卡尔曼滤波实战：MATLAB实现

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"《Kalman filter theory and practice using Matlab》是由加州理工学院的教授Mohinder S. Grewal和已退休的Rockwell Science Center成员Angus P. Andrews合著的一本国际知名的教材，专门讲解卡尔曼滤波的理论与实践，并结合Matlab进行深入的探讨。本书详细介绍了卡尔曼滤波的各种概念、算法和应用，旨在帮助读者理解和掌握这一重要的信号处理和估计理论工具。" 卡尔曼滤波是一种在噪声存在的情况下，用于估计动态系统状态的最优线性滤波器。它基于概率统计理论，利用系统模型和观测数据，通过递推方式计算出系统状态的最优估计。卡尔曼滤波器的核心是其数学框架，包括状态方程和观测方程，以及卡尔曼增益的计算。 1. 状态方程：描述了系统状态随着时间如何演变。这通常是一个线性微分或差分方程，包含系统状态和控制输入的影响。 2. 观测方程：将系统状态转换为可测量的输出，通常也是一条线性关系，包含了观测噪声。 3. 卡尔曼增益：根据当前的预测误差和观测误差的协方差来调整，决定了如何结合系统状态预测和新观测数据来更新估计。本书的第三版可能涵盖了以下内容： - 基本卡尔曼滤波理论的介绍，包括高斯分布、最小均方误差估计和贝叶斯滤波原理。 - 单变量和多变量卡尔曼滤波器的详细分析。 - 扩展卡尔曼滤波（EKF）适用于非线性系统的滤波方法。 - 无偏卡尔曼滤波和平方根卡尔曼滤波等变种的讨论。 - 与Matlab相结合的实例，提供实际应用的代码和仿真结果，帮助读者加深理解。 - 可能还包括粒子滤波和其他现代滤波器的比较，以及在导航、控制系统、图像处理、通信和金融领域的应用案例。通过阅读这本书，读者可以学习到如何设计和实现卡尔曼滤波器，解决实际工程问题，以及如何利用Matlab进行滤波器的建模和仿真。对于从事信号处理、自动控制、航空航天、机器人技术等相关领域的工程师和科研人员来说，这是一本不可多得的参考书。

that you want to control, and the Kalman ﬁlter provides a means for inferring the

missing information from indirect and noisy measurements. The Kalman ﬁlter is

also used for predicting the likely future courses of dynamic systems that people

are not likely to control, such as the ﬂow of rivers during ﬂoods, the trajectories of

celestial bodies, or the prices of traded commodities.

From a practical standpoint, these are the perspectives that this book will present:

†

It is only a tool. It does not solve any problem all by itself, although it can make

it easier for you to do so. It is not a physical tool but a mathematical one.

Mathematical tools make mental work more efﬁcient, just as mechanical

tools make physical work more efﬁcient. As with any tool, it is important to

understand its use and function before you can apply it effectively. The

purpose of this book is to make you sufﬁciently familiar with and proﬁcient

in the use of the Kalman ﬁlter so that you can apply it correctly and efﬁciently.

†

It is a computer program. It has been called “ideally suited to digital computer

implementation” [95], in part because it uses a ﬁnite representation of the esti-

mation problem—by a ﬁnite number of variables. It does, however, assume that

these variables are real numbers—with inﬁnite precision. Some of the problems

encountered in its use arise from the distinction between ﬁnite dimension and

ﬁnite information and from the distinction between ﬁnite and manageable

problem sizes. These are all issues on the practical side of Kalman ﬁltering

that must be considered along with the theory.

†

It is a consistent statistical characterization of an estimation problem.Itis

much more than an estimator, because it propagates the current state of knowledge

of the dynamic system, including the statistical inﬂuence of random dynamic per-

turbations and the effects of all past measurements. These properties are extremely

useful for statistical analysis and the predictive design of sensor systems.

If these answers provide the level of understanding that you are seeking, then there is

no need for you to read the rest of this book. If you need to understand Kalman ﬁlters

well enough to use them effectively, then please read on!

1.1.2 How It Came to Be Called a Filter

It might seem strange that the term ﬁlter would apply to an estimator. More com-

monly, a ﬁlter is a physical device for removing unwanted fractions of mixtures.

(The word felt comes from the same medieval Latin stem, for the material was

used as a ﬁlter for liquids.) Originally, a ﬁlter solved the problem of separating

unwanted components of gas-liquid-solid mixtures. In the era of crystal radios and

vacuum tubes, the term was applied to analog circuits that ﬁlter electronic signals.

These signals are mixtures of different frequency components, and these physical

devices preferentially attenuate unwanted frequencies.

This concept was extended in the 1930s and 1940s to the separation of signals

from noise, both of which were characterized by their power spectral densities.

