Index

1-norm, 1922-norm, 192, 226, 2293 Identification of a Parametric Model, 94

A finite duration signal, 16actuator fault, 448, 449, 451additive fault, 379, 380, 383, 388–394, 399, 400, 414,

420, 421Akaike information criterion, 307, 318anti-causal, 2, 4, 5, 50ARMAX model, 302, 303, 307, 327, 330Asymptotic behavior, 369, 377Asymptotic normality, 119Asymptotic properties, 312, 321asymptotic tracking, 237, 239–241, 243, 244, 246, 247,

249, 252, 261augmented state space model, 490auto correlation, 57, 82, 87, 94, 107Auto Regressive, 2Auto Regressive and Moving Average, 2auto-correlation function, 39

Band-pass filter, 36Band-stop filter, 36bank of Kalman filters, 459, 461, 464, 473batch processing, 305, 307, 308, 316, 336, 368, 376Bayes decision, 380, 390, 391, 394, 398, 414, 417, 420Bayes decision strategy, 464, 465, 475, 495, 506Bayesian approach, 167, 380binary arithmetic, 39binary hypothesis testing, 379, 391binary logic approach, 465bounded, 2, 3, 26, 27, 50

Cauchy, 117, 118, 122, 134, 135, 144–148, 155, 157,159–161

Cauchy-Schwartz inequality, 210

causal, 2, 4, 5, 10, 35, 50causally invertible, 390characteristic polynomial, 365Chebyshev fit, 192Classification of the random waveform, 31closed loop identification, 357, 359, 374closed loop sensor network, 362coherence spectrum, 58, 75, 76, 85, 86colored noise, 30, 289, 290, 293, 295, 296, 300, 301,

304, 305, 311, 313–315, 318, 322, 330, 333complementary sensitivity, 490, 491complementary sensitivity function, 431, 432Completing the square approach, 170composite hypotheses testing, 380, 401, 422condition monitoring, 2condition number, 297, 315, 316, 326, 344condition number of the matrix, 217conditional expectations, 172conditional mean, 169, 172, 174, 177, 178Consistency, 119consistent estimator, 127, 156constant signal, 18control system, 232, 233, 236, 237, 243, 244controllable, 235, 239, 242, 252, 275, 279convolution, 1, 61, 62, 69, 70, 72, 91, 109correlation, 57–62, 64–72, 74–92, 94, 98, 99, 101, 103,

106–112correlation coefficient, 66, 67, 110Correlation function, 29covariance function, 61covariance of the estimation error, 195–197, 201,

206–209, 211, 216, 221, 228, 369, 377Cramer-Rao inequality, 190, 205, 313, 349Cramer-Rao lower bound (CRLB), 119, 126, 127, 129,

130, 138, 139, 154, 157, 209–212, 220, 290, 313,318, 327

cross correlation, 57, 58, 65, 83, 84, 88, 104

Identification of Physical Systems: Applications to Condition Monitoring, Fault Diagnosis, Soft Sensor andController Design, First Edition. Rajamani Doraiswami, Chris Diduch and Maryhelen Stevenson.© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.

510 Index

dad data, 117damped sinusoidal, 85data matrix, 190, 193–195, 197, 198, 200, 201, 203,

209, 212, 216, 217, 227, 399, 404–406data vector, 46, 289, 307, 308, 329, 330, 332, 345, 382,

388deconvolution, 319, 322, 323delta function, 1, 2, 12, 14, 28–30, 73, 77, 200, 202,

203, 219, 389, 396deterministic approach, 2, 231diagnostic model, 289, 380, 384–386, 389–391,

398–400, 417, 418diagnostic parameters, 380, 381, 384–386, 397–403,

406–412, 414, 416, 447, 449, 451, 453, 456, 457Difference equation model, 5, 6, 12, 30, 43, 49, 50, 54direct approach, 300, 357–360, 373–375, 432, 439direct transmission, 233, 278Discrete-Time Fourier Transform, 31discriminant function, 403, 404, 406, 417, 418, 422disturbance, 1, 231–237, 248–254, 258, 260, 262, 264,

