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Tables of Contents for Estimation With Applications to Tracking and Navigation
Chapter/Section Title
Page #
Page Count
Preface
xvii
Acronyms
xxi
Mathematical Notations
xxii
Introduction
1
88
Background
1
14
Estimation and Related Areas
1
2
Applications of Estimation
3
1
Preview of Estimation/Filtering
4
6
An Example of State Estimation: Vehicle Collision Avoidance
10
5
Scope of the Text
15
4
Objectives
15
1
Overview and Chapter Prerequisites
16
3
Brief Review of Linear Algebra and Linear Systems
19
12
Definitions and Notations
19
1
Some Linear Algebra Operations
20
1
Inversion and the Determinant of a Matrix
21
2
Orthogonal Projection of Vectors
23
1
The Gradient, Jacobian and Hessian
24
1
Eigenvalues, Eigenvectors, and Quadratic Forms
25
2
Continuous-Time Linear Dynamic Systems - Controllability and Observability
27
2
Discrete-Time Linear Dynamic Systems - Controllability and Observability
29
2
Brief Review of Probability Theory
31
41
Events and the Axioms of Probability
31
2
Random Variables and Probability Density Function
33
2
Probability Mass Function
35
1
Mixed Random Variable and Mixed Probability-PDF
36
1
Expectations and Moments of a Scalar Random Variable
37
1
Joint PDF of Two Random Variables
38
3
Independent Events and Independent Random Variables
41
1
Vector-Valued Random Variables and Their Moments
41
3
Conditional Probability and PDF
44
1
The Total Probability Theorem
45
2
Bayes' Formula
47
3
Conditional Expectations and Their Smoothing Property
50
1
Gaussian Random Variables
51
1
Joint and Conditional Gaussian Random Variables
52
2
Expected Value of Quadratic and Quartic Forms
54
1
Mixture Probability Density Functions
55
2
Chi-Square Distributed Random Variables
57
3
Weighted Sum of Chi-Square Random Variables
60
1
Random Processes
61
4
Random Walk and the Wiener Process
65
1
Markov Processes
66
3
Random Sequences, Markov Sequences and Markov Chains
69
1
The Law of Large Numbers and the Central Limit Theorem
70
2
Brief Review of Statistics
72
13
Hypothesis Testing
72
2
Confidence Regions and Significance
74
5
Monte Carlo Runs and Comparison of Algorithms
79
3
Tables of the Chi-Square and Gaussian Distributions
82
3
Notes and Problems
85
4
Bibliographical Notes
85
1
Problems
85
4
Basic Concepts in Estimation
89
32
Introduction
89
1
Outline
89
1
Basic Concepts - Summary of Objectives
89
1
The Problem of Parameter Estimation
90
2
Definitions
90
1
Models for Estimation of a Parameter
91
1
Maximum Likelihood and Maximum a Posteriori Estimators
92
6
Definitions of ML and MAP Estimators
92
1
MLE vs. MAP Estimator with Gaussian Prior
92
2
MAP Estimator with One-Sided Exponential Prior
94
1
MAP Estimator with Diffuse Prior
95
1
The Sufficient Statistic and the Likelihood Equation
96
2
Least Squares and Minimum Mean Square Error Estimation
98
3
Definitions of LS and MMSE Estimators
98
2
Some LS Estimators
100
1
MMSE vs. MAP Estimator in Gaussian Noise
100
1
Unbiased Estimators
101
3
Definition
101
1
Unbiasedness of an ML and a MAP Estimator
102
1
Bias in the ML Estimation of Two Parameters
102
2
The Variance and MSE of an Estimator
104
4
Definitions of Estimator Variances
104
1
Comparison of Variances of an ML and a MAP Estimator
105
1
The Variances of the Sample Mean and Sample Variance
106
1
Estimation of the Probability of an Event
107
1
Consistency and Efficiency of Estimators
108
6
Consistency
108
1
