Introduction to Optimal Estimation

by Kamen, Edward W.; Su, Jonathan

• ISBN13: 9781852331337
• ISBN10: 185233133X
• Format: Paperback
• Copyright: 1999-06-01
• Publisher: Springer Verlag

Summary

Introduction to Optimal Estimation is an introductory but comprehensive treatment of the important topics of Kalman and Wiener filtering. In addition, least-squares, maximum-likelihood and maximum a posteriori (based on discrete-time measurements) estimation are developed, covering a broad range of techniques in a single textbook. Emphasis is placed on showing how these different approaches can be fitted together to form a systematic rationale for optimal estimation. The different matters to be addressed in actually computing estimates and characterizing the properties of estimates viewed as random variables are explained and underlined throughout. The text also incorporates study of nonlinear filtering, focusing on the extended Kalman filter and on a recently-developed nonlinear estimator based on a block-form version of the Levenberg-Marquardt algorithm. Introduction to Optimal Estimation is for use in a single course (or, with judicious pruning, a one-quarter course) on estimation by senior undergraduates or first-year graduate students. A number of the examples in this text were fashioned using MATLAB® and some of the homework problems require it. Students using this book will need to have completed a standard course on probability and random variables and at least one course in signals and systems including state-space theory for linear systems.

Table of Contents

Introduction	1	(27)
Signal Estimation	1	(8)
State Estimation	9	(4)
Least Squares Estimation	13	(14)
Problems	22	(5)
Random Signals and Systems with Random Inputs	27	(42)
Random Variables	27	(17)
Random Discrete-Time Signals	44	(7)
Discrete-Time Systems with Random Inputs	51	(18)
Problems	61	(8)
Optimal Estimation	69	(32)
Formulating the Problem	69	(4)
Maximum Likelihood and Maximum a posteriori Estimation	73	(7)
Minimum Mean-Square Error Estimation	80	(7)
Linear MMSE Estimation	87	(7)
Comparison of Estimation Methods	94	(7)
Problems	96	(5)
The Wiener Filter	101	(48)
Linear Time-Invariant MMSE Filters	101	(4)
The FIR Wiener Filter	105	(9)
The Noncausal Wiener Filter	114	(5)
Toward the Causal Wiener Filter	119	(11)
Derivation of the Causal Wiener Filter	130	(9)
Summary of Wiener Filters	139	(10)
Problems	141	(8)
Recursive Estimation and the Kalman Filter	149	(42)
Estimation with Growing Memory	150	(4)
Estimation of a Constant Signal	154	(6)
The Recursive Estimation Problem	160	(1)
The Signal/Measurement Model	160	(3)
Derivation of the Kalman Filter	163	(6)
Summary of Kalman Filter Equations	169	(2)
Kalman Filter Properties	171	(4)
The Steady-state Kalman Filter	175	(7)
The SSKF as an Unbiased Estimator	182	(2)
Summary	184	(7)
Problems	185	(6)
Further Development of the Kalman Filter	191	(34)
The Innovations	191	(7)
Derivation of the Kalman Filter from the Innovations	198	(2)
Time-varying State Model and Nonstationary Noises	200	(5)
Modeling Errors	205	(5)
Multistep Kalman Prediction	210	(1)
Kalman Smoothing	211	(14)
Problems	219	(6)
Kalman Filter Applications	225	(44)
Target Tracking	225	(10)
Colored Process Noise	235	(10)
Correlated Noises	245	(7)
Colored Measurement Noise	252	(1)
Target Tracking with Polar Measurements	253	(4)
System Identification	257	(12)
Problems	263	(6)
Nonlinear Estimation	269	(44)
The Extended Kalman Filter	269	(6)
An Alternate Measurement Update	275	(6)
Nonlinear System Identification Using Neural Networks	281	(4)
Frequency Demodulation	285	(3)
Target Tracking Using the EKF	288	(5)
Multiple Target Tracking	293	(20)
Problems	307	(6)
A The State Representation	313	(10)
A.1 Discrete-Time Case	314	(2)
A.2 Construction of State Models	316	(2)
A.3 Dynamical Properties	318	(1)
A.4 Discretization of Noise Covariance Matrices	319	(4)
B The z-transform	323	(14)
B.1 Region of Convergence	324	(4)
B.2 z-transform Pairs and Properties	328	(2)
B.3 The Inverse z-transform	330	(7)
C Stability of the Kalman Filter	337	(24)
C.1 Observability	337	(2)
C.2 Controllability	339	(1)
C.3 Types of Stability	340	(2)
C.4 Positive-Definiteness of P(n)	342	(3)
C.5 An Upper Bound of P(n)	345	(3)
C.6 A Lower Bound for P(n)	348	(4)
C.7 A Useful Control Lemma	352	(3)
C.8 A Kalman Filter Stability Theorem	355	(3)
C.9 Bounds for P(n)	358	(1)
C.10 Independence of P-(n)	358	(3)
D The Steady-State Kalman Filter	361	(6)
D.1 An Upper Bound on P-(n)	361	(1)
D.2 A Stabilizability Lemma	362	(1)
D.3 Preservation of Ordering	363	(1)
D.4 Convergence when P-(0) = 0	364	(1)
D.5 Existence and Stability	365	(2)
E Modeling Errors	367	(4)
E.1 Inaccurate Initial Conditions	367	(1)
E.2 Nonlinearities and Neglected States	367	(4)
References	371