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Vidyasagar M. — Nonlinear systems analysis

Vidyasagar M. — Nonlinear systems analysis

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Название: Nonlinear systems analysis

Автор: Vidyasagar M.

Аннотация:

When M. Vidyasagar wrote the first edition of Nonlinear Systems Analysis, most control theorists considered the subject of nonlinear systems a mystery. Since then, advances in the application of differential geometric methods to nonlinear analysis have matured to a stage where every control theorist needs to possess knowledge of the basic techniques because virtually all physical systems are nonlinear in nature.

The second edition, now republished in SIAM’s Classics in Applied Mathematics series, provides a rigorous mathematical analysis of the behavior of nonlinear control systems under a variety of situations. It develops nonlinear generalizations of a large number of techniques and methods widely used in linear control theory. The book contains three extensive chapters devoted to the key topics of Lyapunov stability, input-output stability, and the treatment of differential geometric control theory. In addition, it includes valuable reference material in these chapters that is unavailable elsewhere. The text also features a large number of problems that allow readers to test their understanding of the subject matter and self-contained sections and chapters that allow readers to focus easily on a particular topic.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 2002

Количество страниц: 520

Добавлена в каталог: 04.11.2010

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Предметный указатель

$L_{1}$ -stability of LTI systems      298 301
$L_{1}$ -stability of LTV systems      304
$L_{2}$ gain of LTI systems      300
$l_{P}$       272
$L_{p}$ norm      272
$L_{p}$ , extension of      274
$L_{p}$ -gain      277
$L_{p}$ -gain with zero bias      277
$L_{p}$ -stability of LTI systems      298 301
$L_{p}$ -stability of LTV systems      306
$L_{p}$ -stability, definition      277 282
$L_{p}$ -stability, small signal      285
$L_{\infty}$       272
$L_{\infty}$ -stability of LTI systems      298 301
$L_{\infty}$ -stability of LTV systems      302
$\hat{b}$       320
$\hat{B}$ , partial fraction expansion in      324
$\mathbf{A}$       293
$\mathbf{A}$ , field of fractions      321 329
$\mathbf{A}$ , unit of      310
$\mathbf{A}$ _      320
$\mathbf{A}_{e}$       298
$\mathbf{A}_{\sigma}$       319
$\mathbf{\hat{A}}$       297
$\mathbf{\hat{A}}$ _      520
$\mathbf{\hat{A}}_{\sigma}$       320
( $\mathbf{A}$ , $\mathbf{B}$ )-invariance      407
Absolute stability, input-output      358
Absolute stability, state-space      221
AD      391
Aizerman's conjecture in input-output setting      357
Aizerman's conjecture in state-space setting      222
Attractivity      140
Attractivity, uniform      140
Averaging method      76
Banach space      13
Bellman's inequality      292
Bendixson's theorem      69
Bezout identity      326
Brunovsky canonical form of linear systems      439
Brunovsky canonical form of nonlinear systems      440
Canonical decomposition of linear systems      424
Canonical decomposition of nonlinear systems      426
Cauchy sequence      12
Causality      275
Center      61
Chetaev's theorem      188
Christoffel symbols      184
Circle criterion, input-output      344
Circle criterion, necessity of      361
Circle criterion, state-space      227
Closed set      469
Closure      470
Comparison principle      256
Computed torque method      466
Continuity      13
Continuity, uniform      14
Continuous dependence      43
Continuous function      13
Contraction mapping theorem, global      28
Contraction mapping theorem, local      30 32
Contraction, global      28
Contraction, local      30 32
Controllability, linear systems      220
Controllability, nonlinear systems      see "Reachability"
Convergence      11
Converse theorem, applications      246
Converse theorem, exponential stability      244 245
Converse theorem, global exponential stability      246
Converse theorem, uniform asymptotic stability      239
Convolution      293
Coprime factorization      327
Coprime factorization, existence of      328
Coprimeness      326
Coprimeness, condition for      327 330
Critical disk      123 226 343
Describing function      95
Describing function of hysteresis      100
Describing function of odd nonlinearity      96
Describing function, bounds on      96
Describing function, independence of frequency      95
Detector      251
Detector, weak      252
Diffeomorphism      377
Diffeomorphism, smooth      377
distribution      395
Distribution, completely integrable      396
Distribution, invariant      401
Distribution, involutive      396
Distribution, regular point of      395
Domain      68
Domain of attraction      154
Domain of attraction, properties of      154
Domain, connected      70
Domain, simply connected      68
Equilibrium      3 55 136
Existence and uniqueness of solutions, global      38
Existence and uniqueness of solutions, local      34 37
Exponential stability, converse theorems      244 245
Exponential stability, definition      142
Exponential stability, theorems      171 246 290
Extended $L_{p}$ -spaces      274
Fast dynamics      133
Fast state variable      133
Feedback linearization of multi-input systems      442
Feedback linearization of single input systems      430
Feedback linearization, input-output      456
Feedback stability of LTI systems      309 326
Feedback stability, definition      282
Feedback stability, relation to Lyapunov stability      286 289
Finite escape time      5 38
Finite gain      277
First category      470
Fixed point      28
Focus      61
FORM      378
Form, exact      382
Fractional representation      327
Frobenius theorem      396 397
Frobenius theorem, proof of      479
Function, class K      144
Function, class l      144
Function, continuous      13
Function, decrescent      147
Function, locally negative definite      148
Function, locally positive definite      147
Function, negative definite      148
Function, positive definite      148
Function, radially unbounded      148
Function, uniformly continuous      14
Global exponential stability, converse theorem      246
Global exponential stability, definition      143
Global exponential stability, theorems      173
Global uniform asymptotic stability, converse theorem      246
Global uniform asymptotic stability, definition      143
Global uniform asymptotic stability, theorems      173 179 290
Graphical stability test for LTI systems      316 335
Gronwall's inequality      236
Harmonic balance      81 94 105
Hierarchical systems      258
Hierarchical systems, stability of      259
Hilbert space      16
Hoelder's inequality      273
INDEX      73
Inertia matrix      183 465
Inner product space      15
Input-output stability      277 282
Instability (Lyapunov), definition      137
Instability (Lyapunov), theorems      186 187 188
Integral curve      379