271, 273, 274, 278Disturbance model, 41, 47

Efficiency, 119, 126efficient, 117, 119, 126, 127, 129–131, 137, 139–144,

149, 150, 154, 156, 165efficient estimator, 119, 154, 209emulator, 380, 406, 408, 422, 447, 448, 452, 455–457,

481, 483–488, 491, 497, 498, 503energy signal, 3, 4, 50, 58, 60, 61, 66, 69, 71–73ensemble, 2, 23, 27, 28ensemble average, 59, 94, 266, 272equation error, 289, 290, 293, 299, 302–305, 311–315,

318, 321, 322, 325, 326, 330–333, 342, 347–351ergodic, 267, 271ergodic in the auto correlation, 28, 53ergodic in the cross correlation, 28ergodic in the variance, 28, 53Ergodic process, 27, 52estimation of a random parameter, 167Estimation theory, 117, 165estimator design, 232, 241, 244, 275Exponential signal, 18

fault detection, 379, 380, 390, 414, 420, 460, 473fault detection and isolation, 2fault detection strategy, 449fault diagnosis, 379, 390, 447, 448, 456, 459–461, 471fault indicator component, 388Fault isolation, 379, 381, 390, 396, 399, 401, 412–414,

417, 421fault size, 405, 416, 418, 452, 455Fault tolerant control system, 496feature vector, 46, 189, 190, 209, 314, 316, 317, 320,

327, 331, 332, 353, 382–385, 387, 388, 400, 403,409, 411, 412, 418, 419

filtered data, 388, 399Finite sample properties, 312, 321Fisher information, 119, 127, 131, 135, 138, 160,

210–212, 316Frequency response and pole-zero locations,

31frequency-weighted least-squares estimator,

290Fuzzy-logic approach, 2

game theoretic approach, 118Gaussian, 117–120, 122, 126, 130, 131, 135, 137–142,

145–148, 151, 152, 154, 155, 165Generalized Likelihood Ratio Test, 380good data, 117, 118, 134, 146

H-infinity, 489, 493, 495, 496, 502, 505High Order Least Squares, 358, 366, 376high order least squares method, 290, 305, 313, 321,

323, 325, 326, 330, 331, 333high order polynomials, 367, 376high-order least squares, 289High-pass filter, 36

i.i.d random variables, 138, 146, 157Identification for fault diagnosis, 379, 380identification model, 289, 297, 301, 304–306, 311, 317,

334Identification of non-parametric model, 93, 112Ill conditioned matrices, 217Improper, 10, 51indirect adaptive control, 495indirect approach, 357, 358infinity norm, 192influence vector, 381, 385–388, 390, 398–401,

404–406, 409–414, 416, 417, 420, 449innovation form, 255, 256, 291, 298, 301–304, 307,

319, 320, 331, 334, 338–340, 347innovation process, 290, 298input, 1, 2, 4–10, 12, 16–18, 30, 31, 35, 36, 41, 42, 46,

47, 49, 50integrated model, 2, 49, 50, 234–236, 251, 253–255,

260, 267, 274internal model, 2internal model principle, 231, 232, 236, 237, 240, 241,

243, 245, 247, 252, 262, 275inverse filtering, 305Isolability, 455

Kalman filter, 2, 9, 17, 50, 231–233, 235–237, 250–253,255–257, 259–277, 279, 280, 285–287, 447–449,453, 457

Kalman filter-based model structure, 364Kalman filter-based structure, 289Kalman gain, 231–233, 256, 258–265, 267, 268,

