Subject Index

A accessibility of nonlinear systems, 6, 454 fT.

to the origin, 459 computational complexity, 461 Lie algebra, 457 Ir. rank condition, continuous, discrete

time, 457 fT. strong, and controllability, 468

Ackermann's formula, 319 adaptive control and identification, 387-450 adaptive control, 2, 6, 19, 437-450

algorithms, 438 fT. direct, indirect, 438 fT. indirect, unexpected properties, 445 fT. industrial use, 448 LS estimator as KaIman filter, 443 fT. microprocessor implementation, 448 model reference, 437 self-oscillating, 437

adaptive filters for learning systems, 580 fT. descent equation, 589 fT. realization, 582

algebraic bundle, 335 algebraic coding, see: coding algebraic canonical form, 335 algebraic geometry, 6, 327 fT. algebraic groups, 322 fT. algebraic system theory, see: module theory

and linear system theory; linear systems; polynomial matrices; realization problem

algebraic varieties and morphisms, 328 fT. affine varieties, 331 c1assification of varieties, 330 quasi-projective varieties, 335

algebraization of system theory, 234 algebras, C*, W*, von Neumann, 137

predictor, splitting, 218 algorithms,

B. L. Ho, 284 Berlekamp-Massey recursive, 201 Nevanlinna-Pick, 201, 202 for triangularizing polynomial

matrices, 363-365 analytic c10sed loop solvability (analytic

CLS),407

applications of system theory, 525-592 approximation, Pade, Cauchy, 202, 504 approximative system models (ASM), 390 fT. AR models, 21 fT. ARMA models, 24fT., 216

observability and continuity, 33 ARMAX models, 395 fT. arithmetic,

integer vs. rational, 360-363 modular, 5, 234-239, 250

assignability, pole and coefficient, 319 assignment of dynamics and invariants,

306-308 automata theory and Nerode principle, 296 auxiliary variables, 5, 216 axiomatic framework, 3, 15-37, 503

B balanced realizations, 248

for all-pass transfer functions, 260 and continued fractions, 256

Banach state space, 208 bandwidth, 102, 184 bang-bang principle, 148 barreled space, 206 bases, minimal or proper, 7, 303 fT., 542 fT.

and poles at infinity, 542 Bayesian sufficient statistic, 218 behavior (Hankel) matrix, 193 fT., 340, 396

partially determined, 197 beha vior of model, 19 fT. behavioral approach to system theory, 3,17-37

behavioral equations, 19,21,36,216 Bellman dynamic programming, 18, 19, 148 Berlekamp-Massey recursive algorithm, 201 Beurling-Lax theorem, 504 Bezout identity (equation), 239, 322, 356, 367

double, 350 Bezoutiah matrix, 84, 250 fT. Bezoutian quadratic form, 250 fT. bicausal maps, 236, 275-277 black box modeling (description), 1 fT., 191 fT.

see also: realization problem Bode and Shannon whitening filter, 225 Bode integral formula, 147

594 Subject Index

Brownian motion (or Wiener process), 64 Bryson-Frazier formula, 70

C calculus of variations, 70 canonical embedding and projection, 271, 299 canonical forms, 5, 23, 36, 239, 280

and continued fractions, 256, 257 canonical forms, global algebraic,

non-existence of, 337 canonical structure (decomposition)

theorem, 192, 196 extensions, 6, 475-488 nonlinear systems, 478 fT. time-varying systems, 476fT.

canonical (controllable, observable) systems, 26, 284, 339-341

see also: minimal systems, minimal realizations

canonical system, weakly, 205 cascade interconnection, 201 category theory and system theory, 279,

284-288 Cauchy approximation, 202 Cauchy index, 253 causality and stability, 301 Cayley-Hamilton theorem, 332, 340, 410 CCD filters as systems over rings, 312 chaotic behavior, 2 characteristic and transfer functions, 503 Cholesky factorization, 406 cIosed orbit lemma, 330 co-state or Lagrange multiplier, 180 coding, algebraic and system theory, 527-557

block codes, 527 catastrophic encoder, 531 convolutional codes, 7, 290, 527

algebraic methods, 7, 290, 550 fT. encoder, 356, 529, 536 fT. dual codes and encoders, 539 fT. generalized minimal encoder, 545 fT. minimal bases and poles at infinity, 542 fT. parity-check matrix, 541 realization of encoders, 546 syndrome, 539 systematic encoders, 541 fT.

code sequence, code, 530 generator matrix, 530 Hamming (free) distance, 530 performance vs. complexity, 527 systems over the integers, 312 system theory and algebra, 532 fT. trellis codes, 539,fT.

co integration, 7, 560 fT. and zero structure, 565

commutant lifting theorem, algebraic version, 241

commutative algebra, 280, 283, 285

commutative diagrams, 246fT., 271 fT., 282fT., 302, 336

compensation, dynamic, see: control theory controllers, feedback, H", problem, LQG problem

complexity, high, 1 McMillan (or Smith-McMillan)

degree, 192 fT., 396 computations, in Wiener vs Kaiman filtering, 46

symbolic and error-free, 356 modern integrated circuit technology, 67 complexity of controllability-accessibility, 461

computer algebra and controller synthesis, 355-367

Macsyma and Mathematica, 357 computer technology, perspective on, 438 consistency and asymptotic normality of ML

estimators, 406-420 consistent (strongly) parametrization, 412 consistent estimators, 434 continued fractions, 5,.239, 244, 256-264, 371

and realization, 201 continuity and observability, 33 control theory,

multiple objective problems, 173 by polynomial matrices, 356 fT. distributed systems, 491-499 assign dynamics to control variable, 215 geometrie, 290 adaptive, 2, 6, 19, 437-450 cut-and-try methods, 153 H", optimal, 4, 159

and LQG 156 frequency-domain/operator-theoretic

methods, 160 state-space methods, 160-174

model-based, 152, 153 systems over rings, 319-322 worst case design, 159

control variable, no dynamical description, 215 co nt roll ability, 2, 3, 6, 17-19,36,46,60, 100,

