Index

a

  • Allan deviation  397
  • Approximation, Gaussian  4
  • Autocorrelation  33
  • Autocorrelation function  40, 42
  • Autocovariance function  40, 41

b

  • Backward KF  78
  • Bayes  35
  • Bayesian
  • Bernoulli polynomial  212
  • Bootstrap  102

c

  • Calculus
    • Itô  50
    • Stratonovich  50
  • cdf  25, 27
  • Chain rule  35
  • Characteristic function (cf)  26, 29
  • Clock synchronization  393
  • Coefficient
    • diffusion  51
    • drift  51
    • time‐varying  59
  • Coherence  44
  • Condition
    • Dirichlet  42, 50
    • Lipschitz  50, 52
  • Convolution  5, 117
    • discrete‐time  5
  • Correlation coefficient  33
  • Correlation function  40, 41
  • Cost function  72, 110
  • Covariance  65, 104
    • innovation  86
    • time‐varying  66
  • Cross‐correlation  33
  • Cross‐correlation function  40, 41
  • Cross‐covariance function  41
  • cross‐PSD  42, 43, 46
  • Cumulants  27

d

  • Data latency  376
    • deterministic  377
    • random  380
  • Dirac delta  103
  • Discrete orthogonal polynomials  215
    • Hahn  215
    • Krawtchouk  215
    • Meixner  215
  • Discrete Shmaliy moments  215
  • Distribution
    • Bernoulli  26
    • heavy‐tailed  10
    • Laplace  10
    • meridian  10
    • normal  44
    • uniform  31
  • Disturbance  2, 3, 275, 276
    • norm‐bounded  302

e

  • Entropy  45
  • Equation
    • backward Kolmogorov  52
    • Chapman‐Kolmogorov  49
    • characteristic  62
    • continuous Lyapunov  422
    • continuous‐time algebraic Riccati (CARE)  422
    • differential nonlinear  3
    • discrete algebraic Riccati (DARE)  108, 124, 171, 422
    • discrete difference Riccati (DDRE)  76, 422
    • discrete Lyapunov  280, 422
    • extended state  119
    • Fokker‐Planck‐Kolmogorov  52
    • forward Kolmogorov  52
    • Fredholm integral  143
    • Langevin  53
    • Lyapunov  108
    • nonsymmetric algebraic Riccati (NARE)  92, 422
    • ordinary differential  62
    • partial differential  52
    • quadratic matrix  92
    • Riccati differential (RDE)  94, 421
    • stochastic differential  8, 50, 59
    • stochastic partial differential  97
    • Stratonovich‐Kushner  97
  • Error covariance  8, 120, 137, 184
  • Error function  31
  • Estimate
    • a posteriori  4
    • a priori  4
    • batch  167, 172
    • filtering  4, 6
    • predicted  4
    • p‐shift UFIR  13
    • smoothed  4
    • smoothing  6
  • Estimation error  74, 176
    • a posteriori  4, 68, 74
    • a priori  4, 68
    • boundary  243
  • Estimator
    • batch  6
    • Bayesian  68
    • extended  8
    • extremum  9
    • FIR  7
    • least squares  71
    • LS  68
    • M  10
    • maximum a posteriori probability  9
    • maximum likelihood  9
    • minimax  11
    • minimum variance unbiased  9
    • ML  70
    • ordinary least squares  9
    • peak‐to‐peak  12
    • recursive unbiased  72
    • robust  4, 11, 275
    • unbiased  13, 72
  • Euler method
    • backward (BE)  3, 64
    • forward (FE)  3, 4, 64

f

  • Filter  59, 117
    • H 2 FIR  11
    • H 2/H∞ FIR  11
    • limited memory  7, 117
    • median  10
    • meridian  10
    • moving average  73
    • myriad  10
    • particle  98, 102, 103
  • Filtering  5, 210
    • FIR  4
    • H∞  109
    • Kalman  1
    • particle  102
    • predictive  5, 210
    • smoothing  5, 210
  • FIR estimate
    • filtering  7
    • p‐shift  7
  • Fourier transform  26, 42
  • Frequentist inference  9
  • Function
    • Hermitian  41
    • Lyapunov  298