270–273, 275–279

Index 511

Kalman polynomial, 290, 291, 299, 300, 322, 365, 366,369

Kaman filter, 480

Laplacian, 117, 118, 121, 122, 132, 135, 137, 142, 144,146, 147, 153, 155

leakage, 448, 449, 451, 452least squares estimate, 485Least squares method, 305, 342left singular matrix, 214, 223likelihood function, 119, 122, 123, 129–137, 139–142,

144, 145, 148, 149, 151–153, 155Linear least-squares estimation, 189linear regression model, 46, 235, 286, 382log likelihood function, 455log likelihood ratio, 391–394, 398, 425Low-pass filter, 36

magnetic levitation system, 427, 429, 430, 433, 434, 445maintenance-free, 479, 502matched filtering, 88Maximum Likelihood, 306, 380, 394maximum likelihood estimator, 119, 139, 157, 210Maximum likelihood Method, 190mean, 2, 23–31, 35, 41, 42, 46, 47, 49, 52mean-ergodic, 28measurement noise, 1, 231–236, 248–254, 258–260,

262–264, 271–274, 278, 279, 281Measurement noise model, 42, 47median, 117, 122, 123, 125, 126, 134, 141, 143, 146,

148MIMO identification, 358min-max approach, 144, 396minimum mean square, 167Minimum phase system, 11, 51mixed sensitivity, 481, 492, 495, 498, 502MMSE, 167–176, 178–184model mismatch, 231–233, 261, 267–273, 276, 285Model of a class of all signals, 17, 53Model of a Random Waveform, 30Model of Random Signals, 28model of the plant, 2, 41, 43model order, 449, 452model order selection, 306, 318, 327model uncertainty, 480–482, 491, 497, 502Model validation, 434, 445model-mismatch, 494, 495, 506model-order selection, 434, 445Modeling of Signals and Systems, 1Moving Average, 2, 6, 7, 50, 51multi-linear, 381, 385–387multiple hypothesis testing, 448

Newton- Raphson, 291Neyman Pearson method, 395, 396Noise annihilating operator, 17

nominal model, 480, 483–486, 488, 495, 499, 504non-causal signals, 4, 5non-Gaussian, 117normal equation, 190, 193, 194numerator-denominator perturbation model, 482, 491,

503numerator-denominator uncertainty model, 480

observability, 236observable, 235, 236, 239, 242, 244–247, 252, 275,

280observer, 2, 9, 17, 50, 232, 236, 237, 246–252, 254, 262,

263Observer canonical form, 9optimal nominal model, 289, 480, 485orthogonal complement projector, 194orthogonality, 232, 233, 266, 267, 273, 284orthogonality condition, 194orthogonality principle, 172, 184, 189, 194over determined, 189, 190, 209, 213, 218, 222, 229

parameter-perturbed experiments, 399, 400, 406,416–418, 484, 501, 502

parametric fault, 383, 406part Gauss-part Laplace, 118, 121, 122, 135, 145–148pattern classification, 403, 406PDF, 23–25, 30, 117–122, 126–128, 130–135, 137–142,

144–148, 151–155, 157, 158, 160, 161, 164Periodic impulse train, 15Periodic signal, 14Periodic waveform, 21persistently exciting, 293–297, 304, 309, 328, 341perturbed-parameter experiment, 449pole-zero cancellation, 240polynomial model, 367, 368, 370position control system, 381, 386, 442, 443, 452power signal, 3, 4, 50, 59, 60, 67, 69–73power spectral density, 29, 31, 39, 40, 57, 58, 67, 68,