234,239,267,282,345-347,390,503,505 and Kronecker indices, 553 and pole cancellation, 346 and strong accessibility, 468 distributed systems, 6, 492 fT. generalization of Hautus test, 27 linear and nonlinear, 472 linear systems using modules, 469 fT. loss of and common factors, 30 matrix (reachability matrix), 196, 332, 454 nonlinear systems, 6, 453-472

(accessibility) to the origin, 459 (accessibility), computational

complexity, 461 and Lie brackets, 455 using difTerential fields, 463-472

observability and minimality, 26 path-, and tracking of targets, 566 fT.

pole-assignability and feedback, 267 controllable systems and transfer functions, 30 controllers,

H"" central and LQG c1assical, 168 PID, 18 stabilizing, YJBK parametrization of, 6,

159, 199, 349-353 synthesis and computer algebra, 355-367

convergence, of polynomial matrices, 32 of systems, 321T.

convolution systems, 194,216, 244, 315 convolutional codes, see: coding coprimeness,

of polynomial matrices, 239-242, 307, 348 and corona theorem, 239 . approximate,211 H"",239 observability, reachability, 248

corona theorem and coprimeness, 239 covariance, 45, 57, 90, 216, 227, 407 cross correlation, 222 cybernetics, 17

D Darlington synthesis, 201, 264 datacompression, realization problem, 193,295 dead-beat control, 147 description, external, input/output, black

box, 4, 191,213 internal, state space, 4, 191

see also: state space models see also: modeling, realization problem,

behavioral approach design theory, 1471T. detectability, 47, 48 detector, optimal, 66 dilTerential algebra and fields, 6, 4641T.

dilTerentially algebraic elements, extensions, 4651T.

dilTerentially transcendental elements, extensions, 4661T.

dimension of state space systems, 194 see also: canonical systems

Diophantine (Bezout) equations, 356 discrete event systems, 356 displacement, structure, 781T.

rank, 821T. distributed and nonlinear systems, 451-500 distributed systems, control of 491-499 distributions, Schwartz, 193, 205, 208

E-module framework, 314 smooth, involutive, integrable, 4791T.

disturbance decoupling problem, almost, 164 division, Euclidean, 3, 5, 201, 202, 239, 256,

261,359 pseudo-, 360 IT.

divisors, common, greatest, left-right, 239-240, 358, 533

Subject Index 595

Dry-Tuned-Gyro (DTG), 96 duality, exact reachability and topological

observability, 207 duality, KaIman filtering and LQG

problem, 47, 60 dynamic programming, 18, 19 dynamics assignment and invariants, 306-308 dynamics, topological, 37

E E-module framework for systems over rings, 314 econometrics,

error correction models, 561 errors-in-variables models, 5691T., 572 factor analysis models, 571 inputs and outputs, 5691T. system-theoretic trends, 7, 559-577 tracking of targets, 5661T.

economic time series, random drift, 560 electronic devices, solid state, 17 engineering,

theory, practice, and art, 296 vs. physics, 154 mathematization of, 296 theoretical, 18, 296

entropy minimization problem, 174 equations,

behavioral, 19, 21, 36, 216 Chandrasekhar, 69, 78, 81, 83 c1assical Wiener-Hopf, 78 delay dilTerential, 211 forward and backward evolution, 76 gyro/accelerometer motion, 97 integral and Riccati equations, 64 integral, Fredholm-type, 63 Lyanunov dilTerential, 74 Wiener-Hopf-type, 63, 78, 79 stochastic dilTerential, 65, 227

equivalence, stochastic, 215 equivalent (linear) systems, 194, 3301T. errors-in-equations models, 423 errors-in-variables (EV) models 216, 423,

5691T., 572 estimates,

conditional mean, 42 least-squares, 6, 56, 65, 89, 4381T. filtered, smoothed, 56, 69

estimators, asymptotic properties of, 393 causal, on-line, optimal, 41 consistent, 434 gyro/accelerometer, 99 maximum likelihood (ML), 400

consistency and asymptotic normality, 406-420

minimum prediction error (MPE), 3911T. optimal state, 183 optimal (ASM), 3911T.