g

  • Gain
  • Game theory  275
  • Game theory H∞ filter  81, 110, 229–231
    • energy‐to‐power  312
  • Game theory H∞ predictor, energy‐to‐power  314
  • Ground truth  196, 197, 262

h

  • H ∞ FIR filter  292, 344
    • a posteriori  294
  • H ∞ FIR predictor  297, 346
  • H 2 FIR predictor  288, 334
  • H 2 /H∞ FIR filter  301, 349
  • H 2 /H∞ FIR predictor  301
  • H 2 OFIR filter  277, 279, 327, 329
    • a posteriori estimate  281
    • error covariance  281
    • gain  281
  • H 2 OFIR predictor  289, 335
  • Horizon  5
  • H 2 OUFIR filter  281, 332
    • error covariance  282
    • estimate  282
    • gain  282
  • H 2 OUFIR predictor  289, 338

i

k

  • Kalman‐Bucy filter  93, 142, 144
  • Kalman filter  4, 68, 73, 137, 190, 230
    • a posteriori  74, 75, 79, 120
    • a priori  76, 109, 138
    • algorithm  73
    • alternate form  81
    • backward  78
    • batch  141
    • distributed  360
    • extended  98, 99, 144
    • fusion  375
    • general  81, 82, 87
    • ill conditioned  83
    • information  80
    • intrinsic errors  77
    • limited memory  141
    • optimality  76
    • recursions  7, 118
    • robust  107, 108
    • robustified  106
    • unscented  98, 100
  • Kalman predictor  252
    • algorithm  253
    • bias correction gain  252
    • error covariance  252
  • Kronecker symbol  52
  • Kurtosis  30

l

  • Lagrange multiplier  88, 128, 129, 264
  • Lemma
    • bounded real  12, 294
    • column matrix rule  278, 296
    • energy‐to‐peak  12, 302
    • matrix inversion  134
    • peak‐to‐peak  308
    • real bounded  298
  • Likelihood  36, 70
  • Likelihood function  69
  • Limited memory predictor  252, 253
  • LMF  68, 139
    • batch  139, 140
    • error covariance  141
    • recursive forms  141
  • ℒ∞‐to‐ℒ∞ FIR filter  310, 354
  • ℒ 2‐to‐ℒ∞ FIR filter  302, 305, 350
  • ℒ∞‐to‐ℒ∞ FIR predictor  311, 355
  • ℒ 2‐to‐ℒ∞ FIR predictor  304, 306, 352

m

  • Marginal  107
  • Markov chain  48
    • discrete‐time  48
  • Matrix
  • Matrix theory overview  419
  • Maximum likelihood  9, 132, 147
  • Mean square error  8, 74, 78
  • Measurement differencing  83
  • Median  28
  • Minimax  106
  • Mismodeling  232
  • ML FIR predictor  267, 268
    • a posteriori  268
    • error covariance  268
  • Mode  28
  • Model
    • continuous‐time  61
    • discrete  65
    • Gauss‐Markov  90
    • non‐Gaussian  102
    • nonlinear  97
    • prediction  64
    • predictive control  3, 5
    • time‐varying  6
  • Moments  27
    • central  28
    • product  31
    • raw  27
  • Monte Carlo method  98

n

  • Noise
    • additive  45
    • Cauchy  10
    • colored  49, 68
    • colored measurement  81, 85
    • colored process  82, 90
    • continuous white Gaussian  45
    • correlated  87, 367
    • de‐correlation  83
    • discrete white Gaussian  46
    • Gaussian  3, 9
    • Gauss‐Markov  4
    • heavy‐tailed  10
    • Laplace  10
    • measurement  3, 60, 65, 68, 118
    • non‐Gaussian  3
    • process  49, 118
    • system  3, 67
    • time‐correlated  83
    • white Gaussian  3, 45, 50
  • Noise power gain  14, 160, 238
  • Norm
  • Norms  425

