卡尔曼滤波详解：最优估计与应用

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"最优过滤（Optimal Filtering）是关于卡尔曼滤波的经典著作，由Brian D.O. Anderson和John B. Moore撰写。这本书深入探讨了最优估计理论，并详细推导了离散卡尔曼滤波器、预报器以及平滑器等关键概念。在信息与系统科学系列丛书中，此书由Thomas Kailath编辑，涵盖了广泛的相关领域，如信号处理、通信工程和控制理论。" 卡尔曼滤波是一种在噪声环境中进行状态估计的统计方法，它在众多领域如航空航天、导航系统、自动控制、图像处理和生物医学信号分析中有着广泛应用。该滤波器基于贝叶斯理论，结合了系统的动态模型和观测数据，以提供对系统状态的最优估计。本书首先介绍了最优估计理论的基础，这是理解卡尔曼滤波的核心。最优估计是指在给定所有可用信息的情况下，对未知变量的估计具有最小均方误差。这个概念是卡尔曼滤波器设计的基石，它允许我们根据已有的知识和新的观测数据不断更新对系统的理解。离散卡尔曼滤波器是卡尔曼滤波理论在离散时间系统中的应用。它通过一系列线性递推公式来更新状态估计，这些公式包括预测步骤（利用系统模型预测未来状态）和更新步骤（结合新观测数据校正预测状态）。预报器则是在没有当前观测数据时，对未来状态的预测。而平滑器则是在所有观测数据都可用的情况下，对过去状态的后向估计，它提供了比滤波器更精确的历史状态估计。书中可能还涵盖了其他相关主题，比如噪声处理、信号检测和识别、线性系统理论以及通信系统工程等。通过对这些概念的深入学习，读者将能够理解和应用卡尔曼滤波技术解决实际问题，提高数据处理的精度和效率。 "Optimal Filtering" 是一本深入学习卡尔曼滤波及其应用的重要教材，适合电气工程、控制理论、信号处理和通信领域的学者及工程师参考阅读。通过阅读此书，读者不仅可以掌握卡尔曼滤波的基本原理，还能了解到这一领域的最新发展和实际应用。

~ INTRODUCTION

Ch. 1

linear systems with associated gaussian random processes; it transpires that

the optimal filter in a minimum mean-square-error sense is then linear. Of

course, man y applications involve nonlinear systems and/or nongaussian

random processes, and for these situations, the optimal filter is nonlinear.

However, the plain fact of the matter is that optimal nonlinear filter design

and implementation are very hard, if not impossible, in many instances. For

this reason, a suboptimal linear filter may often be used as a substitute for an

optimal nonlinear filter, or some form of nonlinear filter maybe derived which

is in some way a modification of a linear filter or, sometimes, a collection of

linear filters. These approaches are developed in this book and follow our

discussion of linear filtering, since one can hardly begin to study nonlinear

filtering with any effectiveness without a knowledge of linear filtering.

Finite u?mensionufity. Itturns out that finite-dimensional filters should

be used when the processes being filtered are associated with finite-dimen-

sional systems. Now most physical systems are not finite dimensional;

however, almost all infinite-dimensional systems can be approximated by

finite-dimensional systems, and this is generally what happens in the modeling

process. The finite-dimensional modeling of the physical system then leads to

an associated finite-dimensional filter. This filter will be suboptimal to the

extent that the model of the physical system is in some measure an inaccurate

reflection of physical reality. Why should one use a suboptimal filter?

Though one can without too much difficulty discuss infinite-dimensional

filtering problems in discrete time, and this we do in places in this book,

finite-dimensional filters are very much to be preferred on two grounds: they

are easier to design, and far easier to implement, than infinite-dimensional

filters.

1.4 OUTLINE OF THE BOOK

The book falls naturally into three parts.

The first part of the book is devoted to the formulation and solution of

the basic Kalman filtering problem. By the end of the first section of Chapter

3, the reader should know the fundamental Kalman filtering result, and by

the end of Chapter 3, have seen it in use.

The second part of the book is concerned with a deeper examination of

the operational and computational properties of the filter. For example, there

is discussion of time-invariant filters, including special techniques for com-

puting these filters, and filter stability; the: Kalman filter is shown to have a

signal-to-noise ratio enhancement property.

In the third part of the book, there area number of developments taking

Ch. 1

REFERENCES 7

off

from the basic theory. For example, the topics of smoothers, nonlinear

and adaptive filters, and spectral factorization are all covered.

There is also a collection of appendices to which the reader will probably

refer on a number of occasions. These deal with probability theory and

random processes, matrix theory, linear systems, and Lyapunov stability

theory. By and large, we expect a reader to know some, but not all, of the

material in these appendices. They are too concentrated in presentation to

allow learning of the ideas from scratch. However, if they are consulted when

a new idea is encountered, they will permit the reader to learn much, simply

by using the ideas.