596 Subject Index

optimal (KaIman filter), 45 unbiased, 165, 186

EucIidean algorithm (division), 3, 5, 201, 202, 239, 256, 261, 359

Euclidean and Gaussian elimination, 356, 360 expectation, conditional, 64

F factor analysis models, 221 ff., 423, 571 factorization and realization, 272 factorization, co prime, 5, 31

see also: coprimeness factorization, spectral, 43, 215, 225 factors, common and loss of controllability, 30 families of linear systems, 3, 325-342

adaptive and robust control, 341 input/output and state space systems, 195 reachable systems, 332-333

fast algorithms, 69, 356 feedback, 2 ff.

algebraic approach, 268, 276 and feedforward strategies, 187 and low sensitivity to uncertainty, 187 and realization, module-theoretic

framework, 268 bandwidth and low sensitivity, 187 bandwidth and non-minimum phase

zeros, 188 controllability, pole-assignability, 267 cycIizability, 320 geometrie approach, 267 output feedback 167 ff., 258 ff., 308, 347 realization and invariants, 295-308 state, 162 synthesis, general problem, 351

and computer algebra, 355-367 controllability and observability, 351 fast algorithms, 356 norm minimization, 353 YJBK parametrization, 349-353

filtering problem, 4, 136, 141 see also: KaIman filtering adaptive filters for learning, 580 ff. anti-aliasing filters, 183 discrete nonlinear, 136 for point processes, 67 H<Xl filtering, 166 lattice filters, 84, 380 ff. matched filters, 65 nonlinear Gaussian, 68 notch filters, 100 recursive nonlinear, 67 whitening filters, 57, 60, 223 ff.

Fokker-Planck equation for Markov density functions, 59

formal power series, 235, 530 see also: Laurent series

Fourier transform, 56, 185,225 fractional transformations, linear, 201

fractions, continued, see: continued fractions fractions,

polynomial matrix, 200, 243, 275-277, 347 ff., 367, 395

proper stable rational, 200, 347 ff. Fredholm operator, 514 Fredholm-type integral equations, 63 frequency response, 29, 30, 36 frequency- vs. time-domain in module

theoretic framework, 268 friction,

solid, 94, 105 ff. compensation, 94, 107 Coulomb, Dahl model, 105 gimbal bearing, 105

Frobenius theorem on signature of Hankel matrix, 257 on integrable and involutive

distributions, 481 Fuhrmann realization, infinite dimensional, 209 fundamental result of realization theory, 272

G gain scheduling, 437 gain, KaIman, 44

optimal,99 static, 163 strategy of high, 155

gain-phase relation, 147 games, linear-quadratic and H", control, 164 gap, theory-practice, 4, 148 Gauss, caIculations of orbits of asteroids, 59 Gauss-Markov model, 44, 90, 93 Gaussian and Euclidean elimination, 356, 360 Gaussian process, 66 gcId-gcrd (greatest common left-right

divisor), 239-242, 307, 348 geometric quotient, 332 ff., 340 geometric structure of orbit spaces, 330 gimbal bearing friction, 105 Gohberg Semencul formulas, 84 GPS receiver KaIman filter, 110 . Gram-Schmidt orthogonalization, 60 gramian of linear system, 260 Granger representation theorem, 561 ff. Grassmannian and KaIman space, 335 guidance and navigation, 4, 89-134 gyro/accelerometer, multisensor, 96

equations, estimation and control, 99 gyroscope, gyrocompass, 92

H H", problem,

standard control problem, 161 output-feedback control problem, 167 state-feedback control problem, 162 H", coprimeness, 239 H", filtering and KaIman filtering, 166 H", filtering and smoothing problem, 165

Hamiltonian, equations, 60, 82 framework, 191 matrix, 70, 180

Hamming (free) distance, 530 Hankel,

matrices (behavior matrices), 193,340,396 signature of, 341 partially determined, 197

functional equation, 242 map or Kaiman input/output map, 271 operators and module

homomorphisms, 242-244 operators, singular values of, 260 quadratic form, 250

Hautus controllability test generalization, 27 heat bath, 217 Hermite form of polynomial matrices, 357,

359, 367 Hermite-Fujiwara, matrix, 256

quadratic form, 252-256 Hermite-Hurwitz stability theorem, 254 Hermitian quadratic form, 251 Hessenberg form, 380 hidden modes (decoupling zeros), 471 fT. Hilbert, Nullstellensatz, 331

state space, 205 uniqueness method (HUM) 491, 495

Hurwitz stability, 372 IT.

I identification and adaptive control, 6, 387-450 identification from noisy data, 3, 423-435

consistent estimators, 433-434 in econometrics, 423, 560 prejudice in identification, 425 problem formulation, 425 static, dynamic cases, 425 static case and Perron-Frobenius theory, 427 the solution set, 426 IT.

identification, linear stochastic, 389-421 ASM-PEM framework, 391 IT. approximative system models (ASM), 390 IT. basic nonidentifiability, 424 consistency and normality result, 390 IT. construction of likelihood function, 400-406 error magnitude vs model complexity, 392 identifiability, of parametrizations, 399

of stochastic models, 228 asymptotic, 391, 408

input hypotheses, asymptotically stationary process, 393

deterministic input, 393 wide-sense stationary process, 394

modeling and parameter estimation, 390 persistent excitation, 407 IT. prediction error methods (PEM), 390 IT. system identifiability result, 390 IT.

