Estimation with applications to tracking navigation 1st Edition by Yaakov Bar Shalom, Rong Li, Thiagalingam Kirubarajan 047141655X 9780471416555

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ISBN 10: 047141655X

ISBN 13: 9780471416555

Author: Yaakov Bar-Shalom, X. Rong Li, Thiagalingam Kirubarajan

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.

Estimation with applications to tracking navigation 1st Table of contents:

1 INTRODUCTION

1.1 BACKGROUND

1.1.1 Estimation and Related Areas

1.1.2 Applications of Estimation

1.1.3 Preview of Estimation/Filtering

1.1.4 An Example of State Estimation: Vehicle Collision Avoidance

1.2 SCOPE OF THE TEXT

1.2.1 Objectives

1.2.2 Overview and Chapter Prerequisites

1.3 BRIEF REVIEW OF LINEAR ALGEBRA AND LINEAR SYSTEMS

1.3.1 Definitions and Notations

1.3.2 Some Linear Algebra Operations

1.3.3 Inversion and the Determinant of a Matrix

1.3.4 Orthogonal Projection of Vectors

1.3.5 The Gradient, Jacobian and Hessian

1.3.6 Eigenvalues, Eigenvectors, and Quadratic Forms

1.3.7 Continuous-Time Linear Dynamic Systems — Controllability and Observability

1.3.8 Discrete-Time Linear Dynamic Systems — Controllability and Observability

1.4 BRIEF REVIEW OF PROBABILITY THEORY

1.4.1 Events and the Axioms of Probability

1.4.2 Random Variables and Probability Density Function

1.4.3 Probability Mass Function

1.4.4 Mixed Random Variable and Mixed Probability-PDF

1.4.5 Expectations and Moments of a Scalar Random Variable

1.4.6 Joint PDF of Two Random Variables

1.4.7 Independent Events and Independent Random Variables

1.4.8 Vector-Valued Random Variables and Their Moments

1.4.9 Conditional Probability and PDF

1.4.10 The Total Probability Theorem

1.4.11 Bayes’ Formula

1.4.12 Conditional Expectations and Their Smoothing Property

1.4.13 Gaussian Random Variables

1.4.14 Joint and Conditional Gaussian Random Variables

1.4.15 Expected Value of Quadratic and Quartic Forms

1.4.16 Mixture Probability Density Functions

1.4.17 Chi-Square Distributed Random Variables

1.4.18 Weighted Sum of Chi-Square Random Variables

1.4.19 Random Processes

1.4.20 Random Walk and the Wiener Process

1.4.21 Markov Processes

1.4.22 Random Sequences, Markov Sequences and Markov Chains

1.4.23 The Law of Large Numbers and the Central Limit Theorem

1.5 BRIEF REVIEW OF STATISTICS

1.5.1 Hypothesis Testing

1.5.2 Confidence Regions and Significance

1.5.3 Monte Carlo Runs and Comparison of Algorithms

1.5.4 Tables of the Chi-Square and Gaussian Distributions

1.6 NOTES AND PROBLEMS

1.6.1 Bibliographical Notes

1.6.2 Problems

2 BASIC CONCEPTS IN ESTIMATION

2.1 INTRODUCTION

2.1.1 Outline

2.1.2 Basic Concepts – Summary of Objectives

2.2 THE PROBLEM OF PARAMETER ESTIMATION

2.2.1 Definitions

2.2.2 Models for Estimation of a Parameter

2.3 MAXIMUM LIKELIHOOD AND MAXIMUM A POSTERIORI ESTIMATORS

2.3.1 Definitions of ML and MAP Estimators

2.3.2 MLE vs. MAP Estimator with Gaussian Prior

