identify directions in a piezoelectric element, three material axes are typically used. These axes, termed 1, 2, and 3, are analogous to X, Y, and Z of the classical three dimensional orthogonal set as shown. As mentioned earlier, the piezo-electric transduction of the film causes the applied load to generate or induce a spatially distributed charge q(x,t) on the surface of the film. The piezoelectric constants relating the electric field produced by a mechanical stress are termed the voltage constants, or the “g” coefficients. A “3-3” subscript indicates that the electric field and the mechanical stress are both along the polarization axis (See Fig. 2.2). Thus we may relate the resulting charge distribution
3 (y)
Charge
q(x1,t)
Exogenous force
f(x1,t)
Piezo-film sensor
1 (x) Rigid Surface
2
(z)
Fig. 2.2 PVDF film adhered to a rigid surface
28
2 Spatial Shading of Distributed Transducers
to the applied load as follows ε (¯xi ,t) =
f (¯xi ,t) AE
(2.1)
and q (¯xi ,t) =
2 k33 g33
·
f (¯xi ,t) AE
(2.2)
where f (¯xi ,t) , ε (¯xi ) , q(¯xi ,t) represent the applied force, induced strain and resulting charge respectively associated with a given location x¯ i on a film of cross-sectional area A and modulus E at time t. The piezoelectric coefficients are the voltage 2 . As mentioned earlier we and electromechanical coupling coefficients g33 and k33 may collect the distributed charge for sensing purposes by depositing a spatially distributed electrode over the surface of the film. Piezoelectric polymer film is inherently a dielectric material and when the electrodes are deposited on its top and bottom surfaces it behaves as a capacitor thus V(t) = C0 Q (t)
(2.3)
where Co is the film capacitance, Q(t) is the total charge collected and V(t) is the resulting voltage. Now the total accumulated charge collected by the surface electrodes may be described as
Q(t) =
b0 (¯xi ,t) · q (¯xi ,t)
(2.4)
i
where bo (¯xi ,t) is the spatial distribution of the charge collection enforced by appropriately spatially weighting or shading the sensing film’s collecting electrodes. Combining Eqs. (2.2), (2.3), and (2.4) reveals the importance of spatial shading on the output performance of the sensing transducer as Q(t) =
2 1 k33 · b0 (¯xi ,t) · f (¯xi ,t) AE g33
(2.5)
i
and V(t) =
b (¯xi ,t) · f (¯xi ,t)
(2.6)
i
with b(¯xi ,t) ≡
2 C0 k33 · AE g33
· b0 (¯xi ,t) .
(2.7)
2.2
Spatial Shading of Distributed Transducers
29
It is apparent from Eq. (2.6) that by appropriately shaping or shading the collecting electrode, on-board signal processing or filtering is possible. In other words, the smart structure designer may now exploit the ability to spatially weight or shade the performance of distributed sensors to meet specified performance goals. This is a powerful design technique and the subject of much of the remaining chapter. In the limit as the charge density of the sensor surface goes to infinity equation (2.6) can be written in integral form as b (x,t) · f (x,t) dx dt
V(t) =
(2.8)
where Ω represents the sensing domain of the aperture. In general the shading function can take on any one of three forms, b(¯x,t) = constant which is typically the case of a uniform electrode which weights charge collection equally across the sensor aperture; b(¯x,t) = (¯x) which represents an electrode that is spatially shaped or shaded over the sensor domain to process the “data” represented by the charge distribution on the film surface into the desired “information” needed for performance assessment, i.e. center of pressure. Finally we have the most general case where b(¯x,t) = (¯x,t), a sensor which has the ability to change its shading both in space and time often referred to as a morphing sensor aperture. Here the sensor can reconfigure itself dynamically to provide new information as needed. We choose to define center of force as follows
x · f (x,t)dx dt COF = f (x,t)dx dt
(2.9)
where the spatial extent of the sensing aperture of interest is assumed to be normalized and thus x varies from 0 to 1. As is evident in Eq. (2.9) the COF metric has been defined as a ratio and has a value of 0 when x = 0 and 1 when x = 1. A close examination of (2.9) reveals that both the numerator and denominator functions can be obtained using the piezoelectric polymer sensor design with an appropriate choice of shading function as revealed in Eq. (2.8), i.e. with the numerator weighting of b(¯x,t) = x and a denominator shading of b(¯x,t) = 1. In the next section we expand on this choice of shading functions and discuss a means of achieving the physical realization of such shadings.
2.2.2 Approximating Shaded Apertures For center of pressure measurements we can define the center of pressure Cp as analogous to the centroid of mass distribution and assuming steady state we have
30
2 Spatial Shading of Distributed Transducers
b Cp =
x¯ · p (¯x) d¯x
a
b
(2.10) p (¯x) d¯x
a
where [a, b] defines the aperture of the physical measurement, x¯ is the spatial coordinate along the longitudinal axis of the sensor according to Fig. 2.1, and p (¯x) is the exogenous spatially distributed pressure distribution of interest. Equation (2.10) is essentially a ratio that defines a spatially shaded distributed integrated pressure along x¯ normalized by the integrated pressure load over the sensing aperture. The pressure distribution p in general can vary along the sensor aperture both longitudinally in x and laterally in z, and the sensing aperture is typically of finite width and hence finite in z therefore bd x · p (x, z) dxdz , Cp, x = a b c d p z) dxdz (x, a c
(2.11)
which defines the center of pressure along the x-axis. Note that center of pressure yields a magnitude (the denominator) and location (the ratio) along a continuously distributed sensor and thus is a high resolution, high fidelity measurement. If we assume that the exogenous pressure varies along x, but is homogenous in the z or the spanwise direction; p(x,z) → p(x) then Eq. (2.11) can be rewritten as the one dimensional measurement b Cp,x =
x · p (x) dx
a
d
.
(2.12)
p (x) dx
c
Equation (2.12) suggest that the center of pressure distribution along a surface may be determined by first measuring two component integrals over the apertures [a, b], [c, d], and then taking their ratio. The first component integral, i.e. the numerator aperture is linearly shaded, while the second, denominator aperture is gain weighted by a constant, in this case unity. Note that this measurement requires that both apertures be spatially coincident. The coincidence requirement is difficult in practice to realize. Note that if it were possible to construct a sensor which had a set of coincident spatially distributed apertures, one of which was linearly shaded, while the other was uniformly shaded, then the center of pressure component integrals in (2.12) could be measured exactly. In the remainder of this section, we discuss methods for approximating (2.12) that allows the center of pressure to be measured for this application using available smart materials and techniques. Aperture shading is simply the gain weighting of a sensor aperture. In sonar for example it is used to shape the far-field beam pattern of sensing arrays to improve
2.2
Spatial Shading of Distributed Transducers
Fig. 2.3 Linearly shaded
31
b(x)
a
b
x
their performance e.g. side lobe suppression, main lobe control etc. For this application it is used to develop the component integrals that will be used to compute the center of pressure along an aperture. The numerator aperture shading in Eq. (2.12) is a linear weighting along the xaxis, depicted conceptually in Fig. 2.3. When this weighting is realized in a spatially distributed sensor, e.g. one that produces a voltage proportional to the integral of the applied normal stress, then the sensitivity or gain of the sensor would vary linearly in x in the domain defined by [a, b]. In actual practice we must approximate the shading using modern materials and sensing technologies. This sensor can be constructed of a material that provides an electrical output that is proportional to an applied normal stress, such as a piezoelectrical, piezoresistive, or piezoconductive polymer material. The distribution in Fig. 2.4 would then be a planform view of the sensor collecting electrode which can be realized by a number of techniques e.g. etching, photolithography etc. In a piezoelectric material, the collecting electrode effectively sums or integrates the “charge” resulting from the applied stress over its area. When the sensing aperture is loaded by a spanwise-homogenous exogenous pressure distribution, its output is then proportional to the integration of this pressure over the shaped or shaded aperture.
z Linearly Weighted Electrode
Piezo-film layer
Fig. 2.4 Linearly shaded charge collecting electrode
x
32
2 Spatial Shading of Distributed Transducers
For the linearly shaded aperture the collecting electrode would induce little or no component output near the end x = a, but would contribute maximally to the sensors output in the vicinity of x = b, and would contribute other differential components in linear proportion to the apertures width at all intermediate locations. Because the apertures width varies linearly, this weights the integration of the spatially distributed pressure over the aperture [a, b] linearly. So the shaded aperture depicted in Fig. 2.4 is an approximation to the ideal linearly-shaded aperture shown in Fig. 2.3. As a result, the output of this aperture approximates the numerator integral in Eq. (2.12) for a pressure distribution that can be assumed spanwise homogeneous. The denominator integral in Eq. (2.12) can be realized in practice by a simple rectangular aperture, as this integral has a kernel equal to a constant. The shaped electrode is shown in Fig. 2.5. If the apertures of Figs. 2.4 and 2.5 were made coincident, then the ratios of there outputs would satisfy Eq. (2.12), yielding a measurement of the center of pressure in the x-direction over the sensing aperture. Because these distributed sensors provide outputs that integrate the applied stress over their apertures, we may now use superposition to develop a composite aperture of electrodes like that shown in Fig. 2.6. Subaperture (3) is the linearly shaped aperture shown earlier in Fig. 2.4, and therefore its output provides the numerator integral for Eq. (2.12). By summing the outputs of apertures (1), (2) and (3), this is equivalent to the output of the uniform electroded aperture shown in Fig. 2.5. Thus this summed output provides the denominator integral for Eq. (2.12). The composite shaped apertures shown in Fig. 2.6 thus z Uniformly Weighted Electrode
Piezo-film layer
x Fig. 2.5 Uniformly shaded aperture (planform view)
Fig. 2.6 Form
2.2
Spatial Shading of Distributed Transducers
33
can provide two coincident apertures with the shadings required to form the center of pressure integral (2.12). Note that the output of this sensor is ratiometric and therefore needs no temperature compensation. In addition the ratio is the location of center of pressure with a normalized range that always varies from 0 to 1 and therefore requires no calibration in this regard. Finally the output is always bounded and independent of the magnitude of the exogenous loading. Figure 2.7 is an illustration of the manufactured composite electrode etched onto a G-4 fiberglass substrate as an example of how simple photolithography techniques can be used to manufacture sensors with the required shadings. The one dimensional example above can be extended to two dimensions and higher moments as bd Cp, x (t) =
n c x · p (x, z, t) dxdz , bd p z, t) dxdz (x, a c
a
(2.13)
and bd Cp, z (t) =
n c z · p (x, z, t) dxdz , bd a c p (x, z, t) dxdz
a
(2.14)
where Eqs. (2.13) and (2.14) defines a weighting of the pressure in the x and z directions respectively which is normalized by the loads integral over the sensing aperture. When n = 1 these expressions define the center of pressure in the x and z directions appropriately. We have introduced and explored the concept of spatial weighting or shading of distributed transducers using the method of design by example. The design of a center of pressure sensor has been presented with aperture shading applied in the form
3
Spatially Coincident Apertures
Li
ne
ar
2
Fig. 2.7 Spatially coincident apertures for COP sensor
1
34
2 Spatial Shading of Distributed Transducers
of shaped charge collecting electrodes for a piezoelectric polymer smart material sensing element without loss of generality of the methods applied. As such smart materials when used as sensors in the manner presented are capable of providing integrated measurements proportional to applied loads. The techniques introduced are general and have broad application and involve the spatial shading of the spatial kernels of the resulting integrated measurements in order to achieve desired measurement metrics. In the next section a comprehensive approach for modeling one and twodimensional transducers of arbitrary spatial distribution and shading are developed. The technique is based upon the theory of multivariable distributions and allows distributed transducer shading to be efficiently incorporated into the smart structure design process for multi-dimensional structures. As shown above, by providing a compact, analytical representation of one and two-dimensional transducers, the method of shading can be used to design new devices requiring such transducers for a variety of applications. The method itself is general and is thus applicable to many types of transducers, including piezoelectric, electrostrictive, and magnetostrictive devices.
2.3 Analytical Modeling of Spatial Shading Functions for Distributed Transducers Spatial shading or weighting is critical to the goal of effective smart structure design. As has been shown, distributed transducers may be shaded by varying their gain over their spatial extent. The primary advantage of distributed transducers is that they allow flexibility in the spatial design and synthesis for structural control. Through the application of shaded transducers, the smart structures transfer functions or filter characteristics can be altered so as to achieve desired temporal and spatial performance goals. In this section a modeling technique for two-dimensional distributed transducers is presented. The spatial shading function is defined using multidimensional distributions with composite functions as arguments. A distributional calculus is employed to determine the spatial differential operator describing the spatial dynamics of induced-strain transducers. This approach allows distributed transducer shading to be incorporated into the control design process for multi-dimensional smart structures as an additional design parameter. The method is applicable to many types of strain-inducing and non-strain inducing transducers, such as piezoelectric, electrostrictive, and magnetostrictive materials. One and two-dimensional transducer shading is presented as a design tool for the smart structure design problem. The derivations are based upon the theory of multivariable distributions and are extended to distributions with composite functions as arguments. In this manner, arbitrary spatial weightings of transducers may be efficiently described.
2.3
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
35
2.3.1 A Compact Analytical Representation of Distributed Transducers In this section we will develop a compact representation for distributed transducers in the design and control of smart structures. Such a representation must include concentrated moments and forces as well as distributed loads in the case of actuation and both discrete and distributed measurements in the case of sensors. The general representation must include devices which can reconfigure themselves over space and time vis; u = u(x, t).
(2.15)
These transducers have the unique ability to change their spatial weighting and or shading dynamically and are sometimes referred to as morphing transducers. As the reader might suspect, these devices may be difficult to realize using existing materials and methods. The more practical and common form of distributed transducers are those whose temporal dynamics do not vary over their spatial aperture. In such cases the spatial and temporal dynamics can be represented as separable in space and time and are referred to as degenerate [20]. Degenerate transducers can be represented as a separable product of spatial and temporal functions and hence take the special form: u (x,t) = (x) · u(t),
(2.16)
where (x) describes the transducer’s spatial shading (or spatial kernel) and u(t) describes the transducer’s temporal dynamics. Many types of distributed transducers, including piezoelectric, magnetostrictive, electrostrictive, and fiber optic, may be classified as degenerate transducers because their temporal dynamics do not vary over their spatial aperture. From the smart structure designers perspective we must determine both the necessary temporal and spatial characteristics of the transducers in order to meet specified performance goals. Toward this end generalized functions are a particularly elegant means of modeling (x) and provide a compact notation when using the theory of distributions in the form of singularity functions [21]. Such functions are commonly used in engineering and physics and were originally applied to structural components by Macauley in 1919, and later by Clebsch in 1919 and by Pilkey in 1964 [22–24]. The rigorous theory underlying singularity functions may be found in the open literature on generalized functions. A brief review is presented here for the purpose of the analysis which follows. We begin by considering the family of singularity functions represented by; fn (x) =< x − a >n ≡
0 x
(2.17)
36
2 Spatial Shading of Distributed Transducers
where fn (x) is defined to posses a value only when its argument is positive. When the argument is positive then the pointed brackets mathematically behave simply as ordinary parenthesis. This is the so-called Macauly notation of singularity functions. These functions also obey the following integration law; x < x − a >n 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
2 Spatial Shading of Distributed Transducers
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
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
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
2 Spatial Shading of Distributed Transducers
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
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
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
2 Spatial Shading of Distributed Transducers
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
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
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
2 Spatial Shading of Distributed Transducers
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
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
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
2 Spatial Shading of Distributed Transducers
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<ν≤
ε2 − (x − x0 )2
<1 (2.40) (b − a)2 on the same interval as (2.39) and ν is also a positive definite constant. Hence we write (1 + ν)n ≤ Pn < 2n ;
x0 − ε ≤ x ≤ x0 + ε
(2.41)
and thus over this interval x 0 +ε
Pn (x) (x) dx > βε (1 + ν)n .
(2.42)
x0 −ε
This integral is positive definite and it is evident that the right hand side of (2.42) approaches ∞ as n → ∞. We now turn our attention to the left hand side of (2.42) noting that 0<1+
ε2 − (x − x0 )2 (b − a)2
≤1
(2.43)
on the segment of the interval [a,b] which determines Pn (x) according to (2.37) in the subset (a ≤ x ≤ x0 − ε) and (x0 + ε ≤ x ≤ b) as bounded. If the absolute value of the continuous function (x) is constrained to be bounded by B, then
2.3
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
b x0 −ε Pn (x) (x) dx + Pn (x) (x) dx a x0 +ε b x0 −ε | (x)| dx + | (x)| dx < B (b − a) ≤ a x0 +ε
47
(2.44)
and this part of the integral In of Eq. (2.38) remains bounded as n → ∞. But it was determined that according to (2.42) the integral over (x0 − ε, x0 + ε) approaches +∞ as n → ∞ thus lim In = ∞.
n→∞
(2.45)
Hence only (x) = 0 satisfies the orthogonality condition (2.38) and we must conclude that the set of polynomials is closed, and thus complete, for continuous
(x). In other words the only function that is orthogonal to members contained in the set is the trivial function (x) ≡ 0 and the system of continuous (x) is therefore closed. We would also like to consider the set of shading functions (x) that are squareintegrable but not necessarily continuous. These functions may be defined according to ξ f (ξ ) dξ ≡ (ξ ).
(2.46)
a
Integration by parts and the orthogonality condition (2.38) yield a
b
Pn (x) (x) dx = 0,
(2.47)
since at the end points (a) = (b) = 0 because the function must be orthogonal to the first member of the polynomial set Pn (x) as well, i.e. the constant 1. Now the derivative Pn is simply another polynomial and all parts of the earlier proof hold accordingly. The continuous function (x) which satisfies (2.47) is identically zero and hence f (x) must vanish almost everywhere in [a,b]. We have thus proven that the system of polynomials, and hence the singularity functions, is complete not just in the space L2 , but in L as well.
2.3.2.2 Nearly Arbitrary Spatial Shading Distributions Using Singularity Functions Now that we have shown singularity functions to form a complete set, we proceed to demonstrate that they may be used to represent spatial shading distributions over multidimensional spaces in orthogonal coordinates. In this context it is possible
48
2 Spatial Shading of Distributed Transducers
y
Fig. 2.19 2D projection of arbitrary shading
y = f1(x)
x2
x1
x
y = f2(x)
to have spatial distributions which act on ordinary functions in n-dimensional real space. Consider the arbitrary two-dimensional shading shown in Fig. 2.19. The shading illustrated by Fig. 2.19 can be described as follows, y = f1 (x) y = f2 (x)
for y > 0 over x ∈ [x2 , x1 ] , for y < 0 over x ∈ [x2 , x1 ]
(2.48)
where x1 and x2 are the defined at the boundary. The bounding functions in (2.48) are assumed to be known and single-valued. [Note that if they instead are doubledvalued, then the region can be decomposed into subregions which admit expressions of the form of (2.48)]. Given (2.48) the two-dimensional shading of Fig. 2.19 can be described by
(x,y) = x − x2 0 − x − x1 0 y − f2 (x) 0 − y − f1 (x) 0 .
(2.49)
This expression has a physical interpretation because the distribution is represented by products of distributions in non-parallel coordinates. Within the boundary, the distribution has unit amplitude, = 1, and outside of the boundary the distribution has zero amplitude, = 0. Equation (2.49) thus describes a two-dimensional distributed transducer of arbitrary shape. More complex representations can of course be achieved by segmenting the desired distribution into subregions which can be described by (2.49) and then the complete shading described through superposition. In addition to an irregular shape, the spatial “gain” or amplitude of the transducer can be varied over space as represented in Fig. 2.20. In such cases the generalized functions in (2.49) may be multiplied by ordinary functions to include this weighting. For example, if we take the special form of the general case presented in Fig. 2.19 of a rectangular 2D shape as shown in Fig. 2.21, then for this distribution Eq. (2.49) simplifies and can be re-written as
(x,y) = x + a 0 − x − a 0 y + b 0 − y − b 0 .
(2.50)
2.3
Analytical Modeling of Spatial Shading Functions for Distributed Transducers
49
Fig. 2.20 Spatially gain weighted shading of arbitrary distribution
y
Fig. 2.21 Rectangular shading
y = f1(x) = b
x2 = − a
x1 = a
x
y = f2(x) = − b
Note that this expression is a product of distributions in orthogonal co-ordinates and can be used to describe rectangular electrodes or transducers of rectangular aperture. Spatial gain weighting can readily be included by adding a linear weighting in each direction vis, x y 1 x + a0 − x − a0 y + b0 − y − b0 1− 1− 4 a b (2.51) and by multiplying each term by an ordinary function. The general representation defined in (2.49) can therefore accommodate twodimensional transducers of both arbitrary shape and gain weighting as it permits products of distributions with composite function arguments. The method is useful for directly modeling non-induced strain transducers such as the center of pressure (x,y) =
50
2 Spatial Shading of Distributed Transducers
sensor presented earlier in Sect. 2.2.1 which was designed to be used in its thickness mode. In order to determine the effects of using induced strain transducers such as (2.26) in physical systems, it is necessary to derive a differentiation theorem for distributions with composite function arguments and then to extend the results to multi-dimensions through the use of partial derivatives. As we have seen, for this class of transducers their spatial input/output characteristics are determined by the Laplacian of the distribution. In the general case, the input/output characteristics of induced strain transducers are described by a spatial differential operator. When such a transducer is two-dimensional and can be described by a spatial differential operator such as the Laplacan, a modified distributional chain rule must be used to calculate the requisite spatial differential operator [26]. 2.3.2.3 Extension of Distributional Chain Rule to Multi-Dimensions The spatial distribution defined in (2.49) is the product of two terms, one solely dependent upon x and the other dependent on both x and y,
(x,y) = 1 (x) 2 (x,y) .
(2.52)
In order to take the partial derivative of this distribution with respect to x, we must make use of the product rule for differentiation. Assuming that y is a constant, ∂ (x,y) ∂ 2 (x,y) = 1 (x) 2 (x,y) + 1 (x) . ∂x ∂x
(2.53)
The partial derivative with respect to y is ∂ (x,y) ∂ 2 (x,y) = 1 (x) . ∂y ∂y
(2.54)
When taking higher-order spatial derivatives, one can apply the product rule successively: ∂ 2 (x,y) ∂ 2 (x,y) ∂ 2 2 (x,y) = 1 (x) + 1 (x) , ∂x∂y ∂y ∂x∂y ∂ 2 (x,y) ∂ 2 (x,y) ∂ 2 2 (x,y) +
=
,
+ 2
(x) (x,y) (x) (x) 2 1 1 1 ∂x ∂x2 ∂x2 ∂ 2 (x,y) ∂ 2 2 (x,y) =
. (x) 1 ∂y2 ∂y2
(2.55) (2.56) (2.57)
Using the distribution chain rule defined in (2.57), where y is again assumed to be a constant, the partial derivative of (x,y) defined in (2.49) with respect to x can be written as,
2.4
Application to Two-Dimensional Shading Using Skew Angle
51
∂ (x,y) = x − x2 −1 − x − x1 −1 y − f2 (x) 0 − y − f1 (x) 0 ∂x + x − x2 0 − x − x1 0 −f2 (x) y − f2 (x) −1 + f1 (x) y − f1 (x) −1 . (2.58) The partial derivative of (x,y) with respect to y is simply, ∂ (x,y) = x − x2 0 − x − x1 0 y − f2 (x) −1 − y − f1 (x) −1 . ∂y
(2.59)
The spatial distributions of distributed transducers may be on occasion more complicated than that defined in Eq. (2.49). For example the spatial distribution may include ordinary functions or higher order generalized functions. In such cases one can simply use the distributional chain rule and the product rule for differentiation as needed when taking partial derivatives.
