Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

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BIBLIOGRAPHY 935

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NonlinearBook10pt November 20, 2007

Index

α-level set, 35

α-limit set, 97

α-sublevel set, 35

[α, β]-sublevel set, 35

Abelian group, 57

absolute nor m, 487

absolute stability theory,

372

absolute sum norm, 15

absolute value theorem

for integrals, 64

accumulation point, 23

action integral, 467

adaptive controller, 1 89

adherent point, 23

Aizerman conjecture, 374

algebraic sys tem, 57

almost everywhere, 60

analysis, 10

analytic function, 362

arcwise connected set, 31

asymptotically stable,

136

discrete-time system,

765, 783

funcitional dynamical

system, 200

periodic orbit, 291

time-varying system,

226

asymptotically stable

matrix, 17 2

asymptotically stable set,

283

asymptotically sta ble

with respect to x

210

asymptotically sta ble

with respect to x

uniformly in x

, 211

attracting set, 102

attractive set, 283

attractor, 10 3

autonomous dynamical

system, 10

available exponential

storage, 329

available geometric

storage, 799

available storage, 329

discrete-time system,

799

time-varying systems,

398

axial ﬂow compression

system, 581

backstepping control, 557

Banach ﬁxed point

theorem, 68

Banach space, 67

Barbalat lemma, 221

Barbashin-Krasovskii-

LaSalle theor em,

147

discrete-time system,

770

Bellman equation, 849

Bellman principle of

optimality, 443

discrete-time system,

846

Bendixson theore m, 106

bijective map, 32

binary oper ation, 56

block cascade system,

456, 562

block structured

uncertainty, 728

Bolzano-Le besgue

theorem, 28

Bolzano-Weierstass

theorem, 27

boundary, 2 3

bounded, 241

discrete-time system,

791

bounded conver gence

theorem, 121

bounded function, 37

bounded operator, 70

bounded real lemma, 367

discrete-time system,

821

nonlinear systems, 355

bounded real transfer

function, 362

discrete-time system,

817

bounded sequence, 25

bounded set, 24

bounded trajectory, 99

discrete-time system,

768

bounded with respect to

NonlinearBook10pt November 20, 2007

940 INDEX

uniformly in x

237

discrete-time system,

786

boundedness, 236

discrete-time system,

786

Brouwer ﬁxed point

theorem, 123

Cartesian product, 21

Casimir functions, 159

Cauchy sequence, 25

normed linear space, 66

vectors, 27

Cauchy-Schwarz

inequality, 16

causal operato r, 473

causality, 3

ceiling function, 44

centrifugal ﬂow

compression system,

587

chain rule, 52

Chetaev instability

theorem, 170

circle criterion, 380

discrete-time system,

841

operator form, 507

circle theorem, 378

class K function, 162

class KL function, 162

class K

∞

function, 162

class L function, 162

closed curve, 106

closed dynamical system,

328

closed set, 23

closure, 23

closure point, 22

cluster point, 23

codomain, 32

compact set, 24, 60

comparison principle,

126, 253

compatible norms, 19

complement, 21

complete space, 67

complete vector ﬁeld, 451

completely null

controllable, 330

discrete-time system,

800

completely reachable, 330

discrete-time system,

799

completeness axiom, 26

complex multiplier

functions, 723

complex uncertainty, 722,

731

complex-µ, 731

connected component, 31

connected se t, 30

construction of Lyapunov

functions, 152

contact point, 23

continuous function, 3 8

continuously

diﬀerentiable

function, 48

control Lyapunov

function, 438

discrete-time system,

833

controlled dynamical

system, 10

convergence lemma, 201

convergent sequence, 25

of functions, 36

convergent system, 261

converse Lyapunov

theorems, 161

asymptotic stability,

163

discrete-time system,

772

exp onential stability,

165

ﬁnite-time stability, 258

partial stability, 223

semistability, 26 5

time-varying systems,

235

converse vector Lyapunov

theorem, 310

convex function, 35

convex hull, 31

convex set, 31

convolution operator, 490

countable set, 33

countably inﬁnite set, 33

cover, 60

critically minimum

phase, 450

cyclo-dissipative, 388

cyclo-lossle ss, 389

decreasing function, 37

dense set, 24

denumerable set, 33

derived set, 23

diﬀeomorphic sets, 49

diﬀeomorphism, 49

diﬀerentiable function, 48

Dirichlet function, 64

discontinuous function,

discrete entropy, 894

discrete trajectory, 764

discrete-time Riccati

equation, 86 7

disjoint sets, 21

disk margin, 434

dissipation inequality,

325, 329, 330

discrete-time system,

798

dissipative system, 329

discrete-time system,

798

dissipativity and robust

stability, 666

dissipativity theor y, 325

distribution, 449

disturbance rejection

discrete-time system,

872

NonlinearBook10pt November 20, 2007

INDEX 941

disturbance rejection

control, 60 3, 617

multilinear

performance criteria,

633

polynomial