GENERAL INFORMATION2

constants, there is hardly anything in the universe that is truly constant. The

orbital parameters of the dwarf planet Ceres are not constant, and even the

“ﬁxed” stars and continents are moving. Nearly all physical systems are

dynamic to some degree. If one wants very precise estimates of their character-

istics over time, then one has to take their dynamics into consideration. The

problem is that one does not always know their dynamics very precisely

either. Given this state of partial ignorance, the best one can do is express

our ignorance more precisely—using probabilities. The Kalman ﬁlter allows

us to estimate the state of dynamic systems with certain types of random beha-

vior by using such statistical information. A few examples of such systems are

listed in the second column of Table 1.1.

2. Performance analysis of estimation systems. The third column of Table 1.1

lists some possible sensor types that might be used in estimating the state of the

corresponding dynamic systems. The objective of design analysis is to deter-

mine how best to use these sensor types for a given set of design criteria.

These criteria are typically related to estimation accuracy and system cost.

The Kalman ﬁlter uses a parametric characterization of the probability distribution

of its estimation errors in determining the optimal ﬁltering gains, and this probability

distribution may be used in assessing its performance as a function of the “design

parameters” of an estimation system, such as

†

the types of sensors to be used,

†

the locations and orientations of the various sensor types with respect to the

system to be estimated,

†

the allowable noise characteristics of the sensors,

TABLE 1.1 Examples of Estimation Problems

Application Dynamic System Sensor Types

Process control Chemical plant Pressure

Temperature

Flow rate

Gas analyzer

Flood prediction River system Water level

Rain gauge

Weather radar

Tracking Spacecraft Radar

Imaging system

Navigation Ship Sextant

Log

Gyroscope

Accelerometer

GNSS



receiver



Global navigation satellite system.

GENERAL INFORMATION

1.2.1 Beginnings of Optimal Estimation Theory

The ﬁrst method for forming an optimal estimate from noisy data is the method of

least squares. Its discovery is generally attributed to Carl Friedrich Gauss (1777–

1855) in 1795. The inevitability of measurement errors had been recognized since

the time of Galileo Galilei (1564–1642), but this was the ﬁrst formal method

for dealing with them. Although it is more commonly used for linear estimation

problems, Gauss ﬁrst used it for a nonlinear estimation problem in mathematical

astronomy, which was part of an interesting event in the history of astronomy. The

following account was put together from several sources, including the account by

Makemson in [19].

On January 1, 1801, the ﬁrst day of the nineteenth century, the Italian astronomer

Giuseppe Piazzi was checking an entry in a star catalog. Unbeknown to Piazzi, it

included an error by the printer. While searching for the “missing” star, Piazzi discov-

ered instead something that moved. It was the dwarf planet Ceres—the largest body in

the asteroid belt and the ﬁrst to be discovered—but Piazzi did not know that yet. He

was able to track and measure its apparent motion against the “ﬁxed” star background

during 41 nights before it moved too close to the sun and disappeared.

On January 24, Piazzi had written of his discovery to Johann Bode. Bode is best

known for Bode’s law, which states that the distances of the planets from the sun, in

astronomical units, are given by the sequence

(4 þ3 2

) for n ¼1,0,1,2,?,4,5,...: (1:1)

Actually, it was not Bode but Johann Tietz who ﬁrst proposed this formula, in 1772.

At that time there were only six known planets. In 1781, Friedrich Herschel

discovered Uranus, which ﬁt nicely into this formula for n ¼ 6. No planet had

been discovered for n ¼ 3. Spurred on by Bode, an association of European astron-

omers had been searching for the “missing” eighth planet for nearly 30 years. Piazzi

was not part of this association, but he did inform Bode of his unintended discovery.

Piazzi’s letter did not reach Bode until March 20. (Electronic mail was discovered

much later.) Bode suspected that Piazzi’s discovery might be the missing planet, but

there was insufﬁcient data for determining its orbital elements by the methods then

available. It is a problem in nonlinear equations that Newton himself had declared

as being among the most difﬁcult in mathematical astronomy. Nobody had solved

it and, as a result, Ceres was lost in space again.

Piazzi’s discoveries were not published until the autumn of 1801. The possible

discovery—and subsequent loss—of a new planet, coinciding with the beginning

of a new century, was exciting news. It contradicted a philosophical justiﬁcation

for the existence of only seven planets—the number known before Ceres and a

number defended by the respected philosopher Georg Hegel, among others. Hegel

had recently published a book in which he chastised the astronomers for wasting

their time searching for an eighth planet when there was sound philosophical justiﬁ-

cation for there being only seven. The new celestial object became a subject of

GENERAL INFORMATION6

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