IEEE Medal of Honor, 453 implicit function theorem, 456, 467

Subject Index 597

impulse response, pseudorational, 208 indices,

of polynomial bases, 291, 554 reachability or Kronecker, 3, 201, 305 stability and module, 303-305

industrial processes, 154 inequalities, quadratic matrix, 164 inertia of a matrix, 80, 84 inertial navigation systems (INS),

Alidade alignment, 121 calibration of a ring laser gyro, 115 gimballed, 115 implementation of calibration Kaiman

filter, 118 SIGAL systems, 96 SIGMA calibration method, 116 strapdown, 115, 116 Inertia-GPS, ULISS systems, 125

Kaiman filter implementation, 131 multisensor hybrid navigation, 125

software, 128 infinity, poles and zeros at, 287 information theory, 18

see also: coding innovations, 4, 50, 228, 406

and spectral factorization, 49 martingales, scattering, 55-88

input/output spaces, 194, 235 IT., 269 IT., 298 input/output, maps (descriptions), 191, 194,

269,299 see also: description, external stable maps, 300

parametrization of, 345-353 inputs, control and exogenous, 161,351 inputs, random, as internal variables, 5, 228 integral equations, singular, state space

solution, 7, 509-523 integral operators, inversion by factorization

methods, 520 fT. inversion by input/output methods, 512 IT.

integral, stochastic, Ito, Stratonovich, 60, 64, 66 internal or state space models (description),

see: state space models internal stability, 302, 349 fT.

see also: YJBK parametrization interpolation and realization, 7, 202 interpolation by state space methods, 7,

503-507 intertwining relation, 244 invariant factor theorem, 7, 529 IT., 534 invariant factors, 280, 284, 307, 367

and pole assignment, 297 invariant imbedding theory, 78 invariant subspaces, geometry of, 239 invariant theory and families of systems, 6,

327-342 invariants, realization and feedback, 295-308

system performance, 5, 296 inverse scattering, 84, 201 irreversibility of stochastic evolutions, 227

598 Subject Index

Itö dilTerential rule, 62 Itö stochastic integral, 60

K KaIman experimental setup, 271 KaIman filtering, 2-4, 6, 7, 18, 19,41-143,

185,400,438,492,559 Bierman factorization algorithms, 119 and H", filtering, 166 adaptive filtering, 53 advance in digital data processing, 134 asymptotic stability and time invariance, 47 Chandrasekhar type algorithms, 53 compensation of solid friction, 94 computational issues, 46, 53 continuous time (Kalman-Bucy filtering), 44 controller design, 53 discrete time, 48 dual of LQG problem, 47 extended or nonlinear, 52, 67, 106, 124 historical comment, 59 implementation on digital computer, 91 infinite time interval, 46 key dilTerences with Wiener filtering, 46 LS estimator in adaptive control, 4431T. navigation and guidance, 89-134 quantum, 4, 135-143 separation of dynamics and

measurements, 48, 49 square root filtering, 53 stability, 60 state-space formulas, 78

Kalman-Bucy or continuous-time KaIman filtering, 44, 56-60, 79, 81

Kalman-Bucy formulas, 63, 64, 69 KaIman,

gain,44,79 controllability (reachability) and

observability matrices, 196, 332, 454 input/output or Hanke! map, 271, 2821T. realization diagram, 282 space of families of systems, 3281T.

and the Grassmannian, 335 construction of, 334-335 geometric structure, 335-337

Karmarkar descent algorithm, 591 Kepler's laws, 20 Krohn-Rodes theory, 284 Kronecker indices, 3, 201, 305 Kyoto Prize, 41

L Lagrange multiplier or co-state, 180 Lagrangian framework, 191 Laplace transforms, 36 LaSalle's bang-bang principle, 148 latent variables, 19,36,424 laUice filters, 84, 380 IT. Laurent series, 282, 2971T., 530

and time-invariance, 269 formal, and z-transform, 281 rational, 236 truncated, 235

Iclm-~crm (least common left-right multiple), 240 learnmg systems, 7, 5791T.

adaptive subspace filters, 580 IT. realization, 582

autoassociation and regression, 5841T. continuous and combinatorial aspects, 590 descent equation implementation, 5891T. fading memory filters, 5871T. singular value decomposition, 580 total least squares approximation, 586

least squares estimation, 56, 65, 89, 4381T. Levinson algorithm, 83 Lie brackets, 455, 457, 463, 479, 482, 582 Lienard-Chipart stability criterion, 253, 375,

378 lifting, 137, 138 likelihood function contruction, 400-406 likelihood matrix, 89 linear systems,

algebraic and analytic object, 5 algebraic-geometric framework, 327 canonical or minimal, 246, 282

see also: canonical realizations discrete, stability of, 371-384 equivalent systems, 194, 330 IT. families of, 5, 325-342 gramian and singular values of, 260 general feedback synthesis problem, 351 infinite dimensional, canonical, 207 local analytic vs. global algebraic

viewpoint, 327 over rings, 311-322 parametrization of input/output

maps, 345-353 poles and zeros, 280, 289-292 state space, definition, 194, 244, 270 stochastic, 399 universal parameter space, 327 see also: AR, MA, ARMA, ARMAX, SSX

models linearity and time invariance, fusion of, 298 localizations of polynomials, 2991T. Löwner interpolation problem, 505 Löwner matrix, 202 log-likelihood function, 403 LQG (linear quadratic gaussian) problem, 3,

4,18,70,96,145-188,347,439 and state-space H", theory, 159-176 impact of, 152 industrial applications, 152 intuitive interpretation, 177 microprocessor implementation, 152 robustncss of, 155 time-domain version of Wiener theory, 148 unified continuous and discrete time

theory, 177-188

LTR (loop transfer recovery), 155, 187 Luenberger form, 382 Lure's theorem, 147 Lyapunov,

equation, 74, 256, 406 function, 379 stability theorem, 255 second method, 377

Lyapunov-Routh-Hurwitz stability, 374

M MA models, 27 Cf. Macsyma and Mathematica, 357, 366 manifest variables, 20, 36 Mansour or discrete Schwarz form, 6,