The Cramer-Rao Lower Bound and the Fisher Information Matrix
109
1
Proof of the Cramer-Rao Lower Bound
110
2
An Example of Efficient Estimator
112
1
Large Sample Properties of the ML Estimator
113
1
Summary
114
1
Summary of Estimators
114
1
Summary of Estimator Properties
115
1
Notes and Problems
115
6
Bibliographical Notes
115
1
Problems
116
5
Linear Estimation in Static Systems
121
58
Introduction
121
1
Outline
121
1
Linear Estimation in Static Systems - Summary of Objectives
121
1
Estimation of Gaussian Random Vectors
122
1
The Conditional Mean and Covariance for Gaussian Random Vectors
122
1
Estimation of Gaussian Random Vectors - Summary
123
1
Linear Minimum Mean Square Error Estimation
123
6
The Principle of Orthogonality
123
4
Linear MMSE Estimation for Vector Random Variables
127
2
Linear MMSE Estimation - Summary
129
1
Least Squares Estimation
129
17
The Batch LS Estimation
129
3
The Recursive LS Estimator
132
3
Examples and Incorporation of Prior Information
135
2
Nonlinear LS - An Example
137
8
LS Estimation - Summary
145
1
Polynomial Fitting
146
8
Fitting a First-Order Polynomial to Noisy Measurements
146
3
Fitting a General Polynomial to a Set of Noisy Measurements
149
3
Mapping of the Estimates to an Arbitrary Time
152
2
Polynomial Fitting - Summary
154
1
Goodness-of-Fit and Statistical Significance of Parameter Estimates
154
7
Hypothesis Testing Formulation of the Problem
154
2
The Fitting Error in a Least Squares Estimation Problem
156
3
A Polynomial Fitting Example
159
2
Order Selection in Polynomial Fitting - Summary
161
1
Use of LS for a Nonlinear Problem: Bearings-Only Target Motion Analysis
161
11
The Problem
161
1
Observability of the Target Parameter in Passive Localization
162
1
The Likelihood Function for Target Parameter Estimation
163
1
The Fisher Information Matrix for the Target Parameter
164
3
The Goodness-of-Fit Test
167
1
Testing for Efficiency with Monte Carlo Runs
168
1
A Localization Example
169
1
Passive Localization - Summary
169
3
Notes, Problems and a Project
172
7
Bibliographical Notes
172
1
Problems
172
4
Project: An Interactive Program for Bearings-Only Target Localization
176
3
Linear Dynamic Systems with Random Inputs
179
20
Introduction
179
1
Outline
179
1
Linear Stochastic Systems - Summary of Objectives
179
1
Continuous-Time Linear Stochastic Dynamic Systems
180
7
The Continuous-Time State-Space Model
180
1
Solution of the Continuous-Time State Equation
181
2
The State as a Markov Process
183
1
Propagation of the State's Mean and Covariance
184
1
Frequency Domain Approach
185
2
Discrete-Time Linear Stochastic Dynamic Systems
187
8
The Discrete-Time State-Space Model
187
2
Solution of the Discrete-Time State Equation
189
1
The State as a Markov Process
190
1
Propagation of the State's Mean and Covariance
191
1
Frequency Domain Approach
192
3
Summary
195
1
Summary of State Space Representation
195
1
Summary of Prewhitening
195
1
Notes and Problems
196
3
Bibliographical Notes
196
1
Problems
196
3
State Estimation in Discrete-Time Linear Dynamic Systems
199
68
Introduction
199
1
Outline
199
1
Discrete-Time Linear Estimation - Summary of Objectives
199
1
Linear Estimation in Dynamic Systems - the Kalman Filter
200
18
The Dynamic Estimation Problem
200
2
Dynamic Estimation as a Recursive Static Estimation
202
2
Derivation of the Dynamic Estimation Algorithm
204
3
Overview of the Kalman Filter Algorithm
207
4
The Matrix Riccati Equation
211
2