2.4 Application to Two-Dimensional Shading Using Skew Angle It has been demonstrated that spatially distributed transducer shading is a powerful design tool and can be accomplished by spatially varying the polarization, profile on active materials e.g. reshaping the surface electrode. As pointed out earlier in Chap. 1, shading may also be achieved by changing the skew angle between the structural principal axis and the material axis of the active lamina [27, 28]. The material axes of the active lamina is defined and described for specific materials. In this section, spatial shadings of piezoelectric materials will be examined without loss of generality as the fundamental approach is valid for a wide range of uniaxial and biaxial materials. The skew angle is introduced to reflect the symmetry that exits in many active materials instilled during manufacture. Typically the poling axis is the z-axis and the x-direction is the stretching or rolling axis for uniaxially stretched (poled) materials. The approach and notation here is that found in Sullivan et al. [29]. The skew angle θ , is illustrated in Fig. 2.22 and is defined with respect to the principle active material axes. We have noted that when an active film is applied to a structure for the purposes of actuation the input, V(x,t) appears in terms of a differential operator, its Laplacian, which is a consequence of the structures moment curvature relationship and describes the actuator spatial dynamics and the resultant loading. For a piezopolymer film such as Polyvynlidene Flouride (PVDF) the differential operator describing its spatial dynamics, including the possibility of skew angle, θ , between the transducer and structure axes, is given by Lee and Moon [27] as 2 2 ∂ 2 (x,y) 0 ∂ (x,y) 0 ∂ (x,y) + e + e , L (x,y) = e031 32 36 ∂x∂y ∂ 2x ∂ 2y
with
(2.60)
52
2 Spatial Shading of Distributed Transducers
z',3' y',2'
Lamina
θ
xx,1 ,1 Skew Angle
x',1' Fig. 2.22 Skew angle relative to structural principal axes
⎡
⎤
⎡ ⎤ cos2 θ sin2 θ −2 cos θ sin θ ⎢ 0 ⎥ ⎣ sin2 θ cos2 θ 2 cos θ sin θ ⎦ ⎣ e32 ⎦ = cos θ sin θ − cos θ sin θ cos2 θ − sin2 θ e036 ⎡ ⎤⎡ ⎤ d30 1 0 Ep / 1 − v2p vp Ep / 1 − v2p ⎢ ⎥⎢ ⎥ ⎣ d0 ⎥ , • ⎢ 0 ⎣ vp Ep / 1 − v2p Ep / 1 − v2p ⎦ 32 ⎦ 0 0 0 Ep /2 1 + vp e031
(2.61)
where e031 ,e032 ,e036 represent the piezoelectric stress/charge constants with respect to the structures axes; d30 1 ,d30 2 are the piezoelectric strain/charge constants with respect to the PVDF material axes; and Ep ,vp are the Young’s modulus and Poisson’s ratio respectively of PVDF. We now consider the effect of a triangular shaped piezoelectric distribution such as the one shown in Fig. 2.23 where the magnitude of the slope of the taper, m is equal to b/a. This distribution may be obtained by either shaping the physical piezoelectric transducer or its electrode distribution where the material axes of the transducer is coincident with the x and y axes as shown in Fig. 2.23, i.e. with a skew angle of zero.
y b m
0 a Fig. 2.23 Triangular shaded distribution
−b
x
2.4
Application to Two-Dimensional Shading Using Skew Angle
53
This triangular shading can be represented using Eq. (2.49), where f1 (x) = −mx + b and f2 (x) = +mx − b or with
(x) = (x,y) = x 0 − x − a 0 y − (mx + b) 0 − y − ( − mx + b) 0 . (2.62) which describes a distribution that is active with unit intensity when x is within the range 0 < x < a and y is within the range (mx − b) < y < (−mx + b) . This shading distribution can be simplified as
(x,y) = x 0 − x − a 0 y − mx + b 0 − y + mx − b 0 .
(2.63)
The differential operator given by (2.60) may be determined for this distribution through the use of Eqs. (2.52), (2.53), (2.54), (2.55), (2.56), (2.57), (2.58), and (2.59) and the distributional chain rule for linear substitution thus, L (x,y) = e031 x −2 − x − a −2 y − mx + b 0 − y + mx − b 0 (2.64) −2me031 x −1 − x − a −1 y − mx + b −1 + y + mx − b −1 2 0 0 0 0 + m e31 + e32 x − x − a y − mx + b −2 − y + mx − b −2 . The physical interpretation of this result, i.e. the actuator spatial dynamics or equivalent loading may be seen in Fig. 2.24. The first term in Eq. (2.64) contributes distributed doublet functions (moments or angular displacement) of magnitude e031 on the lines x = 0 and x = a, but only within the lines y = mx–b and y = −mx+b. Because the sloped lines intersect at (a,0), no doublet functions exist on the right hand boundary of the triangle. The third term contributes spatially distributed doublet functions (moments or angular displacements) of magnitude m2 e031 +e032 along the lines y = mx−b and y = −mx+b, but only within the lines x = 0 and x = a. The second term in Eq. (2.64) gives downward delta functions (forces or linear displacement) of magnitude 2me031 at the points (0,b) and (0,–b) and an upward delta function of 4me031 at the point (a,0).
0 2me31
y 0 0 m2e31 + e32
0 e31
0 4me31
x 0 2me31
Fig. 2.24 Equivalent loading for triangular distribution
54
2 Spatial Shading of Distributed Transducers
This result essentially indicates that doublet functions appear and line the boundaries, where there is a discontinuity in amplitude of the chosen distribution, and delta functions appear at discontinuities in boundary slope. Also note that this two-dimensional description approaches the one-dimensional Laplacian as the two-dimensional structure collapses to a one-dimensional system. The structure effectively becomes very “stiff” in the y-direction as this limit is approached. Thus, the two delta functions on the left vertices in Fig. 2.24 sum to one which is equal in magnitude but opposite in direction from the delta function at the right vertex. Due to stiffness in the y-direction, the bending effect of the doublet functions along the sloped lines becomes negligible in comparison to that of the distributed moment along the left boundary. The distributed doublet functions along the left boundary can be summed to give one resultant doublet function acting in opposition to the delta function at the right vertex. The differential operator for this shading also reduces to the result of that of a uniform distribution when taking the limit as the slope of the taper, m, approaches zero and the distribution approaches a rectangular distribution,
(x,y) = x 0 − x − a 0 y + b 0 − y − b 0 ,
(2.65)
in which case the differential operator approaches − x − a −2 y + b 0 − y − b 0 L (x,y) = e031 x −2 +e032 x 0 − x − a 0 y + b −2 − y − b −2 .
(2.66)
The magnitude of the distributed doublet functions along the lines y = mx − b and y = −mx + b approaches e032 and the magnitudes of the delta functions at the vertices approach zero as m approaches zero. Therefore, as the slope of the taper, m, approaches zero, the differential operator result approaches that of a uniform rectangular distribution with distributed moments along the boundaries and no delta functions at any of the comers. In the proceeding analysis the specific polarization of the piezoelectric material, i.e. uniaxial versus biaxial, used to create the tapered distribution was not specified. The difference between the two lies in the values of their respective strain/charge constants, d30, 1 d30 2 . The stress/charge constant, calculated from Eq. (2.60) for zero skew angle, is related to the strain/charge constants by the following relation, ⎡
⎤ ⎡ 0 ⎤ e031 qd3 1 + vp qd30 2 ⎣ e0 ⎦ = ⎣ vp qd0 + qd0 ⎦ , 32 31 31 0 e036
(2.67)
where q = Ep / 1 − v2p .
(2.68)
2.4
Application to Two-Dimensional Shading Using Skew Angle
55
For uniaxial, piezoelectric film typically d30 1 = 24x10−12 m/V (or C/N) and d30 2 = 3x10−12 m/V (or C/N) so that the stress/charge constants are ⎡
⎤ ⎤ ⎡ e031 2.40x10−11 ⎣ e0 ⎦ = ⎣ 1.07x10−11 ⎦ q. 32 0 e036
(2.69)
For biaxial, piezoelectric film typically d30 1 = d30 2 = (6 − 10)x10−12 m/V (or C/N) and the maximum stress/charge constants are ⎤ ⎤ ⎡ e031 1.33x10−11 ⎣ e0 ⎦ = ⎣ 1..33x10−11 ⎦ q. 32 0 e036 ⎡
(2.70)
Using these values, the ratio of the magnitudes of the doublet functions, e031 m2 + along the sloped lines of the taper is
e032 ,
12 2m2 + 1 Muniaxial 24m2 + 11 ≈ . = Mbiaxial 13.3 m2 + 1 13.3 m2 + 1
(2.71)
The ratio varies between 0.83 and 1.3 for slope, m between 0 and 1 respectively. With uniaxial film it is therefore possible to obtain the same magnitude of distributed doublet functions along the sloped edges of the taper as with biaxial film. It is also possible to achieve a two-dimensional loading/sensing effect with uniaxial film by cutting the film or shaping the electrodes in a direction non-orthogonal to the rolling axis of the film. As a practical matter, uniaxial film is simpler to manufacture than biaxial film and is therefore more readily available and cost-effective as a transducer.
2.4.1 Applications Including Finite Skew Angle of Material Axes In this section we generalize the modeling techniques above for inclusion of skew angle shadings for application to the class of generalized distributions shown in Fig. 2.25. In other words to distributions in which the material axes are coincident with the geometric x and y axes, but the geometric axes of the rectangle, xr and yr , are skewed with respect to the x and y axes. This distribution has stress charge constants given by (2.60) where the skew angle is equal to zero as shown in Eqs. (2.67) and (2.68). Since the geometric boundary itself in Fig. 2.25 is skewed, multi-dimensional distributions, with cos θ and sin θ included in the arguments, are needed to describe the shading. The piezoelectric laminate theory reveals that the e036 stress/charge constant in the piezoelectric differential operator (2.60) may be generated by skewing the material axes of the transducer with respect to the laminate by an angle θ . This occurs because the stress/charge constants weight the cross term in (2.60) and causes the
56
2 Spatial Shading of Distributed Transducers
Fig. 2.25 Rectangular distribution with skewed boundary
yr y rolling axis x
θ
a
Fig. 2.26 Rectangular distribution with skewed material axis
xr
b
y rolling axis θ
b
x
a
equivalent of delta functions to be produced at the corners of a rectangular distribution. In addition, varying the skew angle causes all of the stress/charge constants to vary in a sinusoidal manner with increasing skew angle. Also considered is the alternate shading distribution shown in Fig. 2.26 where the material axes are skewed with respect to the x and y axes but the geometric axes of the rectangle are coincident those axes. This distribution has stress/charge constants given by Eq. (2.60) where the skew angle is a finite θ . Differentiation of these composite distributions will produce the cross-terms, resulting in delta functions at the corners, and appropriately weights the doublet functions acting along the boundary. We begin development of the appropriate generalized piezoelectric differential operators by defining the distribution shown in Fig. 2.25 using the geometric axes with the following coordinate transformation to the material axes, xr = x cos θ − y sin θ yr = y cos θ + x sin θ
(2.72)
the distribution (or shading function) can now be written in terms of these coordinates as
(xr ,yr ) = xr + a/2 0 − xr − a/2 0 yr + b/2 0 − yr − b/2 0 . (2.73)
2.4
Application to Two-Dimensional Shading Using Skew Angle
57
Because the choice of coordinates does not affect the amplitude of the distribution,
(xr ,yr ) = (x,y) .
(2.74)
The distribution may now be written in terms of x, y and skew angle θ using the coordinate transformation of (2.72) vis.
(x,y) = x cos θ − y sin θ + a/2 0 − x cos θ − y sin θ− a/2 0 x y cos θ + x sin θ + b/2 0 − y cos θ + x sin θ − b/2 0 .
(2.75)
Note: This definition is valid since the limit of (x,y) as θ approaches zero reduces to (2.76)
(x,y) = x + a/2 0 − x − a/2 0 y + b/2 0 − y − b/2 0 , and the limit as θ approaches π/2 reduces to
(x,y) = y + a/2 0 − y − a/2 0 x + b/2 0 − x − b/2 0 .
(2.77)
The distribution (2.75) involves two terms which are both dependent on x and y,
(x,y) = 1 (x,y) 2 (x,y) .
(2.78)
Due to the added dependence of y in 1 , the second derivative with respect to x becomes ∂ 2 (x,y) ∂ 2 1 (x,y) ∂ 1 (x,y) ∂ 2 (x,y) ∂ 2 2 (x,y) +
=
, + 2 (x,y) (x) 2 1 ∂x ∂x ∂x2 ∂x2 ∂x2 (2.79) and the second derivative with respect to y is, ∂ 2 (x,y) ∂ 2 (x,y) ∂ 2 1 (x,y) ∂ 1 (x,y) =
+ 2 (x,y) 2 ∂y ∂y ∂y2 ∂y2 + 1 (x,y)
∂ 2 2 (x,y) . ∂y2
(2.80)
The piezoelectric differential operator, which reveals the effective loading of the chosen spatial shading function, for the case of zero skew angle of material axes as shown in Fig. 2.25 can now be written as 2 ∂ 2 (x,y) 0 ∂ (x,y) L (x,y) = e031 + e , 32 ∂ 2x ∂ 2y
where
(2.81)
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2 Spatial Shading of Distributed Transducers
∂ 2 (x,y) = cos2 θ x cos θ − y sin θ + a/2 −2 − x cos θ − y sin θ − a/2 −2 2 ∂x 0 0 • y cos θ + x sin θ + b/2 − y cos θ + x sin θ − b/2 +2 cos θ sin θ x cos θ − y sin θ + a/2 −1 − x cos θ − y sin θ − a./2 −1 • y cos θ + x sin θ + b/2 −1 −0 y cos θ − y sin θ − b/2 −1 0 2 x cos θ − y sin θ + a/2 − x cos θ − y sin θ − a/2 + sin θ • y cos θ + x sin θ + b/2 −2 − y cos θ + x sin θ − b/2 −2 , (2.82) and ∂ 2 (x,y) = sin2 θ x cos θ − y sin θ + a/2 −2 − x cos θ − y sin θ − a/2 −2 ∂y2 0 0 • y cos θ + x sin θ + b/2 − y cos θ + x sin θ − b/2 −2 cos θ sin θ x cos θ − y sin θ + a/2 −1 − x cos θ − y sin θ − a/2 −1 • y cos θ + x sin θ + b/2 −1 − y cos +x sin θ − b/2 −1 2 θ x cos θ − y sin θ + a/2 0 − x cos θ − y sin θ − a/20 + cos • y cos θ + x sin θ + b/2 −2 − y cos θ + x sin θ − b/2 −2 . (2.83) The stress/charge constants are given by Eq. (2.60) with skew angle θ = 0: ⎡
e031
⎤
⎡
qd30 1 + vqd30 2
⎤
⎥ ⎢ 0 ⎥ ⎢ ⎣ e32 ⎦ = ⎣ vp qd30 1 + qd30 2 ⎦ . 0 e036
(2.84)
where q = Ep / 1 − v2p .
(2.85)
The results of this operation on the distribution are shown in Fig. 2.27 where Fc = 2 sin θ cos θ e031 − e032 ma = e031 sin2 θ + e032 cos2 θ mb = e031 cos2 θ + e032 sin2 θ
(2.86)
Similarly the result of the piezoelectric operator acting on the distribution shown in Fig. 2.26 is given by L (x,y) = q cos2 θ d30 1 + vp d30 2 + sin2 θ vp d30 1 + d30 2 • x + a/2 −2 − x − a/2 −2 y + b/2 0 − y − b/2 0 0 −2 cos θ sin θ q d3 1 + vp d30 2 − vp d30 1 − d30 2 • x + a/2 −1 − x − a/2 −1 y + b/2 −1 − y − b/2 −1 +q sin2 θ d30 1 + vp d30 2 + cos2 θ vp d30 1 + d30 2 • x + a/2 0 − x − a/2 0 y + b/2 −2 − y − b/2 −2 ,
(2.87)
2.4
Application to Two-Dimensional Shading Using Skew Angle
Fig. 2.27 Piezoelectric equivalent loading on a rectangular distribution with skewed material axes
59
Fc
y
rolling axis θ
x mb ma
Fig. 2.28 Piezoelectric equivalent loading on a rectangular distribution with skewed boundary
and is shown in Fig. 2.28 where ma = q sin2 θ d30 1 + vp d30 2 + cos2 θ vp d30 1 + vp d30 2 mb = q cos2 θ d30 1 + vp d30 2 + sin2 θ vp d30 1 + vp d30 2 Fc = 2 sin θ cos θ q d30 1 + vp d30 2 − vp d30 1 − d30 2
(2.88)
The cos θ and sin θ terms multiplying the generalized functions in (2.87) were obtained from the nonzero skew angle assumption in Eq. (2.60). The cos θ and sin θ terms in Eqs. (2.82) and (2.83) were obtained when taking distributional derivatives of generalized functions with composite arguments involving cos θ and sin θ . Note that the magnitudes and orientation of the doublet functions along the boundary and the delta functions at the corners match for both distributions.
60
2 Spatial Shading of Distributed Transducers
2.5 Summary An analytical modeling technique for one and two-dimensional spatially distributed transducers has been presented. This approach allows distributed transducer shape or shading to be incorporated into the smart structure control design process for multi-dimensional structures as an additional design parameter. The method is applicable to both strain-inducing and non-strain inducing transducers, with uniaxial and biaxial orientation such as piezoelectric, electrostrictive, and magnetostrictive materials. The methods and analytical techniques presented are based upon the theory of multivariable distributions and were extended to distributions with composite functions as arguments. In this manner, nearly arbitrary shadings of transducers may be described. A differentiation theorem for such distributions was used to calculate the spatial differential operator for strain-inducing transducers. Several example shadings were presented to show the utility of this technique. It was also shown that two-dimensional delta and doublet function distributions could be obtained even when using a uniaxial oriented transducer if the boundaries of the transducer (or electrode) were shaped to be non-orthogonal to the material axes. Guidelines for defining spatial transducer distributions using generalized functions with composite arguments were also established. In the next chapter, this modeling technique will be used to design smart structure vibration controllers the using transducer shading to sense and control BernoulliEuler type structures.
Notes The modeling techniques for nearly arbitrary shading of multi-dimensional transducers presented herein was based on the work of Sullivan et al. [29]. Generalized functions (distributions) defined over multidimensional spaces in orthogonal coordinates are discussed in Zemanian [26] and Strang [30]. References on transducer shading techniques are described in a number of additional papers that can be found in the open literature. For example research conducted by Lee et al., Burke and Hubbard, and Dimitriadis et al. [31, 32, 33] has also utilized generalized functions for modeling two-dimensional transducers. This work, however, was restricted to generalized functions defined in orthogonal coordinates, and thus only rectangular shaded transducers could be examined. Burke and Hubbard [34] and Miller and Hubbard [35] have developed a design methodology for shading to accomplish all-mode sensing and control of beams with arbitrary boundary conditions. Lee and Moon [36] and Lee [37] demonstrated modal damping in beams using shaped PVDF actuators and sensors. Miller et al. [38] presented the use of shaped PVDF sensors to create desired spatial filtering properties for beams. Clark and Fuller [39] investigated the shaping of sensors for detecting acoustically significant modes in plates. Transducer shading can also provide useful modal coupling for plates and Burke and Hubbard [32] have extended the concept of continuous shading for beams to plates.
Problems
61
Burke and Hubbard [40] also examined the use of two-dimensional transducers for vibration control of plates with arbitrary boundary conditions. They assumed that the spatial gain of the transducer could be continuously varied, or shaded, over its surface area. In this work, the transducer shading was assumed to be rectangular with boundary axes coincident with the plate axes. Burke and Hubbard [32] and Miller and Hubbard [35] cut PVDF to approximate the linear shading in one dimension. In Lee and Moon [36] and Burke and Hubbard [34], products of generalized functions in orthogonal coordinates were used to model the shading of transducers. This procedure, however, could only accommodate transducers of rectangular shape.
Problems (2.1) Consider the system shown in Fig. 2.1a, which is a cantilever beam with a tip mass augmented with a thrust force T (1,t) and an acceleration measurement A (xa ,t), where x ∈ [0,1]. The transfer function from the thrust force to the accelerometer measurement is
∞ s2 A (xa ,s) φn (1) φn (xa ) = 1 2 T (1,s) s2 + μ4n n=1 0 φn (x) dx where μn is an Eigenvalue, φn (x) is an Eigenfunction, and s is a Laplace variable. a. Using the first four modes, generate a Bode magnitude plot for the case when the accelerometer is placed at the beam tip xa = 1. Which modes can be seen in the response? b. Repeat the previous question for when the accelerometer is placed very close to the root of the beam, e.g. xa = 0.05. Which modes can be seen in the response? How does this plot compare with that found in the previous question? c. Generate a surface showing the Bode magnitude as a function of both the forcing frequency and the accelerometer location. Explain the results
Fig. 2.1a Cantilever beam with tip mass and exogenous load
62
2 Spatial Shading of Distributed Transducers
and discuss the implications of using a discrete sensor to observe the dynamics of a distributed system. (2.2) For the Fig. 2.2a below, derive the spatial distribution using the McCaulay singularity notation. Determine the Laplacian of the distribution and provide a schematic of the equivalent loadingresulting from an induced strain actuator with this spatial distribution.
Fig. 2.2a Arbitrary load distribution
(2.3) Discuss the differences and similarities between McCauley notation described in this text and Heaviside notation. a. Using either Heaviside notation or McCauley notation, write a computer code to display the normalized 2D boxcar distribution. b. Using either Heaviside notation or McCauley notation, write a computer code to display the normalized 2D ramp distribution. c. Using either Heaviside notation or McCauley notation, write a computer code to display an arbitrary normalized 2D distribution. (2.4) Derive the equation for the output voltage of a center of pressure sensor mounted to a cantilever beam. What effect does the beam motion have on the output of the sensor? (2.5) Consider a thin beam with a trapezoidal aspect ratio and a 3-segment pennant-shaped center of pressure sensor applied to its surface, as shown in Fig. 2.5a below. Derive the equation for the center of pressure along the length of the beam. Assume a uniform loading p (x,t) along the beam width. (2.6) Consider the linearly-tapered center of center of pressure sensor shown in Fig. 2.6a. a. Suppose that a small transverse error ∈y is introduced during the manufacturing of the sensor, as shown in Fig. 2.6b. What is the effect on the voltage output? b. Suppose instead that a longitudinal error ∈x occurs, as shown in Fig. 2.6c. What is the effect on the voltage output?
Problems
63
Fig. 2.5a Tapered center of pressure sensor aperture
Fig. 2.6a Pressure sensor
Fig. 2.6b Pressure sensor with transverse error
c. Suppose the sensor is manufactured perfectly, but the applied load is not uniform across the sensing aperature and can be described by the positions of the vertices a, b, and c, as shown in Fig. 2.6d. Find the resulting voltage output and discuss implications. d. Repeat the previous questions using the linear taper realization of Fig. 2.6e. Discuss advantages and disadvantages of using this realization over that in Fig. 2.6a.
64
2 Spatial Shading of Distributed Transducers
Fig. 2.6c Pressure sensor with longitudinal error
Fig. 2.6d Pressure sensor with non-uniform loading
Fig. 2.6e Alternative realization of a pressure sensor
(2.7) a. Suppose a simply supported thin beam is fitted with four uniformly shaded sensors of equal dimension, as shown in Fig. 2.7a. Weight the output voltages to best approximate the voltage which would result from using a single, one-dimensional linearly shaded sensor. b. Suppose a simply supported plate is fitted with a four by four matrix of uniformly shaded sensors of equal dimension, as shown in Fig. 2.7b. Weight the output voltages to approximate best approximate the voltage which would result from using a single, two-dimensional linearly shaded sensor.
Problems
65
Fig. 2.7a One-dimensional sensor with discrete uniform patches
Fig. 2.7b Two-dimensional sensor with discrete uniform patches
(2.8) We wish to explore spatially distributed sensing using a piezo-film sensor mounted on a beam. We can write an expression between the charge produced by the film as a result of an exogenous strain field as Q (t) =
−1
e31 b (x) ∈ (x,t) dx
0
where e31 is the active strain constant, and ∈ (x,t) is the exongenous applied strain field, b (x) is the spatial distribution of the collected charge, enforced by appropriately shaping the electrodes applied to the sensing film. From principles of strength of materials we can write the strain field in terms of the beam curvature as ∈ (x,t) = −Elyxx where EL is the beam flexural rigidity and y is the beam displacement. a. Assuming the film has capacitance Co , develop a relation for the sensor output voltage Vo (t) as a function of spatial weighting b (x), beam displacement y (x,t), and boundary conditions. b. Assuming that b (x) is identically zero at the boundaries and that we use a uniform spatial weighting on the sensor, write a final expression for the sensor output Vo (t) for this case. a. Write the Lyapunov energy functional for this 2D system. Note: we may neglect all terms associated with any passive lumped boundary elements without loss of generality as they are separable from the active control design.
