Vibration control of active structures : an introduction.v. 96. : 2nd

Preface to the second edition xv Preface to the first edition xvii 1 Introduction 1 5 7 7 9 10 12 13 15 17 18 20 20 22 25 26 30 31 33 34 37 38 40 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.3.1 1.3.2 Active versus passive Smart materials and structures Control strategies Feedback Feedforward The various steps of the design Organization of the book References Problems 2 Some concepts of structural dynamics 2.1 2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.5 2.6 2.7 2.8 3 Actuators, piezoelectric materials, and active structures 3.1 3.2 3.3 Introduction Proof-mass actuator Reaction wheels and gyrostabilizers vii Equation of motion of a discrete system Vibration modes Modal decomposition Structure without rigid body modes Structure with rigid body modes Example Transfer function of collocated systems Continuous structures Guyan reduction References Problems 1 17 37 viii Vibration control of active structures 3.4 Piezoelectric actuators 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.5 3.6 3.7 3.8 3.8.1 3.8.2 3.9 3.10 3.11 4 Collocated versus non-collocated control 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5 Active damping with collocated pairs 5.1 5.2 5.2.1 5.3 5.3.1 5.3.2 5.3.3 5.3.4 5.4 5.4.1 5.4.2 5.5 5.5.1 Constitutive equations Linear actuator Laminar actuator Laminar sensor Example: Tip displacement of a cantilever beam Spatial modal filters Passive damping with piezoceramics Active cantilever beam Active truss Piezoelectric shell Two-dimensional constitutive equations Kirchhoff shell Finite element formulation References Problems Introduction Pole-zero flipping Collocated control Non-collocated control Notch filter Pole-zero flipping in the structure Effect on the Bode plots Relation to the mode shapes The role of damping References Problems Introduction Direct Velocity Feedback Lead compensator Acceleration feedback Direct Velocity Feedback Second order filter SISO system with many modes Multidimensional case Positive Position Feedback SISO system Multidimensional case Integral Force Feedback Modal damping 40 41 46 48 51 52 53 54 55 58 63 64 65 68 70 73 75 76 78 80 80 82 84 84 87 88 88 91 91 93 94 96 97 97 98 100 101 101 102 103 104 75 CONTENTS ix 5.6 5.7 5.8 5.6.1 5.6.2 Remarks Controllability, observability Actuator and sensor dynamics References Problems 6 Active vibration isolation 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.5.1 6.7.1 6.8 6.9 6.10 6.11 7 State space approach 7.1 7.2 7.2.1 7.2.2 7.2.3 7.3 7.4 7.5 7.5.1 7.5.2 7.6 7.7 7.7.1 7.8 7.8.1 7.8.2 7.9 7.10 7.10.1 7.11 Introduction Passive isolator The “sky-hook” damper Force feedback Flexible clean body Free-free beam with isolator 6 d.o.f. isolator Decentralized control of the 6 d.o.f. isolator Remarks Pointing control Vehicle suspension References Problems Introduction State space description Single degree of freedom oscillator Flexible structure Inverted pendulum System transfer function 7.3.1 Poles and zeros Pole placement by state feedback 7.4.1 Example: oscillator Linear Quadratic Regulator Symmetric root locus Inverted pendulum Observer design Kalman Filter Inverted pendulum Reduced order observer Oscillator Inverted pendulum Separation principle Transfer function of the compensator The two-mass problem References 107 107 107 109 110 113 114 116 118 119 121 124 127 128 129 131 133 134 137 139 139 140 141 142 144 145 146 148 149 149 151 153 154 155 156 156 157 158 159 162 137 113 x 7.12 Problems 163 8 Analysis and synthesis in the frequency domain 8.1 8.2 Gain and phase margins Nyquist criterion 8.2.1 8.2.2 Cauchy’s principle Nyquist stability criterion 8.3 8.4 Nichols chart Feedback specification for SISO systems 8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 Sensitivity Tracking error Performance specification Unstructured uncertainty Robust performance and robust stability 8.5 8.6 8.7 8.8 Bode gain-phase relationships The Bode Ideal Cutoff Non-minimum phase systems Usual compensators 8.8.1 8.8.2 8.8.3 8.8.4 8.8.5 System type Lead compensator PI compensator Lag compensator