Properties of the Innovations and the Likelihood Function of the System Model
213
1
The Innovations Representation
214
1
Some Orthogonality Properties
215
1
The Kalman Filter - Summary
215
3
Example of a Filter
218
14
The Model
218
1
Results for a Kalman Filter
219
1
A Step-by-Step Demonstration of DynaEstTM
219
13
Consistency of State Estimators
232
13
The Problem of Filter Consistency
232
2
Definition and the Statistical Tests for Filter Consistency
234
3
Examples of Filter Consistency Testing
237
6
Absolute Errors
243
1
Filter Consistency - Summary
244
1
Initialization of State Estimators
245
3
Initialization and Consistency
245
1
Initialization in Simulations
246
1
A Practical Implementation in Tracking
247
1
Filter Initialization - Summary
248
1
Sensitivity
248
13
Model Mismatch
249
5
Reduced-Order Filters
254
2
Suboptimal Gains
256
1
Examples of Modeling Errors and Filter Approximations
256
5
Notes and Problems
261
6
Bibliographical Notes
261
1
Problems
261
4
Computer Applications
265
2
Estimation for Kinematic Models
267
34
Introduction
267
1
Outline
267
1
Kinematic Models - Summary of Objectives
267
1
Discretized Continuous-Time Kinematic Models
268
4
The Kinematic Models
268
1
Continuous White Noise Acceleration Model
269
1
Continuous Wiener Process Acceleration Model
270
2
Direct Discrete-Time Kinematic Models
272
4
Introduction
272
1
Discrete White Noise Acceleration Model
273
1
Discrete Wiener Process Acceleration Model
274
1
Kinematic Models - Summary
275
1
Explicit Filters for Noiseless Kinematic Models
276
1
LS Estimation for Noiseless Kinematic Models
276
1
The KF for Noiseless Kinematic Models
276
1
Steady-State Filters for Noisy Kinematic Models
277
17
The Problem
277
1
Derivation Methodology for the Alpha-Beta Filter
278
2
The Alpha-Beta Filter for the DWNA Model
280
6
The Alpha-Beta Filter for the Discretized CWNA Model
286
3
The Alpha-Beta-Gamma Filter for the DWPA Model
289
3
A System Design Example for Sampling Rate Selection
292
1
Alpha-Beta and Alpha-Beta-Gamma Filters - Summary
293
1
Notes and Problems
294
7
Bibliographical Notes
294
1
Problems
295
6
Computational Aspects of Estimation
301
18
Introduction
301
2
Implementation of Linear Estimation
301
1
Outline
302
1
Computational Aspects - Summary of Objectives
303
1
The Information Filter
303
5
Recursions for the Information Matrices
303
3
Overview of the Information Filter Algorithm
306
1
Recursion for the Information Filter State
307
1
Sequential Processing of Measurements
308
3
Block vs. Sequential Processing
308
1
The Sequential Processing Algorithm
309
2
Square-Root Filtering
311
6
The Steps in Square-Root Filtering
311
1
The LDL' Factorization
312
1
The Predicted State Covariance
312
2
The Filter Gain and the Updated State Covariance
314
1
Overview of the Square-Root Sequential Scalar Update Algorithm
315
1
The Gram-Schmidt Orthogonalization Procedure
315
2
Notes and Problems
317
2
Bibliographical Notes
317
1
Problems
317
2
Extensions of Discrete-Time Linear Estimation
319
22
Introduction
319
1
Outline
319
1
Extensions of Estimation - Summary of Objectives
319
1
Autocorrelated Process Noise
320
4
The Autocorrelated Process Noise Problem
320
1
An Exponentially Autocorrelated Noise
321
1
The Augmented State Equations
322
2
Estimation with Autocorrelated Process Noise - Summary
324
1
Cross-Correlated Measurement and Process Noise
324
3
Cross-Correlation at the Same Time
324
2
Cross-Correlation One Time Step Apart
326
1