2.3.3 MAP Estimator with One-Sided Exponential Prior

2.3.4 MAP Estimator with Diffuse Prior

2.3.5 The Sufficient Statistic and the Likelihood Equation

2.4 LEAST SQUARES AND MINIMUM MEAN SQUARE ERROR ESTIMATION

2.4.1 Definitions of LS and MMSE Estimators

2.4.2 Some LS Estimators

2.4.3 MMSE vs. MAP Estimator in Gaussian Noise

2.5 UNBIASED ESTIMATORS

2.5.1 Definition

2.5.2 Unbiasedness of an ML and a MAP Estimator

2.5.3 Bias in the ML Estimation of Two Parameters

2.6 THE VARIANCE AND MSE OF AN ESTIMATOR

2.6.1 Definitions of Estimator Variances

2.6.2 Comparison of Variances of an ML and a MAP Estimator

2.6.3 The Variances of the Sample Mean and Sample Variance

2.6.4 Estimation of the Probability of an Event

2.7 CONSISTENCY AND EFFICIENCY OF ESTIMATORS

2.7.1 Consistency

2.7.2 The Cramer-Rao Lower Bound and the Fisher Information Matrix

2.7.3 Proof of the Cramer-Rao Lower Bound

2.7.4 An Example of Efficient Estimator

2.7.5 Large Sample Properties of the ML Estimator

2.8 SUMMARY

2.8.1 Summary of Estimators

2.8.2 Summary of Estimator Properties

2.9 NOTES AND PROBLEMS

2.9.1 Bibliographical Notes

2.9.2 Problems

3 LINEAR ESTIMATION IN STATIC SYSTEMS

3.1 INTRODUCTION

3.1.1 Outline

3.1.2 Linear Estimation in Static Systems — Summary of Objectives

3.2 ESTIMATION OF GAUSSIAN RANDOM VECTORS

3.2.1 The Conditional Mean and Covariance for Gaussian Random Vectors

3.2.2 Estimation of Gaussian Random Vectors — Summary

3.3 LINEAR MINIMUM MEAN SQUARE ERROR ESTIMATION

3.3.1 The Principle of Orthogonality

3.3.2 Linear MMSE Estimation for Vector Random Variables

3.3.3 Linear MMSE Estimation — Summary

3.4 LEAST SQUARES ESTIMATION

3.4.1 The Batch LS Estimation

3.4.2 The Recursive LS Estimator

3.4.3 Examples and Incorporation of Prior Information

3.4.4 Nonlinear LS — An Example

3.4.5 LS Estimation — Summary

3.5 POLYNOMIAL FITTING

3.5.1 Fitting a First-Order Polynomial to Noisy Measurements

3.5.2 Fitting a General Polynomial to a Set of Noisy Measurements

3.5.3 Mapping of the Estimates to an Arbitrary Time

3.5.4 Polynomial Fitting — Summary

3.6 GOODNESS-OF-FIT AND STATISTICAL SIGNIFICANCE OF PARAMETER ESTIMATES

3.6.1 Hypothesis Testing Formulation of the Problem

3.6.2 The Fitting Error in a Least Squares Estimation Problem

3.6.3 A Polynomial Fitting Example

3.6.4 Order Selection in Polynomial Fitting — Summary

3.7 USE OF LS FOR A NONLINEAR PROBLEM: BEARINGS-ONLY TARGET MOTION ANALYSIS

3.7.1 The Problem

3.7.2 Observability of the Target Parameter in Passive Localization

3.7.3 The Likelihood Function for Target Parameter Estimation

3.7.4 The Fisher Information Matrix for the Target Parameter

3.7.5 The Goodness-of-Fit Test

3.7.6 Testing for Efficiency with Monte Carlo Runs

3.7.7 A Localization Example

3.7.8 Passive Localization — Summary

3.8 NOTES, PROBLEMS AND A PROJECT

3.8.1 Bibliographical Notes

3.8.2 Problems

3.8.3 PROJECT: An Interactive Program for Bearings-Only Target Localization

4 LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS

