Index

ℓ₂([k₀, K], ℜⁿ), 19

ℓ₂-gain, 5

ℱ-derivative, 344

${\mathcal{H}}_2$-norm, 293

${\mathcal{H}}_{\infty }$ suboptimal control problem, 5

${\mathcal{H}}_{\infty }$-norm, 5, 293

ℒ₂([t₀, T], ℜⁿ), 19

ℒ₂-gain, 4, 287

state-delayed systems, 127

definition, 104

ℒ₂-gain almost-disturbance-decoupling, 133

ℒ₂-norm, 305

ℒ_∞-norm, 305

1-DOF filter, 217

discrete-time, 232

2-DOF filter, 220

discrete-time, 235

2-DOF proportional-derivative filter, 220

discrete-time, 235

Admissible filter, 228, 313, 324

Algebraic-Riccati equation, 111

ARE, 59, 111, 123, 124, 143, 336, 346

filtering, 213

stabilizing solution, 114

AREs

coupled, 290, 314

discrete-time, 301

Autocorrelation, 284

Available-storage, 120

Basis functions, 338

Bounded-power signal, 283

Bounded-real condition, 244

Bounded-real inequality, 146

Bounded-real lemma, 50, 111, 140

discrete-time, 70

Bounded-spectral signal, 283

Canonical transformation, 1, 82

Cascaded systems, 260

CEF, 315, 329

Central controller, 147

Certainty-equivalent controller, 139

Certainty-equivalent filter, 217

2-DOF, 220

discrete-time, 232

Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\infty }$

discrete-time, 329

mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\infty }$, 315

Characteristic curve, 93

Characteristic index, 133

control, 133

disturbance, 133

Class $\mathcal{KL}$-function, 23

Class $\mathcal{K}$-function, 23

Closed 1-form, 353

Congruence-transformation, 346

Conservative systems, 81

Constraints, 351

Contact transformation, 1

Continuously-differentiable, 343

Controller parametrization, 115

time-varying systems, 126

discrete-time

state-feedback, 184

discrete-time systems

general class, 196

general class of nonlinear systems, 162

output measurement-feedback, 147

state-delayed systems, 129

Coprime-factorization, 115

Cost function, 6

Cost functional, 6, 11, 13, 120, 176, 207, 225, 285

Curl condition, 344

DARE, 71, 181, 192, 245

DHJIE, 177

Differentiable manifold, 18

Differential game, 27

discrete-time, 30

dynamic, 27

linear-quadratic, 32

N-player continuous-time, 34

noncooperative, 27

nonzero-sum, 27

discrete-time, 31

two-player

nonzero-sum, 294, 307

zero-sum, 31, 32, 207, 285

zero-sum, 27

discrete-time, 176

Differential-form, 19

Discrete-algebraic-Riccati equation

(DARE), 231

Discrete-algebraic-Riccati equations (DAREs), 329

Discrete-Riccati equation, 247

Discrete-time Riccati inequality, 193

Discriminant-equation, 345

Discriminant-factor, 345

Disipativity, 140

Dissipation-inequality, 209, 227, 310, 322, 328

discrete-time, 66, 230

Dissipative Analysis, 111

Dissipative system, 8, 44

ℒ²-gain analysis, 54

discrete-time, 9

required-supply, 46

available-storage, 45

bounded-real lemma

for nonlinear systems, 55

discrete-time, 65

ℓ₂-gain analysis, 69

dissipation-inequality, 45

infinitesimal, 50

dissipation-rate, 44

local dissipativity, 44

losslessness, 48

Luré problem, 53

stability, 51

stability of feedback-systems, 52

storage-function, 44

supply-rate, 44

Dissipativity, 180

Dynamic controller, 7, 153

Dynamic feedback-controller, 138

Dynamic model, 350

Dynamic programming, 27

discrete, 29

Dynamic-programming equation, 6

Equilibrium-point, 19