performance criteria,

633

disturbe d dynamical

system, 10

divergence oper ator, 106

divergent sequence, 25

domain, 32

domain of attraction, 143

discrete-time system,

767

set, 284

domain of semistability,

261

dominated convergence

theorem, 121

dual norm, 487

Duﬃng equation, 108,

417

Dulac theorem, 109

dynamic, 2

dynamical invariants, 159

dynamical system, 1, 2,

element of a set, 21

empty set, 21

energy function, 140

energy-based control, 420

energy-Ca simir method,

158

energy-Ca simir theore m,

159

entropy, 638

equi-induced norm, 19

equilibrium point, 11

discrete-time system,

765

equivalent norms , 16

equivalent sets, 33

essentially nonnegative

matrix, 18 3

essentially nonnegative

vector ﬁeld, 198

Euclidean nor m, 15

Euler-Lagrange equation,

340

with dissipation, 341

exp onential storage

function, 331

exp onentially

cyclo-dissipative, 388

exp onentially dissipative

system, 329

exp onentially

nonexpansive, 352

exp onentially passive,

352

exp onentially stable, 136

time-varying system,

226

exp onentially stable w ith

respect to x

uniformly in x

, 211

extendability of

solutions, 85

extended L

space, 474

feedback equivalent

systems, 448

feedback interconnections

of dissipative

systems, 411

discrete-time system,

825

feedback linearizable, 447

feedback linearizable

system, 449

feedback pas sive, 452

ﬁeld, 57

ﬁnite c over, 60

ﬁnite e scape time, 75

ﬁnite set, 33

ﬁnite-time convergence,

254

ﬁnite-time stability, 252

ﬁnite-time stable, 194,

254

time-varying systems,

318

ﬁrst law of

thermodynamics, 404

ﬁxed point, 68

ﬂoor function, 43

ﬂow, 72

frequency-dependent

scaling, 723

Frobenius nor m, 18

function, 32

functional dynamical

system, 199

gain mar gin, 434

generalized energ y

balance equation, 350

generalized global

invariant set theorem,

289

generalized invariant set

theorem, 288

generalized Lyapunov

function, 271

generalized Lyapunov

function candidate,

271

generalized Lyapunov

surfaces, 276

generalized Lyapunov

theorem

for sets, 284

generalized Lyapunov

theorems, 268

generalized momenta,

341

generalized power

balance equation, 350

geometric dissipation

inequality, 798

geometric storage

function, 800

geometrically dissipative

system, 798

NonlinearBook10pt November 20, 2007

942 INDEX

geometrically stable, 765,

784

global diﬀeomorphism,

469

global existence of

solutions, 88

global invariant set

theorem, 151

globally asymptotically

stable, 136

discrete-time system,

765, 784

time-varying system,

226

globally asymptotically

stable set, 283

globally asymptotically

stable with respect to

, 211

globally asymptotically

stable with respect to

uniformly in x

211

globally bounded, 241

discrete-time system,

791

globally bounded with

respect to x

uniformly in x

, 237

discrete-time system,

786

globally exponentially

stable, 136

time-varying system,

226

globally exponentially

stable with respect to

uniformly in x

211

globally ﬁnite-time

stable, 194, 254

globally geometric ally

stable, 784

discrete-time system,

765

globally Lipschitz

continuous function,

globally s e mistable

equilibrium, 261

globally s e mistable

system, 261

globally ultimately

bounded with bound

ε, 241

discrete-time system,

791

globally ultimately

bounded with respect

to x

uniformly in x

with bound ε, 2 37

discrete-time system,

786

globally uniformly

asymptotically sta ble

discrete-time system,

784

time-varying system,

226

globally uniformly

bounded, 239

discrete-time system,

789

globally uniformly

ﬁnite-time stable

time-varying systems,

318

globally uniformly

ultimately bounded

with bound ε, 2 39

discrete-time system,

789

gradient, 52

gradient dynamical

system, 199

graph, 36

Gronwall lemma, 80

Gronwall-Bellman

lemma, 1 25

group, 57

Hardy space , 513

discrete-time system,

851

norm, 51 4

discrete-time system,

852

/positivity a nalysis,

613

∞

control theory, 603

∞

norm, 48 5

Hamilton principle of

least action, 467

Hamilton-Jacobi-Bellman

equation, 19 1, 446,

521

nominal system, 671

Hamilton-Jacobi-Is aacs

equation, 60 5

Hamiltonian dynamical

system, 340, 344

Hamiltonian function,

343

Hankel norm, 498

higher-order Lie brackets,

449

higher-order Lie

derivative, 449

Hilbert space, 122

H¨older constant, 117

H¨older continuous

function, 117

H¨older ineq uality, 15

for functions, 506

H¨older norms, 14

homeomorphic sets, 49

homeomorphism, 49

homogeneous function,

116

Hurwitz matrix, 172, 515

hyperplane, 36

identity function, 32

image, 33

implicit function

theorem, 53

impulse response matrix

function, 472

increasing function, 37

indexed family of sets, 22

NonlinearBook10pt November 20, 2007

INDEX 943

induced convolution

operator norms, 489

induced matrix norm, 19

