اسم المؤلف
Ashish Tewari
التاريخ
16 نوفمبر 2017
المشاهدات
777
التقييم

Modern Control Design With MATLAB and SIMULINK
Ashish Tewari
Indian Institute of Technology, Kanpur, India
Contents
Preface xiii
1. Introduction 1
1.1 What is Control? 1
1.2 Open-Loop and Closed-Loop Control Systems 2
1.3 Other Classifications of ControlSystems 6
1.4 On the Road to Control System Analysis and Design 10
1.5 MATLAB, SIMULINK, and the Control System Toolbox 11
References 12
2. Linear Systems and ClassicalControl 13
2.1 How Valid is the Assumption of Linearity? 13
2.2 Singularity Functions 22
2.3 Frequency Response 26
2.4 Laplace Transform and the Transfer Function 36
2.5 Response to Singularity Functions 51
2.6 Response to Arbitrary Inputs 58
2.7 Performance 62
2.8 Stability 71
2.9 Root-Locus Method 73
2.10 Nyquist Stability Criterion 77
2.11 Robustness 81
2.12 Closed-Loop Compensation Techniques for Single-Input, Single-Output Systems 87
2.12.1 Proportional-integral-derivative compensation 88
2.13 Multivariable Systems 105
Exercises 115
References 124
3. State-Space Representation 125
3.1 The State-Space: Why Do I Need It? 125
3.2 Linear Transformation of State-Space Representations 140viii CONTENTS
3.3 System Characteristics from State-Space Representation 146
3.4 Special State-Space Representations: The Canonical Forms 152
3.5 Block Building in Linear, Time-Invariant State-Space 160
Exercises 168
References 170
4. Solving Hie State-Equations 171
4.1 Solution of the Linear Time Invariant State Equations 171
4.2 Calculation of the State-Transition Matrix 176
4.3 Understanding the Stability Criteria through the State-Transition Matrix 183
4.4 Numerical Solution of Linear Time-Invariant State-Equations 184
4.5 Numerical Solution of Linear Time-Varying State-Equations 198
4.6 Numerical Solution of Nonlinear State-Equations 204
4.7 Simulating Control System Response with SIMUUNK 213
Exercises 216
References 218
5.1 Design: Classical vs. Modern 219
5.2 Controllability 222
5.3 Pole-Placement Design Using Full-State Feedback 228
5.3.1 Pole-placement regulator design for single-input plants 230
5.3.2 Pole-placement regulator design for multi-input plants 245
5.3.3 Pole-placement regulator design for plants with noise 247
5.3.4 Pole-placement design of tracking systems 251
5.4 Observers, Observability, and Compensators 256
5.4.1 Pole-placement design of full-order observers and compensators 258
5.4.2 Pole-placement design of reduced-order observers and compensators 269
5.4.3 Noise and robustness issues 276
Exercises 277
References 282
6. Linear OpHmal Control 283
6.1 The Optimal Control Problem 283
6.1.1 The general optimal control formulation for regulators 284
6.1.2 Optimal regulator gain matrix and the riccati equation 286
6.2 Infinite-Time Linear Optimal Regulator Design 288
6.3 Optimal Control of Tracking Systems 298
6.4 Output Weighted Linear Optimal Control 308
6.5 Terminal Time Weighting: Solving the Matrix Riccati Equation 312
Exercises 318
References 321CONTENTS ix
7. Kalman Filters 323
7.1 Stochastic Systems 323
7.2 Filtering of Random Signals 329
7.3 White Noise, and White Noise Filters 334
7.4 The Kalman Filter 339
7.5 Optimal (Linear, Quadratic, Gaussian) Compensators 351
7.6 Robust Multivariable LOG Control: Loop Transfer Recovery 356
Exercises 370
References 371
8. Digital Control Systems 373
8.1 What are Digital Systems? 373
8.2 A/D Conversion and the ^-Transform 375
8.3 Pulse Transfer Functions of Single-Input, Single-Output Systems 379
8.4 Frequency Response of Single-Input, Single-Output Digital Systems 384
8.5 Stability of Single-Input, Single-Output Digital Systems 386
8.6 Performance of Single-Input, Single-Output Digital Systems 390
