Contents

Series Preface vii

Preface ix

List of Symbols xix

1 Introduction to Structures 1

1.1 Examples 1

1.1.1 A Simple Structure 1

1.1.5 The Deep Space Network Antenna 3

1.1.6 The International Space Station Structure 6

1.2 Definition 6

1.3 Properties 7

2 Standard Models 13

2.1 Models of a Linear System 14

2.1.1 State-Space Representation 14

2.1.2 Transfer Function 15

2.2 Second-Order Structural Models 16

2.2.1 Nodal Models 16

2.3 State-Space Structural Models 29

2.3.1 Nodal Models 29

2.3.2 Models in Modal Coordinates 31

3 Special Models 41

3.1 Models with Rigid-Body Modes 41

3.2 Models with Accelerometers 45

3.2.1 State-Space Representation 45

3.2.2 Second-Order Representation 48

3.2.3 Transfer Function 49

3.3 Models with Actuators 50

3.3.1 Model with Proof-Mass Actuators 50

3.3.2 Model with Inertial Actuators 53

3.4 Models with Small Nonproportional Damping 54

3.5 Generalized Model 58

3.5.1 State-Space Representation 59

3.5.2 Transfer Function 59

3.6 Discrete-Time Models 60

3.6.1 State-Space Representation 61

3.6.2 Transfer Function 63

4 Controllability and Observability 65

4.1 Definition and Properties 65

4.1.1 Continuous-Time Systems 66

4.1.2 Discrete-Time Systems 68

4.1.3 Relationship Between Continuous- and

Discrete-Time Grammians 69

4.2 Balanced Representation 71

4.3 Balanced Structures with Rigid-Body Modes 73

4.4 Input and Output Gains 74

4.5 Controllability and Observability of a Structural Modal Model . . . 76

4.5.1 Diagonally Dominant Grammians 76

4.5.2 Closed-Form Grammians 79

4.5.3 Approximately Balanced Structure in Modal Coordinates . . . 80

4.6 Controllability and Observability of a Second-Order Modal Model . . 85

4.6.1 Grammians 85

4.6.2 Approximately Balanced Structure in Modal Coordinates . . . 87

4.7 Three Ways to Compute Hankel Singular Values 91

4.8 Controllability and Observability of the Discrete-Time

Structural Model 91

4.9 Time-Limited Grammians 94

4.10 Frequency-Limited Grammians 99

4.11 Time- and Frequency-Limited Grammians 103

4.12 Discrete-Time Grammians in Limited-Time and -Frequency Range 107

5 Norms 109

5.1 Norms of the Continuous-Time Systems 109

5.1.3 The Hankel Norm 112

5.2 Norms of the Discrete-Time Systems 113

5.2.3 The Hankel Norm 114

5.3 Norms of a Single Mode 115

5.3.3 The Hankel Norm 118

5.3.4 Norm Comparison 119

5.4 Norms of a Structure 120

5.4.3 The Hankel Norm 123

5.5 Norms of a Structure with a Filter 124

5.5.3 The Hankel Norm 127

5.6 Norms of a Structure with Actuators and Sensors 127

5.6.3 The Hankel Norm 132

5.7 Norms of a Generalized Structure 135

5.8 Norms of the Discrete-Time Structures 137

5.8.3 The Hankel Norm 140

5.8.4 Norm Comparison 140

6 Model Reduction 143

6.1 Reduction Through Truncation 143

6.2 Reduction Errors 145

6.2.1 H2 Model Reduction 145

6.2.2 H and Hankel Model Reduction 146

6.3 Reduction in the Finite-Time and -Frequency Intervals 147

6.3.1 Reduction in the Finite-Time Interval 148

6.3.2 Reduction in the Finite-Frequency Interval 150

6.3.3 Reduction in the Finite-Time and -Frequency Intervals . . . 151

6.4 Structures with Rigid-Body Modes 155

6.5 Structures with Actuators and Sensors 159

6.5.1 Actuators and Sensors in a Cascade Connection 159

6.5.2 Structure with Accelerometers 161

6.5.3 Structure with Proof-Mass Actuators 162

6.5.4 Structure with Inertial Actuators 165

7 Actuator and Sensor Placement 167

7.1 Problem Statement 168

7.2 Additive Property of Modal Norms 168

7.2.2 The H and Hankel Norms 169

7.3 Placement Indices and Matrices 170

7.3.1 H2 Placement Indices and Matrices 170

7.3.2 H and Hankel Placement Indices and Matrices 172

7.3.3 Actuator/Sensor Indices and Modal Indices 173

7.4 Placement for Large Structures 180

7.4.1 Actuator Placement Strategy 182

7.4.2 Sensor Placement Strategy 182

7.5 Placement for a Generalized Structure 187

7.5.1 Structural Testing and Control 187

7.5.2 Sensor and Actuator Properties 189

7.5.3 Placement Indices and Matrices 192

7.5.4 Placement of a Large Number of Sensors 193

