Estimation with Applications to Tracking and Navigation Theory Algorithms and Software

, by Bar-Shalom, Yaakov; Li, X. Rong; Kirubarajan, Thiagalingam

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ISBN: 9780471416555 | 047141655X
Cover: Hardcover
Copyright: 6/25/2001

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Expert coverage of the design and implementation of state estimation algorithms for tracking and navigation Estimation with Applications to Tracking and Navigation treats the estimation of various quantities from inherently inaccurate remote observations. It explains state estimator design using a balanced combination of linear systems, probability, and statistics. The authors provide a review of the necessary background mathematical techniques and offer an overview of the basic concepts in estimation. They then provide detailed treatments of all the major issues in estimation with a focus on applying these techniques to real systems. Other features include: * Problems that apply theoretical material to real-world applications * In-depth coverage of the Interacting Multiple Model (IMM) estimator * Companion DynaEst(TM) software for MATLAB(TM) implementation of Kalman filters and IMM estimators * Design guidelines for tracking filters Suitable for graduate engineering students and engineers working in remote sensors and tracking, Estimation with Applications to Tracking and Navigation provides expert coverage of this important area.

Preface

xvii

Acronyms

xxi

Mathematical Notations

xxii

Introduction

(88)

Background

(14)

Estimation and Related Areas

(2)

Applications of Estimation

(1)

Preview of Estimation/Filtering

(6)

An Example of State Estimation: Vehicle Collision Avoidance

(5)

Scope of the Text

(4)

Objectives

(1)

Overview and Chapter Prerequisites

(3)

Brief Review of Linear Algebra and Linear Systems

(12)

Definitions and Notations

(1)

Some Linear Algebra Operations

(1)

Inversion and the Determinant of a Matrix

(2)

Orthogonal Projection of Vectors

(1)

The Gradient, Jacobian and Hessian

(1)

Eigenvalues, Eigenvectors, and Quadratic Forms

(2)

Continuous-Time Linear Dynamic Systems - Controllability and Observability

(2)

Discrete-Time Linear Dynamic Systems - Controllability and Observability

(2)

Brief Review of Probability Theory

(41)

Events and the Axioms of Probability

(2)

Random Variables and Probability Density Function

(2)

Probability Mass Function

(1)

Mixed Random Variable and Mixed Probability-PDF

(1)

Expectations and Moments of a Scalar Random Variable

(1)

Joint PDF of Two Random Variables

(3)

Independent Events and Independent Random Variables

(1)

Vector-Valued Random Variables and Their Moments

(3)

Conditional Probability and PDF

(1)

The Total Probability Theorem

(2)

Bayes' Formula

(3)

Conditional Expectations and Their Smoothing Property

(1)

Gaussian Random Variables

(1)

Joint and Conditional Gaussian Random Variables

(2)

Expected Value of Quadratic and Quartic Forms

(1)

Mixture Probability Density Functions

(2)

Chi-Square Distributed Random Variables

(3)

Weighted Sum of Chi-Square Random Variables

(1)

Random Processes

(4)

Random Walk and the Wiener Process

(1)

Markov Processes

(3)

Random Sequences, Markov Sequences and Markov Chains

(1)

The Law of Large Numbers and the Central Limit Theorem

(2)

Brief Review of Statistics

(13)

Hypothesis Testing

(2)

Confidence Regions and Significance

(5)

Monte Carlo Runs and Comparison of Algorithms

(3)

Tables of the Chi-Square and Gaussian Distributions

(3)

Notes and Problems

(4)

Bibliographical Notes

(1)

Problems

(4)

Basic Concepts in Estimation

(32)

Introduction

(1)

Outline

(1)

Basic Concepts - Summary of Objectives

(1)

The Problem of Parameter Estimation

(2)

Definitions

(1)

Models for Estimation of a Parameter

(1)

Maximum Likelihood and Maximum a Posteriori Estimators

(6)

Definitions of ML and MAP Estimators

(1)

MLE vs. MAP Estimator with Gaussian Prior

(2)

MAP Estimator with One-Sided Exponential Prior

(1)

MAP Estimator with Diffuse Prior

(1)

The Sufficient Statistic and the Likelihood Equation

(2)

Least Squares and Minimum Mean Square Error Estimation

(3)

Definitions of LS and MMSE Estimators

(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

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Estimation with Applications to Tracking and Navigation Theory Algorithms and Software

Summary

Author Biography

Table of Contents

Supplemental Materials