卡尔曼滤波：理论与MATLAB实践

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"卡尔曼滤波是用于处理随机噪声数据的一种高级统计方法，广泛应用于工程、控制理论、信号处理和许多其他领域。本书《卡尔曼滤波：理论与实践使用MATLAB》第三版由Mohinder S. Grewal和Angus P. Andrews合著，详细介绍了卡尔曼滤波的基本概念、数学理论以及实际应用。书中涵盖的内容可能包括卡尔曼滤波器的设计、实现及在MATLAB环境中的应用实例。" 正文：卡尔曼滤波是一种递归的估计算法，它利用了线性系统理论和高斯概率分布假设，能够对动态系统的状态进行最优估计。这一方法特别适用于处理包含噪声的测量数据，可以有效地从噪声中提取出真实信号。卡尔曼滤波器的两个主要阶段是预测（Prediction）和更新（Update），在每个时间步长，滤波器会结合系统模型和新测量数据来不断优化对系统状态的估计。 1. **基本原理**：卡尔曼滤波基于贝叶斯定理，它将系统状态的当前估计看作是上一时刻估计和当前测量值的加权平均。这个权重取决于系统噪声和测量噪声的协方差，从而使得滤波器能够自动调整对新数据的信任度。 2. **数学模型**： - **状态方程**：描述了系统状态从一个时间步长到下一个时间步长的变化，通常是一个线性动态系统。 - **观测方程**：定义了如何从系统状态中获得观测值，也可能是线性的，并且通常包含了观测噪声。 - **协方差矩阵**：包含了系统过程噪声和观测噪声的信息，决定了滤波器的性能。 3. **卡尔曼增益**：卡尔曼增益是滤波器的关键组成部分，它调整了新测量值对状态估计的影响。当增益高时，滤波器更倾向于依赖新的测量数据；当增益低时，滤波器则更信任上一时刻的估计。 4. **MATLAB实现**： MATLAB作为一个强大的数值计算工具，提供了方便的环境来实现和模拟卡尔曼滤波。用户可以使用MATLAB的内置函数或者自定义代码来构建和调试滤波器，同时，通过MATLAB的可视化功能，可以直观地理解滤波效果。 5. **应用领域**：卡尔曼滤波在许多领域都有应用，例如： - **导航系统**：用于GPS定位，结合多传感器数据提高位置估计精度。 - **控制系统**：优化控制器性能，减少噪声影响。 - **信号处理**：从噪声中恢复信号，如语音识别、图像处理等。 - **金融预测**：预测股票价格或经济指标。 - **生物医学工程**：如心率监测、呼吸监测等。《卡尔曼滤波：理论与实践使用MATLAB》第三版提供了全面的理论知识和实践指导，对于想要深入理解和应用卡尔曼滤波技术的读者来说是一本宝贵的资源。通过阅读此书，读者不仅可以掌握卡尔曼滤波的基本概念，还能学习如何在实际项目中有效利用这一技术。

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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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