66
2 Spatial Shading of Distributed Transducers
b. Write the expression for the power flow for this system. c. Combine the governing equation and the power flow equation to obtain an expression for power flow that contains boundary integrals into which appropriate boundary conditions can be substituted. d. Substitute the boundary conditions given for a plate which is simply supported on all four edges, i.e. all for edges are pinned. e. Write a modified power flow expression of that obtained in part d, neglecting all terms which are independent of the control voltage and all terms which are conservative, i.e. which are not contributing to power flow into or out of the system. (2.9) We wish to design a vibration controller for a Bernoulli-Euler beam with clamped-clamped boundary conditions using a thin spatially distributed active film. The film distribution is applied to a beam whose non-dimensional governing equations are given as wtt + wxxxx − wxx = Vxx ,
x ∈ [0,1]
where W is the plate non-dimensional deflection and V = ∧ (x) φ (t) is the non-dimensional control voltage. A beam with clamped-clamped boundary conditions will have modes of vibration with either vanishing displacement or slope at its center, but never both. a. Sketch a spatial distribution, which will control all modes of the beam using the graph in Fig. 2.9a. b. Write an expression for the load intensity function (x) for the chosen spatial distribution. c. Sketch the effective loading profile given by this distribution. (Be careful to show the correct sign convention for positive and negative forces and moments.) d. Justify your answer in part a in detail. (Hint: The power flow for the system has the following proportionality F˙ ∼ V
max φ(t) wt 12 c +wxt 12 c
Fig. 2.9a
Problems
67
(2.10) The n th mode of a pinned-pinned beam has the form w (x,t) = A sin (nπ x) cos n2 π 2 t Assume that the instantaneous power flow over half a cycle that an active vibration controller removes is given by Econtrol T = −2nπ AVmax 2
where W is the lateral beam displacement, n is the mode number, x is the length scale, and Vmax is the maximum control voltage. The inherent passive viscous damping in the beam is given by Eviscous
1 T =1 ηn2 π 2 A2 2 4
where η is the beam loss factor due to viscous damping. Derive the condition for the control voltage when the controller will remove more energy from the beam that the passive damping inherent in the beam. (2.11) Consider the governing LNS equations mentioned in question 1.9. Write out the Lyapunov functional for this system. (Hint: in most cases the disturbance kinetic energy is a natural choice for the measure of the disturbance size. Consider the exchange of energy with the base flow and energy dissipation due to viscous effects). (2.12) Consider the beam vibration suppression example of Sect. 3.3. Use the same procedure to synthesize a nonlinear vibration control law using the Lyapunov control function F (t) =
2
1 0
1 w2t dx
which contains only kinetic energy terms. How does the performance of this new control law compare with the control law presented in the chapter? (2.13) Consider the beam example presented in this chapter. Compare the performance of the distributed control law of Eq. (3.11) and the discrete control law of Eq. (3.1).
Chapter 3
Active Vibration Control with Spatially Shaded Distributed Transducers
3.1 Introduction Smart structures are inherently control systems that are required to meet stringent performance metrics. These structural components must be lightweight, compact, low power and low cost in order to justify their incorporation into more complex systems that have traditionally met performance goals by passive means. Target systems are often lightly damped, have low mass densities, slow structural response and long decay times. Without active compensation these characteristics tend to degrade the performance and undermine the stability of what otherwise would be high performance systems. Much of the research done in smart structures to date has focused on the design of the temporal filter characteristics of the control system making use of output feedback principles, state variable feedback, the linear quadratic regulator and its augmentation with loop transfer recovery, optimal control, and classical PID methods. But active structural control falls under the general heading of distributed parameter control, i.e. the control of systems described using space and time variables, and this class of systems can have significant spatial dynamics. The spatial filter characteristics of such systems are often addressed in an ad hoc manner, if at all. The continuum nature of vibratory systems makes it challenging to apply traditional lumped parameter control approaches and methods. Modal decomposition and expansions are frequently employed to approximate structural behavior. In order to capture the true behavior of real systems, all terms of the series expansion must be included. In practice, computational limitations mandate a truncation of modal expansions in order to yield a more tractable plant model. As shown in Chap. 1, these reduced order models can lead to control and observation spillover and/or the need for high order compensators. In the previous chapter unified models of linear distributed and discrete transducers were developed using the theory of generalized functions and singularity functions. In this chapter these models are combined with energy based distributed parameter control design methods to synthesize stabilizing compensators for the control of vibrations in elastic beams and plates.
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_3,
69
70
3 Active Vibration Control with Spatially Shaded Distributed Transducers
3.2 Control System Synthesis Based on the Lyapunov Direct Method The second or direct method of Lyapunov is based on system global energy and power flow considerations and provides a convenient means of synthesizing stabilizing compensators for structural control. This method produces low order controllers that can readily be implemented and does not require a modal plant representation or associated model truncation. Lyapunov’s second or direct method can be used as a design method for control systems which can easily deal with bounded inputs and can be extended to distributed parameter systems. The design goal for vibration control is to synthesize a transducer shading (x), or distribution, using the representations of the previous chapter, and a feedback control law u(t) that removes energy from the system, therefore damping its resonant response. Using this method, we define a functional that resembles the energy of the system and chooses the control to minimize (or make as negative as possible) the time rate of change of the functional at every point in time. This so called control-Lyapunov function is a generalization of the ordinary Lyapunov function that is used in stability analysis to test whether a dynamic system is stable, that is whether the system started in a state x = 0 will eventually return to x = 0. By contrast the control-Lyapunov function is used to test whether a system is feedback stabilizable, in other words whether for any state x there exists a control u (x,t) such that the system can be brought to the zero state by applying the control u. The flexural vibrations of an elastic beam having an active film actuator bonded to one face was derived in Chap. 2 and is described by the non-dimensional equation ∇ 4 w + wtt = ∇ 2 V;
0<x<1
(2.26)
which is repeated here for convenience. The boundary conditions to be considered are further generalized to include possible linear lumped elements as w (ξ ,t) = wx (ξ ,t) = 0
Clamped (3.1a)
wxx (ξ ,t) = V (ξ ,t) − It wxtt (ξ ,t) − κt wx (ξ ,t) − βwxt (ξ ,t) (3.1b) w (ξ ,t) = 0
Pinned
wxx (ξ ,t) = V (ξ ,t) − It wxtt (ξ ,t) − κt wx (ξ ,t) − βwxt (ξ ,t) (3.1c) wxxx (ξ ,t) = Vx (ξ ,t) + Mt wtt (ξ ,t) + kt w (ξ ,t) + bwt (ξ ,t)
Free
wxxx (ξ ,t) = V (ξ ,t) − It wxtt (ξ ,t) − κt wx (ξ ,t) − βwxt (ξ ,t) wx (ξ ,t) = 0 total system energy,
Sliding
(3.1d)
3.3
Control System Synthesis for Beams
71
where ξ is the non dimensional boundary point x = 0 or x = 1. The linear lumped elements included are translational masses b, rotary inertias I, translational and rotary springs k and κ, translational and rotary viscous dampers b and β, respectively. Exogenous inputs located at the boundaries can be included but are left out here for the sake of simplicity. The Lyapunov design methodology may now be applied to the system described in (2.26) by first constructing a control-Lyapunov function. We choose an energy based functional that describes the total system energy 1 F(t) = 2
1 0
1 (wxx )2 + (wx )2 + (wt )2 dx + Mt [wt (ξ ,t)]2 2 (3.2)
1 1 1 + It [wxt (ξ ,t)]2 + κt [wx (ξ ,t)]2 + kt [w (ξ ,t)]2 . 2 2 2 Equation (3.2) represents both the structural strain and kinetic energies in addition to energy storage mechanisms of any lumped parameter elements which appear at the boundaries. Vibration damping may now be based upon total system energy considerations and deals directly with the distributed plant model with no truncations or approximations other than those associated with the Bernoulli Euler model (2.26). Consequently, conditions sufficient for asymptotic stability can be derived using this distributed parameter formulation. Note that this is in sharp contrast to the modally truncated plant models generally used in smart structure control systems design as described in Chap. 1. The time derivative of the control-Lyapunov function (3.2) when combined with the non-dimensional governing equation (2.26) and any pair of boundary conditions (3.1) yields an expression for power flow out of the system that is stabilizing i.e. ˙ = F(t)
1
wxxt (x,t) V (x,t) dx − β [wxt (x,t)]2 − b [wt (x,t)]2
(3.3)
0
Note that the spatially distributed control V(x,t) only appears in the spatial integral. In (3.3) the lumped boundary terms are decoupled from the control input and always act to remove energy from the system since negative power means power flow out of the system by convention. The smart structure controller design simply ˙ is negative semi-definite for all time t so as to involves choosing V(x,t) so that F(t) insure that (3.3) is as negative as possible in order to extract the most energy.
3.3 Control System Synthesis for Beams As an example of control system synthesis using Lyapunov’s Direct Method consider a cantilever beam with tip mass that has a uniformly shaded active film actuator bonded to one face. The non-dimensional governing equations and associated boundary conditions may be formulated as
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
wxxxx + wtt = 0
0≤x≤1
(3.4)
w(0,t) = 0 wx (0,t) = 0 (3.5) wxx (1,t) = −It wxtt + V wxxx (1,t) = Mt wtt The control-Lyapunov function chosen for this system consist of the energy in the system and is described as follows, 1 F(t) = 2
1 0
[w2 + w2t ] dx.
(3.6)
Differentiating (3.6) yields an expression for the power flow from the system as ˙ = F(t)
1
1
(w · wt ) dx −
0
(wxxxx · wt ) dx.
(3.7)
0
The second integral in (3.7) may be evaluated by introducing the boundary conditions appropriately,
1
(wxxxx · wt ) dx
0
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
=
|wt wxxx |x=1 x=0
1
−
(wxxx · wt )dx
0
1
= [Mt wt wtt ]x=1 − [wt wxxx ]x=0 −
(wxxx wt ) dx
.
0 1
= [Mt wt wtt ]x=1 −
(wxxx wt ) dx 0
(3.8)
The second integral in (3.8) may be evaluated similarly as
1 0
(wxxx · wt ) dx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
= |wxt wxx |x=1 x=0 −
1
(wxx · wxxt ) dx
0
= [wxt (−It wxtt + V (x,t))]x=1 − [wxt wxx ]x=0 −
1
(wxx wxxt ) dx
0 1
= [−It wxt wxtt ]x=1 + [V wxt ]x=1 −
(3.9)
(wxx wxxt ) dx
0
Combining the results of (3.7), (3.8), and (3.9) yields the desired power flow as
3.3
Control System Synthesis for Beams
˙ = F(t)
1
73
(w wt − wxx wxxt ) dx − [Mt wt wtt ]x=1 − [It wxt wxtt ]x=1
0
(3.10)
+ [V (x,t) wxt ]x=1 . Note that the control voltage only appears in the last term in (3.10) and this term should be chosen so that it is always as negative as possible, thus V(1,t) = − Vmax sgn [wxt (1,t)]
(3.11)
where wxt (1,t) is the dimensionless angular velocity at the tip of the beam. The control voltage should be chosen with as large a magnitude as possible and generates a control moment that always opposes the angular motion of the tip of the beam. In this manner, the maximum amount of work is being done against the beam at all times, removing as much energy as possible from the system at every point in time thus actively damping the structure in a time optimal fashion. The resulting control law (3.11) has several desirable characteristics. First, no modes have been truncated. This control law will (theoretically) work on any and all modes of vibration of the cantilever beam since every mode has some angular motion at the tip of the beam. Secondly, the control law depends only on the angular velocity at the tip of the beam, not an integral along its length. This means that in principle only one discrete sensor measurement is needed to implement this distributed-parameter control law. The effect of spatial shading can be observed explicitly by considering the generalization of the power flow as presented in (3.3). For the case of uniform shading on a cantilever beam V(x,t) takes the form V(x,t) = Vxx u(t)
(3.12)
and for uniform shading we know from results of Chap. 2 that V(x,t) = Vmax x −2 − x − 1 −2 u(t).
(3.13)
This means that the shading provides two opposed moments at the ends of the distribution. Utilizing the boundary conditions for a cantilever beam the power flow equation becomes ˙ = −u (t) wxt (1,t) . F(t)
(3.14)
˙ Once again to maximize F(t) and extract the most energy from the beam we choose the exogenous control time signal u(t), to be that given in (3.11) as the time optimal solution. One potential disadvantage of this time optimal control law is that the sgn (·) function is nonlinear and discontinuous when its argument is zero. This nonlinear control law could lead to problems such as limit-cycling and/or sliding modes. An alternate control law that may be more practical is the constant gain controller
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
V(t) = −Kp · ( wxt |x=1 ) ,
|V(t)| ≤ Vmax .
(3.15)
This constant-gain controller is both linear and continuous. It can be derived from physical insight (negative velocity feedback tends to stabilize the system) or more rigorously from a modal control viewpoint. The drawback to this controller is that as the amplitude of the velocity decays, so does the feedback voltage amplitude resulting in a quasi-optimal solution. This will reduce the effectiveness of the damper at low vibration levels, for a given voltage limit. There is also the question of sensing in that (3.11) requires the real-time feedback of wxt (1,t) or angular velocity at the tip of the beam. While one might surmise from the material presented in Chap. 2 that a uniformly shaded sensor can also provide the needed measurement, in the next section we provide a generalization of sensor and actuator requirements for Lyapunov control.
3.3.1 Collocated Distributed Transducers and Lyapunov Control Collocated sensors and actuators offer significant robustness and design benefits in smart structure vibration control. Collocated spatially distributed transducers can be used to construct stabilizing Lyapunov vibration controllers for Bernoulli-Euler beams. The stabilizing effect of collocated spatially discrete actuators and rate sensors is well known. Smart structural components which contain embedded, collated sensing and actuation can be integrated into more complex structures as part of a high performance control system. An alternate form of (3.3) from Lyapunov beam analysis which considers the power flow from a Bernoulli-Euler beam with strain induced spatially distributed active film transducers bonded to each of its faces, 1 d ˙ = u(t)
(x) wxx (x,t) dx . (3.16) F(t) dt 0 The spatially shaded sensor and actuator distributions b(x) and (x) respec↔
tively, are chosen to be identical. This collocated distribution is designated as (x). If the temporal piece of the control input u(t) is chosen to be proportional to the time derivative of the distributed sensors’ output then u(t) =
−Kp2
d dt
1 ↔ 0
(x)wxx (x,t) dx ,
(3.17)
where Kp is a nonzero constant of proportionality. The power flow equation now takes the form & ˙ = −Kp2 F(t)
d dt
1 ↔ 0
(x) wxx (x,t) dx
'2 .
(3.18)
3.3
Control System Synthesis for Beams
75
The power flow expression given by (3.18) is negative semi-definite for all time t, for collocated sensor and actuator distributions. No matter what the sensor and actuator shadings of the transducers bonded to the beam are, if they are identical, i.e. b(x) = (x), and the two are interconnected via using rate feedback according to (3.17), then the resulting closed loop control system will be stable in the sense of Lyapunov. This collocation result has important implications for the designer of smart structure components. Smart material transducers such as piezoelectric, magnetostrictives, shape memory alloys etc. are self reacting devices and thus require no fixed inertial frame to react against. This means that distributed transducers can be embedded into individual smart structural components during manufacture along with the appropriate Lyapunov compensators implemented “on-board” the component via modern microprocessor technology. These individual components can then be assembled into more complex structures to extract vibration energy with the guarantee that the resulting system will always be stable. There are additional restrictions on transducer types that also need to be satisfied, particularly with regard to degenerate devices other than spatially distributed film transducers. This can be made evident by considering (3.16) written in the generic form of 1 d ˙F(t) = u(t)
(x) w(x,t) dx . (3.19) dt 0 The sensor output can be represented by the expression
1
V0 (t) = 0
w(x,t) Lxm [b(x)] dx
(3.20)
where Lxm [◦] is a linear differential operator which models the transducers operation. This representation allows for a compact description of the transducer in terms of its aperture and spatial derivative order. Using nomenclature adapted from deLafontaine and Steiber [44] we designate devices such as force actuators as type 0 for their zeroth order spatial derivative operator and strain induced devices such as piezo actuators as type 2 devices for their second order spatial derivative operator. This is evident if we consider a point force actuator located at x = xn and is thus represented as Lxm [b(x)] =< x − xn >−1
(3.21)
where m = 0 is the operator order. Similarly for uniform shading on a beam we know from (3.12) that Lxm [b(x)] = Vxx
(3.22)
which has a spatial derivative order of m = 2 making this a type 2 device. The collocation requirement when applied to (3.19) and (3.20) means that the sensor distribution Lxm [b(x)] must be equal to the actuator distribution (x) and therefore
76
3
Active Vibration Control with Spatially Shaded Distributed Transducers
the transducers must have both the same physical distribution (shading) over the beam as well as the same spatial derivative order. As an example point moment actuators may not be collocated with linear velocity sensors and interconnected via rate feedback in accordance with (3.3). They can however be collocated with point angular velocity sensors and interconnected with rate feedback to produce a stabilizing controller in the sense of Lyapunov. In this section a Lyapunov control design method was shown to produce simple, effective and physically realizable active control systems for smart structures. The control design is physically motivated as the synthesis method is based on intuitive energy and power flow considerations. Spatially distributed transducers may be incorporated into a smart structure a priori permitting the construction of control systems with a minimum of input/output channels and compensators of low order. The technique presented here is equally adaptable to spatially discrete transducers. The control compensators design introduced here has been focused on the removal of energy from the entire system at all points in time and only deals with global stability and damping. In subsequent chapters of this text we will develop techniques for multi-input multi-output systems that target specific performance goals such as settling time, tracking error and/or overshoot. In the remaining sections of this chapter we discuss the impact of transducer shading on Lyapunov vibration controllers.
3.3.2 Performance Limitations of Control Designs with Shaded Distributions One of the principle advantages of spatially varying control distributions (shading) discussed thus far is that they are simple spatial configurations that can be implemented on real systems. In this section we will demonstrate that they also can provide insight into a design methodology for controllability and observability that can appeal to intuition. To understand this we must re-write the power flow equation (3.3) as ˙ = V (x,t) wxt (x,t) | 10 − Vx (x,t) wt (x,t) | 10 + F(t)
1
wt (x,t) Vxx (x,t) dx
(3.23)
0
3.3.3 Performance Limitations of Uniformly Shaded Transducers In Chap. 2 a spatially uniform control distribution was defined in terms of singularity functions as V(x,t) = (x) u(t) = Vmax [< x > − < x − 1 >] u(t)
(2.21)
3.3
Control System Synthesis for Beams
77
where u(t) is the time portion of the control, e.g. rate feedback. The Laplacian of the distribution is given by
xx (x,t) = Vmax < x >−2 − < x − 1 >−2
(2.22)
and reveals that the uniform shading will only produce boundary control in (3.23) in terms of linear and angular velocities at x = 0 and x = 1, because the uniform shading produces a pair of point moments at the boundaries. The uniform shading and its effective loading interpretation were given earlier in Chap. 2 and is repeated here for convenience (See Figs. 3.1 and 3.2). Beams with clamped ends where (3.1a) holds do not display boundary control in terms of linear or angular velocities because wx (x,t) and w (x,t) must vanish at the boundaries, thus a clamped–clamped beam is not controllable using spatially uniform shading. Similarly, a beam with clamped-sliding boundary conditions will not be controllable with uniform shading because the sliding end of the beam, like the root, must have wx (x,t) = 0 in accordance with 3.1d. In contrast a beam that is clamped at one end with a pinned or free boundary at the other end is controllable with uniform shading and (3.23) takes the form ˙ = u(t) wxt (1,t) F(t)
(3.24)
where x = 1 is the pinned or free boundary. As presented earlier in this chapter to ˙ in the case of (3.24) and hence extract the most energy for vibration maximize F(t) damping we choose the Lyapunov controller V (1,t) = −sgn [wxt (1,t)] Vmax .
(3.25)
1
Λ(x) Fig. 3.1 Spatially uniform control distribution (shading)
0 0
Λ xx(x)
Fig. 3.2 Loading interpretation for a uniformly shaded control distribution
< x >−2
0
1
< x −1>−2
1
78
3 Active Vibration Control with Spatially Shaded Distributed Transducers
For a pinned-pinned beam with uniform shading (3.23) becomes ˙ = Vmax [wxt (1,t) − wxt (0,t)] u(t). F(t)
(3.26)
This suggests that for a pinned-pinned beam with uniform shading the instantaneous power flow is non-zero when the angular velocities or slopes at the ends of the beam are not equal. While the control formulation is non-modal based, the pinned-pinned beam offers an opportunity to gain an intuitive insight into the impact on performance due to shading by examine modal symmetries. For a simply supported beam (pinnedpinned), the mode shapes are sinusoids and hence the displacement profiles can be written as w=
∞
An sin (ωn t) sin (nπ x)
(3.27)
n=1
and therefore wx = wxx =
∞
An nπ sin (nπ x)
(3.28)
−An n2 π 2 sin (nπ x)
(3.29)
n=1 ∞ n=1
where An is a modal amplitude, ωn is the modal frequency with mode number n. A uniform shading can be represented in terms of sinusoids as well by its Fourier series expansion as f (x) =
∞ 4 sin ([2n + 1]) π x . π 2n + 1
(3.30)
0
If we consider the application of a uniformly shaded sensor to a simply supported beam then we can write a sensor equation as V(t) = −
Q C
1
wxx (x,t) f (x) dx
(3.31)
0
where Q C represents the appropriate transduction constants, for example charge and capacitance in the case of a piezoelectric sensor. Combining (3.30) and (3.31) yields the sensor voltage output 4Q V(t) = πC
1 0
(
∞ n=1
An n2 π 2 sin (ωn t) sin (nπ x) · )* 1
+ (
∞ sin ([2n + 1] π x) n=0
2n + 1 )*
dx. +
2
(3.32)
3.3
Control System Synthesis for Beams
79
Expressions (1) and (2) represent infinite series and yield the series (1) =A1 π 2 sin (ω1 t) sin (π x) + 4A2 π 2 sin (ω2 t) sin (2π x) + 9A3 π 2 sin (ω3 t) sin (3π x) + . . . (3.33) (2) = sin (π x) +
1 1 sin (3π x) + sin (5π x) + . . . 3 5
(3.34)
From the principle of orthogonality we know that
1
sin (aπ x) sin (bπ x) =
0
⎧ ⎨
0
⎩1 2
if a = b (3.35)
−
sin(2π c) 4π c
if a = b, where c = a = b
which indicates that all of the terms in the product of sines above evaluate to zero with the exception of the sin2 terms. In general (3.32), (3.33), (3.34), and (3.35) combine to yield the output voltage of the sensor as V(t) =
∞ n=1, 3, 5,...
Q An sin (ωn t) [2nπ − sin (2nπ )] . C
(3.36)
The significance of (3.36) is that for a simply supported beam with a uniformly shaded sensor, only the odd numbered or symmetric modes are observed. Odd order modes have even symmetry and thus produce motions for which the beams slopes ˙ = 0 in accordance with (3.36). Even ordered at the boundaries are unequal and F(t) modes have odd symmetry (anti-symmetric) as indicated in Fig. 3.3, and the slopes ˙ at the boundaries are equal and thus F(t) = 0 for these modes. Because the time derivative of the energy functional (3.36) represents the instantaneous power flow, this means that no energy can be extracted from or added to the system for these vibration modes using a uniformly shaded actuator and is thus also uncontrollable. As shown for even numbered modes the uniformly shaded distribution results in the product of even and odd functions (with respect to the beam mid-span), and the ensuing integral must vanish over the beam length for a simply supported beam. Fundamentally the uniformly shaded distribution (sensor or actuator) can be expressed as a Fourier sine series containing only odd harmonics. When this series is substituted into the spatial integral in (3.3), it essentially becomes a restatement of modal orthogonality for a simply supported beam for all odd-numbered modes.
odd
Fig. 3.3 First and second mode symmetries for a simply supported beam
x 0
1 even
80
3 Active Vibration Control with Spatially Shaded Distributed Transducers
The result is that only symmetric modes lead to non-zero contributions to the beam’s control. This same result holds for free–free boundary conditions in that anti-symmetric modes are not controllable using measurements of angular velocity at the boundaries.