8.7 Closed-Loop Compensation Techniques for Single-Input, Single-Output Digital
Systems 393
8.8 State-Space Modeling of Multivariable Digital Systems 396
8.9 Solution of Linear Digital State-Equations 402
8.10 Design of Multivariable, Digital Control Systems Using Pole-Placement:
Regulators, Observers, and Compensators 406
8.11 Linear Optimal Control of Digital Systems 415
8.12 Stochastic Digital Systems, Digital Kalman Filters, and Optimal Digital
Compensators 424
Exercises 432
References 436
9. Advanced Topics in Modern Control 437
9.1 Introduction 437
9.2 Hoq Robust, Optimal Control 437
9.3 Structured Singular Value Synthesis for Robust Control 442
9.4 Time-Optimal Control with Pre-shaped Inputs 446
9.5 Output-Rate Weighted Linear Optimal Control 453
9.6 Nonlinear Optimal Control 455
Exercises 463
References 465
Appendix A: Introduction to MATLAB, SIMULINK and the
Control System Toolbox 467х CONTENTS
Appendix В: Review of Matrices and
Linear Algebra 481
Appendix C: Mass, Stiffness, and Control Influence Matrices of
the Flexible Spacecraft 487
Index
Index
Terms in italic refer to MATLAB commands
abs, 34
Abstol, 208
Acceleration:
angular, 21
normal, 20, 21, 192
acker, 236
Ackermann’s pole-placement formula,
233
Active circuit, 95
Active control, 453
Active flutter-suppression, 440, 444
Actuator, 122-123, 228
Aerodynamic:
control surface, 201
forces and moments, 201
model, 463
parameters, 461-463
properties, 201, 211
Aeroelasticity, 191
Aerospace vehicles, 11
Aileron, 98, 163
Aircraft:
bomber, 191
fighter, 98, 211
lateral dynamics, 217
longitudinal dynamics, 43, 143
roll dynamics, 98
wing-rock phenomenon, 211
Algorithm, 185
Aliasing, 384
Amplifier, operational, 95
Amplifier-motor, 139
Amplitude, 27
Analog system, 9
Analog-to-digital (A/D) converter,
374-376
angle, 34
Angle of attack, 43, 144, 211
Antenna, 65, 67
Arbitrary function, 26
are, 290
Asymptotic stability, 72
Attenuation, 329
augtf, 441
Automatic control, 1
Autopilot, 15
Azimuth angle, 18, 320
Backlash, 215
Balanced bridges, 227
Band-pass filter, 329
Bandwidth, 34, 334
Bandwidth, loop transfer recovery,
364-366
Bang-bang input, 447
Bank angle, 211
Bellman, 456
Bellman equation, 458
Bilinear form, 416, 485
Blakelock, 45
Block-diagonal structure, 442
Block diagram, 2-4
bode, 31
Bode plot, 30
Boundary conditions, 19
Bounded input, bounded output, 71
Breaks, 47
Burn-out time, 201, 202
Butterworth pattern, 240-243496 INDEX
c2d,c2dm, 188
Canard, 192
Cancellation of poles with zeros, 99,
127, 220, 221
canon, 155
Canonical forms:
controller companion form, 136-137,
152
Jordan canonical form, 152-154
observer companion form, 137-138,
152
Center of gravity, 163, 201
Capacitor, 5, 15, 32, 116
Cauchy’s theorem, 79
Causal system, 335
Cayley-Hamilton theorem, 485
Chaotic system, 7
Characteristic equation, 40-44
Characteristic polynomial, 44, 137
Characteristic modes, 45
Chemical plant:
large, 100
multivariable, 112, 113
Classical control theory, 27
Closed contour, 78
Closed-loop system, 3-5
Closure velocity, 316
Cofactor, 484
Colored noise, 336
Commutation of matrices, 175
Compensator, 64
Compensation techniques:
lag compensation, 96-100
102-105
PID compensation, 88-96
Complimentary solution, 28
Complementary sensitivity matrix, 439
Complex differentiation, 37
Complex integration, 37
Complex space, 27
cond, 159
Condition number, 159
Conditional mean, 340
Conformal mapping, 79
conj, 53
Conjugate pair, 42, 53
Constraint, 284
Constrained minimization, 284
Control, 1
Control cost matrix, 284
Control input, 1
Control system, 1
Controllability, 222-228
Controllability test matrix, 224
Controllability test theorem, 224
Control-law, 228
Controller, 1
conv, 44
Convergent series, 185
Convolution integral, 59
Corrected state, 410, 426
Correlation, 324
