卡尔曼滤波基础实践指南：第四版

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"Fundamentals of Kalman Filtering: A Practical Approach 第四版是一本关于卡尔曼滤波理论与实践的权威书籍，由Paul Zarchan和Howard Musoff合著，并被收录在《航天与航空学进展》系列（Volume 246）中。这本书由美国航空航天学会(AIAA)出版，主编是Timothy C. Lieuwen，来自乔治亚理工学院。" 卡尔曼滤波是一种广泛应用在信号处理、控制理论和估计理论中的数学方法，尤其在导航、控制系统、图像处理和许多其他领域中具有重要价值。它提供了一种最优的线性估计方法，能够通过融合不同来源的数据，消除噪声并精确估计系统状态。本书第四版深入浅出地介绍了卡尔曼滤波的基本原理，包括滤波方程的推导、数学背景以及实际应用案例。内容可能涵盖以下几个关键知识点： 1. **基本概念**：首先，书中会解释滤波问题的背景，介绍随机过程和随机变量的基础知识，这是理解卡尔曼滤波的前提。 2. **卡尔曼滤波器理论**：详细阐述卡尔曼滤波器的数学框架，包括状态空间模型、预测和更新步骤、协方差矩阵的更新规则等。此外，还可能讨论扩展卡尔曼滤波（EKF）和无迹卡尔曼滤波（UKF），这些是处理非线性问题的变体。 3. **实例分析**：通过具体的应用例子，如目标跟踪、传感器融合或控制系统，展示如何将理论应用于实践。这些实例有助于读者理解如何在实际工程问题中设计和实施卡尔曼滤波器。 4. **Matlab和Simulink**：鉴于Matlab和Simulink在信号处理和控制领域的广泛使用，书中有可能包含使用这些工具实现卡尔曼滤波算法的指导，帮助读者进行数值模拟和原型设计。 5. **高级主题**：除了基础内容，可能还会涉及一些高级话题，如多传感器数据融合、自适应卡尔曼滤波、鲁棒卡尔曼滤波，以及与粒子滤波的比较等。 6. **实用技巧**：作者可能会分享一些在实际应用中调整滤波参数、优化性能和解决滤波稳定性问题的经验和技巧。通过对本书的学习，读者不仅可以掌握卡尔曼滤波的基本概念和理论，还能获得在真实世界问题中应用这些知识的能力。无论是初学者还是有经验的工程师，都能从中受益匪浅，提升在相关领域的专业技能。

from the fourth edition, interested readers can ﬁnd the third edition chapters and

Appendix in PDF format on AIAA’s Web site.

On a personal note, it continues to be very gratifying for me to learn that many

people working with or needing to learn about Kalman ﬁltering have found Fun-

damentals of Kalman Filtering: A Practical Approach both nonintimidating and

useful. Over the years, many readers have contacted me and have asked questions

when the book’s material was not clear to them. Material in the existing chapters

has been clariﬁed so that all will beneﬁt from my interaction with the readers. It is

still my hope that this fourth edition will be of value not only to new readers, but

will also be worthwhile to those who have read previous editions.

Paul Zarchan

August 2015

xvi PREFACE TO THE FOURTH EDITION

Downloaded by NORTHWESTERN POLYTECHICAL UNIV. on November 14, 2021 | http://arc.aiaa.org | DOI: 10.2514/5.9781624102776.0000.0000 | Book DOI: 10.2514/4.102776

INTRODUCTION

It has been more than ﬁve decades since Kalman introduced his systematic

approach to linear ﬁltering based on the method of least-squares (Kalman,

R. E., “A New Approach to Linear Filtering and Prediction Problems,” Journal

of Basic Engineering, Vol. 82, No. 1, March 1960, pp. 35–46). Although his orig-

inal journal article was difﬁcult to read and understand, the results of the paper

were applied immediately in many different ﬁelds by individ uals with a variety

of backgrounds because the ﬁltering algorithm actually worked and was easy to

implement on a digital computer. People were able to apply Kalman ﬁltering

without necessarily understanding or caring about the intricacies of its derivation.

Because of the ease of implementation of the original recursive digital Kalman

ﬁlter, engineers and scientists were able to ﬁnd out immediately that this new ﬁl-

tering technique was often much better than existing ﬁltering techniques in terms

of performance. Both performance improvements and ease of implementation

rather than analytical eleg ance made the Kalman ﬁlter popular in the world of

applications. However, the Kalman ﬁlter was usually much more computationally

expensive than existing ﬁltering techniques, which was an issue in many cases for

the primitive computers that were available at that time. In addition to improved

performance, this new ﬁltering technique also provided a systematic approach to

many problems, which was also an improvement over some of the ad hoc schemes

of the day. Today, because of the popularity and proliferation of Kalman ﬁltering,

many individuals either do not know (or care) about any other ﬁltering tech-

niques. Some actually believe that no ﬁltering took place before 1960.