3.3.4 Performance Limitations of Linearly Shaded Transducers Based on the results of the previous section it is apparent that the symmetry of a chosen spatial shading function is important when compared to the modal symmetries of the structure being controlled. Distributions which have even symmetry about the mid-span of a beam can only observe and control modes which have the same symmetry. In the case of a simply supported beam for example, in order to target and control all modes of the structure we must select a spatial shading function that has both even and odd symmetry about the beam mid-span. A close examination of the linear or ramp distribution function reveals that it has both even and odd spatial symmetry and thus meets the requirements for all mode control of a simply supportedbeam. The spatial shading of Fig. 3.4 is conveniently described using singularity functions by
(x) =< x >0 − < x >1 + < x − 1 >1 .
(3.37)
The equivalent loading for this spatial distribution when applied to a BernoulliEuler beam as a strain induced transducer is given by its Laplacian as
xx (x) =< x >−2 − < x >−1 + < x − 1 >−1 .
(3.38)
This means that the distribution produces a concentrated control moment at x = 0, and point forces at the two boundaries which sum to satisfy static equilibrium requirements (See Fig. 3.5). To extract the maximum amount of energy per cycle in vibration damping of a Bernoulli Euler type smart structure using this spatial shading we choose the control V (0,t) = −sgn [wxt (0,t)]Vmax .
(3.39)
y 1
Λ(x) Fig. 3.4 Linearly shaded distribution with even and odd symmetries
0
x 0
1
3.3
Control System Synthesis for Beams
81 y
Fig. 3.5 Linearly shaded distribution equivalent loading
1
Λ xx(x) 0
x 1 0
The control forces involve point forces and a point moment. Because translational motion is denied by the pinned boundary conditions the terms that involve delta functions will vanish in the time derivative of the Lyapunov energy functional. Therefore the spatially shaded strain induced control offered by the smart material film appears to the smart structure system as a boundary controller at one end. Bringing the spatial distribution to zero gracefully (without discontinuity in amplitude) at the boundary x = 1 will not generate a point moment there.
3.3.5 Design Guidelines on Spatial Shading for Vibration Control At this juncture we can make some general observations regarding spatial shading and beam vibration control: • Spatially uniform transducer shadings are simple in form and can provide effective damping for many beam configurations. • The Lyapunov design method presented here results in film transducer shadings that have discontinuities in amplitudes and/or slopes as they are superposition’s of step and ramp functions. Using this method yields control laws in terms of linear and angular velocity feedback. • Discontinuities in amplitude will generate point moments on the beam at the location of the discontinuity. The resultant control amplitude will be proportional to the magnitude of the amplitude discontinuity. • Discontinuities in slope produces point forces that result in control laws in terms of linear velocity at the location along the beam of the discontinuity. The resultant control amplitude will be proportional to the magnitude of the slope discontinuity. • The film spatial shading function (x) is the spatially weighted moment distribution corresponding to the effective loading given by xx (x). (Strain induced actuators, for example, exert a spatially distributed moment on the beam). • As a result of the above the smart structure designer can simply synthesize a film actuator shading function by prescribing a desired loading xx (x), and then integrating to obtain (x). The collocation requirement can then be enforced to determine the concomitant sensor.
82
3 Active Vibration Control with Spatially Shaded Distributed Transducers
• The location of the shading discontinuities can be varied so as to implement a control which targets only certain modes. For example modes with even symmetry can be targeted with shadings that have even symmetry and vice versa. • The film actuator will be most effective at low frequencies and low amplitudes for Bernoulli Euler type beams as this is the energy regime in which the system is “most flexible and lightly damped”. To illustrate the last point consider the simply supported beam which has resonant characteristics typical of all Bernoulli Euler beams but whose modes are described by simple sinusoids. The n-th mode of a simply supported beam has the form w (x,t) = A sin (nπ x) cos n2 π 2 t .
(3.40)
Assuming fairly low control authority typical of most modern smart film actuators, the modes may be taken to be uncoupled for the purposes here. Also the temporal frequency in (3.40) is taken so as to satisfy the beams dispersion relation. We can now compare the beams active energy extraction to its inherent damping characteristics. At t = 0 the beam’s initial strain energy can be written as Estrain
1 = 2
1
[wxx (x,0)]2 dx =
0
n4 π 4 A2 . 4
(3.41)
The control energy that a switching controller such as (3.39) can extract over half a cycle, using the shading (3.37) can be obtained by integrating the expression for the instantaneous power flow over half a period to yield
E
control
T/ = −2nπ AVmax . 2
(3.42)
Assuming that the damping inherent in the structure can be represented as an equivalent viscous loss proportional to the beam’s lateral velocity, the energy removed over half cycle is E
viscous
T/ 2
=−
η n2 π 2 A2 4
(3.43)
where η is the loss factor associated with the particular mode of interest. From the above it is apparent that the controller will remove more energy from the beam than that provided by the inherent passive damping when 8Vmax > 1. ηnπ A
(3.44)
3.3
Control System Synthesis for Beams
83
This is essentially a modal bandwidth limit and indicates that the controller will be most effective at low frequencies and low amplitudes where inherent damping is generally insufficient to adequately damp modes. As an example of the intuitive application of the design guidelines presented in this section consider the vibration damping of a non-dimensional Bernoulli Euler beam structure with clamped–clamped boundary conditions at x = 0 and x = 1 respectively. Such a structure will admit modes which have either vanishing slope or vanishing displacement at its center, but never both. This leads the designer to choose a shading which results in a point moment at mid span for even numbered asymmetric modes and a point force at mid span for odd numbered symmetric modes as illustrated in Fig. 3.6. A shading (x) which has a discontinuous amplitude at x = 12 (yielding a point moment and angular velocity term) and a discontinuous slope there as well (yielding a point force and linear velocity term) will produce the loading of Fig. 3.6. This spatial shading distribution is illustrated in Fig. 3.7. The Lyapunov control law which extracts the most energy per cycle for vibration damping is
V(t) = sgn wt
1 1 ,t − wxt ,t . 2 2
(3.45)
The methods and synthesis procedures presented here can be extended to thin plates and biaxial active film transducers. The Lyapunov control system analysis and synthesis for plates is presented in the next section. Λxx (x) x−
1 2 −2
x − 1 −1
1/2 x−
1 2
− −1
1 x −1 −2 2
Fig. 3.6 Effective loading for a clamped–clamped beam
Λ (x) Fig. 3.7 Spatial shading function for a clamped–clamped beam
1/2
84
3 Active Vibration Control with Spatially Shaded Distributed Transducers
3.4 Control System Synthesis for Plates In this section we consider the Lyapunov vibration control of a thin elastic plate which has an active layer of biaxial film as a distributed actuator adhered to one face (For reference recall Fig. 2.16). The active film layer is such that it responds to an applied electric field across its faces via a longitudinal strain in both in plane dimensions. Assuming that the deflections of the plate are small the governing partial differential equation for the composite plate/film structure in the absence of in plane tension is ˆ xˆ ∈ , D∇ˆ 4 wˆ + ρ wˆ ˆtˆt = m∇ˆ 2 V,
(3.46)
where wˆ is the lateral deflection of the plate, m is the film electro-mechanical coupling, D is the flexural rigidity of the plate/film structure, ρ is its area density, and Vˆ is the spatially distributed control voltage applied across the faces of the active film. The spatial coordinate xˆ = (x,y) and (•)t denote 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 with its boundaries contained in . We wish to investigate the selection of spatial shading for the distributed strain induced actuator commensurate with the development of a control law for V xˆ ,ˆt that will result in the dampening of vibrations in the plate. Introducing the following non-dimensionalizations, wˆ xˆ , w= , L L , ˆ ˆt D VD ,V= , t= 2 ρ L mL2 x=
(3.47)
where L is the characteristic length of the plate, allows (3.46) to be written as ∇ 4 w + wtt = ∇ 2 V.
(3.48)
We now choose a control Lyapunov function that is proportional to the energy of the system which takes the form F=
1 2
2 2 ∇ 2 w + wxy + wxx wyy + (wt )2 dx.
(3.49)
The factor of two is added here for mathematical convenience. The first three terms in (3.49) describe the system’s strain energy with the latter two arising due to the plate’s Gaussian curvature. The plate’s kinetic energy is represented by the last term and (3.49) is the system’s total energy within the constraints and assumptions of the governing equation and its Bernoulli Euler representation.
3.4
Control System Synthesis for Plates
85
The system power flow is found by differentiating (3.49) with respect to time to obtain F˙ =
1 wxx wyyt + wyy wxxt dx ∇ 2 w ∇ 2 wt + wtt wt + wxy wxyt + 2
(3.50)
When F˙ < 0 for all time the system is asymptotically stable in the sense of Lyapunov as power will always be removed. Boundary conditions may be appropriately introduced using the following integral identity -
a∇ 2 u ∇ 2 v d =
a
∂u 2 ∇ vds − ∂n
u
∂ 2 a∇ v ds + u∇ 2 a∇ 2 v d. ∂n
(3.51) Here the functions u and v are arbitrary functions and a is prescribed. The variable s describes the length along the boundary while n is the outward normal of the boundary. For the analysis presented here a = 1, u = wt , v = w, s = and n = n. The expression for the power flow now becomes F˙ =
.
+
∂(wt ) 2 ∂n ∇ w d wt wtt dx +
+
wt ∇
4 w dx −
wxt wxyt +
1 2
.
2 ∂ ∇ w d wt ∂n
wxx wyyt + wyy wxxt dx.
(3.52)
Substituting the governing equation (3.48) into the power flow equation yields the important result F˙ =
.
∂(wt ) 2 ∂n ∇ w d
+
+
wxt wxyt +
wt ∇
1 2
2V
dx −
.
2 ∂ ∇ w d wt ∂n
wxx wyyt + wyy wxxt
(3.53) dx.
Note that now Eq. (3.53) contains an integral involving the Laplacian of the plate control voltage, as well as boundary integrals into which the appropriate boundary conditions of interest may be applied. This analysis is valid for any natural co-ordinate system in which ∇ 4 and ∇ 2 are defined e.g. non-rectangular plates. This is quite similar to the beam formation and analysis. The boundary conditions for the plate/film smart structure system can be assumed to be homogeneous with regard to any lumped or distributed elements and with respect to the control voltage as well. We may describe clamped, pinned, free and sliding boundary conditions respectively as
86
3 Active Vibration Control with Spatially Shaded Distributed Transducers
w (,t) =
∂ ∂n w (,t)
= 0,
w (,t) = ∇ 2 w (,t) = 0, ∇ 2 w (,t) =
∂ ∂n
∂ ∂n w (,t)
∂ ∂n
=
. ∇ 2 w (,t) = 0,
(3.54)
2 ∇ w (,t) = 0.
Using any combination of these boundary conditions along with (3.54) results in
F˙ =
wt ∇ 2 V dx
1 wxx wyyt + wyy wxxt dx + wxt wxyt + 2
(3.55)
The second integral in (3.55) is independent of the control voltage. Assuming that the uncontrolled plate is stable, the second integral is conservative in the sense that it does not contribute to power flow into or out of the system regardless of the boundary conditions. The control design therefore reduces to the selection of the appropriate active film transducer shading that guarantees that the first integral in (3.55) is negative semi-definite. In the following sections on design synthesis we will therefore neglect the second integral in the ensuing analysis.
3.4.1 Performance Limitations of Uniformly Shaded Actuators for Plates Consider the spatially uniform shading distribution as illustrated in Fig. 3.8. This distribution may be defined according to the following: 0 x 0 − x − lx 0 y 0 − y − ly u(t) (3.56) where we have assumed the control input to be of degenerate form i.e. separable in space and time with lx and ly being the nondimensional lengths of the sides parallel to the x and y axis. The Laplacian of this distribution is given by V (x,t) = Vmax (x) u(t) = Vmax
Vmax ∇2 V =
0 x −2 − x − lx −2 y 0 − y − ly
. + x 0 − x − lx 0 y −2 − y − ly −2 u(t)
(3.57)
3.4
Control System Synthesis for Plates
87
1.0
0.5 1.0 0.0 0.0 0.5
y
0.5 x
1.0
0.0
Fig. 3.8 Two dimensional uniform shading distribution
The first integral of (3.55) may now be written as ˙ = F(t) −Vmax u(t)
ly 0
wt ∇
2V
dx =
. l wx |x=0 − wx |x=lx dy + 0x wy y=0 − wy y=l dx+ y (3.58)
It now becomes apparent that the spatially uniform shading distribution provides control in terms of spatially distributed angular moments along its boundaries. Intuition tells us that for example this choice of shading would not be appropri˙ =0 ate for a plate that is clamped along all of its edges because for that case F(t) and therefore this actuator shading can not extract energy from nor add energy to a clamped plate. Equation (3.58) can, in like manner, be used to analyze distributions applied to plates with other types of boundary conditions. To provide further insight consider the case of a plate that is simply supported along all sides. If opposing sides of the plate have identical angular velocity distributions, then modal vibration components in the direction normal to those sides will not be controllable with a uniform shading ˙ = 0 and no energy can be extracted from or added to the because once again F(t) plate. Similarly according to (3.58) if the integrated angular velocity distributions are equal for two opposing sides, the same result occurs.
88
3 Active Vibration Control with Spatially Shaded Distributed Transducers
Consider the fact that the response of a simply supported plate can be written as w (x,t) = A sin (kx x) sin ky y eiωt+φ
(3.59)
with eigenvalues expressed as kx =
mπ lx
ky =
nπ ly
.
(3.60)
Here m and n are integers chosen to satisfy the appropriate boundary conditions. The associated angular velocity components are wxt = −iωkx A cos (kx x) sin ky y e−i (ωt+φ) , (3.61)
wyt = −iωky A sin (kx x) cos ky y e−i (ωt+φ) .
For the case where m is even and n is odd as represented by the 2-1 mode shown in Fig. 3.9, the angular velocities evaluated along the sides x = 0 and x = lx are equal for odd n and that contribution to the functional (3.58) will vanish (See Fig. 3.10). As shown the angular velocity distributions along the sides x = 0 and x = lx are equal. Similar results can be observed for the case when m is even and n is even as in the 2-2 mode shown in Figs. 3.11 and 3.12. For the 1-2 mode depicted in Fig. 3.13 when m is odd, the first integral in (3.58) vanishes when n is even as the functional involves the line integral of the angular velocity distribution at the boundaries in the y direction, which is identically zero due to the fact that the integrand is an odd function as shown in Fig. 3.14.
1.0 0.5 0.0 –0.5 0.0
–1.0 1.0
0.5
0.5 x
y
0.0
1.0
Fig. 3.9 Non dimensional simply supported plate 2-1 mode w displacement field
3.4
Control System Synthesis for Plates
89
5 0 –5 1.0
0.0 0.5
0.5
x
0.0
y
1.0
Fig. 3.10 Non dimensional simply supported plate 2-1 mode wx field
1.0 0.5 0.0 –0.5 0.0
–1.0 1.0
0.5
0.5
y
x
0.0
1.0
Fig. 3.11 Non dimensional simply supported plate 2-2 mode w displacement field
5 0 –5
1.0
0.0 0.5
0.5 x
1.0
0.0
Fig. 3.12 Non dimensional simply supported plate 2-2 mode wx field
y
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
1.0 0.5 0.0 –0.5 –1.0 1.0
0.0
0.5
0.5
y
x
1.0
0.0
Fig. 3.13 Non dimensional simply supported plate 1-2 mode w displacement field
2 0 –2 0.0
1.0
0.5
0.5
y
x
0.0
1.0
Fig. 3.14 Non dimensional simply supported plate 1-2 mode wx field
The case when m is odd and n is odd yields angular velocities at the opposing sides which are not equal and the integrand is an even function thus the first integral in (3.58) will be non-vanishing as evidenced in Figs. 3.15 and 3.16. An examination of (3.58) for the angular velocity wyt yields similar results in that n even, or n odd with m even, will not be controllable using spatially uniform shading for the control voltage. The symmetry arguments developed and presented earlier in this chapter for beams must be extended to include not only the symmetry of mode shapes with respect to linear and angular velocities, but also with the symmetries of the distributions along the boundaries. As demonstrated here, a simply supported plate is only controllable with a uniform shading distribution if both m and n are odd. For the sake of further discussion we define such modes as “strictly
3.4
Control System Synthesis for Plates
91
1.0 0.5 0.0 1.0
0.0
0.5
0.5
y
x
0.0
1.0
Fig. 3.15 Non dimensional simply supported plate 1-1 mode w displacement field
2 0 –2
0.0
1.0
0.5
0.5
y
x
0.0
1.0
Fig. 3.16 Non dimensional simply supported plate 1-1 mode wx field
odd” while those with m and n both even will be referred to as “strictly even” and finally all others “mixed order” vis; • n-odd with m-even Not Controllable: “Mixed order” • m and/or n are even Not Controllable: “Strictly even” • m and n both odd Controllable: “Strictly odd”.
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
3.4.2 Performance Limitations of Non-uniformly Shaded Actuators for Plates The preceding section discussed the performance limitations of spatially uniform distributions for plates with nearly arbitrary boundary conditions. As presented, the uniform shading can be very restrictive for many significant boundary conditions of interest. The uniform shading was shown to either render all modes unobservable and uncontrollable or at best target modes with restricted symmetries. There are certain exceptions such as clamped-free boundary conditions in which all modes can be controlled with a spatially uniform shading. To mitigate these restrictions we must, of course employ non-uniform shadings. For example in order to synthesize a controller that will control all modes of a simply supported plate, we introduce a shading that is a product of “ramp” functions in the two plate dimensions
(x) =
0 y x x y x 0 − 1 − x − lx 0 y 0 − 1 − 1− y − ly . 1− lx lx ly ly (3.62)
This distribution is plotted below in Fig. 3.17. The effective loading provided by this shading is given by its Laplacian components 1 x −1 − x − lx −1
xx (x) = x −2 − lx
0 y y 0 y − 1 − y − ly , × 1− ly ly
and
yy (x) =
x x x 0 − 1 − x − lx 0 1− lx lx 1 y −1 − y − ly −1 × y −2 − ly
(3.63)
(3.64)
The distribution described by (3.63) and (3.64) exerts a spatially distributed moment along the sides x = 0 and y = 0 in addition to point loads distributed along the boundaries. As we have seen previously, the net loading satisfies static equilibrium in accordance with the self reacting, strain induced film actuation. Using (3.63) and (3.64) along with (3.58) determines the power flow for the system with the shading distribution described in (3.62) as
3.4
Control System Synthesis for Plates
93
1.0
1.0
0.5
0.0 0.5
0.0
y
0.5 x
0.0 1.0
Fig. 3.17 Non dimensional “ramp” shading distribution
˙ = −Vmax u(t) F(t)
&
ly 0
' lx y x dy + dx . wxt |x=0 1 − wyt y=0 1 − ly lx 0 (3.65)
Note that (3.65) has only terms that involve angular velocity distributions along x = 0 and y = 0 thus no cancellation of component integrands occurs. In addition the line integrals of the angular velocity distributions are now weighted by linear “ramp” functions. It was demonstrated earlier that for the uniform spatial shading, mixed order modes were uncontrollable due to the vanishing line integrals, while strictly even modes were uncontrollable both because of asymmetry of the plate slopes at the boundaries and vanishing line integrals. The double ramp or “snow plow” shading of Fig. 3.17 can be expressed as a sum of both even and odd functions and thus even if the distribution of angular velocity along the boundaries has odd symmetry (about mid-span), the integrands of (3.65) will have remaining components and hence will be non-vanishing. This means that the shading distribution (3.62) guarantees controllability for the simply supported plate for all modes. Structural control based on Eq. (3.65) requires a measurement of the line integrals of the angular velocity distributions along the boundaries x = 0 and y = 0 for implementation. There are no commercially available transducers with this capability. In the next section we discuss employing a biaxial sensing distribution to achieve the requisite measurement.
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
3.4.3 The Unique Compatibility of Distributed Transducers for Arbitrary Spatial Shadings In this section we investigate the compatibility of spatially distributed smart material transducers for sensing and actuation. For the purposes of illustration we assuming a smart structure that can be generically represented as having a layer of biaxial active film bonded to a plate on the face opposite that with the actuator distribution. The film sensor will produce a voltage output that is proportional to the integrated strain on the plate’s surface according to Vsens = β
s (x)∇ 2 w dx
(3.66)
where β is a constant that will depend on plate and film geometric and constitutive properties and s (x) is a sensor shading distribution. We now employ the “second form” of Green’s theorem in the plane of vibration in accordance with Hildebrand [45] - ∂ψ ∂ϕ 2 2 ϕ −ψ d. (3.67) ϕ ∇ ψ − ψ ∇ ϕ dx = ∂n ∂n Equation (3.66) now becomes Vsens = β
w∇ s dx + 2
∂w d −
s ∂n
∂
d . w ∂n
(3.68)
The boundary conditions for a simply supported plate dictate that the displacement at the boundaries must vanish and hence the last integrals in (3.68) equal zero. Now choosing the sensor spatial shading function s to be the same as the actuator shading function (3.62) transforms the sensed output (3.68) into &
ly
Vsens (t) = β 0
' lx y x dy + dx . (3.69) wxt |x=0 1 − wyt y=0 1 − ly lx 0
With the exception of the time dependence, this is the desired sensed parameter for the Lyapunov vibration controller suggested in (3.65)! One need only differentiate the output from the shaded sensor distribution. This means that shaded sensor distributions can allow vibration damping of all modes of a plate using a single feedback channel. The required sensed parameter can only be measured using spatially distributed sensors. This result is true for any spatial shading, including those that control/sense targeted plate modes. The result also holds for any strain induced transducer. If a mode is not controllable with a chosen shading, then it is not observable using the same sensor shading. Because the Bernoulli-Euler smart structures discussed here are assumed open loop stable, all modes are detectable/stabilizable with
3.5
Summary
95
respect to any distribution. If an actuator distribution is suitable chosen such that all modes are controllable, then choosing a sensing shading with the same spatial shading distribution allows for the simple construction of a closed loop vibration controller for all modes of vibration. This is the unique compatibility of distributed transducers.
3.5 Summary Spatially varying shadings are simple spatial weightings that can be implemented on real physical systems. They can also provide insight into observability and controllability of smart structure systems that appeal to the designers’ intuition. The required states that must be measured in order to implement these Lyapunov controllers are also realizable. The techniques presented here require no modal truncation as the method is non-modal based. The approach described here provides insight and a design methodology to apply distributed film smart material transducers to nearly arbitrary Bernoulli Euler beam configurations in order to control the vibrations of all modes. The method can also be applied to Rayleigh and Timoshenko beam models as well [41]. In addition, techniques are presented for individual or group “mode targeting” for specialized applications. While the actuators given in the design examples have lumped parameter counterparts as indicated by their effective loading profiles, film distributions can readily be synthesized that have no discrete equivalent. The symmetry arguments presented here for both smart structure beam and plate analysis an synthesis can be extended to systems with more general boundary conditions. However it should be noted that there are some 21 combinations of boundary conditions for rectangular plates. The problem of a plate that is simply supported on all sides as presented here for the purposes of illustration is the simplest case to solve exactly. In actuality there a total of 6 combinations of rectangular plate boundary conditions for which 2 sides are simply supported and in which exact solutions can be obtained. The point being that closed form solutions exist for only a subset of all possible rectangular plate boundary conditions and numerical and or analytical approximations need to be used to analyze and synthesize the appropriate spatial shading distributions for observation and control.
Notes There have been other approaches to the design and synthesis of the required spatial shading needed for the vibration control of beams and plates using distributed transducers. Most notably is the work of Lee et al. [42], Isler and Washington [43] and Collins et al. [44]. In these methods the transducer spatial distribution is generally chosen such that its dimensions and distribution in a given direction are proportion to the plate curvature function. These techniques can be effective for mode targeting but fall short as a design methodology that provides insight into the selection of
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
spatial shading distributions, which can be effective and realizable for the control of many modes simultaneously.
Problems (3.1) a. Write a computer code to calculate and plot the mode shapes of a simply supported rectangular plate for at least the first eight modes. b. Modify your code from part a to plot the mode shapes of a clamped rectangular plate.