4.1 INTRODUCTION

4.1.1 Outline

4.1.2 Linear Stochastic Systems — Summary of Objectives

4.2 CONTINUOUS-TIME LINEAR STOCHASTIC DYNAMIC SYSTEMS

4.2.1 The Continuous-Time State-Space Model

4.2.2 Solution of the Continuous-Time State Equation

4.2.3 The State as a Markov Process

4.2.4 Propagation of the State’s Mean and Covariance

4.2.5 Frequency Domain Approach

4.3 DISCRETE-TIME LINEAR STOCHASTIC DYNAMIC SYSTEMS

4.3.1 The Discrete-Time State-Space Model

4.3.2 Solution of the Discrete-Time State Equation

4.3.3 The State as a Markov Process

4.3.4 Propagation of the State’s Mean and Covariance

4.3.5 Frequency Domain Approach

4.4 SUMMARY

4.4.1 Summary of State Space Representation

4.4.2 Summary of Prewhitening

4.5 NOTES AND PROBLEMS

4.5.1 Bibliographical Notes

4.5.2 Problems

5 STATE ESTIMATION IN DISCRETE-TIME LINEAR DYNAMIC SYSTEMS

5.1 INTRODUCTION

5.1.1 Outline

5.1.2 Discrete-Time Linear Estimation — Summary of Objectives

5.2 LINEAR ESTIMATION IN DYNAMIC SYSTEMS — THE KALMAN FILTER

5.2.1 The Dynamic Estimation Problem

5.2.2 Dynamic Estimation as a Recursive Static Estimation

5.2.3 Derivation of the Dynamic Estimation Algorithm

5.2.4 Overview of the Kalman Filter Algorithm

5.2.5 The Matrix Riccati Equation

5.2.6 Properties of the Innovations and the Likelihood Function of the System Model

5.2.7 The Innovations Representation

5.2.8 Some Orthogonality Properties

5.2.9 The Kalman Filter — Summary

5.3 EXAMPLE OF A FILTER

5.3.1 The Model

5.3.2 Results for a Kalman Filter

5.3.3 A Step-by-Step Demonstration of DynaEst™

5.4 CONSISTENCY OF STATE ESTIMATORS

5.4.1 The Problem of Filter Consistency

5.4.2 Definition and the Statistical Tests for Filter Consistency

5.4.3 Examples of Filter Consistency Testing

5.4.4 Absolute Errors

5.4.5 Filter Consistency — Summary

5.5 INITIALIZATION OF STATE ESTIMATORS

5.5.1 Initialization and Consistency

5.5.2 Initialization in Simulations

5.5.3 A Practical Implementation in Tracking

5.5.4 Filter Initialization — Summary

5.6 SENSITIVITY

5.6.1 Model Mismatch

5.6.2 Reduced-Order Filters

5.6.3 Suboptimal Gains

5.6.4 Examples of Modeling Errors and Filter Approximations

5.7 NOTES AND PROBLEMS

5.7.1 Bibliographical Notes

5.7.2 Problems

5.7.3 Computer Applications

6 ESTIMATION FOR KINEMATIC MODELS

6.1 INTRODUCTION

6.1.1 Outline

6.1.2 Kinematic Models — Summary of Objectives

6.2 DISCRETIZED CONTINUOUS-TIME KINEMATIC MODELS

6.2.1 The Kinematic Models

6.2.2 Continuous White Noise Acceleration Model

6.2.3 Continuous Wiener Process Acceleration Model

6.3 DIRECT DISCRETE-TIME KINEMATIC MODELS

6.3.1 Introduction

6.3.2 Discrete White Noise Acceleration Model

6.3.3 Discrete Wiener Process Acceleration Model

6.3.4 Kinematic Models — Summary

6.4 EXPLICIT FILTERS FOR NOISELESS KINEMATIC MODELS

6.4.1 LS Estimation for Noiseless Kinematic Models

6.4.2 The KF for Noiseless Kinematic Models

6.5 STEADY-STATE FILTERS FOR NOISY KINEMATIC MODELS

6.5.1 The Problem