Correlation matrix, 324
Co-state vector, 419, 457
Coulomb friction, 215
cov, 328
Covariance matrix, 324
Covariance matrix, conditional, 340
Critical damping, 41
Critical frequencies, 401
Cross-correlation matrix, 343,
Cross-spectral density matrix, 343
ctrb, 225
Current observer, 409
Cut-off frequency, 329
damp, 44
Damping, 41
Damping ratio, 41
dbode, 385
DC gain, 66
degain, 66
DC gain, digital, 392
ddamp, 389, 393
ddegain, 392
Decoupling the state-equations, 141,
151, 158, 172, 173, 175-177
Delay element, 397
demux block, 215
Denominator polynomial, 40
Design of closed-loop systems:
frequency domain, single-input,
single-output, 88-105
pole-placement, single-input,
regulator, 230-245
pole-placement, multi-input, regulator,
245-247INDEX 497
pole-placement, single-output,
observers and compensators,
258-275
optimal, multi-input, multi-output,
regulator, 284-298, 308-312
optimal, multi-input, multi-output,
tracking system, 298-308
optimal, multi-input, multi-output,
terminal time problem, 312-318
optimal, multi-input, multi-output,
LQG compensator, 356-369
frequency domain, single-input,
single-output, digital, 393-395
pole-placement, regulator, observer,
and compensator, digital
406-415
optimal, multi-input, multi-output,
digital, 415-432
Design parameters, 219, 228
Design tolerance, 441
Desired output, 2-5
det, 144, 234
Determinant, 112, 143, 483
Deterministic system, 7, 323
Detectable system, 258
Diagonal matrix, 154, 483
Dirac delta function, 22
Direct connection, 147
Difference equation, 397
Differential equation, 9, 13
Differential operator, 28
Direct transmission matrix, 147
Digital:
approximation to analog system,
187-188, 400-402
Bode plot, 385
filter, 425
Kalman filter, 425
multivariable systems, 396
Nyquist plot, 387-389
state-equations, 397
Digital signal, 187
Digital system, 9, 187, 373
Digital-to-analog (D/А) converter, 374
dimpulse, 404
dinitial, 404, 408, 423
Discontinuous function, 36
Distillation column, 278
Distinct eigenvalues, 176
Distortion, filtered signal, 329-332
Distributed parameter system, 9, 295
dlqe, 428
dlqew, 428
dlqr, 421
dlsim, 404
dnyquist, 387
Dominant poles, 64, 92, 221
doub, 208-210
Double pendulum, 7,8, 208
dreg, 414, 415
Dryden turbulence model, 337
dsigma, 432
dstep, 404
Dynamic programming, 416
Eccentricity of orbit, 19
eig, 150
Eigenstructure assignment, 246
Eigenvalues, 44, 148, 484
Eigenvalue problem, 148
Eigenvectors, 148, 485
Electrical network:
active, 95
nonlinear, 116
passive, 95-97
second order, 15, 32
Elevator, 43, 144, 192
Elliptic filter, 332, 333
Encirclement, 79
Engine RPM governor, 6
Ensemble statistical properties, 325
Equilibrium point, 15-17
Ergodic system, 325
Error signal, 4, 62
estim, 262
Estimated state-vector, 259
Estimation error, 259
Excessive stability, 250
Excitation frequency, 26, 27
exp, 179
Expected value, 324
Exponential decay, 25, 41
Exponential of a matrix, 174
expm, expml, expm2, 185
ехртЗ, 179
Feedback controller, 76
Feedback loop, 3, 4
Feedback signal, 107
Feedback transmission, 107
Feedforward controller, 5, 6, 452
Feedforward gain matrix, 263
ffb 327498 INDEX
Fictitious controller, 443
Filter, 329
Final value theorem, 37, 378
Finite-element model, 295
First-order actuator, 216
First-order hold, 188
Flexible modes, 448
Flexible aircraft, 191, 440
Flexible spacecraft, 295
Flutter, 440
Forward transmission, 107
Fourier transform:
Imaginary axis, 27, 28
Imaginary sampler, 379
impgen, 196, 197
impresp, 54, 55
impulse, 56, 196
Impulse response, 51
Impulse response matrix, 334
Inductor, 5, 15, 32, 116
inf, 91
Infimum function, 443
Infinite series, 184
Infinite-time optimal control, 288
definition, 38-39
discrete, 326
inverse, 39
inverse discrete, 326
Frequency domain, 39
Frequency prewarping, 188, 401