o

  • Observation  3, 60, 63
  • Observation function
    • nonlinear  3
  • OFIR Estimator  7
  • OFIR filter  117, 118, 147, 190
    • a posteriori  10
    • backward  166
    • backward recursive  168
    • batch a posteriori  120, 122
    • batch a priori  138
    • continuous‐time  142, 144
    • error covariance  122
    • extended  144–146
    • forced gain  122
    • homogeneous gain  121
    • iterative algorithm  124
    • recursive forms  122
  • OFIR predictor  252, 259
    • batch  259
    • error covariance  260
    • extended  268
    • extended algorithm  269
    • iterative algorithm  260
  • OFIR smoother  159
    • backward filter/backward model  161, 172
    • backward filter/forward model  161, 177
    • error covariance  164, 174, 177, 180
    • forward‐backward  180
    • forward filter/backward model  161, 174
    • forward filter/forward model  161
    • q‐lag  177
    • two‐filter  161, 180
  • Operator
    • nonlinear  6
    • time‐invariant  6
    • time‐varying  6
  • Orthogonality  33, 140
  • OUFIR filter  9, 117, 126, 128, 129, 132, 134, 136, 147
    • a posteriori  126
    • a priori  138
    • batch a posteriori  127
    • error covariance  127, 138
    • gain  127, 129
    • recursive forms  127, 130

p

  • pdf  26, 27
    • conditional  36
    • Gaussian  34
    • nonstationary  53
    • ramp  215
    • Rayleigh  39
  • Performance
    • energy‐to‐peak  302
    • generalized H 2  12
    • H∞  11
    • H 2  11
    • H 2/H∞  12
    • ℒ 1  12
    • localization  406
    • ℒ 2‐to‐ℒ∞  302
  • PF  102
  • pmf  26
  • Pochhammer symbol  215
  • Polynomial model  209
    • filtering  214
    • prediction  217
    • predictive filtering  217
    • smoothing  216, 217
    • smoothing filtering  216
    • system matrix  210
  • Prediction  4–6, 95, 211
    • one‐step  5
  • Predictive filter  20
  • Predictor  251
  • Probability  25
    • conditional  34
    • integral  31
    • marginal  35
    • total theorem  35
    • transitional  49
  • Problem
    • computational  68
    • minimization  12, 198, 286
    • optimal filtering  8
    • optimization  12, 13, 196
    • smoothing  159
  • Process  1
    • continuous stochastic  39
    • diffusion  51
    • discrete stochastic  39
    • ergodic  40
    • Gaussian  9, 44, 45
    • Gauss‐Markov  43, 81, 98
    • Markov  47
    • Markovian  51
    • nonstationary  39
    • Ornstein‐Uhlenbeck  43
    • Poisson  48
    • random  25
    • stationary  39, 41
    • stochastic  1, 39
    • strictly stationary  39
    • wide‐sense stationary  39, 42
    • Wiener  48, 50
  • Property
    • computational complexity  15, 16
    • deadbeat  13, 137, 148
    • denoising  14
    • Gaussianity  44
    • Hermitian  43
    • Markov  104
    • memory use  16
    • optimality  8, 77
    • robustness  14, 148
    • stability  14, 147
    • suboptimality  11
    • unbiasedness  13, 137
    • unbiased optimality  9, 77, 137, 182
  • PSD  42, 46, 65

r

  • Random variable  25
    • Gaussian  44
    • Laplacian  10
    • scalar  25
    • transformation  37, 38
    • uncorrelated  45
    • vector  33
  • Receding horizon  5
  • Receding horizon FIR filter  251, 263
    • minimum variance  263, 265
  • Receding horizon H 2 MVF filter  291
  • Resampling  105
  • Residual
    • measurement  74, 79, 86
    • measurement covariance  74
  • RH MVF H 2 filter  287
  • Robustness  148, 191, 275
  • Robust state estimation  105
  • Root mean square error  78