375 Cf. map, continuously invertible, 205, 207 margin, gain and phase, 102, 183 Markov,

chains, quantum, 135, 141 density functions, Fokker-Planck equation

for, 59 diCfusions, 60 parameters, 192, 195, 244, 269, 270 process,90 property, 74, 217 splitting property, 219 Cf.

martingales, 4, 64, 66, 223 match between dynamical intuition and

algebra, 287 mathematical models,

behavior and behavioral equations, 19 Cf. extern al (phenomenological), internal, see:

description see also: modeling, realization problem

mathematics and system theory, 501-523 mathematization of engineering, 296 matrices,

behavior or Hankel, see: Hankel matrices

Löwner, 202 polynomial, see: polynomial matrices scattering, 71

matrix factorization, 83, 504 matrix fractions for rational matrix

functions, 200, 243 Cf., 275, 313, 347 Cf. coprime fractions, see: coprimeness

matrix pencils, 510 matrix rational interpolation

problems, 201 Cf., 503 Cf. maximum likelihood estimators, 390, 400

consistency and asymptotic normality, 406-420

maximum principle, 18, 19, 148 Maxwell, J. c., 17,233 Mayne-Frazer smoothing formula, 73 McMillan degree, 192

see also: invariant factors memory span, 22, 32

Subject Index 599

memoryless instrument (quantum probability), 140

minimal (controllable, observable) systems see: canonical systems

minimum phase plant, 187 minimum variance control, 439, 442 model reduction of discrete time systems, 381 Cf. model-based control, 152, 153 modeling, see also: realization problem

state construction, 191 Cf. issue of modeling, 153 Cf. stochastic exact, 393 stochastic, distributional, 214 Cf.

models, see also: realization problem see also: AR, MA, ARMA, ARMAX, SSX

models black box 1 Cf., 191 Cf. dynamical, for learning, 579 errors-in-variables, 6, 216 from first principles, 36, 154 Gauss-Markov,44 identifiability of stochastic, 228 modules and algebraic system

theory, 279-292 most powerful unfalsified (MPUM), 37 sampled, 183 scattering, 71 stochastic, 183, 216

modes, hidden, unstable, 347 modular arithmetic, 234-239, 250 module structure, stochastic context, 224 module theory and linear system theory, 3,

5, 233-327, 469 Cf. dynamics, 288 stability framework, 302 Cf.

module and stability indices, 303-305 modules, strict observability, 302 modules over the polynomials, see:

polynomial modules and matrices modules, pole and zero, 287, 289 modules, torsion, and finite dimensional

realizations, 274 moduli space of dynamical systems, 330

classification, 390 fine, 337

moving average control, 440 Cf. MPUM (most powerful unfalsified model), 37 multiple, common, least, left-right, 239-240,

358,533 Mumford's theorem, 332, 334, 335

N Navier-Stokes equations, 492 navigation and guidance, 4, 89-134

navigation Kaiman filters, implementation, 94-134

hybrid systems, 93 inertial navigation, "strapdown" system, 91 Cf.

600 Subject Index

navigation and guidance (cant.) inertia-GPS multisensor system, 94 ring laser gyrosystem, 94 SIGAL systems, 96 Super-Etendard system, 94

radio navigation, 91 radio satellite navigation, 94 radionav systems, 93 NA VSTAR GPS (global positioning

system), 110 Nehari problem,

state space solution, 504 algebraic version, 242

Nerode equivalence, 205, 209, 211, 283 Nerode theorem, 389 networks, analysis, synthesis, 17, 192 neural models, 579 Nevanlinna-Pick,

algorithm, 201, 202 interpolation problem, state space

solution, 7, 504 fT. nice selection, 334 N oetherian domain, 313 fT. noise, variables, 218, 222-227

additive, 68 covariance matrix, 89, 427 measurement,45 modeling, 423 multiplicative, 68 process, 41, 167 white, 44, 65, 90

non-Gaussian processes, 67 nonlinear and distributed systems, 451-500

accessibility, 454 fT. controllability, 453 fT. local controllability and Lie brackets, 455 nonlinear dynamics using difTerential fields,

4661T. real-analytic dynamics, 454

normal equation, 89, 91 numerical sensitivity of floating-point

methods, 357 Nyquist, 17 Nyquist stability test, 147

o observability, 2 fT., 17 fT., 28 fT., 46, 60, 92, 100,

116, 192, 196 245, 267, 282, 284, 339, 345 fT., 390, 476, 486, 503, 505

and continuity, 33 and injectivity, 273, 302 and reachability indices, 198 and right coprimeness, 248 matrix, 196 controllability and minimality, 26 gramian,72 topologieal, 207, 208

observability/reachability decomposition, 192, 196

see also: canonical structure theorem observables, algebra of, 136 observer, 19, 44, 348 operation valued measures, 138 operator model theory, 503 operator, Markovian, 137 operator, Volterra, 63 fT. optimal control, 145-188