State Estimation with Decorrelated Noise Sequences - Summary
327
1
Autocorrelated Measurement Noise
327
3
Whitening of the Measurement Noise
327
2
The Estimation Algorithm with the Whitened Measurement Noise
329
1
Autocorrelated Measurement Noise - Summary
330
1
Prediction
330
3
Types of Prediction
330
1
The Algorithms for the Different Types of Prediction
330
2
Prediction - Summary
332
1
Smoothing
333
5
Types of Smoothing
333
1
Fixed-Interval Smoothing
334
3
Overview of Smoothing
337
1
Smoothing - Summary
338
1
Notes and Problems
338
3
Bibliographical Notes
338
1
Problems
338
3
Continuous-Time Linear State Estimation
341
30
Introduction
341
1
Outline
341
1
Continuous-Time Estimation - Summary of Objectives
341
1
The Continuous-Time Linear State Estimation Filter
342
13
The Continuous-Time Estimation Problem
342
1
Connection Between Continuous - and Discrete-Time Representations and Their Noise Statistics
343
2
The Continuous-Time Filter Equations
345
2
The Continuous-Time Innovation
347
2
Asymptotic Properties of the Continuous-Time Riccati Equation
349
2
Examples of Continuous-Time Filters
351
2
Overview of the Kalman-Bucy Filter
353
1
Continuous-Time State Estimation - Summary
354
1
Prediction and The Continuous-Discrete Filter
355
3
Prediction of the Mean and Covariance
355
1
The Various Types of Prediction
356
1
The Continuous-Discrete Filter
357
1
Duality of Estimation and Control
358
4
The Two Problems
358
1
The Solutions to the Estimation and the Control Problems
359
1
Properties of the Solutions
360
2
The Wiener-Hopf Problem
362
4
Formulation of the Problem
362
1
The Wiener-Hopf Equation
362
4
Notes and Problems
366
5
Bibliographical Notes
366
1
Problems
367
4
State Estimation For Nonlinear Dynamic Systems
371
50
Introduction
371
1
Outline
371
1
Nonlinear Estimation - Summary of Objectives
371
1
Estimation in Nonlinear Stochastic Systems
372
9
The Model
372
1
The Optimal Estimator
373
1
Proof of the Recursion of the Conditional Density of the State
374
2
Example of Linear vs. Nonlinear Estimation of a Parameter
376
3
Estimation in Nonlinear Systems with Additive Noise
379
2
Optimal Nonlinear Estimation - Summary
381
1
The Extended Kalman Filter
381
14
Approximation of the Nonlinear Estimation Problem
381
2
Derivation of the EKF
383
2
Overview of the EKF Algorithm
385
2
An Example: Tracking with an Angle-Only Sensor
387
7
The EKF - Summary
394
1
Error Compensation in Linearized Filters
395
9
Some Heuristic Methods
395
1
An Example of Use of the Fudge Factor
396
1
An Example of Debiasing: Conversion from Polar to Cartesian
397
5
Error Compensation in Linearized Filters - Summary
402
2
Some Error Reduction Methods
404
3
Improved State Prediction
404
1
The Iterated Extended Kalman Filter
404
3
Some Error Reduction Methods - Summary
407
1
Maximum a Posteriori Trajectory Estimation Via Dynamic Programming
407
2
The Approach
407
1
The Dynamic Programming for Trajectory Estimation
408
1
Nonlinear Continuous-Discrete Filter
409
5
The Model
409
1
The Fokker-Planck Equation
410
3
Example
413
1
Notes, Problems and a Project
414
7
Bibliographical Notes
414
1
Problems
414
5
Project - Nonlinear Filtering with Angle-Only Measurements
419
2
Adaptive Estimation and Maneuvering Targets
421
70
Introduction
421
3
Adaptive Estimation - Outline
421
2
Adaptive Estimation - Summary of Objectives
423
1
Adjustable Level Process Noise
424
3
Continuous Noise Level Adjustment
424
1
Process Noise with Several Discrete Levels
