Spatial Filtering for the Control of Smart Structures

An Introduction
1. Auflage 2009
ISBN: 978-3-642-03804-4
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 209 Seiten

What follows is my personal perspective on early events that played a signi?cant role in the formation of the ?eld now known as Smart Structures. It is by no means meant to be all inclusive or de?nitive 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 ?exible mirror concepts. These proposed systems r- resented 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 ap- tures on the order of 100 m.

Autoren/Hrsg.

Inhaltsverzeichnis

1;An Historical Prologue and Preface;6
2;Contents;11
3;1 Smart Structure Systems;14
3.1;1.1 Introduction;14
3.2;1.2 Smart Structure Architecture and Performance;14
3.3;1.3 Smart Material Transducer Considerations;18
3.4;1.4 Continuum Representation of Smart Structures;22
3.5;1.5 Time Domain Representation of Smart Structure Models;28
3.6;1.6 Organization of the Book;31
3.7;1.7 Problems;32
4;2 Spatial Shading of Distributed Transducers;38
4.1;2.1 Introduction;38
4.2;2.2 Spatial Shading of Distributed Transducers;39
4.2.1;2.2.1 Design by Example: A Center of Pressure Sensor;39
4.2.2;2.2.2 Approximating Shaded Apertures;42
4.3;2.3 Analytical Modeling of Spatial Shading Functions for Distributed Transducers;47
4.3.1;2.3.1 A Compact Analytical Representation of Distributed Transducers;48
4.3.2;2.3.2 Two Dimensional Representation of Distributed Transducers with Nearly Arbitrary Spatial Shading;58
4.3.2.1;2.3.2.1 The Completeness of Singularity Functions;58
4.3.2.2;2.3.2.2 Nearly Arbitrary Spatial Shading Distributions Using Singularity Functions;60
4.3.2.3;2.3.2.3 Extension of Distributional Chain Rule to Multi-Dimensions;63
4.4;2.4 Application to Two-Dimensional Shading Using Skew Angle;64
4.4.1;2.4.1 Applications Including Finite Skew Angle of Material Axes;68
4.5;2.5 Summary;72
4.6;2.6 Notes;73
4.7;2.6 Problems;74
5;3 Active Vibration Control with Spatially Shaded Distributed Transducers;81
5.1;3.1 Introduction;81
5.2;3.2 Control System Synthesis Based on the Lyapunov Direct Method;81
5.3;3.3 Control System Synthesis for Beams;83
5.3.1;3.3.1 Collocated Distributed Transducers and Lyapunov Control;86
5.3.2;3.3.2 Performance Limitations of Control Designs with Shaded Distributions;88
5.3.3;3.3.3 Performance Limitations of Uniformly Shaded Transducers;88
5.3.4;3.3.4 Performance Limitations of Linearly Shaded Transducers;92
5.3.5;3.3.5 Design Guidelines on Spatial Shading for Vibration Control;93
5.4;3.4 Control System Synthesis for Plates;95
5.4.1;3.4.1 Performance Limitations of Uniformly Shaded Actuators for Plates;98
5.4.2;3.4.2 Performance Limitations of Non-uniformly Shaded Actuators for Plates;104
5.4.3;3.4.3 The Unique Compatibility of Distributed Transducers for Arbitrary Spatial Shadings;106
5.5;3.5 Summary;107
5.6;3.6 Notes;107
5.7;3.6 Problems;108
6;4 Multi-Dimensional Transforms and MIMO Representations of Smart Structures;112
6.1;4.1 Introduction;112
6.2;4.2 Convolution and the Spatially Distributed Plant;116
6.2.1;4.2.1 Green's Function Representations for Temporally Stationary Systems;118
6.2.1.1;Right Convolution Form ;119
6.2.1.2;Right and Left Convolution Form;119
6.2.1.3;Example: Green's Function for a String;120
6.2.1.4;Damped Convolution Form;121
6.2.1.5;Example: Green's Function for a Bernoulli-Euler Beam;122
6.2.1.6;The Symmetric Form of the Green's Function;122
6.3;4.3 Multi-Input Multi-Output (MIMO) Representations of Smart Structures;124
6.3.1;Example: String on an Elastic Foundation ;128
6.4;4.4 Problems;129
7;5 Performance Measures for Smart Structures with MIMO Representations;131
7.1;5.1 Introduction;131
7.2;5.2 Performance Metrics;131
7.3;5.3 Assessment of Performance Metrics Using Singular Values;134
7.3.1;5.3.1 Command Following;134
7.3.2;5.3.2 Disturbance Rejection;137
7.3.3;5.3.3 Sensor Noise;138
7.4;5.4 Metrics for Controllability and Observability;138
7.4.1;5.4.1 Controllability;139
7.4.2;5.4.2 Observability;140
7.5;5.5 Example: Active Damping of a Simply Supported Beam;140
7.5.1;5.5.1 Spatially Uniform Actuator Distributions;141
7.5.2;5.5.2 Linear or ''Ramp'' Actuator Distributions;143
7.6;5.6 Metrics for Achieving Stability and Robustness for Control of Smart Structures;146
7.6.1;5.6.1 Additive Error Uncertainty;149
7.6.2;5.6.2 Multiplicative Error Uncertainty;151
7.7;5.7 Summary;151
7.8;5.8 Notes;152
7.9;5.8 Problems;152
8;6 Shape Control: Distributed Transducer Design;155
8.1;6.1 Introduction;155
8.2;6.2 Shape Control and the Notion of Discrete Spatial Bandwidth;156
8.2.1;6.2.1 Orthonormal Expansions and the Discrete Spatial Transform;157
8.2.2;6.2.2 Minimization of the Integrated Mean Square Profile Error;159
8.3;6.3 Plant Representations in Terms of an Expansion Basis Set;161
8.3.1;6.3.1 The Generic Green's Function Representation;161
8.3.2;6.3.2 The Symmetric Green's Function Representation;163
8.4;6.4 Input/Output Coupling and Transducer Shading;165
8.4.1;6.4.1 The Singular Value Decomposition and Performance Metrics for Shape Control;166
8.4.1.1;Example: Shape Control of a Simply Supported Beam (Actuator Design) ;168
8.5;6.5 Spatially Distributed Sensors and Shape Estimation;172
8.5.1;Example: Shape Control of a Simply Supported Beam (Sensor Design) ;174
8.6;6.6 Summary;175
8.7;6.7 Problems;176
9;7 Shape Control, Modal Representations and Truncated Plants;177
9.1;7.1 Introduction;177
9.2;7.2 Shape Error and Feed Forward Correction;177
9.2.1;7.2.0 Example 1: Shape Control of a Non-dimensional String (Feed Forward Correction);181
9.3;7.3 A Complete Dynamic Shape Control Case Study;181
9.3.1;7.3.1 Case Background;183
9.3.2;7.3.2 Airfoil Shapes and the Discrete Spectrum Parameterization;184
9.3.3;7.3.3 The Concept of Eigenfoils;186
9.3.4;7.3.4 Morphing Airfoil Design Considerations;188
9.3.5;7.3.5 Actuator Placement and Input/Output Coupling;189
9.3.6;7.3.6 Morphing Airfoil Rib: Discrete Parameterization and the System Model;192
9.3.7;7.3.7 State Space Canonical Form;193
9.3.8;7.3.8 Morphing Airfoil Closed Loop Shape Controller Synthesis;194
9.3.9;7.3.9 Morphing Airfoil Closed Loop Shape Control Simulation;201
9.4;7.4 Summary;205
9.5;7.5 Problems;205
10;References;208
11;Index;211