inverse problem of, 148 and H"" 164

control performance vs. input power, 151 optronic systems, 105 orbit space, 340 order chain and list, 304 orthogonality condition, 57, 62, 69 orthogonality of subspaces, conditional, 220 output and input spaces, see: input/output

spaces output-feedback canonical form and

invariants, 258-259 outputs, regulated and measured, 161, 351

P Pade approximation, 202, 256, 504 parameter identifiability, 399 parameter space, 396 fT. parameter space, universal, of dynamical

systems, 337-339 parametrization,

asymptotically identifiable, 408 and continuity, 29 fT. of stable input-output maps (YJBK

parametrization), 199,345-353 strongly consistent, 412

partial realizations, 5, 84, 193, 197fT., 256 and properness of compensator, 200

passive network synthesis, 192 PD Es, evolution type, 491

boundary and initial conditions, 491 performance vs. robustness, 173 Perron-Frobenius theory of positive

matrices, 430 persistency of excitation, 407 fT., 444 physics vs. engineering, 154 physics, classical and quantum, 156 point processes, filtering for, 67 Poisson processes, 67 pole assignment,

controllability and feedback, 267 control, 439 and invariant factor assignment, 297

pole module, 283 fT. indices, degree, 305

pole cancellation and controllability, 346 pole-zero modules, 2891T. pole-zero cancellation,

interconnected systems, 290

reachability, observability, 440 unstable, 346, 348

pole-zero exact sequence, 290 poles and zeros of linear systems, 268, 280,

289-292 poles at infinity and minimal bases, 542 Ir. polynomial (module) action and time-shift, 281 polynomial ba ses and indices, minimal, 291, 554 polynomial matrices, 5, 21

terminology of 357-359 column, row reduced, 552 convergence of, 32 coprimeness, 239-242, 307, 348 elementary row and column operations, 358 fractions of, 200, 243, 275-277, 347 fT., 367, 395

poles, zeros at infinity, 552 Hermite forms, 357 predictable degree property, 538 triangularization of, 356, 359 fT. unimodular, 29, 358

polynomial models, 5, 235, 247 polynomial modules, 235, 268, 280

and dynamical structure, 280 and input/output spaces, 281

polynomials, 3, 235, 281 computation, 348 error-free computation, 6, 357 fT. generalized, 349 localizations of, 299 fT.

Pontryagin maximum principle, 18, 19, 148 positive definiteness and stability, 253 positive real lemma, 3, 192, 215 power series, formal, 235

rationality of, 313 see also: Laurent series

prediction, 46, 389 prediction error methods (PEM), 390 fT.

loss and cost functions, 391 recursive construction, 403

predictor algebras, 218 prejudice in identification, 425 principal ideal domain, 300, 315, 532 probability,

spaces, measures, distributions, 215 fT. axiomatic framework for modeling, 216 conditional, 217

processes, random (stochastic), 41, 45, 215 fT. Gaussian stationary, 400 non-Gaussian, 67 Poisson, 67, 68 purely non-deterministic, 224 state-space description of nonstationary, 57 wide sense or second order, 216, 217, 394

properness of compensator and partial realization, 200

pseudo-division lemma, 361 pseudorationality, 208 Ptak space, 206

Subject Index 601

Q quadratic forms: Bezoutian, Hankel,

Hermite, Hermite-Fujiwara, 250-256 quadratic stabilization theory, 156 quadrature, 80

R radiative transfer theory, 76, 78 Radon-Nikodym derivative, 65 random drift in economic time series, 560 random inputs as interna I variables, 228 random processes and variables, 41, 45, 215 fT.

see also: processes rational interpolation, 201-203, 503-507 rational functions and matrices,

field of rational functions, 235 proper stable fractions of, 200 factorization problems, 504 interpolation problems, 201 fT., 503 fT. Smith-McMillan form, 7, 289, 367, 395,

532, 534, 553 state space methods, 504 see also: transfer functions, power series

rational matrix symbol, 509 rational model, 248 rational vector spaces, minimal (proper)

bases, 7, 303, 542-548 and poles at infinity, 542

rationality, and finite dimensional realizations, 274

reachability, 192, 196,245, 284, 332, 475, 486 and feedback cyclizability, 320 and lert coprimeness, 248 and observability indices, 198, 305 and surjectivity, 273, 302 approximate, 205, 207 input/state map, 276, 277 matrix (controllability matrix), 196, 332,454 pairs, 227, 338 reachable family of systems, 337 set, 455

reachability /observability canonical decomposition, 192fT., 475fT.

see also: canonical structure theorem realizability and separability, 487 realization (modeling) problem, 3 fT, 19, 37,

189-229, 244-249, 272, 339, 389, 511 abstract realization, 301

and factorization, 246, 272 and continued fractions, 201, 256

and feedback, module-theoretic framework, 268

and interpolation, 7, 193,201-203 and stability modules, 301-303 and synthesis problems, 199-200 canonical (mirninal) realizations, 193,267,

273, 282, 284, 302, 345 compressibility of infinite data set, 193, 295

602 Subject Index

realization (modeling) problem (cant.) continuity in the parameters, 341 continuous-time linear systems, 193 dia gram of Kaiman, 282 feedback synthesis, 193 finite-dimensional, 274 finiteness = finite rank ofbehavior matrix, 196 finiteness = finite support of denominator

distribution, 208 infinite dimensional environment, 217 infinite dimensional Fuhrmann realization,