PID compensator 8.9 8.10 References Problems 9 Optimal control 9.1 9.2 9.3 9.4 9.4.1 9.5 9.6 9.7 9.8 9.9 9.9.1 9.10 9.11 9.12 9.13 9.13.1 9.14 Introduction Quadratic integral Deterministic LQR Stochastic response to a white noise Remark Stochastic LQR Asymptotic behaviour of the closed-loop Prescribed degree of stability Gain and phase margins of the LQR Full state observer Covariance of the reconstruction error Kalman-Bucy Filter (KBF) Linear Quadratic Gaussian (LQG) Duality Spillover Spillover reduction Loop Transfer Recovery (LTR) 165 165 166 166 167 170 171 171 172 173 174 175 178 181 183 185 185 187 187 189 189 189 190 193 193 193 194 196 197 197 198 200 201 202 204 204 205 205 206 209 210 Vibration control of active structures 9.15 Integral control with state feedback 9.16 Frequency shaping 9.16.1 Frequency-shaped cost functionals 9.16.2 Noise model References Problems 10 Controllability and Observability 10.1 Introduction 211 212 212 215 216 217 221 222 222 224 224 226 226 227 228 229 230 230 231 232 233 234 235 235 237 237 237 238 240 241 245 246 247 248 249 249 249 251 251 252 10.1.1 Definitions 10.2 10.3 Controllability and observability matrices Examples 10.3.1 10.3.2 10.3.3 A cart with two inverted pendulums Double inverted pendulum Two d.o.f. oscillator 10.4 State transformation 10.4.1 Control canonical form 10.4.2 10.4.3 Left and right eigenvectors Diagonal form 10.5 10.6 10.7 10.8 10.9 PBH test Residues Example 9.17 9.18 Sensitivity Controllability and observability Gramians Relative controllability and observability 10.10.1Internally balanced coordinates 10.11 Model reduction 10.11.1 10.11.2 10.11.3 Transfer equivalent realization Internally balanced realization Example 10.12 10.13 References Problems 11 Stability 11.1 Introduction 11.1.1 Phase portrait 11.2 Linear systems 11.2.1 Routh-Hurwitz criterion 11.3 Liapunov’s direct method 11.3.1 11.3.2 11.3.3 11.3.4 11.3.5 Introductory example Stability theorem Asymptotic stability theorem Lasalle’s theorem Geometric interpretation CONTENTS xi 10.10 221 245 11.3.6 Instability theorem xii Vibration control of active structures 11.4 11.5 11.6 11.7 11.8 11.9 Liapunov functions for linear systems Liapunov’s indirect method An application to controller design Energy absorbing controls References Problems Semi-active control 253 254 255 256 257 259 259 12.1 12.2 12.3 12.4 12.5 12.6 Introduction Magneto-rheological (MR) fluids MR devices Semi-active control Open-loop control Feedback control 12.6.1 12.6.2 12.6.3 Continuous control On-off control Force feedback 12.7 12.8 References Problems 13 Applications 263 263 264 266 267 268 268 268 273 274 275 275 13.1 Digital implementation 13.1.1 13.1.2 13.1.3 13.1.4 Sampling, aliasing and prefiltering Zero-order hold, computational delay Quantization 13.2 Active damping of a truss structure 13.2.1 13.2.2 Actuator placement Implementation, experimental results 13.3 Active damping generic interface 13.3.1 13.3.2 13.3.3 Active damping Experiment Pointing and position control 13.4 Active damping of a plate 13.4.1 Control design 13.5 Active damping of a stiff beam 13.5.1 System design 13.6 The HAC/LAC strategy 13.6.1 13.6.2 13.6.3 Wide-band position control Compensator design Results 13.7 Volume displacement sensors 13.7.1 QWSIS sensor 277 277 278 279 280 281 282 282 284 286 287 288 290 290 291 293 294 295 297 299 299 302 303 Discretization of a continuous controller 12 CONTENTS xiii 13.7.2 13.7.3 13.7.4 Discrete array sensor Spatial aliasing Distributed sensor 13.8 13.9 References Problems 14 Tendon Control of Cable Structures 14.1 14.2 14.3 14.4 14.5 14.6 Introduction Tendon control of strings and cables Active damping strategy Basic Experiment Approximate linear theory Application to space structures 14.6.1 14.6.2 14.6.3 14.6.4 Guyed truss experiment JPL-MPI testbed Free floating truss experiment Microvibrations 14.7 Application to cable-stayed bridges 14.7.1 14.7.2 14.7.3 Laboratory experiment Control of parametric resonance Large scale experiment 14.8 References Bibliography Index 347 361 305 308 309 316 319 321 321 323 324 325 327 329 329 331 331 334 335 335 336 338 344