Fig. 3.1a Simply supported plate
(3.2) Consider the 2D normalized ramp distribution shown in Fig. 3.1a. This distribution is applied to a simply supported plate for the purpose of active vibration control. The non-dimensional governing equation for the system can be written as wtt + wxxxx = f (x,t) ,
x ∈ [0,1]
where w is the plate non-dimensional normal deflection, V is the nondimensional control voltage applied across the film, and D is the domain
Problems
97
Fig. 3.2a Spatial load distribution
of the system with boundary . The boundary conditions for the plate/film system are assumed to be homogeneous with respect to any (lumped or distributed) elements and with respect to the control voltage as well. The requisite boundary conditions for a simply supported plate are w (,t) = wxx (,t) = 0 a. Write the Lyapunov energy functional for this 2D system. Note: we may neglect all terms associated with any passive lumped boundary elements without loss of generality as they are separable from the active control design. b. Write the expression for the power flow for this system. c. Combine the governing equation and the power flow equation to obtain an expression for power flow that contains boundary integrals into which appropriate boundary conditions can be substituted. d. Substitute the boundary conditions given for a plate which is simply supported on all four edges, i.e. all for edges are pinned. e. Write a modified power flow expression of that obtained in part d, neglecting all terms which are independent of the control voltage and all terms which are conservative, i.e. which are not contributing to power flow into or out of the system. (3.3) We wish to design a vibration controller for a Bernoulli-Euler beam with clamped–clamped boundary conditions using a thin spatially distributed active film. The film distribution is applied to a beam whose non-dimensional governing equations are given as wtt + wxxxx − wxx = Vxx
x ∈ [0,1]
where w is the plate non-dimensional deflection and V = (x)φ(t) is the non-dimensional control voltage. A beam with clamped–clamped boundary conditions will have modes of vibration with either vanishing displacement or slope at its center, but never both.
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3 Active Vibration Control with Spatially Shaded Distributed Transducers
a. Sketch a spatial distribution, which will control all modes of the beam using the graph in Fig. 3.2a. b. Write an expression for the load intensity function (x) for the chosen spatial distribution. c. Sketch the effective loading profile given by this distribution. (Be careful to show the correct sign convention for positive and negative forces and moments.) d. Justify your answer in part a in detail. (Hint: The power flow for the system has the following proportionality F˙ ∼ V
max φ(t) wt 12 ,t +wxt 12 ,t
(3.4) The nth mode of a pinned–pinned beam has the form w (x,t) = A sin (nπ x) cos n2 π 2 t Assume that the instantaneous power flow over half a cycle that an active vibration controller removes is given by Econtrol, T = −2nπ AVmax 2
where w is the lateral beam displacement, n is the mode number, x is the length scale, and Vmax is the maximum control voltage. The inherent passive viscous damping in the beam is given by Eviscous, T = − 2
1 ηn2 π 2 A2 4
where η is the beam loss factor due to viscous damping. Derive the condition for the control voltage when the controller will remove more energy from the beam that the passive damping inherent in the beam. (3.5) Consider the governing LNS equations mentioned in question 1.9. Write out the Lyapunov functional for this system. (Hint: in most cases the disturbance kinetic energy is a natural choice for the measure of the disturbance size. Consider the exchange of energy with the base flow and energy dissipation due to viscous effects). (3.6) Consider the beam vibration suppression example of Sect. 3.3. Use the same procedure to synthesize a nonlinear vibration control law using the Lyapunov control function F(t) =
2
1 0
1 w2t dx
Problems
99
which contains only kinetic energy terms. How does the performance of this new control law compare with the control law presented in the chapter? (3.7) Consider the beam example presented in this chapter. Compare the performance of the distributed control law of Eq. (3.11) and the discrete control law of Eq. (3.15).
Chapter 4
Multi-Dimensional Transforms and MIMO Representations of Smart Structures
4.1 Introduction Earlier in Chap. 1 we introduced the concept of smart structures as self contained feedback control systems which are designed to meet certain performance goals. These goals may include but are not limited to “good”, stability, tracking, command following, disturbance rejection and noise attenuation. While in subsequent chapters we have demonstrated the elegance and utility of the Lyapunov method as a design synthesis technique for vibration control; the method has some notable limitations. The method as presented made extensive use of a governing partial differential equation. The method did not specifically incorporate or address performance measures to quantify tracking, disturbance rejection and robustness to model error and noise. Finally the method did not address synthesis for systems described by finite element analysis, state space representations, modal analysis or plant integral equations which form the basis for a large class of systems of interest. One approach to the design of feedback systems that meet the required metrics is that of frequency domain analysis and synthesis. Many of the modern robust control synthesis techniques such as H∞ optimal control theory have risen entirely as a result of “frequency domain thinking”. For smart structure applications that involve shape control or structural “morphing” the notion of spatial frequency or spatial bandwidth is tantamount to describing and achieving the desired spatial performance. This notion of spatial bandwidth is used routinely in acoustics and Fourier optics. The sensing apertures of hydrophones for example are typically transformed in space to reveal their spatial filtering characteristics in “steered” arrays. In optics deformable mirrors can be employed for image enhancement and real time wavefront correction or phase conjugation. In this case the control task can be posed in terms of correcting for a spatially limited basis set of wavefront aberrations e.g. power, focus, and astigmatism by changing the shape of the mirror in real time. A classical lumped parameter, time domain analysis will only address these performance metrics in a limited fashion and in many cases not at all. Transform methods can be applied to spatially distributed systems in order to characterize their spatial response, but few of them are suitable for incorporation
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_4,
101
102
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Multi-Dimensional Transforms and MIMO Representations of Smart Structures
into modern robust control analysis, design and synthesis techniques. In addition such methods must readily lend themselves to a reduction to practice and subsequent physical realization in order to be useful for smart structure design. Wavenumber transforms have been particularly effective in the describing the spatial response characteristics of certain classes of distributed parameter systems including spatially distributed transducers. Consider for example a “one-dimensional” sensor that has an aperture length, L. This sensing aperture may be represented using the classical “boxcar” aperture defined according to
(x) = h(x) − h(x − L),
(4.1)
where h(x) is a generalized function. The spatial frequency response of such a device is described by the Fourier transform of (4.1) i.e.
(k) =
i 1 − eikL k
(4.2)
where the variable k is called a wavenumber and represents the spatial frequency response of the aperture and is inversely proportional to spatial wavelength. The normalized boxcar aperture weighting and corresponding wavenumber response is illustrated in Figs. 4.1 and 4.2. The spatial frequency response of the aperture is seen to be constant for exogenous inputs whose wavelength is less than that of the sensing aperture. Note that when the incident wave has a wavelength that is equal to the sensing aperture, the average response across the sensor sums to zero and creates the first null response, giving rise to the concept of spatial bandwidth. Nulls at higher wavenumbers correspond to inputs that have spatial harmonics or periodicities which are integer multiples of the characteristic aperture length, L. The aperture can be seen to filter the spatial response or “roll off” as the wavenumber increases thus defining the transducer spatial filter characteristics. Fundamentally all transducers of finite dimension exhibit such spatial filtering characteristics. It should now become apparent that the spatial shading techniques presented in Chaps. 2 and 3 involved the design of transducers with desired spatial filtering properties. This is re-enforced
Λ(x)
Fig. 4.1 “Boxcar” aperture
4.1
Introduction
103
Fig. 4.2 Wavenumber response for the boxcar aperture
by observing that (4.1) is simply a special case of the shading representation given earlier in (2.31). All physical structures have inherent spatial filtering properties and characteristics e.g. string, beams, membranes and plates. Martin [45] for example has extended spatial filtering concepts to beams and plates by representing them as space-time filters which include both the spatial and temporal response characteristics of the system. The forced resonant response that we targeted for control in Chap. 3 using the Lyapunov controller is in fact a combination of temporal and spatial coincidence. For each resonant mode of a structure there is a corresponding spatial frequency or wavenumber response. Figure 4.3 is an experimentally determined plot of the spatial frequency response of the (9,1) mode of a rectangular plate that has simply supported/clamped boundary conditions as presented by Martin. The apparent “spatial resonance” characteristics are due to the spatial decomposition of the corresponding mode shape. The lobing in wavenumber k reflects the plates “spatial aperture” or characteristic length scale. Observe that the wavenumber response above the “resonant” wavenumber rolls off with increasing k. As we observed earlier, this is determined by the structures boundary conditions and strain curvature relation. The forced resonant response of a structure is a combination of temporal and spatial coincidence, without such a coincidence there will be no response. Consider for example the response of a beam to a discrete excitation at a point, with the exciter being driven at some temporal resonant frequency of the beam but physically located at a spatial node of the resonance. In this case there is a temporal coincidence but no spatial coincidence and thus the beam does not respond. Each mode has both a preferred temporal frequency and a spatial wavelength associated with it. This is illustrated by the Fourier integral wavenumber/frequency response function for a non-dimensional simply supported beam as shown in Fig. 4.4.
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Fig. 4.3 Wavenumber acceptance of S-C-S-C plate’s (9,1) mode after Martin [45]
This three dimensional frequency response shows that the forced response of this distributed parameter system is indeed a combination of both temporal and spatial phenomena. At constant wavenumber we encounter the resonant peaks of the beam response as we move along in temporal frequency, the so-called modal response. At constant temporal frequency we encounter the system nulls as we move along in wavenumber. The locus of peaks follows the beams dispersion relation
Amplitude
Wavenumber Frequency Fig. 4.4 Wavenumber/frequency transform of a non-dimensional simply supported beam
4.2
Convolution and the Spatially Distributed Plant
105
which is essentially the space/time characteristic equation and defines the systems space/time filter characteristics or transfer function. Wavenumber transforms are useful in describing the spatial response of smart structures. In the sections that follow we develop and present an input /output representation of the smart structure in a temporal and spatial frequency domain. This representation can readily be augmented with spatially distributed transducers, as presented in earlier chapters, to construct a Multiplicative Multi-Input Multi-Output or MIMO relation. With this MIMO formulation modern well developed tools for the synthesis of MIMO based control synthesis and analysis can be employed. We can also combine this formulation with elements of modern MIMO control analysis to develop rigorous performance measures for smart structure control systems thereby supplementing the perceived shortfalls of the single input single output Lyapunov approach presented earlier.
4.2 Convolution and the Spatially Distributed Plant Smart structures can be designed as multi-input multi-output control systems. Often in the design of such systems it is necessary to employ many discrete and/or spatially distributed transducers in order to achieve specific performance goals. Choosing the placement, type and shading of such transducers allows the engineer to synthesize a MIMO control architecture. The analytical tools of modern MIMO control analysis are primarily focused on lumped parameter, linear time invariant systems and rarely address the issue of spatial performance. The spatially distributed nature of the system is typically ignored, neglected or treated using heuristic ad hoc methods. There are many smart structure systems whose performance goals must be specified in spatial terms e.g. morphing structures or structures that need to achieve radical shape changes under closed loop control. Modern techniques incorporate time domain information using transform methods such as the Fourier or Laplace transform, but do not extend such methods to include spatial performance. In this section we develop techniques for analyzing smart structures as distributed parameter control systems using an input/output plant representation in a spatially and temporally transformed frequency space. The classes of distributed parameter systems which readily lend themselves to such analysis are described in terms of their Green’s functions. It is well known that linear time invariant systems can be completely characterized in terms of their impulse response. This is due in part to the fact that an impulse can be represented as a superposition of infinite sinusoids. The response therefore of a linear time invariant system to an impulse is a broadband response spanning all possible frequencies thus exciting the linear system in such a manner as to reveal its full harmonically rich profile. This is illustrated graphically in Fig. 4.5 where h(t) is defined as the so-called impulse response function. For linear systems we can use the shifting theorem to show that the response of the system to a time shifted impulse δ (t − τ1 ), is simply
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Multi-Dimensional Transforms and MIMO Representations of Smart Structures
Fig. 4.5 Response of an LTIS to an impulse
δ (t)
LTI
h (t)
a time shifted response function h(t − τ1 ). Also for Linear Time Invariant Systems (LTIS), the laws of superposition and scaling can be used to show that the response of the system to a weighted sum of impulses is simply a weighted sum of impulse response functions vis x(t) =
Input:
/
δ (t − τi ) x (τi )
i
Output:
y(t) =
/
.
(4.3)
x (τi ) h (t − τi )
i
The summation of impulse responses in (4.3) can be viewed as a convolution sum. In the limit as the spacing between samples (i) becomes infinitely small, the output relation becomes exact and turns into a convolution integral, y (t) =
∞ −∞
x(τ ) h (t − τ ) dτ .
(4.4)
The convolution in (4.4) may be thought of as a filtering operation on the input x (τ ) where the impulse response function h (t − τ ) determines which properties of the input are filtered; in this case h (t − τ ) acts as a temporal filter. There is an analogy in distributed parameter systems to the impulse response function h (t − τ ) called the Green’s function and it provides us with a means to design space/time filters. Thus far we have chosen to describe smart structures as distributed parameter systems whose governing behavior can be represented by partial differential equations with the appropriate initial and boundary conditions. A more convenient alternative is the use of integral equations to analyze problems often described with differential equations. Integral equations contain all of the necessary information about the distributed parameter plant and include initial and boundary conditions. We will use an integral approach which characterizes distributed parameter plants by their Green’s functions. For a distributed parameter system the exogenous input excitation u (x,t) and corresponding system response y (x,t) are generally related via the so-called composition integral which has the form y (x,t) =
∞ −∞
u (ξ ,τ ) h (x,ξ ,t,τ ) dξ dτ
(4.5)
where u (x,t) and y (x,t) are scalar signals spatially distributed over the domain . The kernal of the integral operator h (x,ξ ,t,τ ) is the plant Green’s function.
4.2
Convolution and the Spatially Distributed Plant
107
Analogous to the impulse response function of the time domain, the Green’s function as employed here is the response of the plant at location x and time t ≥ 0, to an impulse input applied at location ξ and time τ ≥ 0. If the composition integral (4.5) is linear, then the laws of superposition and scaling can be used to show that the response of the system to a weighted superposition of spatially distributed impulses is simply a weighted superposition of Green’s functions. It should also be noted here that while the response of the system to a specific input is unique, the specific form of the Green’s function is not. Equation (4.5) can always be made homogenous with respect to its initial and boundary conditions by suitable standardizing functions [46]. A direct extension of (4.4) to the spatial domain can be achieved by assuming that the system is stationary and thus (4.5) assumes a temporal convolution form and becomes y (x,t) =
∞ −∞
u (ξ ,τ ) h (x,ξ ,t − τ ) dξ dτ .
(4.6)
For a large class of distributed parameter systems, the Green’s function can be manipulated such that the space/time Fourier transform of (4.6) yields a multiplicative input/output relation in the transformed space as y (k,ω) = h (k,ω) u (k,ω) ,
(4.7)
where k is the spatial transform variable or wavenumber, ω is the temporal frequency, y (k,ω) is the transformed output, u (k,ω) the transformed input and h (k,ω) is the space/time response function. Now the distributed parameter plant (e.g. smart structure) can be clearly seen to act as a space/time filter which filters the spatially distributed input u. Once put in the form (4.7) extensions to modern MIMO control analysis and synthesis can be used to design modern robust smart structure architectures. In the next section we define the class of systems for which the multidimensional transform model is applicable.
4.2.1 Green’s Function Representations for Temporally Stationary Systems Throughout this section it is assumed that the temporal properties of linear distributed parameter plants of interest are stationary and thus the Green’s function has the general form h (x,ξ ,t − τ ). As demonstrated, this allows the associated composition integral to have a temporal convolution form and thus a product form in temporal frequency ω in the Fourier transformed space. This also requires that the plants Green’s functions be Fourier-transformable and thus square integrable. As discussed earlier in Chap. 2, a function is square-integrable on an interval if the integral of the square of its absolute value, over that interval, is finite. The set of all measurable functions that are square-integrable form the so-called L2 space.
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Similarly the input and output signals are assumed square integrable and only continuous Fourier spatial transforms are considered. In addition we assume that the Green’s functions of interest can be static or dynamic and the inputs and outputs that appear in the composition integrals can be vectors with matrix kernals. We now consider the following Green’s function forms: Right Convolution Form This is the case when the Green’s function takes on the form h (x,ξ ,t − τ ) = h (x − ξ ,t − τ )
(4.8)
and the associated composition integral becomes y (x,t) =
∞ −∞
u (ξ ,τ ) h (x − ξ ,t − τ ) dξ dτ
(4.9)
i.e. a convolution in space and time. Note that the kernal of (4.9) is a function of the difference between the spatial variables i.e. x − ξ and not the absolute location in the plant or on the structure and thus this form of the Green’s function also exhibits a type of spatial and temporal stationarity. This form of the spatial convolution is commonly encountered in the theory of Fourier optics. The form of the Green’s function depends only on the sum and/or difference of the input/output spatial parameters and as such is termed an isoplanatic convolution. When (4.9) is Fourier transformed in space and time it assumes a multiplicative form in wavenumber/frequency space, y (k,ω) = h (k,ω) u (k,ω) .
(4.7)
Right and Left Convolution Form Some distributed parameter plants will admit Green’s functions which are sums of right and left convolution forms e.g. h (x,ξ ,t − τ ) = h0 (x − ξ ,t − τ ) ± h0 (x + ξ ,t − τ ) .
(4.10)
In a spatially bounded distributed parameter system such as a smart structure of finite dimensions, the response (4.10) at any point in space can be affected by information on both sides of it due to the nature of the excitation, boundary conditions etc. Information propagates in multiple directions and the x + ξ contribution is sometimes referred to as anti-causal in analogy with lumped parameter systems. The corresponding composition integral becomes y (x,t) =
∞ −∞
u (ξ ,τ ) [h0 (x − ξ ,t − τ ) ± h0 (x + ξ ,t − τ )] dξ dτ .
(4.11)
4.2
Convolution and the Spatially Distributed Plant
109
When (4.11) is transformed in space and time the first term of the integral becomes ⎧ ⎫ ⎨ ∞ ⎬ h0 (x − ξ ,t − τ ) u (ξ ,τ ) dξ dt = h0 (k,ω) u (k,ω) (4.12) ⎩ −∞ ⎭
i.e. a multiplicative form in wavenumber frequency space. Similarly the second term of the transformed integral is
⎧ ⎨ ⎩
∞ −∞
h0 (x + ξ ,t − τ ) u (ξ ,τ ) dξ dt
⎫ ⎬ ⎭
= h0 (k,ω) u∗ (k,ω)
(4.13)
where the (◦)∗ asterisk denotes the complex conjugate. We can combine the results of (4.12) and (4.13) to give a multiplicative input/output relation in the transformed space for (4.11) as y (k,ω) = h0 (k,ω) u (k,ω) ± u∗ (k,ω) .
(4.14)
This can be rewritten in a more compact form as y (k,ω) = h0 (k,ω) ueffective (k,ω) .
(4.15)
Example: Green’s Function for a String Consider the Green’s function for a string of length L which has the form
h (x,ξ ,ω) =
∞ an sin n=1
nπ x L ω2
+ c2
nπ ξ sin L nπ 2
(4.16)
L
where c is the wave speed and whose Green’s function is observed to be an expansion in orthogonal functions. Equation (4.16) can be written in Right and Left Convolution Form by considering its term by term expansion where each term can be written as 3 nπ 4 nπ ξ sin L L an cos nπ L (x − ξ ) − cos L (x + ξ ) = . 2 2 2 ω2 + c2 nπ ω2 + c2 nπ L L
an sin
nπ x
The transformed response function of (4.17) will have components
(4.17)
110
4
Multi-Dimensional Transforms and MIMO Representations of Smart Structures ⎫ ⎧ ⎨ an sin nπL x sin nπL ξ ⎬ = 2 ⎭ ⎩ ω2 + c2 nπ L
⎧ ⎪ ⎨ an 4
⎪ ⎩
⎡ ⎢ (−1)n sin (kL) + i ⎣
⎤ ⎤⎫ ⎡ ⎬. n n ⎪ 1 1 ⎥ cos − cos (kL) (−1) (kL) (−1) ⎥ ⎢ + − + i ⎦ ⎦ ⎣ nπ nπ nπ nπ ⎪ ⎭ k− k+ k− k+ 2 2 2 2 nπ 2 ω 2 + c2 L (4.18)
Now h (k,ω) may be written in the form (4.15) using a superposition of these components. Damped Convolution Form In this case the Green’s function takes the form h (x,ξ ,t − τ ) = h0 (x − i ξ ,t − τ ) ± h0 (x + i ξ ,t − τ )
(4.19)
with the associated composition integral y (x,t) =
∞ −∞
u (ξ ,τ ) [h0 (x − iξ ,t − τ ) ± h0 (x + iξ ,t − τ )] dξ dτ .
(4.20)
The space/time transform of the first component of (4.20) is defined using a modified convolution theorem to yield the result
⎧ ⎨ ⎩
∞ −∞
h0 (x − iξ ,t − τ ) u (ξ ,τ ) dξ dt
⎫ ⎬ ⎭
= ih0 (k,ω) u (ik,ω)
(4.21)
and in a similar manner the second component yields
⎧ ⎨ ⎩
∞ −∞
h0 (x + iξ ,t − τ ) u (ξ ,τ ) dξ dt
⎫ ⎬ ⎭
= ih0 (k,ω) u∗ (ik,ω) .
(4.22)
Now these may be combined to give the compact form of the desired multiplicative input/output relation in wavenumber/frequency space, y (k,ω) = h0 (k,ω) iu (k,ω) ± iu∗ (k,ω) = h0 (k,ω) ueffective (k,ω) .
(4.23)
4.2
Convolution and the Spatially Distributed Plant
111
Example: Green’s Function for a Bernoulli-Euler Beam As an example of the Damped Convolution Form of the Green’s function consider the Green’s function for a free–free Bernoulli-Euler beam which contains terms of the form an sin (μn x) cos (μn ξ ) . ω2 + c2 μ4n
(4.24)
Thru the use of the trigonometric identity cosh (a) = cos (ia) we can re-write (4.24) as an sin (μn x) cos (μn ξ ) an {sin [μn (x − iξ )] + sin [μn (x + iξ )]} = . 2 ω2 + c2 μ4n ω2 + c2 μ4n
(4.25)
Again these terms may be combined to give the compact form (4.23) of the desired multiplicative input/output relation in wavenumber/frequency space. Bernoulli-Euler beams as well as plates with nearly arbitrary boundary conditions will have Green’s functions of the forms defined thus far. Consequently these transform methods can be applied to a wide range of smart structures for control synthesis and analysis. The Symmetric Form of the Green’s Function When the input and output spatial arguments of the Green’s function are interchangeable as depicted in Eq. (4.26), the Green’s function is said to be symmetric
h (x,ξ ,t − τ ) = h (ξ ,x,t − τ )
(4.26)
This symmetry is a consequence of the property of reciprocity in linear time invariant systems. In this case reciprocity states that the relationship between an applied impulsive load and the resulting response function is unchanged if one interchanges the locations where the impulse is applied and where the response is measured. Symmetric Green’s functions can be found in systems which are described using self-adjoint partial differential operators. A self-adjoint operator is one which is equal to its own conjugate transpose, e.g. x = x∗ . For example Bernoulli-Euler structures exhibit the property of self-adjointness as they are a part of a broad class of systems known as Sturm-Liouville systems. In mathematics the classical Sturm-Liouville equation is a (real) second order differential equation which has the general form −
dY (x) d T (x) + q (x) y = λw (x) y dx dx
(4.27)
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4
Multi-Dimensional Transforms and MIMO Representations of Smart Structures
where y is a function of the free variable x and the functions p (x) > 0, q (x) , and w (x) > 0 are specified and are generally continuous throughout the domain. The function Y is typically required to satisfy the system boundary conditions and w (x) is a weighting function. Finding the values of λ for which there exists a non-trivial solution of (4.27) that satisfies the boundary conditions form a part of the so-called Sturn Liouville or eigenvalue problem. The corresponding solutions (for each value of λ) are the eigenfunctions of the problem. Symmetric Green’s functions can always be expressed in terms of the systems eigenvalues and eigenfunctions as a bilinear Hilbert-Schmidt expansion or bilinear expansion, vis h (x,ξ ,ω) =
N ϕi (x) ϕi (ξ ) , λi (ω)
(4.28)
i=1
where for simplicity the temporal transform has already been applied. Here ϕ i are the eigenfunctions of the kernel h, and λi (ω) the corresponding eigenvalues as discussed above. It is important to note that for self-adjoint distributed parameter systems, eigenfunctions of h are the same as the system eigenfunctions and eigenvalues. Also the summation in (4.28) may often have the index N → ∞. The bilinear form of the Green’s function only requires that it be symmetric and is applicable to a large number of structures. Butkovskiy [47] has published a tabulation of more than 500 bilinear expansions for a number of distributed parameter systems. The composition integral (4.5) corresponding to the bilinear expansion (4.28) becomes y (x,ω) =
N ϕ (x) ϕ (ξ ) / i i u (ξ ,ω) dξ λ (ω) i i=1
N ϕ (x) / i = ϕi (ξ ) u (ξ ,ω) dξ λ i=1 i (ω)
.