209 invariants and feedback, 295-308 linear deterministic, 191-212

complete data, 193-197 finite-dimensional systems, 193-204 infinite-dimensional systems, 193,204-212 partial data (partial realization), 5, 84,

197-199 miminal (canonical) realizations, 193, 267,

273, 282, 284, 302, 345 modeliI~g from impulse response and

transfer function, 191 module approach, 205 noise variables, 218 of matrix fractions, 247-248 of rational matrices, 503, 546 over a commutative ring, 313 I

recursiveness issues, 193, 200-201 shift, with polynomial models, 247

with rational models, 248 splitting variables, 218-222 state space, 270-273 state-space system and recursive

computability, 214 stochastic, 5, 213-229, 394

factor analysis models, 222 axiomatic probability framework, 216 prototype problem, 218

symmetry, 249 torsion modules and rationality, 274 transfer function with simple and multiple

poles, 192 uniqueness of canonical realizations, 192,

196,204-207 recursive algorithms, 239

Berlekamp-Massey, 201 Redheffer, star product, 75 reflection coeflicients, left and right, 71 regression, 427, 584 ff. regulation of industrial process, 441 regulator,

inverse problem, 155 self-tuning, 437 ff.

direct,442 indirect 440

relation function, 424 representations, ARMAX, SSX, 395 ff. residue of matrix function, 562 Riccati equation (RE), 2, 4, 18, 46, 47, 49, 62,

72, 81, 91, 151,406,492

RE and spectral factorization, 49 RE framework for systems over rings, 322 ARE, algebraic RE, 76, 99, 160, 181

similarity between LQG and H"" 164 RE asymptotic behavior, 76 RE differential, difference, 77, 160, 181, 184 time-varying RE and time-varying spectral

factorization, 50 square-root or array versions, 81

Riemann-Hilbert problem, 554 robustness issues, 154, 159, 185 robustness vs. performance, 173 ff. RollNix, ship steering autopilot, 448 root location of polynomials, 249-256 root-Iocus method, 147 Rosenbrock theorem, 305 Routh array, 373 Routh-Hurwitz criterion, 84, 147, 373

S sampling period in LQG problem, 4, 178 ff. scattering,

approach, 77 matrix, 71,72 model, 69, 71, 75 pairs, 221 representation, 226 inverse, 84, 201

Schrödinger equations, 497 Schuler period, 92 Schur algorithm, 83, 84 Schur-Cohn stability criterion, 84, 375 ff. Schur-Cohn-Jury table, 383 Schur-Hadamard product, 582 Schwarz matrix, 373 ff. Schwarz, discrete, or Mansour form, 6, 375 ff. sciences,

descriptive, 1, 7 of the artificial, 7, 154 prescriptive, 1,7,17,19

scientific activities and methodology, 1 self-tuning regulator, 2, 6, 437 ff.

direct,442 indirect, 440

sensitivity minimization trade-offs, 188 sensor fusion, 94 separability and realizability, 487 separation theorem and certainty

equivalence, 185 separation, state observers-state feedback

controllers, 19 sheaf theory, 331 shift,

group,220 operator, 178, 236

compression of, 237 eigenvalues and eigenfunctions, 237

realization, 258 with polynomial models, 247 with rational models, 248

short exact sequence, 236 fundamental pole-zero, 291

SIGAL strapdown inertial systems, 96 signal,

estimation and detection, 68 processing and systems over the integers, 312 processing, statistical, 214 space, 20

singular values of linear systems, 260 singular value decomposition, 580 Smith canonical form and invariant factors,

284, 367 Smith-McMillan, form, 7, 289, 367, 395, 532,

534, 553 degree, 396

Smith-McMillan-Yoo form, 562 smoothing, 46, 501T., 701T. Sobolev space, 496 solid friction, adaptive compensation, 105 spaces,

input, output, see: input/output spaces state, see: state, state space realizations probability, 215 Ptak, barreled, 206

spectral densities, 178, 184, 394, 424 spectral factorization, 4, 43, 90, 215, 225

of Wiener and Hopf, 56, 57 innovations, Riccati equation, 49 time-varying, 50

spectral matrix, 50 spectral problems, 239 splitting algebra, variables, 218-222 Sputnik, 59 SSX models, 3951T. stability,

margin, 378-379 interna!, 297 of input/output map, 300

stability, module theoretic framework, 2971T. stability and causality, 301 stability module, 303

stability and module indices, 303-305 stability rings, 297

stability criteria, Hermite-Hurwitz, 254 Lienard-Chipart, 253, 375, 378 Lyapunov, 255, 3731T. Routh-Hurwitz, 372 ff. Schur-Cohn, 375 ff. unified treatment, 250

stability theory, linear systems, 249-256 discrete linear systems, 371-384

stabilizability, 47, 48 stabilizer subgroup, 333 stabilizing controllers, YJBK

parametrization, 349-353 state, 3, 135, 159,214,219,270

see also: modeling or realization construction, 4, 191 ff. interface between past and future, 152

Subject Index 603

minimal generalized, 467 quantum mechanical, 135 ff.

state feedback, 275-277 state space and pole module, 283 state space models (description), 19, 79, 191,

194, 214, 222, 226, 345 state space realizations, see: realization problem state variable, sufficient statistic of, 214 state-transition matrix, 70, 72, 77 stationary problems, 90 stationary wide sense (or Gaussian) setting, 220 statistic, sufficient of state variable, 214