72, 73, 77, 81, 84, 88, 90, 92–94, 101, 102,106–109

PRBS, 38–40, 54prediction error method, 289, 305, 327, 330, 331, 333,

345, 358, 372, 374predictor form, 255, 256, 291, 298, 302–304, 306, 331,

334, 335, 339, 347probabilistic approach, 2probability density function, 117, 158, 160process control system, 469Product quality, 479projection operator, 194, 198, 199, 221, 222Proper, 10, 51Pseudo Random Binary Sequence, 54Pseudo Random Binary Signal, 293

quasi-stationary, 26, 27, 31, 309, 328Quasi-stationary ergodic process, 25

512 Index

random processes, 1random signal, 1–3, 24, 25, 27–29, 36, 41, 51, 52, 59,

70, 92, 94, 96–99, 103, 106, 112rational spectrum, 238Rayleigh theorem, 67rectangular pulse, 82recursive processing, 305, 307, 316reduced-order model, 290, 323, 324, 326, 352Reliability, 479residual, 189–192, 194–203, 205–209, 213, 216, 219,

221, 228, 379–382, 384, 388–391, 393, 395–399,402–405, 414, 416, 418, 419, 459, 462–465, 468,471, 473, 475, 480, 481, 488, 489, 494–496, 499,500, 502, 506

Residual generation, 379, 380residual model, 290, 298, 304, 305, 365, 366Riccati equation, 231–233, 258, 260, 277–279right singular matrix, 214, 223robust controller, 481, 489, 493–495, 498, 502, 505robust estimator, 121, 144, 145

Selection of the high order, 369, 377sensitivity, 481, 490–492, 495, 498, 502, 505sensitivity function, 240, 431–434, 436, 439, 442, 445,

446sensor fault, 448, 449, 451sensor network, 438, 439, 459–462, 466, 467, 470Singular Value Decomposition, 189, 193, 213, 214, 220,

223, 315singular values, 214, 215, 217, 223, 225, 226sinusoid, 2, 17, 19, 20, 22, 26, 27, 57, 74, 77, 78, 80, 81,

85, 87, 88, 90, 108, 109soft sensor, 479–482, 484, 488, 489, 493, 495, 496, 500,

502, 505square wave, 74, 81, 83–85Square waveform, 21Stability, 10stabilizer, 237, 239, 240, 242, 243, 246, 247, 249, 250,

252, 255State space model, 5, 7, 14, 19, 30, 44, 45, 51, 52,

54Stationary random signal, 23stochastic estimation theory, 231stochastic process, 2, 23–28stochastic signals, 3strict sense stationary, 23, 24Strict sense stationary of order, 25, 52Strictly proper, 10, 51

suboptimal Kalman filter, 232, 261subspace method, 289–291, 305, 306, 335, 339, 340,

358superposition principle, 46, 50SVD, 189, 193, 213–217, 220, 221, 225–229system identification, 2, 6, 27, 35, 39, 50

test statistics, 381, 394, 395, 398, 416The log-likelihood function, 123, 135thicker tails, 117, 121thin tails, 117threshold value, 455time delay, 57, 58, 63, 82–84, 87, 89Toeplitz matrix, 96Triangular wave, 22triangular waveform, 90, 111truth table, 466two tank process control system, 436, 437two-stage approach, 357–359, 374two-stage identification, 432, 433, 439two-tank process control, 448

unbiased estimator, 119, 124–129, 143, 156–161, 189,195, 205, 210, 211, 218, 220

unbounded, 2, 3under determined, 190, 212, 213, 218, 222uniform PDFs, 122unitary matrix, 214, 223, 224

Vector calculus approach, 170, 171velocity control system, 481, 496–500, 502

weighted least-squares method, 119, 139, 141, 189, 210,211, 216, 220

white noise process, 28, 30, 31, 35, 41, 42, 49, 231–233,250, 252, 256, 267, 270, 271, 273, 278

whitening, 318, 364, 365whitening filter, 256Wide Sense Stationary, 65Wide-sense stationary random signal, 24, 52worst case PDF, 118, 121, 134, 144, 145, 147WSS, 65, 66, 70

Yule-Walker, 96, 98

zero input response, 237, 238, 241, 243, 254zero mean white noise process, 57, 87, 89, 92, 94, 99,

101, 103–106, 363, 365, 366, 369, 370, 372