(4.29)
If the input to the system is now assumed to be degenerate according to our definition given earlier in (2.16) then it takes the special separable form u (ξ ,ω) = q (ξ ,ω) u (ω)
(4.30)
where q (ξ ,ω) can represent the most general case of a time varying spatial shading or morphing function as well as any inherent dynamics associated with the spatially distributed transducer. For MIMO systems the input may also be a superposition of exogenous inputs from multiple actuators. Combining (4.29) and (4.30) yields y (x,ω) = u (ω)
N ϕi (x) i=1
λi (ω)
ϕi (ξ ) q (ξ ,ω) dξ .
(4.31)
If we restrict the exogenous inputs to those actuator distributions to those that interact with the system via temporal control signals only (in accordance with the
4.3
Multi-Input Multi-Output (MIMO) Representations of Smart Structures
113
spatially degenerate transducers introduced in Chap.2), then in (4.31) the integral term becomes purely a function of ω for a given (fixed) spatial shading and thus we can write (4.32) ai (ω) ≡ ϕi (ξ ) q (ξ ,ω) dξ
and therefore y (x,ω) = u (ω)
N i=1
ai (ω)
ϕi (x) . λi (ω)
(4.33)
As before to obtain the multiplicative form of the input/out relation in the transformed space we have &
' N ϕi (x) ϕi (k) y (k,ω =) u (ω) ai (ω) ai (ω) = u (ω) = h (k,ω) u (ω) λi (ω) λi (ω) i=1 i=1 (4.34) and again we have the compact form (4.34) of the desired multiplicative input/output relation in wavenumber/frequency space for this class of systems. N
4.3 Multi-Input Multi-Output (MIMO) Representations of Smart Structures The multidimensional frequency response functions developed in this chapter clearly demonstrate that the response of smart structures as distributed parameter systems are a combination of both temporal and spatial phenomena. The Green’s function forms part of multiplicative input/output relation in a transformed wavenumber/frequency space. For a scalar Green’s function, this relationship will also be scalar. For MIMO systems we can construct a multiplicative MIMO relation in the transformed space. This will allow the smart structure design engineer to exploit the power of modern MIMO control synthesis and analysis tools for improved performance. Consider the input to multidimensional smart structure systems to consist of a superposition of N inputs, with each input shaded or filtered in space and time. We represent these inputs according to (4.30) by u (k,ω) =
N
qi (k,ω) ui (k,ω)
(4.35)
i=1
where qi (k,ω) is the space/time transform of each actuator shading function and its associated dynamics. For degenerate actuators which have fixed spatial shading and whose temporal dynamics do not vary over its spatial aperture (4.35) takes the special form
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Multi-Dimensional Transforms and MIMO Representations of Smart Structures
qi (k,ω) = qi (k) γi (ω) .
(4.36)
Similarly we can represent a superposition of M sensor outputs as y (k,ω) =
M
pj (k,ω) yj (k,ω)
(4.37)
j=1
and for the case of degenerate sensors (4.37) assumes the special form pj (k,ω =) pj (k) ϑi (ω)
(4.38)
For the smart structure as a spatially distributed system with N actuators and M sensors convolved with the plants response function, each sensor’s output in the transformed wavenumber/frequency space becomes y1 (k,ω) =
p1 (k,ω) h (k,ω) q1 (k,ω) u1 (k,ω) + ···
.. . yM (k,ω) =
p1 (k,ω) h (k,ω) qN (k,ω) uN (k,ω)
.. . pM (k,ω) h (k,ω) q1 (k,ω) u1 (k,ω) + ···
.
(4.39)
pM (k,ω) h (k,ω) qN (k,ω) uN (k,ω)
Equation (4.39) may be written in a more compact form as a matrix multiplicative MIMO relation ⎡
y1 (k,ω)
⎤
⎡
p1 (k,ω) h (k,ω) q1 (k,ω) · · · p1 (k,ω) h (k,ω) qN (k,ω)
⎤ ⎡
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ y2 (k,ω) ⎥ ⎢ p2 (k,ω) h (k,ω) q1 (k,ω) · · · p2 (k,ω) h (k,ω) qN (k,ω) ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ .. .. ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ . . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣ yM (k,ω) ⎦ ⎣ pM (k,ω) h (k,ω) q1 (k,ω) · · · pM (k,ω) h (k,ω) qN (k,ω) ⎦
u1 (k,ω)
⎤
⎢ ⎥ ⎢ ⎥ ⎢ u2 (k,ω) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ uN (k,ω) ⎦
(4.40)
or more compactly as y (k,ω) = G (k,ω) u (k,ω) .
(4.41)
Equation (4.41) is the desired MIMO representation of a distributed parameter system. Recall from Chap. 1 that smart structures are distributed parameter control systems and in the context of (4.41) can be represented in block diagram form as shown
4.3
Multi-Input Multi-Output (MIMO) Representations of Smart Structures d i (k,ω) r(k,ω)
e(k,ω) + –
d 0 (k,ω)
up K(k,ω)
115
yp G(k,ω)
uc
y 0 (k,ω)
n 0 (k,ω)
Fig. 4.6 The smart structure as a MIMO distributed parameter system
in Fig. 4.6 where G (k,ω), represents the “Plant” response matrix or physical system to be controlled, e.g. a flexible wing, bridge, component beam etc. Because we have chosen to include actuator and sensor dynamics in the form of qi (k,ω) and pi (k,ω) in the plant response matrix, we will hereafter refer to G (k,ω) as the augmented plant response matrix. ⎡ ⎤ yp1 (k,ω) ⎢ yp2 (k,ω) ⎥ ⎢ ⎥ The plant output is yp , in the MIMO representation where yp = ⎢ ⎥ .. ⎣ ⎦ . ypM (k,ω) ⎤ up1 (k,ω) ⎢ up2 (k,ω) ⎥ ⎢ ⎥ up = ⎢ ⎥ and contains control .. ⎣ ⎦ . ⎡
up is the plant input vector where
upN (k,ω) commands K (k,ω) represents the controller or compensator gain matrix and our goal in part is to learn how to design K (k,ω) to ⎡ meet specified ⎤ performance. uc1 (k,ω) ⎢ uc2 (k,ω) ⎥ ⎢ ⎥ uc , is the input or “control vector”, uc = ⎢ ⎥ generated by the con.. ⎣ ⎦ . ucN (k,ω) ⎡ ⎤ r1 (k,ω) ⎢ r2 (k,ω) ⎥ ⎢ ⎥ troller partially in response to a reference command vector r = ⎢ ⎥ which .. ⎣ ⎦ . rm (k,ω) reflects the desired behavior of the structure. Finally ⎡ y 0 is the ⎤actual output or behavy01 (k,ω) ⎢ y02 (k,ω) ⎥ ⎢ ⎥ ior of the structure as represented by y0 = ⎢ ⎥ . As discussed earlier .. ⎣ ⎦ . y0 M (k,ω) we will generally seek to achieve a robust performance in the presence of noise, disturbances and model uncertainties. This is reflected by the inclusion of sensor
116
4
Multi-Dimensional Transforms and MIMO Representations of Smart Structures
⎡ ⎢ ⎢ noise, n0 = ⎢ ⎣
n01 (k,ω) n02 (k,ω) .. .
⎤ ⎥ ⎥ ⎥ which is associated with the measurement of the sys⎦
n0 M (k,ω) tem 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. The distributed nature of the system is evident in that both spatial and temporal dynamics are reflected throughout the entire system. The form presented in (4.41) and Fig. 4.6 allows all of the analytical tools of modern control system design and analysis to be exploited e.g. • • • • • •
Singular Value Decomposition Performance Robustness Metrics Stability Robustness MATLAB SIMULINK LQG/LTR, H∞ , μ -synthesis . . .
In subsequent chapters of this text we will explore these tools in considerable detail as applied to smart structure design, synthesis and analysis. If the system elements i.e. sensors, actuators etc. were not spatially distributed but instead traditional lumped parameter elements that were independent of wavenumber k, then (4.41) reduces to the familiar time domain matrix response y (ω) = G (ω) u (ω) .
(4.42)
It is well known that Eq. (4.42) can be written in the form y (ω) =
1 G (ω) u (ω) (ω) ( )* + ( )* + Matrix
(4.43)
Scalar
where (ω) contains information about the poles of G (ω) and is also the determinant relation which defines the systems characteristic equation. The transfer matrix G (ω) describes the interaction of the system inputs and outputs. There is an equal analogy for the distributed parameter representation as we can rewrite (4.41) in the form y (k,ω) = h (k,ω) G (k,ω) u (k,ω) ( )* + ( )* + Scalar
(4.44)
Matrix
where h (k,ω) contains information about the distributed temporal and spatial poles of the system. The transfer matrix G (k,ω) describes the space/time interaction of the system inputs and outputs. The invariant scalar in (4.44) will be the inverse of
4.3
Multi-Input Multi-Output (MIMO) Representations of Smart Structures
117
the plants dispersion relation relating its temporal and spatial poles in the manner discussed earlier in this chapter defining the temporal and spatial coincidences (See Fig. 4.4 for example) which define the systems resonances.
Example: String on an Elastic Foundation We illustrate the MIMO relations developed in this section using the simple example of an infinite string on an elastic foundation shown in Fig. 4.7. In this system we have two degenerate point displacement sensors located spatially at x 1 and x 2 along with two degenerate force actuators at locations x 3 and x 4 . The lateral string displacement is determined by y (x,t) and the governing equation T
∂ 2y ∂ 2y − m 2 − κ y = u (x,t) 2 ∂x ∂t
(4.45)
where m is the mass density of the string, κ is the foundation stiffness, u (x,t) the systems exogenous input and T is the tension in the string. The space/time transform of (4.45) is given by −T k2 + mω2 − κ y (k,ω) = u (k,ω)
(4.46)
where we have employed the Fourier integral space/time transform ∞ ∞ y (k,ω) =
ei ( kx+ωt) y (x,t) dx dt.
(4.47)
−∞ −∞
The plants space/time response or Greens function is thus h (k,ω) =
1 . −Tk2 + mω2 − κ
(4.48)
In accordance with Eq. (4.40) we wish to augment (4.48) with the appropriate sensor and actuator filter functions to obtain the augmented plant response matrix. For the degenerate point actuators these filter functions take the form
y
x = x1
x = x2 x = x3
Sensor
Sensor
Actuator Actuator
x
...
x = x4
y = y(x,t)
Base Rigid Fig. 4.7 Infinite string on an elastic foundation
118
4
Multi-Dimensional Transforms and MIMO Representations of Smart Structures
q1 (k,ω) = e ikx3 γ1 (ω) (4.49) q2 (k,ω) = e ikx4 γ2 (ω) and we can write u (k,ω) in (4.46) as u (k,ω) = e i kx3 γ1 (ω) u1 (ω) + e i kx4 γ2 (ω) u2 (ω) .
(4.50)
Similarly the degenerate point displacement sensors take the form p1 (k,ω) = e ikx1 θ1 (ω) .
(4.51)
p2 (k,ω) = e ikx2 θ2 (ω) Combining the results of (4.48), (4.49), (4.50), and (4.51) yields the desired MIMO input/output relation: ⎡
⎤ e i kx1 θ1 (ω) e i kx3 γ1 (ω) e i kx1 θ1 (ω) e i kx4 γ2 (ω) ⎡ ⎤ ⎢ ⎥ u1 (ω) y1 (ω) −Tk2 + mω2 − κ ⎢ −Tk2 + mω2 − κ ⎥ ⎥ ⎣ ⎦=⎢ ⎦. ⎣ ⎢ ⎥ i kx i kx i kx i kx 2 3 1 4 ⎣ ⎦ y2 (ω) u2 (ω) θ2 (ω) e γ1 (ω) e θ2 (ω) e γ2 (ω) e 2 2 2 2 −Tk + mω − κ −Tk + mω − κ ( )* + ⎤
⎡
≡ G(k,ω)
(4.52) We can also rewrite this in the special form (4.44) as ⎡ ⎣
y1 (ω) y2 (ω)
⎤ ⎦=
1 −Tk2 + mω2 − κ ( )* + Scalar
⎡ i kx1 ⎤ e θ1 (ω) e i kx3 γ1 (ω) e i kx1 θ1 (ω) e i kx4 γ2 (ω) ⎡ ⎤ ⎢ ⎥ u1 (ω) ⎢ ⎥ ⎣ ⎦ ×⎢ ⎥ ⎣ i kx2 ⎦ u2 (ω) e θ2 (ω) e i kx3 γ1 (ω) e i kx1 θ2 (ω) e i kx4 γ2 (ω) (
)*
+
Matrix
(4.53)
where the scalar term is the inverse of the plants dispersion relation relating its temporal and spatial poles as a continuous locus of resonances along two wavenumber branches defining the temporal and spatial coincidences.
Problems
119
Problems (4.1) The equation of motion for the torsional vibration of a circular shaft is given by JGφxx (x,t) − JQφtt (x,t) = τ (x,t) where J is the polar moment of inertia, G is the shear modulus, φ is the rotation angle, Q is the density of the shaft, and τ is the applied torque. Determine this plants space/time response function or Green’s function. (4.2) a. Derive the Green’s function for the 1D wave equation on an unlimited domain, i.e. a string. b. Derive the 2D Green’s function for a membrane with circularly clamped boundary conditions. (4.3) Consider the transverse vibration of a normalized free–free beam having the governing equation wtt + wxxxx = f (x,t) and boundary conditions wxx (0,t) = wxx (1,t) = 0 wxxx (0,t) = wxxx (1,t) = 0 This system has the Green’s function
h (x,ξ ,t) = 4
∞
φn (x) φn (ξ ) n=1
μ2n φn2 (1)
sin μ2n t
which describes the response of the beam at t and x to an impulsive loading at ζ, where μn are the Eigenvalues and φn (x) are the Eigenfunctions. a. Generate plots showing the excitation of the system with discrete impulses at the center and tip of the structure. b. Use the Green’s function to form the composition integral for an arbitrary forcing and simulate the system to a force doublet at the tip of the beam. (4.4) Generate a bode magnitude surface showing the displacement response of the system in question 4.3 as a function of both forcing frequency and wave number. (4.5) Write a computer code to plot the 1D wave equation as a function of frequency f and wave number κ, i.e. generate the 3D Bode plot.
Chapter 5
Performance Measures for Smart Structures with MIMO Representations
5.1 Introduction Thus far we have explored the concept of smart structures as self contained feedback control systems designed to meet specific, predetermined performance goals. These goals included such metrics as “good” stability, tracking, command following, disturbance rejection and noise attenuation. In this chapter we will develop rigorous criteria for achieving both spatial and temporal performance as related to the above metrics. The criteria are presented in the context of smart structures as distributed parameter systems which have Multi-Input Multi-Output representations as illustrated using the transform methods of Chap. 4. More specifically the performance measures presented herein represent the logical extensions of modern robust multivariable control techniques to spatially distributed systems. The measures developed are synergistic with and applicable to both discrete and continuous systems. We will combine the design methods presented thus far in discrete and distributed sensors and actuators, using continuous and discrete spatial transforms.
5.2 Performance Metrics We began with the smart structures architecture of Fig. 5.1. Here K(k,ω) represents the spatially distributed MIMO compensator matrix which is convolved with the physical structure whose space/time filter characteristics are given by the distributed plant G(k,ω). Similarly are shown the space/time d (k,ω) r (k,ω)
e (k,ω)
K (k,ω)
G (k,ω)
y (k,ω) n (k,ω)
Fig. 5.1 Smart structure topology
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_5,
121
122
5 Performance Measures for Smart Structures with MIMO Representations
transformed reference input, disturbance, plant output and sensors noise, r (k,ω), d (k,ω), y (k,ω), n (k,ω) respectively. Here we have assumed that any and all disturbances which occur at the plant input are simply reflected thought the plant and are incorporated in the output disturbance as shown. Similarly the reader is reminded of the fact that the plant may incorporate spatially distributed sensor and actuator characteristics as represented by the fact that G (k,ω) = h (k,ω) p (k,ω) q (k,ω) as discussed in considerable detail in the previous chapter. Using conventional block diagram algebra the transfer function from the error signal e (k,ω) to the plant output y (k,ω), in the absence of disturbances, can be obtained as; T (k,ω) = G (k,ω) K (k,ω)
(5.1)
with T (k,ω) defined as the “loop transfer matrix”. The closed loop input/output relation containing T (k,ω) is now given as; y (k,ω) = [1 − T (k,ω)]−1 d (k,ω) + [1 − T (k,ω)]−1 T (k,ω) [r (k,ω) − n (k,ω)] . (5.2) Equation (5.2) may be written in a more compact form by adapting a few parametric definitions, thus we define the Sensitivity Transfer Matrix as, S (k,ω) = [1 + T (k,ω)]−1
(5.3)
and similarly the Closed Loop Transfer Matrix as, C (k,ω) = S (k,ω) T (k,ω) .
(5.4)
Now (5.2) can be re-written as y (k,ω) = S (k,ω) d (k,ω) + C (k,ω) [r (k,ω) − n (k,ω)] .
(5.5)
At this point the closed loop input/output relation for the controller as given by (5.5) allows us to develop and investigate our requisite performance metrics as discussed earlier. An output error e0 (k,ω) can be defined as the difference between a reference command input r (k,ω) and the control system output y (k,ω) vis. e0 (k,ω) = r (k,ω) − y (k,ω) .
(5.6)
Combining (5.6) and (5.5) yields e0 (k,ω) = S (k,ω) [r (k,ω) − d (k,ω)] + C (k,ω) n (k,ω) .
(5.7)
Equation (5.7) is important to performance in that it allows us to explore the impact of noise on how well the controller accomplishes command following [48]. We are now in a position to define and quantify performance in terms of spatially distributed metrics.
5.2
Performance Metrics
123
1) Good Command Following For good command following the smart structure as a spatially distributed controller must be designed such that y (k,ω) ∼ = r (k,ω) ;
ω ∈ r , k ∈ Kr ,
(5.8)
where as previously discussed r and Kr represent the temporal and spatial domains of interest, as commanded by the reference r (k,ω). We desire good command following over specified “spatial bandwidths” and so the frequency and wavenumber bands r and Kr are indeed those in which r (k,ω) will have significant energy content. From Eq. (5.5) “good command following” in the absence of noise and disturbances is achieved when C (k,ω) ∼ = I ; ω ∈ r ,
k ∈ Kr .
(5.9)
2) Good Tracking For “good tracking” the output error expressed in (5.7) must be “small” and thus this places limits on the parameters S (k,ω) in accordance with (5.4), hence S (k,ω) ∼ = [0] ; ω ∈ r , k ∈ Kr .
(5.10)
As can be seen in (5.3) this is achieved when the loop transfer matrix T (k,ω) is “large”. The exact nature and meaning of the subjective terms “small” and “large” will be developed later in this chapter. 3) Good Disturbance Rejection Disturbance rejection, according to (5.5) requires that S (k,ω) ∼ = [0] ; ω ∈ d , k ∈ Kd .
(5.11)
where now we are concerned with the domains in which the wavenumber and frequency bands of the disturbance have significant energy content. This also has an impact on the output error defined in (5.7) in that good tracking and disturbance rejection yield small output error. 4) Immunity to Noise Immunity to noise n (k,ω) means that C (k,ω) ∼ = [0] ; ω ∈ n ,
k ∈ Kn .
(5.12)
again making note that the domains of interest are those in which the frequency and wavenumber have significant energy. As one can observe there is a tradeoff between the command following requirement of (5.9) and (5.12) meaning that the command input must not intrude into the high wavenumber frequency domains and the closed loop transfer matrix must be vanishingly small in the bands Kn and n .
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5 Performance Measures for Smart Structures with MIMO Representations
It is worth noting that “noise” n (k,ω) may in fact include high wavenumber modeling errors associated with the choice and achievement of sensor spatial distributions. Likewise “disturbances” d (k,ω) can include modeling errors as well. In the next section we will address the determination of “small” and “large” for the performance metrics developed above.
5.3 Assessment of Performance Metrics Using Singular Values In this section we will explore the concept of “small” versus “large” for in the context of the performance measures and matrices developed earlier. The reader is familiar with the concept as applied to vector magnitudes. The notion of the length or magnitude of a vector such as x = x1 , x2, ... xn is captured by the equation x =
x12 + · · · xn2
(5.13)
often referred to as the vector norm. In mathematics the matrix norm represents a natural extension of the concept to matrices. The Euclidean norm is among the most commonly used and can be generally represented by z =
|z1 |2 + |z2 |2 + · · · |zn |2 .
(5.14)
which is equivalent to the Euclidean norm on R2n . The Euclidean norm is also called the L2 distance or L2 norm vis x =
√
xT x .
(5.15)
5.3.1 Command Following Considering the command following criteria, in the absence of noise and disturbances, the tracking error can be written as; e0 (k,ω) = S (k,ω) r (k,ω) .
(5.16)
Now taking the conjugate transpose of (5.16), then multiplying by the original expression and finally taking the square root of both sides yields the Euclidean norm of the error as; e0 (k,ω)2 = S (k,ω) r (k,ω)2 .
(5.17)
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Assessment of Performance Metrics Using Singular Values
125
Judicious application of the Cauchy-Schwartz inequality to (5.17) results in the expression e0 (k,ω)2 ≤ S (k,ω)2 r (k,ω)2 .
(5.18)
In the special case of the Euclidean norm and square matrices, the induced matrix norm is the spectral norm. The spectral norm of a matrix A is the largest singular value of A or the square root of the largest eigenvalue of the positive-semidefinite matrix A∗ A. This spectral norm defined in terms of its maximum singular value can thus be rewritten as e0 (k,ω)2 ≤ σmax [S (k,ω)] r (k,ω)2
(5.19)
We now conclude that good command following is achieved in the presence of non-zero reference input when σmax [S (k,ω)] << 1;
ω ∈ r ,
k ∈ Kr .
(5.20)
We conclude then that a matrix is “small” when its maximum singular value is small. A brief word is perhaps in order here regarding singular values. The singular values of a matrix A are the non-negative square roots of the eigenvalues of AH A where AH represents the Hermitian of the matrix A. Thus we can mathematically represent the singular values as; σ (A) =
λ AH A .
(5.21)
where the Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose and λ is a scalar eigenvalue of A. The maximum singular value of a matrix may be thought of as its maximum (multiplicative) gain. In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix and results in a rank ordering of the singular values from maximum to minimum. The singular value decomposition (SVD) of the matrix A may be written as A = U V.
(5.22)
where the diagonal entries of are necessarily equal to the singular values of A. The matrix is an m x n matrix whose off diagonal elements are 0, ⎡ ⎢ =⎣
σ1
0 ..
0
. σn
⎤ ⎥ ⎦
(5.23)
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5 Performance Measures for Smart Structures with MIMO Representations
and whose singular values are real, non-negative and appear in descending i.e. σ1 ≥ σ2 ≥ · · · σmin (m,n) ≥ 0.