Bayesian sufficient, 218 statistical,

filters, 89 optimization theory of Wiener, 148 regularity in modelling, 217 -mechanical setup, 217

steel rolling, 154 stochastic,

control algorithm, 440 control theory, 66 equivalence, 215 exact modeling, 393 identification, 389-421 linear systems, ARMAX, SSX, 399 models, 214ff.

and physical systems, 228 identifiability of, 228

process, 215, 217 stationary Gaussian, 400

Stokes iden ti ti es, 76 Stratonovich stochastic integral, 60 superposition principle, 20 symbolic methods in computer algebra, 357 symplectic matrices, 180 synthesis,

c1assical and model-based, 4 of feedback systems, 346 of passive networks, 192,315 problems without observers, 348 problems with generalized polynomials,

349 problems, properness of controller, 348

system component, 1 system description, externa!, internal, see:

description system identification,

from noisy data, 6, 423-435 linear stochastic, 6, 389-421

system invariants and performance, 296 system stability and performance, 349 system theoretic,

activity,1 trends in econometrics, 559-577

system theory, and convolutional codes, 7, 527-557 algebraization of, 234 applications, 525-592 influence in mathematics, 501-523 general, 17

604 Subject Index

systems, see also: linear systems; AR, MA, ARMA, ARMAX, SSX models

approximatively reachable, 207 at infinity, theory of, 288 autonomous, 27, 215 canonical, see: canonical systems controllable, see: controllability convergent, 32 convolution, 194 delay, 5, 209, 316 deterministic and stochastic (heat bath), 217 dynamical, 17, 20, 36, 41 families of, 5, 6, 287 free, 319 latent variable, 20, 28 learning,7 linear, see: linear systems man-made, 17 neutral, 211 nonlinear and distributed, 451-500 over a ring of operators, 315 over rings, 3, 287 fT.

continuous-time case, 314-316 E-module framework, 314 pole and coefficient assignability, 319 fT. Riccati equation framework, 322

parameter-dependent,318 singular, 287 state space, see: state space models stochastic dynamical, 215 fT., 389 fT. topologically observable, 207 two-dimensional (2-D) 5, 316 weakly canonical, 205

T theorem,

canonical decomposition (structure), 6, 192, 196, 475-488

commutant lifting, algebraic version, 241 Nehari, algebraic version, 242 open mapping/c1osed graph, 206 uniqueness of canonical realizations, 192, 196

theory of distributions, 193 theory of the artificial, 17 theory-practice gap, 4, 148 time invariance and Laurent series, 269 time series, integrated and cointegrated, 561 time- vs. frequency-domain in module

theoretic framework, 268 Toeplitz matrices, 510 topological aspects of continued fractions, 256 topological observability in bounded time, 208 topology of pointwise convergence, 32 torsion modules and rationality, 274 fT. tracking of targets and path controllability,

566 fT. transfer functions (matrices), 5, 18, 29, 36,

161 fT., 195 fT., 208 fT., 241 fT., 270, 282, 299 fT., 312, 315, 317, 345, 393, 397, 503 fT.

and controllable subsystems, 30 as rational and complex-valued functions, 249 trapezoid diagrams, 283 entropy of c10sed loop, 174 number of poles and zeros of, 290 and state feedback, 277 behavior at infinity, 287 coprime factorizations, 356 Smith-McMillan form of, 289 stable and localization, 288

transition expectation, 138 transmission coefficients, forward and

backward, 71 triangular form, 359 triangularization of polynomial matrices,

359,363 fT. two-fiIter formulas, 73-75

U ULISS navigation systems, 125 uncertainty, difTerent types of, 187 unimodular,

group, 23, 32 map,301 matrices, 29, 358

unique factorization domain (UFD), 360 unobservable subspace, 196

V valuation theory, 542 variables,

auxiliary, 5, 216 external (measurement), inputs and

outputs, 3, 191,213 internal and random inputs, 228 internal, latent or state, 36, 191,213 manifest, 36 noise, 218 random,217 splitting, 218-222

variance, 184 Viterbi algorithm, 527 Volterra operator, 63 fT. Volterra series, 487

W wave equation, 491 wave, forward and backward, 71 weather forecasting, 152 Wedderburn-Forney construction, 291,

542-548 white noise, 50, 222

disturbances, 177 representation, 223 functionals of, 223

whitening filter, 60, 223 wide sense stationary processes, 424

Wiener N., 17 Wiener filtering, 3, 4, 41, 43, 49, 90

calculation burden, 46 and KaIman filtering, key dilTerences, 46 signal model for, 42, 44

Wiener process (or Brownian motion), 64, 66, 68, 178, 223

causal, anticausal, 223, 225 Wiener-Hopf factorization, 56, 57, 570 Wiener-Hopf integral equations, 63, 510, 554 Wiener-Paley physical realizability theorem,

147 Wiener-Volterra expansion, 67 Wold representation, 224

Subject Index 605

Y Yakubovic-Kalman-Popov lemma, 192,215 YJBK parametrization of stabilizing

controllers, 349-353

Z Zariski open subset, 3321T. Zariski sheaf, 339 Zeiger's lemma, 284 zero and pole modules, 287, 289 zero order hold input, 178 zeros and poles, of linear systems, 5, 289-292

at infinity, 544 zeros, input-output decoupling, 464, 4711T.