Last, we make the point that there are many ideas developed in the

problems. Many are not routine.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

REFERENCES

SKILLING,H. H.,

Electrical Engineering Circuits, John Wiley & Sons, Inc.,

New York, 1957.

STORER,J. E.,

Pamive Network Synrhesis, McGraw-Hill Book Company, New

York, 1957.

TERMAN,F. EL,

Electronic and Radio Engineering, McGraw-Hill Book Com-

pany, New York, 1955.

VITERBI,A. J.,

Principles of Coherent Communication, McGraw-Hill Book Com-

pany, New York, 1966.

GOLD, B., and C. M. RADER,

Digital Processing of Signals, McGraw-Hill Book’

Company, New York, 1969.

RABINER,L. R., and B.

GOLD, Theory and Application of Digital Signal Pro-

cessing,

Prentice-Hall, Inc., Englewood Cliffs, N. J., 1975.

OPPENHEIM,A. V., and R. W. SCHAFER,

Digital Signal Processing, Prentice-

Hall, Inc., Englewood Cliffs, N.J., 1975.

WIENER,N.,

Extrapolation, Interpolation & Smoothing of Stationary Time Series,

The M.I.T. Press, Cambridge, Mass., 1949.

KOLMOGOROV,A. N.,

“Interpolation and Extrapolation,”

Bull. de l’acadkmie

des sciences de U.S. S. R.,

Ser. Math. 5, 1941, pp. 3-14.

WAINSTEIN,L. A., and V. D. ZUBAKOV,

Extraction of Signals from Noise,

Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962.

KALMAN,R. E., and R. S. BUCY,“New Results in Linear Filtering and Predic-

tion Theory,”

J. of Basic Eng., Trans. ASME, Series D, Vol. 83, No. 3, 1961,

pp. 95-108.

KALMAN, R. E., “A New Approach to Linear Filtering and Prediction Prob-

lems,”

J. Basic Eng., Trans. ASME, Series D, Vol. 82, No. 1, 1960, pp. 35-45.

FILTERING, LINEAR SYSTEMS, AND ESTIMATION

Ch. 2

Later, we shall pose specific mathematical models for some systems, and

even formally identify the system with the model.

In discussing filtering and related problems, it is implicit that the systems

under consideration are noisy. The noise may arise in a number of ways. For

example, inputs to the system may be unknown and unpredictable except for

their statistical properties, or outputs from the system may be derived with

the aid of a noisy sensor, i.e., one that contributes on a generally random

basis some inaccuracy to the measurement of the system output. Again,

outputs may only be observed via a sensor after transmission over a noisy

channel.

In virtually all the problems we shall discuss here, it will be assumed that

the output measurement process is noisy. On most occasions, the inputs also

will be assumed to be noisy.

Now let us consider exactly what we mean by

jiltering. Suppose there is

some quantity (possibly a vector quantity) associated with the system opera-

tion whose value we would like to know at each instant of time. For the sake

of argument, assume the system in question is a continuous time system, and

the quantity in question is denoted by s(. ).* It may be that this quantity is

not directly measurable, or that it can only be measured with error. In any

case, we shall suppose that noisy measurements Z(.) are available, with Z(. )

not the same as S(.).

The term filtering is used in two senses. First, it is used as a generic term:

filtering is the recovery from Z(.) of s(.), or an approximation to s(.), or

even some information about S(.). In other words, noisy measurements of a

system are used to obtain information about some quantity that is essentially

internal to the system. Second, it is used to distinguish a certain kind of

information processing from two related kinds, smoothing and prediction.

In this

sense, filtering means the recovery at time tof some information about

s(t)using measurements up till time t.The important thing to note is the

triple occurrence of the time argument t.First, we are concerned with obtain-

ing information about S(. ) at time t,i.e., s(t).Second, the information is

available at time t,not at some later time. Third, measurements right up to,

but not after, time t are used. [If information about s(t) is to be available at

time t, then causality rules out the use of measurements taken later than time

t in producing this information.]

An example of the application of filtering in everyday life is in radio

reception. Here the signal of interest is the voice signal. This signal is used to

modulate a high frequency carrier that is transmitted to a radio receiver. The

received signal is inevitably corrupted by noise, and so, when demodulated,

it is filtered to recover as well as possible the original signal.

●Almost without exeeption throughout the book,

x(t) will denote the value taken by

a function at time I, and x(. ) will denote that function. Therefore, x(t) is a number, and

x(. ) an infinite set of pairs, {t, x(t)}, for / ranging over all possible values.

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