(5.24)
U is an m x n orthogonal matrix with column vectors Vi and are the left singular vectors of A. V is an m x n orthogonal matrix with column vectors Vi and are the right singular vectors of A. It is clear then that the SVD is a useful tool in the determination of the performance metric given in (5.20) and other subsequent measures. When the maximum singular value of a matrix is viewed as its maximum (multiplicative) gain, then (5.20) extends the classical frequency domain specification for discrete lumped parameter systems to the spatially distributed domain. In other words the matrix Σ contains the singular values, which can be thought of as scalar “gain controls” by which each corresponding input is multiplied to give a corresponding output. The worst tracking error occurs when r (k,ω) “points” along the right singular vector of S (k,ω) in accordance with σmax [S (k,ω)]. A similar result holds for the best tracking error and now it should be apparent that (5.20) essentially provides a measure of the “size” of S (k,ω) to meet the performance requirement in space and time. Additional insight into the impact of sensor and actuator placement can be gained by re-writing (5.20) in terms of the loop transfer matrix. This can be done by using additional properties of the singular value [49] along with the earlier definition of the sensitivity transfer matrix as σmax [S (k,ω)] =
1 << 1; σmin [I + T (k,ω)]
ω ∈ r , k ∈ Kr .
(5.25)
This suggest the need to make σmin [I + T (k,ω)] >> 1 over the relevant domain, however σmin [I + T (k,ω)] < σmin [T (k,ω)] + 1
(5.26)
and thus the requirement for good command following becomes σmin [T (k,ω)] >> 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 .
(5.32)
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5 Performance Measures for Smart Structures with MIMO Representations
which now quantifies the requirement that S (k,ω) be “small” over the relevant temporal and spatial bandwidths. Again the maximum disturbance sensitivity occurs at a given k, ω when d (k,ω) points along the right singular vector of S (k,ω) as associated with σmax [S (k,ω)]. A similar result holds for the minimum disturbance sensitivity. Additional insight is gained by re-writing (5.32) in terms of the loop transfer matrix σmin [T (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)] . << 1 ;
ω ∈ n,
k ∈ Kn .
(5.36)
which now quantifies (5.12) and results in high sensitivity to disturbances and low loop gain in the temporal/spatial bands n and Kn . This of course requires that a tradeoff be made with respect to the requirement for good tracking error given in (5.9). In practice the requirements are applied over the separate bandwidths n , Kn and r , Kr and if these are well separated then performance can be readily achieved; i.e. the command input can not have significant energy content in the band contained by n and Kn . Typically the noise n (k,ω) occurs at bandwidths associated with the higher wavenumbers associated with sensor modeling errors and the smart structure designer must take care and not require response outside of the temporal and spatial bandwidth of the sensors.
5.4 Metrics for Controllability and Observability The concept of controllability is important in the design of smart structures as control systems, and the property of controllability is critical in control system design and the stabilization of unstable systems by feedback, or optimal control.
5.4
Metrics for Controllability and Observability
129
Controllability and observability are dual aspects of the smart structure design problem. For degenerate, temporally based linear time invariant systems, state controllability requires that it be possible, using admissible inputs, to take the states from any initial value to any final value within a given time window. If this is true for all times t0 and all initial states x (t0 ) the system is declared to be completely state controllable. The notion of observability parallels that of controllability. Whereas controllability ties together the input u (t) and the state of a system, observability ties together the output and the state of a system. A linear system is said to be state observable at t0 if the state x (t0 ) can be determined from the output y (t). Again if the system is observable for all t0 , then the system is said to be completely state observable. In what follows we will extend these concepts to distributed parameter systems using the methods presented earlier in this chapter. The controllability and observability of a distributed parameter system must be assessed in terms of its spatially distributed sensors and actuators.
5.4.1 Controllability Without loss of generality we will consider inputs u (k,ω) that are spatially and temporally “white” so that all system responses can be excited and hence y (k,ω) = p (k,ω) h (k,ω) q (k,ω) .
(5.37)
For controllability we wish to assess the impact of the actuator distribution q (k,ω) on the systems ability to achieve a non-zero output at every k,ω and thus we consider sensors that are also spatially and temporally “white” i.e. p (k,ω) is essentially a scalar filter that has infinite spatial and temporal bandwidth and unit magnitude. Under these assumptions (5.37) becomes y (k,ω) = h (k,ω) q (k,ω) .
(5.38)
According to our earlier definition of controllability should there exist a wavenumber kc and frequency ωc where y (k,ω) vanishes then the system will not be controllable at that wavenumber/frequency for a given actuator distribution q (k,ω). Again we began by considering the spectral norm of (5.38) y (k,ω)2 = h (k,ω) q (k,ω)2
(5.39)
which must under the conditions stated above. Equation (5.39) can be re-written as y (k,ω)2 = σ h (k,ω) q (k,ω)
(5.40)
noting the fact that because h (k,ω) q (k,ω) is a vector only one singular value is involved. From above the system will not be controllable at wavenumber Kc and frequency ωc if
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5 Performance Measures for Smart Structures with MIMO Representations
σ h (kc ,ωc ) q (kc ,ωc ) = 0.
(5.41)
Equation (5.41) becomes the metric for controllability assessment for distributed parameter systems. Note that for a smart structure; if wavenumber kc and frequency ωc correspond to the structures eigenvalues and natural frequencies, then the corresponding “mode” of the structure will not be controllable (assuming the structure admits a modal expansion).
5.4.2 Observability The performance criteria for observability is developed in a manner similar t that above for controllability. We consider actuator distributions that are spatially and temporally “white” so that q (k,ω) consist of a scalar filter function that has infinite spatial and temporal bandwidth and unit magnitude thus y (k,ω) = p (k,ω) h (k,ω)
(5.42)
As before we seek to determine if there exist a wavenumber k0 and frequency ω0 where the output y (k,ω) vanishes and thus the systems output will not be observable at that wavenumber/frequency using the sensor distribution p (k,ω). Similarly the associated spectral norm of the output will vanish in accordance with σ h (k0 ,ω0 ) q (k0 ,ω0 ) = 0
(5.43)
and (5.43) becomes the criteria for loss of observability. Again we note that for a smart structure; if wavenumber k0 and frequency ω0 correspond to the structures eigenvalues and natural frequencies, then the corresponding “mode” of the structure will not be observable (assuming the structure admits a modal expansion). It is difficult to determine numerically when the singular value vanishes and in practice the positive definite left hand side of (5.41) and (5.43) become a measure of controllability and observability respectively for the plant input and output for all wavenumbers and frequencies for given sensor/actuator distributions.
5.5 Example: Active Damping of a Simply Supported Beam In this section we illustrate the performance metrics discussed above using a simple Single Input Single Output (SISO) system. More specifically we consider the active damping of a pinned–pinned Bernoulli-Euler beam. We chose a SISO system for its simplicity as the singular values, which can be thought of as scalar “gain controls” by which each corresponding input is multiplied to give a corresponding output, are reduced to a single scalar gain. It is noted that while this particular example represents a single input, single output system, it contains all of the essential elements for analysis.
5.5
Example: Active Damping of a Simply Supported Beam
131
The non-dimensional governing equation of motion for a thin beam is given by: f (x,t) =
∂ 2y ∂ 3y ∂ 4y + + λ = u (x,t) ; ∂x2 ∂x2 ∂t ∂t2
0<x<1
(5.44)
where λ is a damping parameter. Here the input is both a function of space and time and will be used to model a smart material piezoelectric polymer film actuator of the type discussed earlier in Chap. 1. Assuming homogeneous boundary conditions yields, y(0,t) = y(1,t) =
∂ 2 y(0,t) ∂ 2 y(1,t) − = 0. ∂x2 ∂x2
(5.45)
The non-dimensional finite Fourier Integral spacial/continuous time transform takes the form, f˜ (kn ,ω) ≡ Fs {f (x,t)} =
1 ∞ 0
f (x,t) sin (kn x) ei ω t dt dx, kn = n π , n = 1,2 . . . ,
0
(5.46) and whose inverse is f (x,t) ≡
Fs−1
∞ ! ˜f (kn ,ω) = 2
∞
n=1 −∞
f˜ (kn ,ω) sin (kn x) e−iω t dω
(5.47)
which must be applied to the governing equation of motion along with the boundary conditions for the pinned–pinned beam to yield the uncompensated and transformed plant response function: 1 y˜ (kn ,ω) = h(kn ,ω) = 4 . u˜ (kn ,ω) kn + iωλkn2 − ω2
(5.48)
Figure 5.3 shows h (kn ,ω) plotted over a band limited portion of its spectrum using the discrete spatial transform and a damping parameter of λ = 0.002. Plots are superimposed for wavenumbers of kn = π , 2π , 3π · · · 6π . Note that the plant response “rolls off” as ω− 2 in temporal frequency and kn− 4 in spatial frequency. The reader is reminded that Fig. 5.3 is a representation of the 3 dimensional response characteristics associated with distributed parameter systems; in this case it shows the discrete wavenumber-continuous time characteristics of this system.
5.5.1 Spatially Uniform Actuator Distributions For active damping we now must augment the plant with an actuator and compensator. Consider the spatially uniform control distribution discussed in detail in
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5 Performance Measures for Smart Structures with MIMO Representations
Fig. 5.3 Uncompensated response function for a damped simply supported beam
Chap. 2. Here the control distribution over the length of the beam can be modeled using generalized functions h (◦) as u(x,t) = Vmax [h(x) − h(x − 1)]u(t).
(5.49)
Close examination of this spatially distributed compensator reveals that it is degenerate in the sense that it is separable in space and time having a maximum gain Vmax and a purely temporally modulated control component u (t). The space/time transform of this actuator is q(kn ,ω) = Vmax kn [( − 1)n − 1] u˜ (ω).
(5.50)
The plant response function (5.48) can now be augmented to give the actuator augmented plant response function, y˜ (kn ,ω) =
Vmax kn [( − 1)n − 1] u˜ (ω). kn4 + iωλkn2 − ω2
(5.51)
Applying controllability metric (5.41) reveals that this system will not be controllable for values of n = 2,4,6, . . . even because for these values h (kc ,ωc ) = 0 and (5.41) is identically satisfied. On the other hand we observe that all odd number “modes” of the beam will be controllable.
5.5
Example: Active Damping of a Simply Supported Beam
133
This confirms our earlier derivation (3.36) and observations concerning the Lyapunov damping controller where it was demonstrated that the spatially uniform distribution acts effectively as two opposed moments at its boundaries and has the same spatial transform as a discrete moment pair which is given by Fs {δ (x) − δ (x − 1)} = kn [( − 1)n − 1].
(5.52)
5.5.2 Linear or “Ramp” Actuator Distributions In Chap. 3 it was shown that a “ramp” distribution can be used to provide controllability for all kn ,ωn and this actuator distribution is represented by u(x,t) = Vmax (1 − x)[h(x) − h(x − 1)]u(t).
(5.53)
which has the space/time transform q(kn ,ω) = −Vmax kn u˜ (ω),
(5.54)
and thus yields the plant response function y˜ (kn ,ω) =
−Vmax kn u˜ (ω). kn4 + iωλkn2 − ω2
(5.55)
In this case it is seen that (5.41) reveals that the system will be controllable over the entire space/ time bandwidth. The actuator augmented plant response in this case (5.55) is shown in the band limited plot Fig. 5.4. This plot reveals the spatial compensation provided by this choice of controller distribution which “rolls the plant characteristics off” as k13 while preserving its temporal or resonant characteristics. The temporally modulated portion of the controller was shown in Chap. 3 to take the form of angular velocity feedback. This results in a stabilizing compensator which takes the form u(t) = −α
∂ 2 y(0,t) , ∂x∂t
(5.56)
where α is simply a non-dimensional positive definite gain. Assuming that the sensor response is essentially temporally “white” i.e. infinite temporal bandwidth, its transform is given by p(kn ) = α kn ,
(5.57)
and the temporal derivative component transforms to K(ω) = −iω.
(5.58)
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5 Performance Measures for Smart Structures with MIMO Representations
Fig. 5.4 Ramp actuator augmented plant response function
Fig. 5.5 Loop transfer function for α = 1 = Vmax
5.5
Example: Active Damping of a Simply Supported Beam
135
The loop transfer function (5.1) of the compensated plant is now T(kn ,ω) =
i (α Vmax ) ω kn2 . + iωλ kn2 − ω2
(5.59)
kn4
and its characteristics are plotted in Fig. 5.5 for the case when α = 1 = Vmax Because the system is SISO the magnitude or gain of the transfer function is simply a singular scalar value. The angular velocity sensor space/time filter characteristics are now convolved with those of the actuator augmented plant to provide additional spatial compensation. The spatial filter characteristics of the augmented plant now rolls off as k12 for large values of wavenumber while the temporal filter characteristics provides additional forward loop gain at higher values of ω. Note that the scalar gain parameters α and Vmax determine the magnitude of forward loop gain available for disturbance rejection and hence vibration damping performance. The Sensitivity transfer functions S (kn ,ω) are presented in Figs. 5.6 and 5.7 for the cases when α Vmax = 1 and 100 respectively. Analysis of the smaller loop gain α Vmax = 1 shown in Fig. 5.6 reveals a controller which only mitigates Sensitivity Response, Vmax = 1
0
–10
–20
–30
kn = π kn = 2π
–40
kn = 3π kn = 4π kn = 5π
–50
–60 100
kn = 6π
101
Fig. 5.6 Sensitivity response function for α = 1 = Vmax
102
103
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5 Performance Measures for Smart Structures with MIMO Representations
0
Sensitivity Response, Vmax = 10 kn = π kn = 2π
–20
kn = 3π kn = 4π –40
kn = 5π kn = 6π
–60
–80
–100
–120 100
101
102
103
Fig. 5.7 Sensitivity response function for α = 100 = Vmax
vibration at the resonances thus yielding relatively poor tracking performance. The use of the higher gain α Vmax = 100 corrects this and results in improved tracking performance as shown in Fig. 5.7. Figures 5.8 and 5.9 show the characteristics of the closed loop response function for the two cases i.e. α Vmax = 1 and 100 respectively. The closed loop response is a measure of reference to output. The smaller the magnitude of the closed-loop response, the closer the output matches the reference. For good command following the closed-loop response is approximately the identity matrix. As is expected, a higher gain results in the closed-loop response being closer to 0 dB over a broader range of frequencies. This shows that a higher gain will show better command following and the output will closely match the reference for most wavenumbers and frequencies. Only higher wavenumbers at lower frequencies or lower wavenumbers at higher frequencies are difficult to control. It should be noted that the control was formulated to damp vibrations, rather than to provide tracking, which would require integral control. The system does have an “infinite” spatial bandwidth, as the control affects the system response at all kn . When the closed-loop response function does not roll off quickly at high (kn ,ω), the control design will be sensitive to high temporal and spatial frequency modeling errors, as contended in the stability/robustness development.
5.6
Metrics for Achieving Stability and Robustness for Control of Smart Structures
0
137
Closed-Loop Response, Vmax = 1
–10
–20
kn = π
–30
kn = 2π kn = 3π
–40
kn = 4π kn = 5π
–50
kn = 6π –60 100
101
102
103
Fig. 5.8 Closed loop response function for α = 1 = Vmax
5.6 Metrics for Achieving Stability and Robustness for Control of Smart Structures In the previous sections we developed performance metrics given models of the plant, sensors and actuator characteristics. In this section we explore the effect on performance of model uncertainties on performance. In actual practice for example the plant model is precisely that, a nominal model or approximation of the true or physical plant. In this regard all linear model approximations will contain various forms of error. In addition the smart structure system designer must employ idealizations of sensor and actuator distributions. The desire is to maintain control system performance and stability in the presence of such uncertainties. Stability as discussed herein means that for any bounded input (over any amount of time), the system output will also be bounded. This is commonly known as BIBO (Bounded Input-Bounded Output) stability. When a system is declared to be BIBO stable the output cannot become infinite in magnitude while the input remains finite. Robust control theory explicitly deals with uncertainties in its approach to controller design. Controllers designed using robust control methods have some capability to maintain performance given small differences between the true system and the
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5 Performance Measures for Smart Structures with MIMO Representations
0
Closed–Loop Response, Vmax = 10
–0.1
kn = π
–0.2
kn = 2π –0.3
kn = 3π kn = 4π
–0.4
kn = 5π kn = 6π
–0.5
–0.6
–0.7 100
101
102
103
Fig. 5.9 Sensitivity response function for α = 100 = Vmax
nominal model used for design. While the early control design methods of Bode and others were fairly robust; state-space methods have been sometimes found to lack robustness. Robust control methods seek to maintain robust performance and/or stability in the presence of small modeling errors. Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design. The measures developed here are presented in terms of the boundedness of the control systems response in the presence of modeling uncertainties. In all cases it is assumed that the forward loop response matrix and the closed loop system are nominally stable. There is no loss of generality in such an assumption as the techniques are readily extended to open loop unstable plants via the introduction of a stabilizing inner-loop. Stability and robustness measures for discrete lumped parameter systems as applied to multi-input multi-output systems (MIMO) have been develop using various forms of error [50, 51]. The robustness performance measures are typically based on magnitude information associated with the modeling errors of the closed
5.6
Metrics for Achieving Stability and Robustness for Control of Smart Structures
139
loop system and tend to result in metrics which may be considered somewhat conservative. The metrics are posed in terms of typical MIMO Nyquist stability criteria using the so called small gain theorem. The techniques presented here are based on a derivation by Lehtomaki [50]. We extend this generalization to smart structures as distributed parameter systems via the multi-dimensionality of the independent transform variables. The transformed input/output representation of a distributed parameter system in the absence of sensor noise and disturbances can be written in accordance with (5.2) as y(k,ω) =
T(k,ω) r(k,ω) [I − T(k,ω)]−1
(5.60)
with T (k,ω) being a representation of the actual or physical true plant loop transfer matrix, r (k,ω) the input command vector, y (k,ω) the filtered output vector. Equation (5.60) is considered to be bounded input bounded output (BIBO) stable if [I + T(k,ω)] is non-singular for y(k,ω) =
adj [I + T(k,ω)] T(k,ω)r(k,ω), det [I + T(k,ω)]
(5.61)
where singularity is interpreted as the case when det (I + T) = 0 which also implies σmin (I + T) = 0. In actual practice the sensors, actuators and compensator filter functions are designed according to a nominal plant model whose forward loop transfer matrix we shall designate as Tnom . Subsequently we now explore the impact of model error as represented by various forms of structured uncertainty in the relationships between T (k,ω) and Tnom , and its impact on the destabilization of the closed loop plant in accordance with (5.61).
5.6.1 Additive Error Uncertainty If the difference between the nominal and true augmented plant models can be determined within an additive error matrix Eadd then T(k,ω) = Tnom (k,ω) + Eadd (k,ω).
(5.62)
This type of error is often encountered in finite element and modal models of smart structures which are by necessity truncated to a finite set of plant modes as discussed in Chap. 1. We wish to determine stability in the sense of whether or not [I + T] and consequently [I + (Tnom + Eadd )] are singular. Now if [I + (Tnom + Eadd )] is singular then there exist a vector quantity x with with x2 > 0 in the null space of [I + (Tnom + Eadd )] which is designated as x ∈ Null such that (I + Tnull + Eadd )x = 0.
(5.63)
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5 Performance Measures for Smart Structures with MIMO Representations
thus we write (I + Tnull ) x = −Eadd x.
(5.64)
Taking the Euclidean norm of both sides of (5.64) yields (I + Tnom )x2 = Eadd x2 ,
(5.65)
Eadd x2 (I + Tnom )x2 = . x2 x2
(5.66)
similarly we can write
If we now choose an arbitrary vector say z with z2 > 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)] <
1 , for all (k,ω) σmax [Em (k,ω)]
(5.79)
reflecting the fact that the closed loop must have small gain in the regions (k,ω) where the modeling error is large, i.e. the designer must not specify a closed loop bandwidth of performance that is greater than the model bandwidth.
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5 Performance Measures for Smart Structures with MIMO Representations
5.7 Summary A method of assessing the performance of linear distributed parameter control systems has been presented in this chapter. The method is based upon the system s mu1tiplicative input/output representation in a transformed wavenumber/frequency space, including spatially-transformed representations of sensor and actuator distributions. The analysis has presented performance measures for distributed parameter control, in both a temporal and spatial sense, for command following, disturbance rejection, noise immunity, and stability-robustness in the presence of modeling errors. Additionally, observability and controllability norms were derived. The formulation readily lends itself to the quantification of spatial performance, such as achievable spatial bandwidth. The performance measures provide a means for evaluating distributed parameter control designs. The distributed parameter control problem was shown to reduce to designing an appropriate space/time compensator. If the compensation is degenerate, then spatial compensation is provided solely by the sensor and actuator distributions. An example problem demonstrated the application of the transform formalism, and reinforced conclusions about a system designed and analyzed in Chap. 1 by Lyapunov methods. This analysis technique must now be extended into a distributed control design methodology. The example problem was seen to have “infinite” spatial bandwidth using the ramp control distribution. In the next chapter discrete spatial transforms will be studied in the context of shape control. The aforementioned example problem will be shown to be inappropriate for shape control applications. Additional performance measures specific to the shape control problem will be introduced. These additional conditions will show that the discrete spatial transform is more appropriate than the continuous (integral) spatial transform for shape control analysis. Further, discrete spatial transforms can be folded into computer-aided design methods directly.
Notes An analysis method is developed in this chapter wherein a distributed plant’s input/output relation is studied in a transformed “wavenumber/frequency” space. The presentation reveals the essential similarities to and extensions from research in MIMO control analysis. Particular emphasis is placed upon utilizing a welldeveloped tool of modern control theory, the singular value decomposition, to analyze compensation and sensor/actuator placement in distributed systems. Criteria for achieving spatial performance goals such as command following and disturbance rejection over “spatial bandwidths” were presented that appeal to physical insight, and are generalized to include temporal performance as well. Controllability and observability measures were derived. A stability-robustness measure was developed. The method is shown to be applicable to both discrete and distributed sensors and actuators, using continuous (integral) or discrete (series) spatial transforms.
Problems
143
Problems (5.1) Consider the pinned–pinned beam given in Eq. (5.44). A new actuator spatial distribution is being considered for use on this plant, given by u (x,t) = Vmax(1−2x)[[h[x]−h[x−0.5]]+[h(x−0.5)−h(x−1)]]u(t) . What is the plant response function for this new configuration. Plot the plant response function for kn = π ,2, . . . ,6π , where λ = 0.002 and Vmax = 1. What are differences between your figure and Fig. 5.4. Using the methodology of Chap. 3 and Eq. (5.52) develop an appropriate feedback controller for this system and plot the resulting loop transfer function for the same spatial frequencies above. (5.2) Modify the code you wrote in question 2.3 to calculate the observability and controllability for vibrational modes in a non-dimensional rectangular plate using distributed sensors/actuators with: a. boxcar distribution b. ramp distribution c. an arbitrary distribution of your choosing (5.3) Consider the results of Problem 1.9. a. Utilize the translation invariance of the A and B in the horizontal directions and write out the equations for the Fourier transformed field. b. If the output of the system is defined as φ = [u,v,w]T , determine the operator C. c. Write out (do not calculate) the expression for the spatio-temporal frequency response of this system. d. Consider that the disturbance is invariant in the streamwise direction, i.e. the streamwise wavenumber is zero. Draw a block diagram of the streamwise constant LNS system including all the A, B, and C operator components. (5.4) We would like to use distributed parameter actuators and distributed parameter control theory to control the disturbances in boundary layer flows. The aim is to stabilize the underlying dynamics of the problem. Consider the results of Problem 5.3. a. Write out the frequency responses from different body force inputs to different velocity outputs as function of spatial frequencies. b. Calculate and plot the transfer function 2-norms and discuss the corresponding three-dimensional flow structures that are most amplified (Hint: 2-norm is the expected RMS value of the output when the input is a unit variance white noise and it may be found by solving a Lyapunov equation).
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5 Performance Measures for Smart Structures with MIMO Representations
c. Based on the wavelength of the most amplified disturbances in the flow define a distributed parameter actuator that you could use to enforce the development of most stable modes. (5.5) Use the results of question 5.4 and convolve the actuators with the given plant. Observe the changes in the frequency response curves. What improvements, if any, should you make to your actuator distribution? (5.6) Comment on the modal observability of the system presented in Problem 2.1 with the accelerometer placed at a location x = xα . (5.7) Comment on the modal controllability of the system presented in Problem 2.1 using the tip force T (1,t).
