Spatial Filtering for the Control of Smart Structures (eBook)

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