Estimator, 319

Exact 1-form, 353

Exact solution, 343

Existence, 344

Exponential-stability, 244, 248

Factorization, 343

Feedback-equivalence, 61

Feedforward-control, 117, 151

Finite-dimensional filter, 246

Finite-gain system, 50

discrete-time, 70

First-order approximation, 336

Fréchet derivative, 19

Gain-matrix, 314

Gaussian white-noise, 12, 212

Generalized-coordinates, 80

Generalized-inverse, 345

Generating function, 1, 84

Guaranteed-cost control, 120

Hamilton, 1

Hamilton’s canonical equations, 81

Hamilton’s characteristic function, 353

Hamilton’s principal function, 1

Hamilton’s principle, 83

Hamilton-Jacobi equation, 1, 55, 86

generalized solutions, 99

Hopf-Lax formula, 97

separation of variables, 87

viscosity solutions, 99

Hamilton-Jacobi inequality, 55

Hamilton-Jacobi-Bellman equation, 1, 336

Hamilton-Jacobi-Isaacs equation, 36, 122, 142, 155, 337

discrete-time, 176, 320

filtering, 207

finite-horizon, 108

infinite-horizon, 108

time-varying, 126

Hamilton-Jacobi-Isaacs equations, 14

coupled, 31, 286, 290, 308

low dimensional, 315

Hamilton-Jacobi-Isaacs inequality, 112, 158

Hamilton-Jacobi-theory (HJT), 1

Hamiltonian canonical system, 1

Hamiltonian function, 80, 106, 131, 148, 178, 185, 208, 297

Hamiltonian matrix, 114

stable eigenspace, 114

Hamiltonian mechanics, 79

Hamiltonian systems, 352

Hamiltonian vector-field, 82

Hessian matrix, 74, 144, 146, 161, 163, 178, 208, 302, 344

Higher-order terms, 336

HJBE, 1, 335

HJE, 1

Homogeneous-function, 338

Immersed-submanifold, 355

Implicit-function Theorem, 87, 93, 343

Implicit-function theorem, 71, 131, 178, 185, 194, 226, 267, 292, 296, 321

Integral-curve, 19

Internal stability, 105, 182, 325

Invariant-manifold, 110, 267

Invariant-set, 19

Invariant-subspace, 346

Inverse-function theorem, 227, 235, 296

Isaac’s equation, 2, 225

Isaacs, 2

Jacobi, 1

Kalman-filter, 205, 305

Kinematic model, 350

Lagrange’s equation, 80

Lagrangian function, 79

Lagrangian mechanics, 1

Lagrangian-submanifold, 355

LaSalle’s invariance-principle, 22, 53, 56, 71, 109, 129, 150, 160, 165, 184, 187, 290, 291, 300, 341

Legendre transform, 83, 98

Lie-bracket, 19

Limit-set, 141

Line-integral, 344

Linear discrete-time system, 243

Linear-matrix-inequality, 181

Linear-quadratic game, 36

Linearized filter, 213

Linearized system, 113

Lipschitz continuous, 100

Local disturbance-attenuation problem, 5, 104

Local zero-input observability, 218, 233

Lyapunov-function, 22, 150, 164

Lyapunov-stability, 20

Maple, 350

Mathematica, 350

Matrix Riccati-ODEs, 36

Measurement noise, 207, 308

Measurement-feedback, 7

Method of characteristics, 92, 357

Cauchy problem, 93

quasi-linear equations, 92

for the Hamilton-Jacobi equation, 96

general first-order equation, 94

Minimax, 6

Minimum-phase system, 57

Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\infty }$ nonlinear control, 12, 285

discrete-time, 293

general class of nonlinear systems, 301

infinite-horizon, 299

linear systems, 297

finite-horizon, 286

for general class of nonlinear systems, 291

infinite-horizon, 289

linear systems, 288

Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\infty }$ nonlinear filtering, 305, 306