s

  • Sample degeneracy  105
  • Sample impoverishment  105
  • Savitzky‐Golay filter  159
    • generalized  217
  • Schur complement  123, 131, 170, 286, 295, 421
  • Sequence  49
    • discrete stochastic  39
    • Gauss‐Markov  49
    • stochastic  39
    • white Gaussian  49
  • Series
  • Simple average  73
  • Skewness  29
  • Smoother, Rauch‐Tung‐Striebel  95, 159
  • Smoothing  4, 5, 94, 198, 205, 211, 317
    • Bryson‐Frazier  96
    • fixed‐interval  95, 180
    • fixed‐lag  95
    • fixed‐point  95
    • optimal  94
    • receding horizon FIR  159
    • receding horizon ML FIR  160
    • two‐filter  96
  • Smoothing filter  20
    • q‐lag  95
  • Stability  14, 107
  • Standard deviation  28
  • State  3
  • State differencing  83, 91
  • State equation  2, 3
    • discrete time  3
    • predictive  3
  • State estimate  2, 4
    • a posteriori  69
    • a priori  78
    • backward  79
    • Bayesian  104
    • ML  71
    • moving average  73
    • p‐shift  7
    • weighted LS  72
  • State estimation  1, 59, 67, 389
    • biosignals  411
    • clock TIE  389
    • colored measurement noise  408
    • delayed and missing data  376
    • distributed  359
    • ECG adaptive smoothing  412
    • ECG features  412
    • EMG envelope extraction  413
    • game theory  312
    • GPS‐based  391
    • H∞ FIR  292, 344
    • H 2 FIR  277, 327
    • H 2 FIR generalized  302, 350
    • H 2/H∞ FIR  301, 349
    • INS/UWB‐Based localization  408
    • Kalman/UFIR  371
    • ℒ 1 FIR  307, 354
    • linear  59
    • localization  404
    • localization over WSN  399
    • master clock TIE  393
    • particle/UFIR  399, 401
    • robust  275, 321
    • self‐localization  403
  • State estimator  1, 5, 59, 68, 117
    • batch  5
    • discrete‐time  5
    • FIR  6
    • limited memory  6, 7
    • linear  1, 117
    • minimax  106
    • nonlinear  6, 97, 98
    • optimal  112
    • p‐shift  6
    • receding horizon  6
    • recursive computation  315
    • robust  105
  • State feedback control  5, 6
  • State space  59
  • State space equations  60, 66, 67
    • extended  119
  • State space model  2, 3, 59
    • augmented  91
    • backward  166
    • BE‐based  276, 322
    • continuous time  3
    • continuous‐time  64
    • discrete‐time  64
    • extended  118, 275, 322, 405
    • extended backward  172
    • extended predictive  253
    • FE‐based  276, 325
    • nonlinear  3
    • time‐invariant  196
    • transformation  382
  • Stochastic integral  51
    • Itô sense  51, 61
    • Lebesgue sense  51
    • Stratonovich sense  51, 61
  • Suboptimal H 2 FIR filter  286, 340
    • bias‐constrained  287, 341
  • Suboptimal H 2 FIR predictor  290, 342
    • bias constrained  291, 343
  • Suboptimal H 2/H∞ FIR predictor  349
  • System  1
    • digital control  3
    • LTI  3, 4, 61, 65, 208
    • LTV  62, 66, 118, 189
    • MIMO  60
    • nonlinear  3
    • oscillatory  60
    • state  1, 2
    • stochastic  1, 2
    • uncertain  321
  • System function
    • nonlinear  3
  • System identification  125

t

  • Tchebychev polynomials  216
  • Temporary uncertainty  235
  • Theorem
    • Cayley‐Hamilton  61
    • Parseval  10, 11, 237
    • Wiener‐Khinchin  42
  • Tracking  282
    • robust  283
  • Transfer function  12, 237, 278
    • DFT domain  241
    • disturbance‐to‐error  275
    • disturbance‐to‐peak  11
    • energy‐to‐peak  12, 275
    • noise‐to‐peak  8, 10
    • peak‐to‐peak  12, 275
    • weighted  279, 280
    • z domain  238

u

v

  • Variance  28
  • Vector, extended  6

w

  • Wireless sensor network (WSN)  359
    • consensus in estimates  364
    • consensus in measurements  360
    • optimal fusion  367
    • RFID‐based  403
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