Frequency response, 29
Frequency response matrix, 336
Full-state feedback, 228
Full-order observer, 258
Full-order compensator, 263
Fuselage bending mode, 191
Inherently digital system, 373, 379
initial, 237
Initial condition, 7, 13, 173
Initial response, 71
Initial-value theorem, 37, 378
Input shaping, 446
inshape, 448-450
Interpolation, 32, 187
inv, 144
Inverse of a square matrix, 112, 141, 484
Inverse transformation, 140
Inverted pendulum on a moving cart, 115
Iterative solution, 205
Gain, 30
Gain crossover frequency, 84
Gain margin, 82, 83
Gravitational constant, 18
Guidance law, 20
Gyroscope, 94, 95
Jerky motion, 453
Jordan block, 154
Kalman filter, 262, 340
kron, 448, 449
Kronecker delta function, 426
H2-norm, 438
Hoc-norm, 438
Hoc optimal control, 437
Hamiltonian function, 457
Hamiltonian matrix, 313, 419
Harmonic input, 26, 27
Hermitian of a complex matrix, 358, 485
High-gain feedback, 251
High-pass filter, 329
hinfopt, 440, 441
Homogeneous differential equation, 17
Homogeneous state-equation, 173
Hybrid methods, 205
Lag circuit, 95, 97
Laplace domain, 39
Laplace transform, 36
Laplace transform, inverse, 39
Laplace variable, 36
Laser-guided missile, 20
Lateral plane, 201
Leibniz rule, 285
Limit cycle motion, 211-213
Linear combination, 13
Linearly dependent, 178
Ideal differentiator, 95
Ideal sampler, 375
Identity matrix, 112, 483
ifft, 327
Ill-conditioned square matrix, 159
compensator, 351
Linear system, 9, 14
Linear, time-invariant system, 14
Linear, time-varying system, 14INDEX 499
Linear transformation, 140, 175
Lipschitz condition, 205
Logarithmic scale, 30
loglO, 34
logspace, 31
Longitudinal axis, 201
Longitudinal plane, 201
Loop shaping, 361
Loop transfer recovery (LTR), 361-368
Low-pass filter, 329
Iqe, lqe2. Iqew, 344
Iqr, lqr2. 290
!qry, 310
Isim, 194
LTI object, 31, 45
Ititr, 189
LTI Viewer, 105, 478
Lumped parameter system, 9, 13, 14
Lyapunov function, 457
Magnitude, 29
Mapping, 77
march, 189, 190
margin, 84
Mass, 16, 18, 117
MATLAB:
beginning, 468-471
Control System Toolbox, 12, 31, 477,
478
introduction, 11, 467
p -Analysis and Synthesis Toolbox,
444, 478
performing matrix operations,
471-474
programming, 475-477
Robust Control Toolbox, 363, 438,
440, 444, 478
matrice, 314
Matrices and their properties, 481 -486
Matrix differential equation, 286
Matrix norm, 185, 358
Maximum overshoot, 4, 5
mean, 328
Mean state-vector, 324
Mean-square value, 326
Mean value theorem, 23
Measurement noise, 87, 247, 294, 311,
312, 339, 357, 369
Measurement noise vector, 339
M-file, 12, 475
Mid-point frequency, 327
Minimal realization, 280
Minimum-phase system, 48, 338, 362
Minimum principle, 420, 447
Miss distance, 316
Missile, 20, 162-168, 201-204
Missile guidance:
beam-rider, 20
command line-of-sight, 21
misstimv, 202, 204
Mixed-sensitivity matrix, 439
Modal matrix, 420
Moments of inertia, 201
Multi-input plant, 245
Multi-output observer, 262
Multiplicity, 40
Multi-step method, 205
Multivariable system, 105
musyn, 444
/z-synthesis, 443
NaN, 91
Natural frequency, 32-35, 41
Negative feedback control system, 107
nichols, 86
Nichols plot, 85
nmass, 450, 451
Noise, 3, 81
Nominal parameters, 462
Nominal plant, 442
Nonlinear feedback control, 455
Nonlinearity, 458
Nonlinear optimal control, 456
Nonlinear state-equations, 205
Nonlinear system, 9, 14, 16, 18, 130
Non-minimum phase system, 48, 337,
365-367
Normal distribution, 327
Numerator polynomial, 31
Numerical accuracy, 205
Numerical stability, 205
Numerical solution:
linear, time-invariant state-equations,
184-198
linear, time-varying state-equations,
198-204
nonlinear state-equations, 204-213
nyquist. 79
Nyquist frequency, 384
Nyquist plot, 77
Nyquist stability criterion, 79500 INDEX
Objective function, 284, 312, 415, 446,