7.6 Simultaneous Placement of Actuators and Sensors 197

8 Modal Actuators and Sensors 203

8.1 Modal Actuators and Sensors Through Modal Transformations . . 204

8.1.1 Modal Actuators 204

8.1.2 Modal Sensors 208

8.2 Modal Actuators and Sensors Through Grammian Adjustment . . 213

9 System Identification 219

9.1 Discrete-Time Systems 220

9.2 Markov Parameters 221

9.3 Identification Algorithm 221

9.4 Determining Markov Parameters 224

9.5 Examples 226

9.5.1 A Simple Structure 226

9.5.3 The Deep Space Network Antenna 232

10 Collocated Controllers 235

10.1 A Low-Authority Controller 236

10.2 Dissipative Controller 237

10.3 Properties of Collocated Controllers 239

10.4 Root-Locus of Collocated Controllers 241

10.5 Collocated Controller Design Examples 245

10.5.1 A Simple Structure 245

10.5.2 The 2D Truss 246

11 LQG Controllers 249

11.1 Definition and Gains 250

11.2 The Closed-Loop System 253

11.3 The Balanced LQG Controller 254

11.4 The Low-Authority LQG Controller 255

11.5 Approximate Solutions of CARE and FARE 257

11.6 Root-Locus 260

11.7 Almost LQG-Balanced Modal Representation 262

11.8 Three Ways to Compute LQG Singular Values 264

11.9 The Tracking LQG Controller 264

11.10 Frequency Weighting 266

11.11 The Reduced-Order LQG Controller 269

11.11.1 The Reduction Index 269

11.11.2 The Reduction Technique 271

11.11.3 Stability of the Reduced-Order Controller 272

11.11.4 Performance of the Reduced-Order Controller 274

11.11.5 Weights of Special Interest 275

11.12 Controller Design Procedure 276

11.13 Controller Design Examples 277

11.13.1 A Simple Structure 277

11.13.2 The 3D Truss 279

11.13.3 The 3D Truss with Input Filter 281

11.13.4 The Deep Space Network Antenna 283

12 Hx and H2 Controllers 287

12.1 Definition and Gains 288

12.2 The Closed-Loop System 291

12.3 The Balanced H Controller 292

12.4 The H2 Controller 294

12.4.1 Gains 294

12.4.2 The Balanced H2 Controller 296

12.5 The Low-Authority H Controller 296

12.6 Approximate Solutions of HCARE and HFARE 298

12.7 Almost H„-Balanced Modal Representation 300

12.8 Three Ways to Compute H Singular Values 301

12.9 The Tracking H Controller 301

12.10 Frequency Weighting 301

12.11 The Reduced-Order H Controller 304

12.11.1 The Reduction Index 304

12.11.2 Closed-Loop Poles 304

12.11.3 Controller Performance 306

12.12 Controller Design Procedure 307

12.13 Controller Design Examples 308

12.13.1 A Simple Structure 308

12.13.2 The 2D Truss 310

12.13.3 Filter Implementation Example 312

12.13.4 The Deep Space Network Antenna with

Wind Disturbance Rejection Properties 313

Appendices 317

A Matlab Functions 319

A.1 Transformation from an Arbitrary State-Space Representation to the Modal 1 State-Space Representation 320

A.2 Transformation from an Arbitrary State-Space Representation to the Modal 2 State-Space Representation 322

A.3 Transformation from Modal Parameters to the Modal 1 State-Space

Representation 324

A.4 Transformation from Modal Parameters to the Modal 2 State-Space

Representation 325

A.5 Transformation from Nodal Parameters to the Modal 1 State-Space

Representation 326

A.6 Transformation from Nodal Parameters to the Modal 2 State-Space

Representation 328

A.7 Determination of the Modal 1 State-Space Representation and the

Time- and Frequency-Limited Grammians 329

A.8 Open-Loop Balanced Representation 331

A.11 Hankel Norm of a Mode 333

A.12 LQG-Balanced Representation 334

A.13 H^-Balanced Representation 335

B Matlab Examples 337

B.1 Example 2.5 337

B.2 Example 3.3 341

B.3 Example 4.11 342

B.4 Example 5.3 344

B.5 Example 6.7 347

B.6 Example 7.2 348

B.7 Example 8.1 353

B.8 Example 9.1 356

B.10 Example 11.13.1 361

B.11 Example 12.13.2 365

C Structural Parameters 371

C.1 Mass and Stiffness Matrices of the 2D Truss 371

C.2 Mass and Stiffness Matrices of the Clamped Beam Divided into

15 Finite Elements 373

C.3 State-Space Representation of the Deep Space Network Antenna . 376

References 379

Index 389

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