6.5.2 Derivation Methodology for the Alpha-Beta Filter

6.5.3 The Alpha-Beta Filter for the DWNA Model

6.5.4 The Alpha-Beta Filter for the Discretized CWNA Model

6.5.5 The Alpha-Beta-Gamma Filter for the DWPA Model

6.5.6 A System Design Example for Sampling Rate Selection

6.5.7 Alpha-Beta and Alpha-Beta-Gamma Filters — Summary

6.6 NOTES AND PROBLEMS

6.6.1 Bibliographical Notes

6.6.2 Problems

7 COMPUTATIONAL ASPECTS OF ESTIMATION

7.1 INTRODUCTION

7.1.1 Implementation of Linear Estimation

7.1.2 Outline

7.1.3 Computational Aspects — Summary of Objectives

7.2 THE INFORMATION FILTER

7.2.1 Recursions for the Information Matrices

7.2.2 Overview of the Information Filter Algorithm

7.2.3 Recursion for the Information Filter State

7.3 SEQUENTIAL PROCESSING OF MEASUREMENTS

7.3.1 Block vs. Sequential Processing

7.3.2 The Sequential Processing Algorithm

7.4 SQUARE-ROOT FILTERING

7.4.1 The Steps in Square-Root Filtering

7.4.2 The LDL’ Factorization

7.4.3 The Predicted State Covariance

7.4.4 The Filter Gain and the Updated State Covariance

7.4.5 Overview of the Square-Root Sequential Scalar Update Algorithm

7.4.6 The Gram-Schmidt Orthogonalization Procedure

7.5 NOTES AND PROBLEMS

7.5.1 Bibliographical Notes

7.5.2 Problems

8 EXTENSIONS OF DISCRETE-TIME LINEAR ESTIMATION

8.1 INTRODUCTION

8.1.1 Outline

8.1.2 Extensions of Estimation — Summary of Objectives

8.2 AUTOCORRELATED PROCESS NOISE

8.2.1 The Autocorrelated Process Noise Problem

8.2.2 An Exponentially Autocorrelated Noise

8.2.3 The Augmented State Equations

8.2.4 Estimation with Autocorrelated Process Noise — Summary

8.3 CROSS-CORRELATED MEASUREMENT AND PROCESS NOISE

8.3.1 Cross-Correlation at the Same Time

8.3.2 Cross-Correlation One Time Step Apart

8.3.3 State Estimation with Decorrelated Noise Sequences — Summary

8.4 AUTOCORRELATED MEASUREMENT NOISE

8.4.1 Whitening of the Measurement Noise

8.4.2 The Estimation Algorithm with the Whitened Measurement Noise

8.4.3 Autocorrelated Measurement Noise — Summary

8.5 PREDICTION

8.5.1 Types of Prediction

8.5.2 The Algorithms for the Different Types of Prediction

8.5.3 Prediction — Summary

8.6 SMOOTHING

8.6.1 Types of Smoothing

8.6.2 Fixed-Interval Smoothing

8.6.3 Overview of Smoothing

8.6.4 Smoothing — Summary

8.7 NOTES AND PROBLEMS

8.7.1 Bibliographical Notes

8.7.2 Problems

9 CONTINUOUS-TIME LINEAR STATE ESTIMATION

9.1 INTRODUCTION

9.1.1 Outline

9.1.2 Continuous-Time Estimation — Summary of Objectives

9.2 THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER

9.2.1 The Continuous-Time Estimation Problem

9.2.2 Connection Between Continuous – and Discrete-Time Representations and Their Noise Statistics

9.2.3 The Continuous-Time Filter Equations

9.2.4 The Continuous-Time Innovation

9.2.5 Asymptotic Properties of the Continuous-Time Riccati Equation

9.2.6 Examples of Continuous-Time Filters

9.2.7 Overview of the Kalman-Bucy Filter

9.2.8 Continuous-Time State Estimation — Summary