inﬁmum, 25

inﬁnite set, 33

inﬁnity norm, 15

injective map, 32

inner product space, 122

input strict passive, 351

input-output

dissipativity, 478

input-output operato r

dissipativity, 481

input-output stability,

472

input-output stability

and Lyapunov

stability, 483

input-output strict

passive, 35 1

input-to-output stable,

638

input-to-state stability,

245

input-to-state stable, 246

instability theorems

time-varying systems,

235

integral equation, 74

integrals of motion, 159

integrator backstepping,

559

interior, 22

interior point, 22

interse c tion, 21

invariant set, 97

discrete-time system,

768

invariant set stability

theorems, 147

discrete-time system,

768

inverse function theorem,

inverse image, 33

inverse optimal control,

522

discrete-time system,

860

inverse optimality of

nonlinear r e gulators,

536

invertible function, 32

involutive dis tribution,

449

Isaacs equation, 604

isocline method, 132

isolated equilibrium

point, 12

isolated point, 22

isolated system, 2

Jacobian, 52

Jordan decomposition,

171

Jordan form, 171

Jurdjevic-Quinn sys tem,

464

Kalman conjecture, 374

Kalman-Yakubovich-

Popov

equations

nonlinear discrete-time

systems, 807

nonlinear systems, 346

Kamke condition, 300

Krasovskii method, 154

Krasovskii theorem, 155

norm, 48 5, 500

space, 65

ℓ

space, 508

-norm, 473

-stable, 47 4

-stable with ﬁnite

gain, 474

-stable with ﬁnite gain

and zero bias, 474

∞

-norm, 474

Lagrange stability, 236

discrete-time system,

786

Lagrange stable, 239, 241

discrete-time system,

789, 791

Lagrange stable with

respect to x

, 237

discrete-time system,

786

Lagrange-Dirichlet

problem, 218

Lagrange-Dirichlet

stability problem, 210

Lagrangian, 341

Lagrangian dynamical

system, 340

Landau order, 13

LaSalle-Yoshizawa

theotem, 234

Leb e sgue integrable, 63

Leb e sgue integral, 65

Leb e sgue integration, 63

Leb e sgue measurable set,

Leb e sgue measure , 61, 65

Leb e sgue normed space,

474

left continuous function,

left invertible function,

left piecewise continuous

function, 45

Legendre transfor mation,

341

Leibnitz integral rule,

385

Lie bracket, 449

Lie derivative, 449

Lienard sys tem, 132, 186

limit cycle, 106

limit inferior, 26

limit point, 23

limit superior , 26

linear autonomous

system, 11

NonlinearBook10pt November 20, 2007

944 INDEX

linear periodic system, 11

linear time-varying

system, 11

linear vector space, 57

linearization of

dissipative

discrete-time systems,

814

linearization of

dissipative systems,

358

Lipschitz constant, 45

Lipschitz continuous

function, 45

locally controllable, 333

loop gain, 463

Lorentz equations, 525

lossless system, 329

discrete-time system,

798

Lotka-Volterra equations,

198

lower semicontinuous

function, 43

Lur´e problem, 374

Lur´e-Postnikov

Lyapunov function,

373

Lyapunov and

asymptotic stability

of sets, 283

Lyapunov equation, 173

Lyapunov ﬁrst instability

theorem, 167

Lyapunov function, 140

discrete-time system,

767

Lyapunov function

candidate, 140

discrete-time system,

767

Lyapunov functional, 2 00

Lyapunov indirect

method

discrete-time system,

836

stabilization, 180

Lyapunov indirect

theorem, 177

Lyapunov instability

theorems, 167

Lyapunov level surfaces,

140

Lyapunov second

instability theorem,

169

Lyapunov stability

theory, 1 36

Lyapunov stable, 136

discrete-time system,

765, 783

functional dynamical

system, 200

periodic orbit, 290

time-varying system,

226

Lyapunov stable matrix,

172

Lyapunov stable set, 283

Lyapunov stable with

respect to x

, 210

Lyapunov stable with

respect to x

uniformly in x

, 210

Lyapunov theorem, 137

discrete-time system,

765

ﬁnite-time stability, 257

linear systems, 173

partial stability, 212

semistability, 26 3

time-varying discrete

systems, 784

time-varying systems,

228

Lyapunov-Krasovskii

functional, 639

Massera lemma, 162

Mathieu equa tio n, 315,

405

matrix norm, 18

maximal interval of

existence, 74

mean value theorem, 53

meaningful cost

functional, 540

measurable function, 61

measurable set, 61

measurable space, 63

measure, 60

measure of a set, 61

minimum input energy,

389

minimum phase, 450

Minkowski inequality, 14

for functions, 506

mixed H

/ℓ

problem,

882

mixed H

/positivity

synthesis, 621

mixed H

problem,

615

mixed-µ analysis , 724

mixed-µ theory, 722

mixed-norm H

∞

problem, 603

mixed-norm H

∞

synthesis, 619

mixed-norm H

∞

analysis, 611

monotone convergence

theorem, 37

for integrals, 121

monotone norm, 487

monotonic function, 37

multilinear function, 515

Nash diﬀerential game

problem, 604

negative limit point, 97

negative limit set, 97

negative orbit, 97

negatively invariant set,

neighborhood, 22

nondecreasing functions,

301

nonexpansive, 351