424
2
Adjustable Level Process Noise - Summary
426
1
Input Estimation
427
4
The Model
427
1
The Innovations as a Linear Measurement of the Unknown Input
428
1
Estimation of the Unknown Input
429
1
Correction of the State Estimate
430
1
Input Estimation - Summary
431
1
The Variable State Dimension Approach
431
4
The Approach
431
1
The Maneuver Detection and Model Switching
432
1
Initialization of the Augmented Model
433
1
VSD Approach - Summary
434
1
A Comparison of Adaptive Estimation Methods for Maneuvering Targets
435
6
The Problem
435
1
The White Noise Model with Two Levels
436
1
The IE and VSD Methods
436
2
Statistical Test for Comparison of the IE and VSD Methods
438
2
Comparison of Several Algorithms - Summary
440
1
The Multiple Model Approach
441
25
Formulation of the Approach
441
1
The Static Multiple Model Estimator
441
3
The Dynamic Multiple Model Estimator
444
3
The GPB1 Multiple Model Estimator for Switching Models
447
2
The GPB2 Multiple Model Estimator for Switching Models
449
4
The Interacting Multiple Model Estimator
453
4
An Example with the IMM Estimator
457
3
Use of DynaEst™ to Design an IMM Estimator
460
5
The Multiple Model Approach - Summary
465
1
Design of an IMM Estimator for ATC Tracking
466
10
ATC Motion Models
466
2
The EKF for the Coordinated Tum Model
468
2
Selection of Models and Parameters
470
1
The ATC Scenario
471
1
Results and Discussion
472
4
When is an IMM Estimator Needed?
476
5
Kalman Filter vs. IMM Estimator
477
4
Use of EKF for Simultaneous State and Parameter Estimation
481
3
Augmentation of the State
481
1
An Example of Use of the EKF for Parameter Estimation
482
2
EKF for Parameter Estimation - Summary
484
1
Notes, Problems, and Term Project
484
7
Bibliographical Notes
484
1
Problems
485
3
Term Project - IMM Estimator for Air Traffic Control
488
3
Introduction to Navigation Applications
491
46
Introduction
491
1
Navigation Applications - Outline
491
1
Navigation Applications - Summary of Objectives
492
1
Complementary Filtering for Navigation
492
3
The Operation of Complementary Filtering
492
1
The Implementation of Complementary Filtering
493
2
Inertial Navigation Systems
495
1
Models For Inertial Navigation Systems
496
5
State Models
496
1
Sensor Error Models
496
1
Single-Axis Models
497
2
Three-Axis Models
499
1
Coordinate Transformation
500
1
The Global Positioning System (GPS)
501
1
The GPS Segments
502
1
GPS Satellite Constellation
502
1
GPS Positioning
502
5
The GPS Principle
502
1
The GPS Signals
503
3
The GPS Observables
506
1
The Accuracy of GPS Positioning
507
4
Dilution of Precision
507
2
GPS Positioning Accuracy
509
2
State-Space Models for GPS
511
4
Models for Receiver Clock State
511
1
Dynamic Models
512
1
Linearized Measurement Model
512
1
A Model for Exponentially Autocorrelated Noise
513
2
Coordinate Transformation
515
1
Example: GPS Navigation With IMM Estimator
515
8
Generation of Satellite Trajectories
516
1
Generation of Trajectories and Pseudorange Measurements
517
1
State-Space Models
518
2
Simulation Results and Discussion
520
3
Do We Need and IMM Estimator for GPS?
523
1
Integrated Navigation
523
7
Integration by Complementary Filtering
524
1
Example
525
2
Integration by Centralized Estimation Fusion
527
1
Integration by Distributed Estimation Fusion
528
2
Notes and Problems
530
7
Bibliographical Notes
530
1
Problems
530
3
Term Project - Extended Kalman Filter for GPS
533
4
Bibliography
537
10
Index
547