continuous-time, 305

discrete-time, 319

approximate solutions, 325

finite-horizon, 320

infinite-horizon, 323

linear systems, 328

infinite-horizon, 312

linear systems, 311

Morse-lemma, 73

Nash-equilibrium, 30, 286, 289, 292, 296, 307, 308, 329

Nash-equilibrium condition, 322, 326

Nash-equilibrium solution, 321

Nash-game, 30

Nonlinear ${\mathcal{H}}_{\infty }$ almost-disturbance-decoupling, 132

state-feedback, 133

Nonlinear ${\mathcal{H}}_{\infty }$ tracking control, 116

output measurement-feedback, 151

state-feedback, 117

Nonlinear ${\mathcal{H}}_{\infty }$-control, 2, 4

discrete-time

linear systems, 184

full-information, 176

state-feedback, 182

discrete-time systems, 175

approximate approach, 198

general class, 194

for singularly-perturbed nonlinear systems, 265

measurement-feedback

linear systems, 142

output measurement-feedback, 137

general class of nonlinear systems, 158

definition, 138

discrete-time systems, 188

output-feedback, 137

state-feedback, 10, 104

general class of nonlinear systems, 131

composite-control, 269

full-information control, 115

linear systems, 110

singularly-perturbed systems, 266

state-delayed systems, 127

time-varying systems, 125

static output-feedback, 169

Nonlinear ${\mathcal{H}}_{\infty }$-filtering, 15, 205

discrete-time, 223

approximate solution, 228

infinite-horizon, 227

robust filtering, 243

finite-horizon, 207

infinite-horizon, 210

linear systems, 209

robust filtering, 214

singularly-Perturbed Systems, 271

Nonlinear benchmark problem, 340

Nonlinear lattices, 88

Nonlinear${\mathcal{H}}_{\infty }$-control

state-feedback

finite-horizon problem, 108

Objective functional, 105

Observer-based controller, 8

Output-feedback, 153

Output-injection, 138, 191

Parametrization, 354

Passive systems, 56

KYP-lemma, 60

definition, 57

discrete-time, 67

bilinear system, 69

bilinear KYP-lemma, 69

feedback-equivalence to a lossless system, 72

losslessness, 67

positive-real-lemma, 69

strictly passive, 67

feedback-equivalence, 62

lossless system, 60

passivity, 60

positive-real lemma, 58

positive-real systems, 60

strictly-passive, 57

Pendulum, 340

Phase-space, 80, 81, 353

Point transformation, 86

Poisson bracket, 83

Polynomial expansion, 338

Positive-definite function, 21

Positive-semidefinite matrix, 20

Potential function, 353

Power-spectral density, 284

Principle of least-action, 83

Projected characteristics, 95

Proportional-derivative filter, 220

Reachability, 5

discrete-time, 5

Relative-degree, 63

Reliable nonlinear ${\mathcal{H}}_{\infty }$-control, 155

Riccati-ODE, 209, 210

Riccati-ODEs, 39

coupled, 289, 311

discrete-time, 298

Robust Nonlinear ${\mathcal{H}}_{\infty }$-Control, 14

definition, 119

Robust Nonlinear ${\mathcal{H}}_{\infty }$-control

state-feedback, 119

Robust nonlinear ${\mathcal{H}}_{\infty }$-control

linear systems, 123

output measurement-feedback, 153

reliable control, 155

Saddle-point, 40, 208, 229

Saddle-point conditions, 122

Saddle-point equilibrium solution, 6

Saddle-point solution, 32, 106

Sattelite, 350

Scaled system, 216

Schur’s complement, 178

Separation of variables, 348

Simple-connectedness, 353

Singular nonlinear ${\mathcal{H}}_{\infty }$-control, 251

state-feedback

high-gain feedback, 255

output measurement-feedback, 257

State-feedback

cascaded systems, 263

state-feedback, 252

linear systems, 256

regular control, 254

static output-feedback, 258

Smooth-stabilizability, 104

Spacecraft, 350

Stability, 20

asymptotic, 20

exponential, 20

uniform, 23

asymptotic, 160, 165, 290, 291, 300

exponential, 24, 125

global uniform-asymptotic, 24

global-asymptotic, 22, 109

local-asymptotic, 109, 187, 341

Static output-feedback control, 166

Strict-feedback systems, 133

Subdifferential, 99

Superdifferential, 99

Symplectic-coordinates, 353

Symplectic-manifold, 82

System noise, 207, 308

Tangent and cotangent bundle, 18

Taylor-series, 71, 76, 165, 178, 185, 296, 297, 338, 344

Taylor-series approximation, 328

Toda lattice, 89

periodic, 91

two-particle, nonperiodic, 355

Tracking control, 116

Uncertainty, 14

matching-conditions, 120

admissible, 119, 153, 214, 243

structured, 120

Value-function, 6, 28, 105

Vector-field, 18

Viscosity subsolution, 100

Viscosity supersolution, 100

Volume-form, 355

Weakly minimum-phase system, 64

Youla (Q)-parametrization, 115, 147

Zero-dynamics, 57, 63

Zero-state detectability, 5, 109, 227, 312, 324

discrete-time, 5

state-delayed systems, 127