With the possible exception of the fast Fourier transform, Kalman ﬁltering is

probably the most important algorithmic technique ever devised. Papers on the

subject have been ﬁlling numerous journals for decades. However, Kalman ﬁlter-

ing is one of those rare topics that is not only popular in academic journals but

also has a history of being rich in practical applications. Kalman ﬁltering has

been used in applications that include providing estimates for navigating the

Apollo spacecraft, predicting short-term stock market ﬂuctuations, and estimating

user location with relatively inexpensive hand-held global positioning system

(GPS) receivers.

The purpose of this text is not to make additional theoretical contributions in

the world of Kalman ﬁltering but is simply to show the reader how actually to

build Kalman ﬁlters by example. It is the authors’ belief that the best way of

learning is by doing. Unlike other texts on Kalman ﬁltering, which devote most

of their time to derivations of the ﬁlter and the theoretical background in under-

standing the derivations, this text does not even bother to derive the ﬁlter. After

all, the ﬁlter has been in use for more than 50 years and being an expert at deri-

vations usually has nothing to do with getting a ﬁlter to work. Instead the Kalman-

ﬁltering equations are simply explained, and the text devotes its time to applying

Kalman ﬁltering to actual problems.

xvii

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Numerous simpliﬁed, but nontrivial, real-world examples are presented in

detail, showing the many ways in which Kalman ﬁlters can be designed. Some-

times mistakes are introduced intentionally to the initial designs to show the inter-

ested reader what happens when the ﬁlter is not working properly. Rarely in real

life is a Kalman ﬁlter working after the ﬁrst try. In fact, it usually takes many tries

just to get the ﬁlter to fail (i.e., even getting the code to compile and give ridiculous

answers is a challenge)! Therefore, we intentionally take the reader through part of

that realistic iteration process. It is hoped that readers with varied learning styles

will ﬁnd the text’s practical approach to Kalman ﬁltering to be both useful and

refreshing.

The text also spends a great deal of time in setting up a problem before the

Kalman ﬁlter is actually formulated or designed. This is done to give the reader

an intuitive feel for the problem being addressed. The time spent understanding

the problem will always be important in later determining if the Kalman ﬁlter

is performing as expected and if the resultant answers make sense.

Often the hardest part in Kalman ﬁltering is the subject that no one talks

about—setting up the problem. This is analogous to the quote from the recent

engineering graduate who, upon arriving in industry, enthusiastically says, “Here

I am, present me with your differential equations!” As the naive engineering

graduate soon found out, problems in the real world are frequently not clear and

are subject to many interpretations. Real problems are seldom presented in the

form of differential equations, and they usually do not have unique solutions.

Therefore, we will often do problems in several different ways to show the reader

that there is no one way of tackling a problem. Each approach will have advantages

and disadvantages, and often the ﬁlter designer will actually do the problem several

different ways before determining the best approach for the application at hand.

The text mostly makes use of the fundamental techniques that were intro-

duced by Kalman ﬁve decades ago. There have been many different implemen-

tations, variations, and innovations of the original Kalman ﬁlter, which were

developed for real applications because of computer hardware constraints.

However, most of those approaches are ignored in this text simply because com-

puter hardware has improved so dramatically since the 1960s that many of those

innovative techniques are no longer required. Using double-precision arithmetic

on today’s 32-bit desktop or ﬂight computers usually negates the need for such

heroic techniques as square-root ﬁltering. It is our belief that discussing such tech-

niques in an introductory text such as this will only confuse readers and possibly

discourage them from ever attempting to build a Kalman ﬁlter because of its per-

ceived complexity. This does not mean that Kalman ﬁlters always work on

modern computers, but most of the time their failure to work is not associated

with subtle numerical problems. Usually the ﬁlter’s inability to work properly

has more to do with mundane issues, such as how the ﬁlter was formulated or

how design parameters were chosen. Simple programming mistakes rather than

computer round-off errors often account for a majority of the initial errors

encountered in debugging a Kalman ﬁlter on today’s computers.

xviii INTRODUCTION

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An attempt has been made to make this book as readable as possible. To make

life easier for the reader, all of the background material required for understanding

the text’s concepts is presented in the ﬁrst chapter. In addition, numerous

examples are presented in each of the chapters in order to illustrate all of the con-

cepts presented in the book. It is our hope that the text will be understandable and

useful to readers with different backgrounds.

Chapter 1 introduce s all of the numerical techniques that are used

throughout the text. Numerical methods for integrating differential equations

are discussed not only because they are important for solving our mathematical

models of the real world but also because they are useful for projecting states

forward in the Kalman ﬁlter. Finally, state-space notation, which is required for

modeling the real world in formulating a Kalman ﬁlter, is simply introduced as

a technique for representing differential equations.

