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

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NonlinearBook10pt November 20, 2007

INDEX 945

nonexpansive control

Lyapunov functions,

631

nonnegative dynamical

system, 183

nonnegative matrix, 836

nonnegative vector ﬁeld,

837

nonparametric

uncertainty, 719

nonsingular transfer

function, 370

normal r ank, 370

normalized norm, 19

normed linear space, 59

nowhere dense set, 24

ω-limit set, 97

oﬀ-axis circle c riterion,

406

one-to-one map, 32

onto map, 32

open ball, 22

open cover, 60

open dynamical system,

328

open set, 22

optimal block

backstepping control

linear systems, 569

nonlinear systems, 573

optimal control, 511

multilinear

performance criteria,

528

polynomial

performance criteria,

526

optimal control for linear

discrete-time system,

866

optimal control problem,

442

discrete-time system,

846

optimal discrete-time

nonlinear contro l, 856

optimal integrator

backstepping control,

566

optimal linea r control,

538

optimal nonlinear

control, 519

orbit, 97

ordered ele ments, 21

orthogonal complement,

119

orthogonal vectors, 119

outer measure, 60

output strict passive, 3 51

p-norms, 14

parabola criterion, 406

parameter-dependent

Lyapunov b ounds,

737

parameter-dependent

Lyapunov function,

720

parametric uncertainty,

719

Parseval theorem, 363

Parseval’s theore m

discrete-time system,

840

partial derivative, 50

partial stability, 207

partial stabilization, 208

partial stabiltiy

discrete-time system,

776

partition of a set, 21

passive, 351

Peano theorem, 76

period, 11, 104

periodic dynamical

system, 11

periodic orbit, 104

periodic s olution, 104

phase plane, 132

phase portrait, 132

Picard method of

successive

approximations, 124

piecewise continuous

function, 44

piecewise continuously

diﬀerentiable

function, 45

Poincar´e-Bendixson

theorem, 109

Poincar´e return map, 291

Poincar´e stability

theorem, 293

polynomial function, 361

Popov criterion, 383

operator form, 507

Popov line, 386

Popov plane, 385

Popov plot, 386

port-controlled

Hamiltonian system,

190, 344, 421

positive limit point, 97

discrete-time system,

768

positive limit set, 97

discrete-time system,

768

positive orbit, 97

positive real lemma

discrete-time system,

818

nonlinear systems, 355

positive real transfer

function, 362

discrete-time system,

817

positively homogeneous

function, 116

positively invariant set,

discrete-time system,

768

positivity theorem, 375

discrete-time system,

840

practical stability, 236

NonlinearBook10pt November 20, 2007

946 INDEX

proper function, 362

proper subset, 21

Pythagorean theorem,

111

quadratic Lyapunov

bounds, 660

quasiconvex function, 758

quasimonotone functions,

300

radially unbounded

function, 144

range, 32

rational function, 362

rational transfer

function, 362

Rayleigh dissipation

function, 341

real Jordan

decomposition, 176

real Jordan form, 176

relative degre e , 450

remainder, 50

required exponential

supply, 336

required geometric

supply, 804

required supply, 336

discrete-time system,

804

time-varying systems,

398

residue matrix, 363

resource norms, 487

reversible ﬂows, 73

Riccati equation, 538,

548

Riemann integration, 61

right continuous

function, 45

right invertible function,

right max imal interval of

existence, 87

right piecewise

continuous function,

ring, 57

robust control

multilinear

performance criteria,

694, 754

polynomial

performance criteria,

686

via

parameter-dependent

Lyapunov functions,

743

robust control Lyapunov

function, 685

robust optimal control,

669

robust performance, 650 ,

651

robust stability, 372, 650,

651

discrete-time system,

884

parameter-dependent

Lyapunov functions,

731

rotating stall, 581

Schauder ﬁxed po int

theorem, 123

Schur matrix, 853

second law of

thermodynamics, 404

sector condition, 373

sector margin, 434

self-dual realization, 395

semigroup, 57

semistability, 260

semistable, 194

discrete-time system,

835