454
Observability, 256
Observability test matrix, 257
Observability test theorem, 257
Observer:
full-order, 256, 258
reduced-order, 256, 269
ode!5s, odelI3, ode23, ode23s, ode23t,
ode23tb, ode45, 208
Open-loop control system, 2, 81
Open-loop frequency response, 77
Open-loop transfer function, 70
Optimal control, 283
Optimal feedback gain matrix, 300
Optimal observer (see Kalman filter)
Optimal regulator gain matrix, 286
Output, 1, 2
Output equation, 126, 131
Output-rate weighted linear optimal
control, 453
Output vector, 105, 131
Output weighted linear optimal control,
308
Overdamped response, 42
parallel, 108
Parallel connection of sub-systems, 107
Parameter estimation, 262
Parameters of a system, 14
Partial differential equation, 9
Partial differentiation, 285
Partial fraction expansion, 50, 51
Particular solution, 28
Passband, 332
Path of a satellite, 18
Peak in gain plot, 32, 34, 47
Peak time, 62, 63
Penalty function, 457
Pendulum, 16, 126
Percentage residual vibration, 452
Perfect measurement, 228
Performance, 62
Performance parameters, 62
Performance robustness, 87
Phase, 29
Phase crossover frequency, 83
Phase-lag, 96
Phase margin, 82, 83
Phasor, 29
Phase-plane plot, 211-213
Phugoid mode, 44-48, 58, 121, 191
Piecewise continuous function, 36
Pitch-acceleration, 454
Pitch-angle, 43, 144
Pitch dynamics, 433
Pitch-rate, 144, 192
Plant, 1
place, 236
Polar coordinates, 30
Polar plot, 30, 35
Poles of a transfer function, 40
Poles of a transfer matrix, 149
Pole-placement approach, 228
pole, 72
poly, 234
Positive definite matrix, 289
Positive semi-definite matrix, 287
Positive feedback control system, 107
Power series expansion, 390
Power spectral density matrix, 325
Power spectrum, 338
Predicted state, 410
Prediction error vector, 426
Predictor-corrector difference equations,
426
Predictor-corrector method, 205
Pre-shaped inputs, 446
Probability distribution, 329
Probability of a state, 323
Probabilistic system (see Stochastic
system)
Process noise, 87, 339, 349
Process noise vector, 339, 357
Product of matrices, 107, 482
Propagation in time, 397
Proper system, 13, 40
Proportional-integral-derivative (PID)
compensator, 88
Proportional-plus-integral (PI)
compensator, 89
Pulse:
rectangular, 24, 25, 194
sawtooth, 25, 194
Pulse transfer function, 379
Pulse transfer matrix, 399
pzmap, 72
Quantization, 432
Rudder, 163, 217, 306
Runge-Kutta method, 205-208
rand, 329
Random system, 323
randn, 327
rank, 225
Rank of a matrix, 224
Rational function, 40, 51, 396
Rational matrix, 146
Real axis, 27
Real differentiation, 37
Real integration, 37
Reduced-order compensator, 273-275
reg, 352
Regulator, 4, 228
RelTol, 208
Remainder series, 184
Repeated poles, 51, 72
Residual, 259
residue, 51
Residues, 50
Resistance, 5, 32
Resolvent, 181
Resonance, 32
response, 60, 61
Return difference, 73, 82
Return difference matrices, 112, 357
Return ratio, 73
Return ratio matrices, 112, 357
ric, 290
Riccati equation:
Sampled data analog system, 379
Sampling interval, 373
Sampling process, 373
Sampling property, 23, 187
Sampling rate, 374
Satellite, 18
Scalar measure of a matrix, 112
Scaling factor, 74
Scaling in the z-plane, 377
Schematic diagram, 132, 397
Sensitivity, 81
Sensitivity matrix of the output, 439
Sensor, 228
Separation principle, 262, 273
series, 108, 161
Series connection of sub-systems, 107
Settling time, 62
Short-period mode, 44, 191
Sideslip angle, 211, 306
Sideslip-rate, 211
sigma, 363
Similarity transformation, 146
band limited white-noise, 276, 294,
306, 311, 431
continuous blocks, 479
discrete blocks, 479
functions and tables, 479
math, 479
nonlinear blocks, 214, 479
procedure of using, 478
algebraic, 289, 301, 454
matrix, 288, 300
Rise time, 62