Chapter 2 introduces Gauss’s method of least-squares as applied to measure-

ments that can be described by a polynomial. All of the ﬁltering techniques of the

text are based on this classical technique. The least-squares method introduced in

Chapter 2 is a batch-processing technique in that all the data must ﬁrst be col-

lected before estimates can be made. The various properties of the method of

least squares when applied to measurements that are polynomial signals corrupted

by noise are demonstrated by numerous examples. Finally, formulas that can be

used to solve for the coefﬁcients of different least-squares polynomial ﬁts to

measurement data are presented.

Chapter 3 shows how the batch-processing method of least-squares can

be made recurs ive. The recursive least-squares ﬁlter is ideal for computer

implementation because estimates can be provided as soon as measurements

are taken. Formulas used for the implementation of different order recursive

least-squares ﬁlters are presented and demonstrated to be equivalent to the

batch-processing method of least-squares presented in Chapter 2. Formulas are

also presented that can be used to predict the errors in the estimate of a least-

squares ﬁlter as a result of both measurement noise and truncation error. All of

the formulas presented in this chapter are empirically validated by numerous

simulation experiments.

Chapter 4 presents the general equations for a discrete Kalman ﬁlter. It is

shown, via simulation experiment, that if the measurement data can be rep-

resented by a polynomial signal corrupted by zero-mean Gaussian noise the dis-

crete Kalman ﬁlter and recursive least-squares ﬁlter are identical. Several examples

are presented illustrating the various possible implementations of a discrete

Kalman ﬁlter for a tracking problem involving a high-speed falling obje ct. It is

shown in this chapter that superior estimation performance can be attained if

the Kalman ﬁlter can make use of a priori information. Finally, it is demonstrated

that the addition of process noise to the Kalman ﬁlter can sometimes be used as a

tool for preventing ﬁlter divergence.

Chapter 5 considers problems in which the real world cannot be represented

by a polynomial. Various polynomial and nonpolynomial Kalman ﬁlters are

INTRODUCTION xix

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designed to illustrate various important concepts. The importance of having an

accurate fundamental matrix for state propagation, but not for ﬁlter gain compu-

tation, is demonstrated via several experiments. It is also demonstrated in this

chapter that if it is impossible to obtain an exact fundamental matrix then the

addition of process noise to the ﬁlter can sometimes be used as a method for

improving performance.

Chapter 6 introduces the equations for the continuous Kalman ﬁlter.

Although the continuous Kalman ﬁlter is usually not implemented, it can be

used as a vehicle for understanding the discrete Kalman ﬁlter from a classical

point of view. The chapter ﬁrst demonstrates, via simulation experiment, how

the continuous and discrete polynomial Kalman ﬁlters are related. Steady-state

solutions are then derived for the Kalman gains and covariance matrix elements

of the continuous polynomial Kalman ﬁlter. These solutions can not only be used

to predict performance of the ﬁlter but can also be used to derive transfer func-

tions of the ﬁlter. The chapter shows how the bandwidth of the steady-state poly-

nomial Kalman ﬁlter is related to the ratio of the process and measurement noise

spectral densities.

Theoretically, the Kalman ﬁlter can only be used if the real world can be

described by a set of linear differential equations. In practice, the model describing

the real world is often nonlinear, and other techniques must be used if we desire to

ﬁlter data and estimate states. Chapter 7 introduces the equations for the extended

Kalman ﬁlter. A numerical example is presented involving estimation of the pos-

ition and velocity of a high-speed falling object in the presence of drag. The

example is similar to the one from Chapter 4, except the addition of drag,

which is assumed to be known, makes the problem nonlinear. The example

chosen illustrates the operation and characteristics of the extended Kalman

ﬁlter. Usually, the extended Kalman ﬁlter is more fragile than a linear Kalman

ﬁlter. Examples are chosen to illustrate some of the things that can go wrong

with the extended Kalman ﬁlter and some engineering ﬁxes that can be used to

improve ﬁlter performance.

Chapter 8 adds one more degree of complication to the high-speed falling

object problem in that drag is now assumed to be unknown. In the new

problem we have to estimate the position, velocity, and ballistic coefﬁcient (or

amount of drag) of the object. Two different extended Kalman ﬁlters, each of

which has different states, are designed to illustrate ﬁlter design and performance

tradeoffs. The use of a linear polynomial Kalman ﬁlter is also explored in this

chapter for this nonlinear problem. At the end of Chapter 8, it is shown how to

use the continuous Riccati equations as a technique for avoiding the computation

of the discrete fundamental and process-noise matrices.

A projectile tracking problem is introduced in Chapter 9 to illustrate how

different measurements from the same sensor can be incorporated into an

extended Kalman ﬁlter. An extended Kalman ﬁlter is ﬁrst design ed in Cartesian

coordinates, where the model of the real world is linear but the measurement

model is nonlinear. Next, an alternative extended Kalman ﬁlter is designed in

xx INTRODUCTION

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