Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

Book

Continuous-time signal, 3, 29

outline, 16

Control conguration, 3, 375

purpose, 11

Control engineering, 1

Boolean

Controllability

actuator, 2

Gramian, 121, 125

algebra of subsets, 127

Control law, 5

sensor, 2

irrelevant specication, 10

Branch-and-bound, 21

specication, 7

Controller, 5

C

1-DOF, 37, 148, 155, 283, 286

CACSD, 10, 19

2-DOF, 39

Cancellation

adaptive, 30, 45, 215

unstable pole-zero, 152

assumptions, 29

Chebychev norm, 99

complexity, 8, 10

Circle theorem, 245

decentralized, 8

Classical

diagnostic output, 28

controller design, 5, 11, 18, 244, 378

digital, 19

optimization paradigm, 57

discrete-time, 2, 380

regulator, 34, 172

estimated-state feedback, 162, 282

Closed-loop

gain-scheduled, 30, 215

controller design, 11, 48, 135, 143,

2-optimal, 279

378

H

bound, 282

convex specication, 135

H

1

implementation, 5, 8, 19

design, 377

LQG-optimal, 5, 279

realizable transfer matrices, 147

LQR-optimal, 277

stability, 147, 150

minimum entropy, 282

stability, stable plant, 155

nonlinear, 30, 45, 246, 286, 381

system, 26

observer-based, 164

transfer matrices achieved by stabi-

open-loop stable, 268

lizing controllers, 154, 353

order, 10, 150

transfer matrix, 33

parameters, 150

Command

parametrization, 353

decoupling, 178

PD, 5, 14, 21

following, 183

PID, 5, 18, 353

input, 28

rapid prototyping, 19

interaction, 178

state-feedback, 277

signal, 28

state-space, 43

Comparing norms, 89, 117

structure, 353

Comparison sensitivity, 203

Controller design, 1

Complementary sensitivity transfer func-

analytic solution, 275

tion, 36

challenges, 10

Complexity

classical, 5, 11, 18, 378

bit, 346

convex problem, 351

controller, 9

empirical, 5

information based, 346

feasibility problem, 51, 335

quadratic programs, 346

goals, 6

Computing norms, 119

\modern", 5

Concave, 139

open vs. closed-loop, 135

INDEX

407

problem, 8, 11, 351

subgradient algorithm, 348

software, 19

upper and lower bounds, 315

state-space, 5

Convolution, 31

trial and error, 20

Crest factor, 91

Control processor, 1, 9, 19

Cutting-plane, 294

DSP chips, 19

algorithm, 313

power, 10

algorithm initialization, 349

Control system, 1

deep-cut, 294

1-DOF, 37, 283, 286

2-DOF, 39

2-DOF MAMS, 40, 202

D

architecture, 25

Damping ratio, 166

classical regulator, 34, 172

dB, 197

continuous-time, 3

Dead-zone nonlinearity, 229, 246

specications, 12, 47

Decentralized controller, 8

testing, 6

Decibel, 197

Control theory

Decoupling

classical, 5

asymptotic, 178

geometric, 45

exact, 180

optimal, 125, 275

Deep-cut, 294

state-space, 5

ellipsoid algorithm, 358

Convex

Descent

analysis, 293

direction, 297, 312

combination, 128

method, 312

controller design problem, 351

Describing function, 220, 246

denition for functional, 129

Design

denition for set, 128

closed-loop, 377

duality, 139

conservative, 210

functional, 128

FIR lters, 380

inner approximation, 207, 239, 262,

Pareto optimal, 55, 281, 337, 369

264

procedure, 63

optimization, 311

trial and error, 20

outer approximation, 267

Design specication, 11, 47

set, 128

achievable, 14, 50

supporting hyperplane for set, 298

as sets, 127

Convexity

boolean, 48

midpoint rule, 129

comparing, 49

multicriterion optimization, 139

consistent, 50

Pareto optimal, 139

families of, 51

performance space, 139

feasible, 50

Convex optimization, 11, 311

hardness, 51

-relaxed problem, 319, 329

inconsistent, 50

complexity, 345

infeasible, 50

cutting-plane algorithm, 313

nonconvex, 205

descent methods, 312

ordering, 49

duality, 292

qualitative, 49

ellipsoid algorithm, 324

quantitative, 54

linear program, 126, 271, 292, 313,

robust stability, 212

319, 348, 379

time domain, 134

quadratic program, 345, 378, 380

total ordering, 54

stopping criterion, 315

Diagnostic output, 28, 45

408

INDEX

Dierential sensitivity, 7, 195

stable transfer matrix, 169

versus robust stability, 210, 233, 244

Failure modes, 7, 45, 214

Digital

False intuition, 89, 108, 117, 379

controller, 19

Feasibility problem, 51, 312, 375

signal processing, 19, 380

families of, 51

Directional derivative, 297

RMS specication, 281, 335

Discrete-time

Feasibility tolerance, 319

plant, 380

Feasible design specications, 50

signal, 2, 29

Feedback

Disturbance, 6, 35

estimated-state, 162, 279, 282

modeling, 187