Right-half s-plane, 48, 73, 78, 79
rlocfind, 75
rlocus, 74
Robotic welding machine, 433
Robustness, 27, 62, 81
Robust system, 81
Rocket thruster, 19
Roll-angle, 98
Rolling moment, 201
Roll-off, 87, 100, 101, 363, 365
Roll-rate, 215, 306
Root-locus method, 73
roots, 34, 72, 75, 76
Round-off, 432
Routh array, 72
signals and systems, 479
simulation parameters, 480
sinks, 214, 480
sources, 214, 480
subsystem block library, 214, 479
Single-input, single-output system, 13
Singularity functions, 22
Singular matrix, 112, 483
Singular value decomposition, 358
Singular values, 114, 358
SISO Design Tool, 105, 478
Slope of gain plot, 42
Spacecraft, 295
Spectral norm, 358
s-plane, 77-79
Square plant, 362
.s\s, 159
ss2ss, 144502 INDEX
ssdata, 159
Stable system, 4, 26, 48, 53, 65, 72
Stability, 17, 62, 71
Stability criteria, 72
Stability margin, 73, 82
Stability robustness, 81, 82
Stabilizability, 222
Stabilizing controller, 438
Starting problem, 205
State, 6, 125
State coefficient matrices, 131
State-dynamics matrix, 149
State equations, 126
State-space, 125
State-space representation, 126
State transformation, 140
State-transition marix, 175, 285, 341
State variables, 125
State vector, 131
State weighting matrix, 284
Static equilibrium, 117
Stationary system, 324
Stationary white noise, 343
Statistics, 323
Steady-state response, 4, 26, 28, 172
step, 57
stepresp, 57
Step response, 52
Stiff equations, 207
Stiffness matrix, 295, 487
Stochastic system, 7, 323
Storage of energy, 5
Strictly proper system, 35, 40, 57, 147
Structured singular value, 442
Structured uncertainty, 442
Subdiagonal, 154
Sufficient conditions, 289
Summing junction, 3
Superdiagonal, 137
Superfluous state variables, 227
Superposition principle, 14, 26, 59
Supremum function, 438
svd, 358
Symmetric matrix, 284, 483
System, 1
Tank-gun turret, 320
Target, 20
Taylor’s series, 16, 174, 206
Terminal condition, 288, 309, 313, 418
Terminal time, 312
Terminal time weighting matrix, 312
tf, 31, 148
Throttle position, 144, 421
Thrust, 201
Thrust vectoring, 201, 202
Time-constant, 211
Time domain, 27
Time-invariant system, 14
Time-marching, 185
Time-shift, 325
Time-varying system, 14
timv, 200
Torque inputs, 295
tpbvlti, 314-316
Tracking system, 4, 65, 251
Transfer function, 3, 38
Transfer matrix, 105
Transformation matrix, 140
Transient energy, 283, 284
Transient response, 4, 28
Translation in Laplace domain, 37
Translation in time, 37, 377
Truncation error, 207
Turbo-generator, 150
Tustin interpolation, 188
Type of a closed-loop system, 70, 71
Uncontrollable system, 222-224
Uncorrelated in time, 334
Uncorrelated signals, 342
Uncorrelated state variables, 324
Undamped system, 32
Undetectable system, 258
Unforced system, 15
Uniform probability distribution, 370
Unit circle, 82, 83, 386
Unit impulse function (see Dirac delta
function)
Unit parabolic function, 71
Unit ramp function, 24
Unit step function, 22
Unmodeled dynamics, 81
Unpredictable system, 7, 323
Unstable system, 42, 48, 72
Unstabilizable system, 222, 223
Upper triangular matrix, 232
van der Pol equation, 115
Vector solution, 172INDEX 503
Velocity:
angular, 21, 67, 139, 163
linear, 163
longitudinal, 43
transverse, 295
Vertical gust velocity, 337
Vibration suppression, 448-452
Viscous damping coefficient, 95
Viscous friction, 215
Voltage amplification ratio, 139
Voltage band, 116
vmarch, 199, 200
Weighted impulse sequence, 448
Weighted sensitivity minimization, 439
White noise, 334
White noise filter, 334
White noise process, 334
White noise vector, 334
Wiener process, 370
Wing-rock motion, 211
wrock, 211, 212
Yawing motion, 211
Yaw-rate, 306
zero, 72
Zeros, 40
Zero-pole-gain representation, 40
zpk, 40
z-transform, 376-379
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