semistable equilibrium,

261

semistable system, 261

sequence of sc alars, 25

sequence of vectors, 26

set, 21

set of measure zero, 60

set theory, 21

settling-time function,

194, 254

shaped Hamiltonian, 346,

432

shaped Hamiltonian

function, 422

shifted Lur´e-Postnikov

Lyapunov function,

406

shifted Popov criterion,

405

simply connected set, 106

small gain theorem, 379,

476, 477

discrete-time system,

840

small gain- type

uncertainty, 655

solution curve, 72

solution of a diﬀerential

equation, 74

solution sequence, 764

spatial nor m, 65

sp e ctral abscissa, 515

sp e ctral radius, 852

sp ool dynamics, 593

stability margins, 434

discrete-time system,

830, 863

stability margins for

nonlinear regulators,

529

stability multipliers, 387

stability of periodic

orbits, 290

stability Popov

multiplier, 386

stability theory

time-varying systems,

225

NonlinearBook10pt November 20, 2007

INDEX 947

star-nor m, 503

state, 1

storage function, 330

discrete-time system,

800

strict bounded real

lemma, 368

nonlinear systems, 357

strict positive real

lemma, 365

discrete-time system,

819

nonlinear systems, 357

strict-feedback system,

563

strictly bounded real

lemma

discrete-time system,

822

strictly bounded real

transfer function, 362

discrete-time system,

817

strictly convex function,

strictly decreasing

function, 37

strictly dissipative, 336

strictly increas ing

function, 37

strictly passive, 351

strictly positive real

transfer function, 362

discrete-time system,

817

strictly proper function,

362

strictly proper transfer

function, 362

strongly bounded real

transfer function, 362

strongly positive real

transfer function, 362

structured disk margin,

435

structured uncertainty,

722

submultiplicative norm,

submultiplicative triple,

subsequence, 27

subset, 21

subspace, 58

sum of sets, 59

supply rate, 328

discrete-time system,

798

supremum, 25

surge, 581

deep sur ge, 587

surjective map, 32

susceptible-infected-

removed model,

197

synthesis, 10

system, 1

Szeg¨o criterion, 841

temporal norm, 65

time-invariant dynamical

system, 10

time-varying dynamical

system, 94

topological mapping, 49

topologically equivalent

sets, 49

topologically equivalent

systems, 125

trajectory, 72

transmission ze ro, 468

triangle inequality, 14

ultimate bo undedness ,

236

discrete-time system,

786

ultimately bounded with

bound ε, 241

discrete-time system,

791

ultimately bounded with

respect to x

uniformly in x

with

bound ε, 237

discrete-time system,

786

uncontrolled dynamical

system, 10

undisturbed dynamical

system, 10

undisturbed system, 2

uniform converge nce , 36

uniformly asymptotically

stable

discrete-time system,

784

time-varying system,

226

uniformly bounded, 239

discrete-time system,

789

uniformly bounded

function, 56

uniformly continuous

function, 40

uniformly ﬁnite-time

stable

time-varying systems,

318

uniformly Lipschitz

continuous function,

uniformly Lyapunov

stable

discrete-time system,

783

time-varying system,

226

uniformly ultimately

bounded with bound

ε, 239

discrete-time system,

789

union, 21

univer sal set, 21

unstable, 137

discrete-time system,

765

NonlinearBook10pt November 20, 2007

948 INDEX

unstable set, 283

upda te law, 189

upper right Dini

derivative, 265

upper semicontinuous

function, 43

van der Pol oscillator,

108

variable gradient

method, 152

vector comparison

principle, 302

vector energy-Casimir

function, 430

vector ﬁeld, 73

for time-varying

systems, 95

vector Lyapunov

function, 307

vector Lyapunov

theorem, 304

discrete-time system,

794

time-varying discrete

systems, 797

time-varying systems,

311

vector norm, 14

virtual available

exp onential storage,

388

virtual available storage,

329, 388

virtual exponential

storage function, 388

virtual stor age function,

388

Weierstrass theorem, 44

worst-case H

norm, 66 4

zero dynamics, 450

zero-state observable, 335

discrete-time system,

803

Zubov method, 157

Zubov theorem, 157