linearization, 30, 45

stochastic, 188, 194

output, 27

unknown-but-bounded, 190

paradigm, 46

Dual

perturbation, 221

functional, 61, 139, 332, 363

plant perturbations, 221

functional via LQG, 281

random, 114

objective functional, 61

state, 27

optimization problems, 142

well-posed, 32, 116

problem, 125, 362

Fictitious inputs, 25

Duality, 61, 139, 332, 362

Finite-dimensional approximation

convex optimization, 292

inner, 352

for LQG weight selection, 332

outer, 362

gap, 142

Finite element model, 19

innite-dimensional, 371

FIR, 380

lter design, 380

E

Fractional perturbation, 202

Ellipsoid, 121, 324

Frequency domain weights, 81, 188

Ellipsoid algorithm, 324

Functional

deep-cut, 358

ane, 128

initialization, 349

bandwidth, 189

Empirical model, 4

convex, 128

Entropy, 136, 282, 291

denition, 53

denition, 112

dual, 61, 139, 363

interpretation, 114

equality specication, 131, 137

state-space computation, 123

general response-time, 177

Envelope specication, 177, 193

inequality specication, 52, 131

Error

level curve, 250

model reference, 186

linear, 296

nulling paradigm, 46

maximum envelope violation, 177

signal, 34, 46, 69

objective, 54

tracking, 252

of a submatrix, 134

Estimated-state feedback, 162, 279, 282

overshoot, 173

controller, 282

quasiconcave, 189

Estimator gain, 162

quasiconvex, 132, 176, 189, 296

Exact decoupling, 180

relative overshoot, 173

Exogenous input, 25

relative undershoot, 173

response-time, 177

F

rise time, 175

Factorization

settling time, 175

polynomial transfer matrix, 169

subgradient of, 293

spectral, 92, 188

sub-level set, 131

INDEX

409

undershoot, 173

I

weighted-max, 62, 131, 139, 268

I/O specication, 172, 250

weighted-sum, 59, 131, 139, 281

Identication, 4, 10, 18, 215

Impulse response, 31, 96, 181, 305, 380

G

Inequality specication, 100

Gain, 94, 115, 231

Infeasible design specication, 50

average-absolute, 98, 111

Information based complexity, 346

cascade connection, 115

Inner approximation, 207, 239, 262, 264

feedback connection, 116

denition, 207

generalized margin, 238, 263

nite-dimensional, 352

margin, 215, 225, 236, 262

Input, 25

peak, 111

actuator, 25

scaling, 235

exogenous, 25

scheduling, 45

ctitious, 25

small gain theorem, 231

signal, 25

variation, 198

Input-output

Generalized

specication, 172, 250

bandwidth, 189

transfer function, 38

gain margin, 238

Integrated

response time, 189

actuators, 18

stability, 165, 232

sensors, 9, 18

Geometric control theory, 45

Interaction of commands, 178

Global

Interactive design, 63

optimization, 21, 345, 347

Internal model principle, 168

positioning system, 9, 18

Internal stability, 150

Goal programming, 63

free parameter representation, 156

Gramian

with stable plant, 155

controllability, 121, 125

Interpolation conditions, 155, 168, 284,

observability, 120, 123, 125

287

-stability, 165

Intuition

G

correct, 118, 140, 287, 362

false, 89, 108, 117, 379

H

ITAE norm, 84

2 norm, 276

H

subgradient, 301

J

2-optimal

Jerk norm, 83, 190

H

controller, 279

norm, 99, 282

H

L

1

bound and Lyapunov function, 245

shifted, 101

Laguerre functions, 370

subgradient, 303

Laplace transform, 30

Half-space, 294

Level curve, 250

Hamiltonian matrix, 121, 277, 282

Limit of performance, 12, 20, 139, 208,

Hankel norm, 103, 118, 121

347

Hardness, 51

Linear

History

fractional form, 150, 157

closed-loop design, 377

functional, 296

feedback control, 18, 208

program, 126, 271, 292, 313, 319,

stability, 169

348, 379

Hunting, 169

time-invariant system, 30

transformation, 81

Linearity consequence, 191, 193

410

INDEX

Linearly ordered specications, 132, 300

Multi-rate plants, 30, 381

Logarithmic sensitivity, 197

MAMS system, 202

Loop gain, 36

N

LQG, 13, 154, 278, 291

Neglected

multicriterion optimization, 280

nonlinearities, 219

optimal controller, 5, 279

plant dynamics, 213, 225, 234

LQG weights

Nepers, 197

selection, 332, 349

Nested family of specications, 132, 178,

trial and error, 350

300

tweaking, 281, 335

Noise, 6

via dual function, 350

white, 96, 106, 110, 193, 275, 278

LQR, 5, 275

Nominal plant, 211

LTI, 8, 28

Nominal value method for weights, 63

Lumped system, 29

Nonconvex

Lyapunov equation, 120, 123, 125, 271,

convex approximation for specica-

371

tion, 207

Lyapunov function, 125, 245

design specication, 205

specication, 205, 245