Integral manifold      396
Interior      470
Invariant set      151
Inverse function theorem      377
Involutivity      396
Isolated subsystem      259
Jacobi identity      390
Jacobian matrix      377
Kalman — Yacubovitch lemma      223
Kalman — Yacubovitch lemma, proof of      474
Kalman's conjecture      222
Kiylov — Boguliubov method      76
Krasovskii — LaSalle theorem      178 179
Kronecker indices of linear systems      438
Kronecker indices of nonlinear systems      440
LaSalle's Theorem      178 179
Lebesgue spaces      see " $L_{p}$ "
Left-coprime factorization      330
Left-coprimeness      329
Leray — Schauder theorem      116
Level set      167
Lie bracket, anti-symmetry of      390
Lie bracket, bilinearity of      390
Lie bracket, definition      382
Lie bracket, interpretation      382 383 385
Lie derivative of a form      389
Lie derivative of a smooth function      381
Lie derivative of a vector field      382
Limit cycle      68
Limit point      71 152
Limit set      71 152
Linear convergence      28
Linear systems, $L_{1}$ -stability      298 301 304
Linear systems, $L_{p}$ -stability      298 301 306
Linear systems, $L_{\infty}$ -stability      298 301 302
Linear systems, asymptotic stability      195
Linear systems, autonomous      196
Linear systems, discrete-time      267
Linear systems, exponential stability      195 196
Linear systems, periodic      206
Linear systems, singularly perturbed      128
Linear systems, stability      194 196
Linear systems, uniform asymptotic stability      196
Linear vector space      6
Linear vector space, finite-dimensional      11
Linearization      210 211
Linearization method      209
Lipschitz constant      34
Lipschitz continuity      34
Lipschitz continuity, conditions for      46
Loop transformation      110 224 234 341
Lur'e problem      219
Lyapunov function      160
Lyapunov function, candidate      161
Lyapunov function, Popov type      232
Lyapunov function, quadratic      199 202
Lyapunov matrix equation      197
Lyapunov matrix equation, "optimal"      214
Lyapunov matrix equation, discrete-time      267
Massera's lemma      236
Matrix measure      22
Matrix measure, conditions for stability      204
Matrix measure, solution estimates      47 52
Matrix, Hurwitz      131 199
Matrix, hyperbolic      131
Maximal solution      471
McMillan degree      336
Meager set      470
Minimal solution      471
Minkowski's inequality      273
Multiplier      231
Node      58
Norm      9
Norm, $l_{1}$       10
Norm, $L_{2}$       11
Norm, $L_{p}$       10
Norm, $l_{\infty}$       9
Norm, Euclidean      11
Norm, induced      20
Norm, submultiplicative      21
Normed linear space      9
Normed linear space, complete      13
Nowhere dense set      470
Nyquist criterion      316 335
Observability, linear systems      220
Observability, nonlinear systems      414 418
Observer-controller stabilization, linear systems      251
Observer-controller stabilization, nonlinear systems      253
Open set      469
Paley — Wiener theorem      309
Passivity      352
Passivity theorem, input-output      350 352 353
Passivity theorem, state-space      223
Passivity, strict      352
Pendulum equation      76 86 138 161
Periodic solutions      68
Periodic solutions, using describing functions      104 109
Phase-locked loop      181
Phase-plane      53
Picard's iterations      42
Picard's method      42
Poincare — Bendixson theorem      77
Popov criterion, input-output      354
Popov criterion, state-space      231 233
Popov plot      234 356
Positively oriented curve      73
Predator-prey equation      74 76
Quasi-linearization method      88
Rayleigh's equation      86
Reachability      286 400
Reachability, conditions for      409
Realization      220
Realization, minimal      220
Regular point      395
Relation      277
Relative degree      458
Relative degree, vector      461
Return difference      311
Right-coprime factorization      330
Right-coprimeness      329
Robot with flexible joints      435 454
Robot, rigid      183 465
Saddle      58
Schwarz' inequality      13
Second category      470
Sector      96 221 339 360
Sector, incremental      110 222
Set, connected      70 154
Set, invariant      757
Set, limit      77 752
Set, negative limit      752
Set, positive limit      152
Set, simply connected      68
Singularly perturbed systems, linear      128
Singularly perturbed systems, nonlinear      278
Slow dynamics      133
Slow state variable      133
Slowly varying systems      248
Small gain theorem      337 340
Solution estimates, linear equations      47
Solution estimates, nonlinearequations      52
Solutions, continuous dependence      43
Solutions, global      38
Solutions, local      34 37
Solutions, maximal      37 471
Solutions, minimal      471
Solutions, prevalence of      469
Spinning body, control of      426
Spinning body, stability of      161 189 216
Stability (Lyapunov), definition      136

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