9.3 PREDICTION AND THE CONTINUOUS-DISCRETE FILTER

9.3.1 Prediction of the Mean and Covariance

9.3.2 The Various Types of Prediction

9.3.3 The Continuous-Discrete Filter

9.4 DUALITY OF ESTIMATION AND CONTROL

9.4.1 The Two Problems

9.4.2 The Solutions to the Estimation and the Control Problems

9.4.3 Properties of the Solutions

9.5 THE WIENER-HOPF PROBLEM

9.5.1 Formulation of the Problem

9.5.2 The Wiener-Hopf Equation

9.6 NOTES AND PROBLEMS 366

9.6.1 Bibliographical Notes

9.6.2 Problems

10 STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS

10.1 INTRODUCTION

10.1.1 Outline

10.1.2 Nonlinear Estimation — Summary of Objectives

10.2 ESTIMATION IN NONLINEAR STOCHASTIC SYSTEMS

10.2.1 The Model

10.2.2 The Optimal Estimator

10.2.3 Proof of the Recursion of the Conditional Density of the State

10.2.4 Example of Linear vs. Nonlinear Estimation of a Parameter

10.2.5 Estimation in Nonlinear Systems with Additive Noise

10.2.6 Optimal Nonlinear Estimation — Summary

10.3 THE EXTENDED KALMAN FILTER

10.3.1 Approximation of the Nonlinear Estimation Problem

10.3.2 Derivation of the EKF

10.3.3 Overview of the EKF Algorithm

10.3.4 An Example: Tracking with an Angle-Only Sensor

10.3.5 The EKF — Summary

10.4 ERROR COMPENSATION IN LINEARIZED FILTERS

10.4.1 Some Heuristic Methods

10.4.2 An Example of Use of the Fudge Factor

10.4.3 An Example of Debiasing: Conversion from Polar to Cartesian

10.4.4 Error Compensation in Linearized Filters — Summary

10.5 SOME ERROR REDUCTION METHODS

10.5.1 Improved State Prediction

10.5.2 The Iterated Extended Kalman Filter

10.5.3 Some Error Reduction Methods — Summary

10.6 MAXIMUM A POSTERIORI TRAJECTORY ESTIMATION VIA DYNAMIC PROGRAMMING

10.6.1 The Approach

10.6.2 The Dynamic Programming for Trajectory Estimation

10.7 Nonlinear Continuous-Discrete Filter

10.7.1 The Model

10.7.2 The Fokker-Planck Equation

10.7.3 Example

10.8 NOTES, PROBLEMS AND A PROJECT

10.8.1 Bibliographical Notes

10.8.2 Problems

10.8.3 Project — Nonlinear Filtering with Angle-Only Measurements

11 ADAPTIVE ESTIMATION AND MANEUVERING TARGETS

11.1 INTRODUCTION

11.1.1 Adaptive Estimation — Outline

11.1.2 Adaptive Estimation — Summary of Objectives

11.2 ADJUSTABLE LEVEL PROCESS NOISE

11.2.1 Continuous Noise Level Adjustment

11.2.2 Process Noise with Several Discrete Levels

11.2.3 Adjustable Level Process Noise — Summary

11.3 INPUT ESTIMATION

11.3.1 The Model

11.3.2 The Innovations as a Linear Measurement of the Unknown Input

11.3.3 Estimation of the Unknown Input

11.3.4 Correction of the State Estimate

11.3.5 Input Estimation — Summary

11.4 THE VARIABLE STATE DIMENSION APPROACH

11.4.1 The Approach

11.4.2 The Maneuver Detection and Model Switching

11.4.3 Initialization of the Augmented Model

11.4.4 VSD Approach — Summary

11.5 A COMPARISON OF ADAPTIVE ESTIMATION METHODS FOR MANEUVERING TARGETS

11.5.1 The Problem

11.5.2 The White Noise Model with Two Levels

11.5.3 The IE and VSD Methods