Noninferior specication, 55

M

MAMS system, 40

Nonlinear

logarithmic sensitivity, 202

controller, 30, 45, 246, 286, 381

Margin, 196

plant, 29, 45, 212, 219, 233, 381

Matrix

Nonlinearity

Hamiltonian, 121, 277, 282

dead-zone, 229

Schur form, 123, 277

describing function, 220

weight, 89, 278

memoryless, 231

Maximum absolute step, 118

neglected, 219

Maximum value method for weights, 63

Non-minimum phase, 283

-circle constraint, 244

Nonparametric plant perturbations, 216

M

Memoryless nonlinearity, 231

Norm, 93

Midpoint rule, 129

2, 96, 111, 120, 276

kH

k

Minimax, 62

, 99, 112, 118, 121, 282

kH

k

1

Minimizer, 311

hankel, 103, 118, 121

kH

k

half-space where it is not, 294

kH

k

gn, 111

L

Model, 4, 10, 18, 215

1

kH

k

gn, 99

black box, 4

L

2

pk gn, 97, 111, 118, 125

detailed, 213

kH

k

pk step, 95, 118, 125

empirical, 4

kH

k

rms , 95, 110

LTI approximation, 45

kH

k

w

rms gn, 98, 112

physical, 4, 10, 215

kH

k

wc, 97, 125

reference, 186

kH

k

1, 81, 88

Modied controller paradigm, 157

kuk

2, 80, 88

for a stable plant, 160

kuk

amusing, 83

observer-based controller, 162

average-absolute value, 74

Multicriterion optimization, 54

Chebychev, 99

convexity, 139

comparing, 89

LQG, 280, 332

comparison of, 117

Multiple

computation of, 119, 125

actuator, 40

denition, 69, 92

sensor, 40

false intuition, 89

INDEX

411

frequency domain weight, 81

Open-loop

gains, 94, 115

controller design, 135

Hankel norm, 103, 118, 121

equivalent system, 203

inequality specication, 100, 135

Operating condition, 215

ITAE, 84

Optimal

jerk, 83, 190

control, 125, 275

maximum absolute step, 118

controller paradigm, 47

MIMO, 110

Pareto, 55, 139, 281, 312, 337, 347,

multivariable weight, 88, 115

369, 374

peak, 70

Optimization

peak acceleration, 83

branch-and-bound, 21

peak gain, 97, 111, 118, 125, 291,

classical paradigm, 57

305

complexity, 345

peak-step, 95, 125

constrained, 311

RMS, 72

convex, 311

RMS gain, 98, 112

cutting-plane algorithm, 313

RMS noise response, 95, 110

descent method, 312, 347

scaling, 88, 115

ellipsoid algorithm, 324

seminorm, 92

feasibility problem, 312

shifted

, 101

global, 21, 345

H

1

signal, 69

linear program, 126, 271, 292, 313,

SISO, 95

319, 348, 379

snap, crackle and pop, 83

multicriterion, 54, 65, 280

stochastic signals, 75

quadratic program, 345, 378, 380

time domain weights, 83

unconstrained, 311

total absolute area, 81

Order of controller, 11, 150

total energy, 80

Outer approximation, 267, 318

using an average response, 94

denition, 267

using a particular response, 93

nite-dimensional, 362

using worst case response, 94

Output, 26

vector average-absolute value, 87

diagnostic, 28

vector peak, 86

referred perturbation, 202

vector RMS, 86

referred sensitivity matrix, 41

vector signal, 86

regulated, 26

vector total area, 88

sensor, 26

vector total energy, 88

Overshoot, 13, 48, 173

weight, 88, 115

worst case response, 97, 306

P

Notation, 30, 383

, 211

P

Nyquist plot, 244

Parametrization

stabilizing controllers in state-space,

162

O

Pareto optimal, 55, 143, 281, 285, 312,

Objective functional, 54

337, 347, 369, 374

dual, 61

convexity, 139

weighted-max, 62

interactive search, 63

weighted-sum, 59, 281

Parseval's theorem, 81, 110

Observability

PD controller, 5, 14, 21

Gramian, 120, 123, 125

Peak

Observer-based controller, 162, 164

acceleration norm, 83

Observer gain, 162

gain from step response, 97

412

INDEX

gain norm, 97, 111, 118, 125, 291

state-space, 43, 164

signal norm, 70

strictly proper, 43

step, 95, 125

time-varying, 212, 233

subgradient of norm, 305

transfer function uncertainty, 216

vector signal norm, 86

typical perturbations, 210

Performance

Pole

limit, 12, 20, 139, 347

plant pole variation, 228, 238

loop, 241

zero cancellation, 152

specication, 6, 171

Polynomial matrix factorization, 169

Performance space, 52

Power spectral density, 75

achievable region, 139

Predicate, 48

convexity, 139

Primal-dual problems, 142

Perturbation, 7

Process noise

fractional, 202

actuator-referred, 35

nonlinear, 212, 233

Programmable logic controllers, 2, 19

output-referred, 202

plant gain, 198

plant phase, 199

Q

singular, 217

, 379

qdes

step response, 205

-parametrization, 162, 169, 353, 379,

Q

time-varying, 212, 233

381

Perturbed plant set, 211

Quadratic program, 345, 378, 380