Chapter 6
Shape Control: Distributed Transducer Design
6.1 Introduction Historically shape control as reflected in the open literature has been driven by the development of adaptive optic systems which began in the 1970s [Hardy 5.1]. These systems were formally designed around temporal bandwidth requirements for atmospheric wavefront aberration corrections and the spatial shape requirements were generally handled informally using ad hoc methods. Spatial performance was driven by the need to determine actuator and sensor sites in order to achieve a stable system and a single shape. There was particular emphasis on maintaining the figure of the system or mirror in the presence of disturbances which caused unwanted vibrations, i.e. shape maintenance. In this chapter we use the techniques introduced in the preceding chapters to develop a method of designing and assessing performance for the control of the shape of a smart structure. In modern times this is sometimes referred to as Morphing or the radical change of the shape of a structure. The methods introduced here are applied to linear time invariant, stationary, distributed parameter systems. Traditional approaches to shape control use discrete transducers and lumped parameter modeling techniques e.g. finite elements. For real time, high fidelity shape control these techniques can only address performance in a limited fashion, if at all. Distributed parameter control as applied to distributed parameter systems, i.e. systems described by space and usually time, can more effectively address the spatial performance requirements dictated by shape control. As we shall see, shape control requires both a prescribed spatial bandwidth and a set of shapes that characterize the control task, e.g. aircraft wing airfoil shapes for efficient flow control over a broad range of flight conditions. The distributed parameter system control techniques developed herein and their application to spatially distributed transducers are well suited to the design and practical implementation of dynamic shape control in modern systems. Given the success and availability of temporal frequency domain tools in classical and robust, multi-variable lumped parameter systems control theory; the use spatial frequency domain transforms applied to distributed parameter systems and control follow naturally. It has been
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_6,
145
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6 Shape Control: Distributed Transducer Design
shown that using Multi-Input Multi-Output techniques one can construct an input– output relation representing a distributed parameter system in both the temporal and spatial frequency domains. Many shapes of interest can be adequately represented using a limited set of orthogonal basis or eigenfunctions. These functions characterize the spatial bandwidth requirements for shape control as well as the set of shapes that characterize the structural morphing problem. In what follows we develop the principles and techniques for the manipulation of a distributed plants output in order to achieve a non-zero set point. This is in fact the distributed or spatial servo problem. From this viewpoint we can distinguish shape control from that of simply damping the resonant response of the plant as discussed in previous chapters, e.g. regulation versus tracking.
6.2 Shape Control and the Notion of Discrete Spatial Bandwidth In this section we address the notion of spatial bandwidth in the context of shape control. We begin by developing a paradigm for the understanding and quantification of “shape” which will allow us to employ useful performance specifications in the shape control design problem. Consider the shape shown in Fig. 6.1. The shape can be readily described using a parameterization in terms of a limited set of orthogonal basis or eigenfunctions vis.,
1 0.8 0.6 0.4 0.2
0 –0.2 –0.4 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Non-dimensional Position Fig. 6.1 Shape profile
0.8
0.9
1
6.2
Shape Control and the Notion of Discrete Spatial Bandwidth
147
.6
Fig. 6.2 Discrete spatial ranform of the shape profile
.5
MAGNITUDE
.4
.3
.2
.1
0
1
2
3
4
FOURIER COEFFICIENT
y (x) = c1 ψ1 (x) + c2 ψ2 (x) + · · ·
(6.1)
More specifically the shape shown can be exactly represented by decomposition on a basis of functions with a discrete spectrum, e.g. a Fourier series as y (x) = 0.1 sin (π x) + 0.5 sin (2π x) + 0.3 sin (3π x) + 0.1 sin (4π x) .
(6.2)
The shape or profile has a discrete transform spectrum and thus a finite spatial bandwidth which shown in Fig. 6.2. This spectral representation appeals to our intuition in that the shape in Fig. 6.1 may be thought of as a linear superposition of distinct functions or shapes. An associated shape control problem thus may be posed in terms of performance requirements that not only include a spatial bandwidth requirement but achieving specific distinct shapes independently or in any linear combination, i.e. a family of shapes. It is particularly convenient to represent this “family of shapes” in terms of orthogonal functions because of their inherent spatial independence e.g. the {ψi (x)} in (6.1) are component profiles of orthogonal shape functions. When the shape control problem is posed in this manner it becomes a matter of deriving an input/output representation of the plant in terms of an expansion in the orthonormal component functions.
6.2.1 Orthonormal Expansions and the Discrete Spatial Transform As the reader may have suspected the profile discussed in Fig. 6.1 was a bit contrived in the sense that it could be represented exactly with the discrete spectrum (6.2). In practice is may be difficult to determine an expansion in orthogonal functions
148
6 Shape Control: Distributed Transducer Design MH20 Airfoil: Pylon racing model aircraft 0.2 0.15
Y/C (Thickness)
0.1 0.05 0 –0.05 –0.1 –0.15 –0.2
0
0.1
0.2
0.3
0.4 0.5 0.6 X/C (Chord)
0.7
0.8
0.9
1
Fig. 6.3 Normalized chord (Y/C) versus thickness (X/C) for the MH20 airfoil
that precisely equals the desired shape, point by point, over the domain of interest. Consider the airfoil profile shown in Fig. 6.3 defined over the domain where x ∈ D. We wish to represent the upper half of the airfoil shape as an expansion in orthogonal functions. The discrete transform spectrum for the first 4 shape components representing the upper profile is plotted in Fig. 6.4. The lines in the figure represents the approximation given by the expansion to the actual profile shown as a solid line. It is clear that while the approximation does not provide “point by point” convergence to the actual shape, it may be a sufficient representation of the profile for control purposes. The criteria of point by point convergence in terms of the profile represented by Eq. (6.1) may be stated as,
lim [y(x,ω) −
n→∞
n
ci (ω)ψi (x)] = 0.
(6.3)
i=1
A more appropriate but somewhat weaker requirement would be one which sought to minimize the integrated mean square error between the actual and approximated profiles over the domain x ∈ D. This so-called limit in the mean criterion when practically applied reduces to a representation determined by finding the coefficients ci that minimize this error in accordance with [52–54] and Eq. (6.4).
6.2
Shape Control and the Notion of Discrete Spatial Bandwidth
149
M2HO Airfoil Upper Profile 0.2
Y/C
0.1 0 –0.1 –0.2 0
0.2
0.1
0.3
0.5
0.4
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. 6.4 Discrete spatial transform of the upper profile of an MH20 airfoil
[y(x,ω) −
lim
n→∞ D
n
ci (ω)ψi (x)]2 dx = 0.
(6.4)
i=1
6.2.2 Minimization of the Integrated Mean Square Profile Error We began our development with the following recapitulations and assumptions: • The desired shape is square integrable over the domain of interest • The desired shape can be expanded via a set of orthogonal shape functions, {ψi (x)} • Orthogonal shape functions are appropriate for the specification of shape as they posses the property of spatial independence • If the system of shape functions is orthogonal then it can be normalized using suitable normalizing coefficients. • The desired shape can be written in terms of a band limited discrete Fourier transform Since the system of shape functions is orthogonal on the domain D and of the unit 2-norm then they satisfy
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6 Shape Control: Distributed Transducer Design
ψi (x)ψj (x)dx = δij ,
(6.5)
D
where δij is the Kronecker delta function; δij = 1 for i = j, δij = 0 otherwise. The system {ψi (x)} is orthogonal and can be normalized using suitable normalizing coefficients [55]. The requirement that the shape be square integrable over D implies y2 (x,ω)dx < ∞.
(6.6)
D
We may now define the mean square error between the desired shape and its truncated, band limited expansion as follows; En (ω) ≡
[y(x,ω) −
n i=1
D
[y(x,ω)2 dx − 2
=
ci (ω)ψi (x)]2 dx
n
y(x,ω)ψi (x)dx +
n
c2i (ω).
(6.7)
The Fourier coefficients of y (x,ω) with respect to ψi (x) are given by ai (ω) ≡ y(x,ω)ψi (x)dx.
(6.8)
ci (ω)
i=1
D
i=1
D
D
Equation (6.8) may now be rewritten as En (ω) =
y2 (x,ω)dx −
n
a2i (ω) +
i=1
D
n
[ai (ω) − ci (ω)]2 .
(6.9)
i=1
Recall that we seek to determine the coefficients ci that minimize this error and note that the first two terms of (6.9) are completely independent of our choice of these. The error will thus be minimized by the proper choice of ci (ω) in the term n
[ai (ω) − ci (ω)]2 .
(6.10)
i=1
The summation is never negative and the mean square error En (ω) is minimized for the desired shape y (x,ω) and functions {ψi (x)} if and only if the coefficients ci (ω) are equal to the Fourier coefficients ai (ω) defined in (6.8), in other words upon the condition that ci (ω) = ai (ω).
(6.11)
6.3
Plant Representations in Terms of an Expansion Basis Set
151
Now a necessary and sufficient condition for the shape y (x,ω) to be approximated in the mean by an orthonormal expansion in the shape functions {ψi (x)} is that they must constitute a complete set in the sense of Parseval, i.e. ∞
c2i (ω) =
i=1
y2 (x,ω)dx.
(6.12)
D
However, even if the system is not complete in this sense, choosing the expansion coefficients ci (ω) to be equal to the Fourier coefficients ai (ω) of y (x,ω) with respect to {ψi (x)} will lead to a representation which minimizes the mean square error over D. As mentioned earlier, the ensuing shape control problem is now a matter of deriving an input/output representation of the plant in terms of the expansion basis set.
6.3 Plant Representations in Terms of an Expansion Basis Set In this section we develop methods of plant representations of the smart structure or spatially distributed plant in terms of shape functions. This can be achieved by equating the controlled systems output with the desired output as described by the expansion (6.2) and then expressing the plant in terms of the expansion coefficients ci (ω) using the Green’s function representations discussed in Chap. 4. It will be shown that there are several advantages to this approach in terms of analytical simplicity particularly if the chosen functions are orthogonal. By way of illustration we investigate two particular forms of the Green’s function i.e. the Generic and Symmetric forms of the Green’s function.
6.3.1 The Generic Green’s Function Representation Assuming a Linear Time Invariant spatially distributed plant, the Generic Green’s function representation is given by h(x,ξ ,t,τ ) = h(x,ξ ,t − τ ), and its associated Fourier time transformed composition integral y(x,ω) = h(x,ξ ,ω)u(ξ ,ω)dξ .
(6.13)
(6.14)
D
As previously discussed we assume that the plant admits a discrete expansion in a set of orthonormal shape functions vis. y(x,ω) = c1 (ω)ψ1 (x) + c2 (ω)ψ2 (x) + . . .
(6.15)
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6 Shape Control: Distributed Transducer Design
which are square integrable over the domain of interest. Using the combination of Eqs. (6.8), (6.11), and (6.14) the requisite expansion coefficients may be written as
h(x,ξ ,ω)u(ξ ,ω)dξ dx
ci (ω) = ψi (x) D D = ψi (x)h(x,ξ ,ω)u(ξ ,ω)dξ dx.
(6.16a) (6.16b)
D D
We now wish to determine a matrix input/output representation between the vector of control inputs and the transformed output response of expansion coefficients representing the desired shape. Equations (6.16) represent the discrete shape integral time transform of the forced system response. Assuming an input which is separable in space and time (i.e. degenerate actuators), the input can be expressed as a superposition of inputs from N actuators. These exogenous inputs can be expressed as a linear superposition of command signals u(x,ω) =
N
qj (x,ω)uj (ω)
(6.17)
j=1
and thus ci (ω) =
N j=1
ψi (x)h(x,ξ ,ω)qj (ξ ,ω)dξ dx uj (ω)
(6.18)
D D
or more succinctly ci (ω) =
N
αij (ω)uj (ω)
(6.19)
j=1
as the desired actuator augmented input/output representation where the actuator dynamics have been absorbed into the integral. We now have a plant representation of the smart structure which is quite amenable to shape control (See Fig. 6.5). Note that since the Matrix M (α) is fixed by actuator types and locations, these requirements become criteria for screening candidate distributions for shape control. This will be covered in considerable detail in subsequent sections of this chapter.
Fig. 6.5 Input/output representation of the plant in terms of an expansion basis set of “shapes”
u
M (α)
c
6.3
Plant Representations in Terms of an Expansion Basis Set
153
Once the actuator distributions have been determined, suitable modern robust control MIMO techniques can be used to synthesize temporal compensators for the desired dynamic performance.
6.3.2 The Symmetric Green’s Function Representation The dynamics of a broad class of smart structures can be characterized in terms of modes and mode shapes which can be described via the systems eigenfunctions ϕi (x) and associated eigenvalues λi (ω). If the spatially distributed system is self-adjoint then the eigenfunctions represent a complete set. A symmetric Green’s function is expressible in terms of such functions as a bilinear expansion of the form,
h(x,ξ ,ω) =
∞ ϕk (x)ϕk (ξ )
λk (ω)
k=1
,
(6.20)
here the temporal transform has been applied for the sake of convenience. This form is also convenient for use in both finite element and modal analyses of structures. By combining (6.20) and (6.16) we can obtain an expression for the discrete spectrum output in terms of the shape function basis as, ci (ω) =
∞ k=1
ψi (x)ϕk (x)dx D
D
ϕk (ξ ) u(ξ ,ω)dξ . λk (ω)
(6.21)
Here we note that the forced response of the system to a distributed input is weighted by the plants mode shapes ϕk and temporal dynamics λk . The first integral in (6.21) represents a transformation from shape space to modal space and is a decomposition of the shape functions in the system eigenfunctions. Because the second integral is independent of x, it can be written as simply a frequency dependant constant: ϕk (ξ ) u(ξ ,ω)dξ (6.22) fk (ω) = D λk (ω) and now (6.21) can be written as ci (ω) =
∞ k=1
ψi (x)ϕk (x)dx.
fk (ω)
(6.23)
D
(Note that (6.23) is convergent under the assumption of the analysis [55]). In accordance with (6.17) the exogenous inputs are degenerate and can be expressed as a superposition of inputs hence,
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6 Shape Control: Distributed Transducer Design
cl (ω) =
∞ N j=1
fjk (ω)hik uj (ω),
(6.24)
k=1
where fjk (ω) = D
ϕk (ξ ) qj (ξ ,ω)dξ , λk (ω)
(6.25)
ψi (x)ϕk (x)dx.
(6.26)
with hik (ω) = D
Once again we have derived in (6.23) an input/output relation between a vector of command inputs and the transformed output space desired e.g. (6.19) with the actuator space-time dynamics being integrated into the integral of (6.25). The space-time filter function qj can also be expanded in terms of the system eigenfunctions, assuming that they are a complete set with respect to the chosen distribution, qj (x,ω) =
∞
ajl (ω)ϕl (x),
(6.27)
l=1
with ajl (ω) =
qj (x,ω)ϕl (x)dx.
(6.28)
D
The orthogonality property of the eigenfunctions yields, ajk (ω) fjk (ω) = λk (ω)
ϕk2 (ξ )dξ ( )* +
(6.29)
D
=consant
which conveniently partitions the actuator transform. Furthermore if the shape functions ψi and the system eigenfunctions ϕk are equal, that is if the desired shapes are expressible in terms of the system mode shapes, (6.26) becomes unity and the input/output relation (6.24) decouples yielding the simpler form ci (ω) =
N
fji (ω)uj (ω).
(6.30)
j=1
It is also interesting to note that this choice represents the natural shapes (modes) of the system and can be achieved with minimal energy. The implications of truncating the plant will be explored in subsequent sections of this chapter. We summarize
6.4
Input/Output Coupling and Transducer Shading
155
by stating that the transformed actuator augmented input/output relations can be written generically in the form ci (ω) =
N
αij (ω)uj (ω),
(6.31)
j=1
where αij (ω) =
ψi (x)h (x,ξ ,ω) qj (ξ ,ω)dξ dx
(6.32)
D D
for the generic Green’s function form, and αij (ω) =
∞
fjk (ω)hik (ω)
(6.33)
k=1
for the bilinear expansion form.
6.4 Input/Output Coupling and Transducer Shading Given the plant representation (6.31) the goal of the shape control problem is to drive the distributed system output to each of the orthonormal shape functions ψi (x) independently or in any linear combination up to a discrete band limit i = L in accordance with a specified integrated mean square error and the desired shape ydesired (x,ω) =
L
βi (ω)ψi (x).
(6.34)
i=1
Equations (6.11), (6.13) and (6.34) lead to the matrix representation, ⎡
α11 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ α21 ⎢ . 1 0 0 ⎢ ⎢0⎥ ⎢1⎥ ⎢ 0 ⎥ ⎢ .. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ αL1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ β1 ⎢ ⎢ . ⎥ + β2 ⎢ . ⎥ + · · · + βL ⎢ . ⎥ = ⎢ ⎢0⎥ ⎢0⎥ ⎢1⎥ ⎢ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎢ ⎢ .. .. .. ⎢ . . . ⎣
α12 α22 .. .
· · · α1N · · · α2N .. .
αL2 · · · αLN .. .. ..
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎡
⎤ u1 ⎢ u2 ⎥ ⎢ ⎥ ⎢ .. ⎥ . ⎣. ⎦
(6.35)
un
As we have seen, the coefficients αij are determined by the spatial and temporal response characteristics of the plant as well as the chosen actuator spatial distributions. The partitioned submatrix in (6.35) or M (α) must be invertible and have full
156
6 Shape Control: Distributed Transducer Design
rank in order for any given actuator to achieve its desired output. This requires that the number of actuators N must equal or exceed the number of degrees of freedom L of the commanded output. Note also that all spatial compensation must be achieved using the chosen actuator distribution since the control vector of exogenous inputs u1 · · · uN is spatially independent (and degenerate). The coefficients of the desired output shape expansion, βi is spanned by a set of orthonormal vectors with unit vectors ei . This means that the output space of the partitioned submatrix must span the L -dimensional vector space in order to independently drive the output to an arbitrary combination of the chosen shape functions and all residual components of the output at the higher spatial frequencies i.e. i > 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)
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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
Input/Output Coupling and Transducer Shading
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
6 Shape Control: Distributed Transducer Design
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
Input/Output Coupling and Transducer Shading
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
6 Shape Control: Distributed Transducer Design
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
6 Shape Control: Distributed Transducer Design
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 Shape Control: Distributed Transducer Design
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 << ωN+1 , i.e. the resonant response of the N+lst mode, is approximated by summing the band limited representation (7.4) and the correction (7.6):
7.2
Shape Error and Feed Forward Correction
y(x,ω) =
169
N K N ϕk (x) f jk (ω)]uj (ω) + dˆ j (x,ω)uj (ω). [ λk (ω) j=1
k=1
(7.8)
j=1
This “quasi-static correction” leads to a feed-forward term in (7.8). Because we posed the shape control task in terms of an expansion of the system response in the shape functions {ψi (x)}, the Fourier coefficients of the expansion (i.e. the discrete space/continuous time transform of the output) are determined for the quasi-statically corrected response as; N K f jk (ω) [ ϕk (x)ψi (x)dx]uj (ω) λk (ω) D j=1 k=1 N + [ dˆ j (x,ω)ψi (x)dx]uj (ω).
ci (ω) =
j=1
(7.9)
D
Utilizing the definition d˜ ij (ω) ≡
dˆ j (x,ω)ψi (x)dx,
(7.10)
D
and the definition of the coefficients hik specifying the transformation from modal to shape space coordinates (6.26), Eq. (7.9) can be expressed more compactly as ci (ω) =
N K N hik f jk (ω) ]uj (ω) + d˜ ij (ω) uj (ω). [ λk (ω) j=1
k=1
(7.11)
j=1
The definition (6.39) for an output vector of Fourier coefficients, and (6.40) for a control vector, permit Eq. (7.11) to take the canonical form y = [C(ω)B + D]u, where
⎡ ⎢ ⎢ C≡⎢ ⎣ ⎡
h11 h12 · · · h1 K h21 h22 · · · h2 K .. .. .. . . . hN1 hN2 · · · hNK
1 ⎢ λ1 (ω) 0 · · · ⎢ 1 ⎢ ⎢ 0 (ω) ≡ ⎢ λ 2 (ω) ⎢ . .. ⎢ . . ⎣ . 0 0 ···
(7.12) ⎤ ⎥ ⎥ ⎥, ⎦ ⎤
0 ⎥ ⎥ ⎥ 0 ⎥ ⎥, .. ⎥ ⎥ . ⎦
1 λK (ω)
(7.13)
(7.14)
170
7 Shape Control, Modal Representations and Truncated Plants
⎡
f 11 f 21 · · · f N1
⎤
⎢ f ⎥ ⎢ 12 f 22 · · · f N2 ⎥ ⎢ ⎥ B≡⎢ . .. .. ⎥ , ⎣ .. . . ⎦ f 1 k f 2 k · · · f NK
⎡
d˜ 11 ⎢˜ ⎢ d21 ⎢ D≡⎢ . ⎢ .. ⎣ d˜ N1
d˜ 12 · · · d˜ 22 · · · .. . ˜dN2 · · ·
⎤ d˜ 1 N ⎥ d˜ 2 N ⎥ ⎥ . .. ⎥ . ⎥ ⎦ d˜ NN
(7.15)
(7.16)
Here the C matrix defines a transformation from the modal space to the shape space. The matrix is the so called modal transition matrix, while B is a matrix of modal coefficients from an expansion of the actuator distributions in the eigenfunctions. This includes actuator temporal dynamics which, if separable, can be absorbed into . The D matrix provides a feed-through quasi-static correction to the modal truncation. The output vector y consists of discrete shape transform coefficients, while the internals (e.g. (ω)) are modal. This form of C permits the shape control formalism to be applied directly to modal state space models of distributed parameter systems. Note that the presence of any mode in the shape-transformed output depends on whether ψi (x)ϕk (x)dx = 0
(7.17)
D
is satisfied, however, this integral can never vanish because of the assumptions of the analysis. This result is of course fortuitous when the shape functions are not equal to the system eigenfunctions. In many cases they are equal and this leads to a second observation. When ψi = ϕk , the orthogonality properties of the eigenfunctions reduce (7.12) to the particularly simple form y = CBu.
(7.18)
The feed-through term vanishes, since hs (x,ξ )u(ξ ,0)dξ − D
=
K ϕk (x)
k=1 ∞
k=K+1
λk (0) ϕk (x) λk (0)
ϕk (ξ )u(ξ ,0)dξ D
ϕk (ξ )u(ξ ,0)dξ , D
(7.19)
7.2
Shape Error and Feed Forward Correction
171
e.g. the quasi-static correction equals the balance of the infinite eigenfunction expansion for k > 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
7 Shape Control, Modal Representations and Truncated Plants
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
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7 Shape Control, Modal Representations and Truncated Plants
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
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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)
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7 Shape Control, Modal Representations and Truncated Plants
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
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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.
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7 Shape Control, Modal Representations and Truncated Plants
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
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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,
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7 Shape Control, Modal Representations and Truncated Plants
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 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”
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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
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7 Shape Control, Modal Representations and Truncated Plants
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)
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A Complete Dynamic Shape Control Case Study
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)
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7 Shape Control, Modal Representations and Truncated Plants
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
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A Complete Dynamic Shape Control Case Study
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
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7 Shape Control, Modal Representations and Truncated Plants
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
A Complete Dynamic Shape Control Case Study
Fig. 7.11 Modes 1 and 2 response of airfoil rib
Fig. 7.12 Modes 3 and 4 response of airfoil rib
187
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7 Shape Control, Modal Representations and Truncated Plants
Fig. 7.13 Input command profile
Fig. 7.14 Influence function due to actuator 1
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A Complete Dynamic Shape Control Case Study
Fig. 7.15 Influence function due to actuator 2
Fig. 7.16 Influence function due to actuator 3
189
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7 Shape Control, Modal Representations and Truncated Plants
Fig. 7.17 Influence function due to actuator 4
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
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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.)
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7 Shape Control, Modal Representations and Truncated Plants
Fig. 7.19 Closed-loop singular value responses for the target and recovered loop
Fig. 7.20 Basis shape 1 step response
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A Complete Dynamic Shape Control Case Study
Fig. 7.21 Basis shape 2 step response
Fig. 7.22 Basis shape 3 step response
193
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7 Shape Control, Modal Representations and Truncated Plants
Fig. 7.23 Basis shape 4 step response
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.
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7 Shape Control, Modal Representations and Truncated Plants
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 .
References
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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