11.5.4 Statistical Test for Comparison of the IE and VSD Methods

11.5.5 Comparison of Several Algorithms — Summary

11.6 THE MULTIPLE MODEL APPROACH

11.6.1 Formulation of the Approach

11.6.2 The Static Multiple Model Estimator

11.6.3 The Dynamic Multiple Model Estimator

11.6.4 The GPB1 Multiple Model Estimator for Switching Models

11.6.5 The GPB2 Multiple Model Estimator for Switching Models

11.6.6 The Interacting Multiple Model Estimator

11.6.7 An Example with the IMM Estimator

11.6.8 Use of DynaEst™ to Design an IMM Estimator

11.6.9 The Multiple Model Approach — Summary

11.7 DESIGN OF AN IMM ESTIMATOR FOR ATC TRACKING

11.7.1 ATC Motion Models

11.7.2 The EKF for the Coordinated Turn Model

11.7.3 Selection of Models and Parameters

11.7.4 The ATC Scenario

11.7.5 Results and Discussion

11.8 WHEN IS AN IMM ESTIMATOR NEEDED?

11.8.1 Kalman Filter vs. IMM Estimator

11.9 USE OF EKF FOR SIMULTANEOUS STATE AND PARAMETER ESTIMATION

11.9.1 Augmentation of the State

11.9.2 An Example of Use of the EKF for Parameter Estimation

11.9.3 EKF for Parameter Estimation — Summary

11.10 NOTES, PROBLEMS, AND TERM PROJECT

11.10.1 Bibliographical Notes

11.10.2 Problems

11.10.3 Term Project — IMM Estimator for Air Traffic Control

12 INTRODUCTION TO NAVIGATION APPLICATIONS

12.1 INTRODUCTION

12.1.1 Navigation Applications — Outline

12.1.2 Navigation Applications — Summary of Objectives

12.2 COMPLEMENTARY FILTERING FOR NAVIGATION

12.2.1 The Operation of Complementary Filtering

12.2.2 The Implementation of Complementary Filtering

12.3 INERTIAL NAVIGATION SYSTEMS

12.4 MODELS FOR INERTIAL NAVIGATION SYSTEMS

12.4.1 State Models

12.4.2 Sensor Error Models

12.4.3 Single-Axis Models

12.4.4 Three-Axis Models

12.4.5 Coordinate Transformation

12.5 THE GLOBAL POSITIONING SYSTEM (GPS)

12.5.1 The GPS Segments

12.5.2 GPS Satellite Constellation

12.6 GPS POSITIONING

12.6.1 The GPS Principle

12.6.2 The GPS Signals

12.6.3 The GPS Observables

12.7 THE ACCURACY OF GPS POSITIONING

12.7.1 Dilution of Precision

12.7.2 GPS Positioning Accuracy

12.8 STATE-SPACE MODELS FOR GPS

12.8.1 Models for Receiver Clock State

12.8.2 Dynamic Models

12.8.3 Linearized Measurement Model

12.8.4 A Model for Exponentially Autocorrelated Noise

12.8.5 Coordinate Transformation

12.9 EXAMPLE: GPS NAVIGATION WITH IMM ESTIMATOR

12.9.1 Generation of Satellite Trajectories

12.9.2 Generation of Trajectories and Pseudorange Measurements

12.9.3 State-Space Models

12.9.4 Simulation Results and Discussion

12.9.5 Do We Need an IMM Estimator for GPS?

12.10 INTEGRATED NAVIGATION

12.10.1 Integration by Complementary Filtering

12.10.2 Example

12.10.3 Integration by Centralized Estimation Fusion

12.10.4 Integration by Distributed Estimation Fusion

12.11 NOTES AND PROBLEMS

12.11.1 Bibliographical Notes

12.11.2 Problems

12.11.3 Term Project — Extended Kalman Filter for GPS

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Yaakov Bar Shalom,Rong Li,Thiagalingam Kirubarajan,Estimation