卡尔曼滤波器：动态系统状态估计的利器

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"卡尔曼滤波器是一种在通信和控制领域广泛应用的统计性质问题的解决方案。它主要用于随机信号的预测、噪声中的信号分离以及已知形式信号（如脉冲、正弦波）在噪声环境中的检测。" 正文: 卡尔曼滤波器，由鲁道夫·卡尔曼所提出，是一种在不确定性和噪声环境中对动态系统状态进行最优估计的算法。它基于概率和线性代数理论，特别适合处理具有线性动态特性和高斯噪声的数据序列。卡尔曼滤波器的工作原理可以分为两个主要步骤：预测和更新。 1. 预测阶段：在这一阶段，滤波器利用上一时刻的系统状态和系统动态模型（通常是一个线性微分方程）来预测当前时刻的状态。预测过程中，考虑到噪声的存在，滤波器会引入一个预测误差，并通过计算协方差来量化这个误差的不确定性。 2. 更新阶段：在接收到新的测量数据后，卡尔曼滤波器会根据这些数据对预测状态进行修正。这个过程称为“更新”或“创新”。更新过程中，滤波器会结合预测状态和测量值，通过一个权重因子（卡尔曼增益）来调整状态估计，使得最终得到的估计更接近实际状态。卡尔曼增益是根据预测误差和测量误差的协方差动态计算的，以确保在噪声大的情况下更多依赖于系统模型，而在噪声小的情况下更多依赖于测量。卡尔曼滤波器的应用广泛，包括但不限于： - 导航系统：如GPS接收机中，用于估计位置、速度和时间。 - 控制系统：如飞机和火箭的自动驾驶，通过对传感器数据的处理优化控制决策。 - 信号处理：从噪声中恢复信号，例如音频和图像信号的去噪。 - 金融分析：预测股票价格或其他时间序列数据。 - 生物医学工程：如心电图信号分析和脑电图信号处理。除了基本的卡尔曼滤波器，还有许多变种和扩展，如扩展卡尔曼滤波器（适用于非线性系统）、无迹卡尔曼滤波器（UKF）和粒子滤波器（PF），它们在处理更复杂系统和非线性问题时展现出强大的能力。尽管卡尔曼滤波器在许多领域都有显著成效，但需要注意的是，它的有效性在很大程度上取决于对系统模型的准确描述和噪声统计的精确估计。如果这些假设不成立，滤波性能可能会受到影响。因此，在实际应用中，理解和调整滤波器参数以适应特定问题是非常关键的。

Pr[x

) ≤ ξ

|y(t

) =

), …, y(t) =

(t)] = F(ξ

) (1)

Evidently, F(ξ

) represents all the information which the meas-

urement of the random variables y(t

), ..., y(t) has conveyed about

the random variable x

). Any statistical estimate of the random

variable x

) will be some function of this distribution and

therefore a (nonrandom) function of the random variables y(t

), ...,

y(t). This statistical estimate is denoted by X

|t), or by just X

)

or X

when the set of observed random variables or the time at

which the estimate is required are clear from context.

Suppose now that X

is given as a fixed function of the random

variables y(t

), ..., y(t). Then X

is itself a random variable and its

actual value is known whenever the actual values of y(t

), ..., y(t)

are known. In general, the actual value of X

) will be different

from the (unknown) actual value of x

). To arrive at a rational

way of determining X

, it is natural to assign a penalty or loss for

incorrect estimates. Clearly, the loss should be a (i) positive, (ii)

nondecreasing function of the estimation error ε = x

) – X

Thus we define a loss function by

L(0) = 0

L(ε

) ≥ L(ε

) ≥ 0 when ε

≥ ε

≥ 0 (2)

L(ε) = L(–ε)

Some common examples of loss functions are: L(ε) = aε

, aε

a|ε|, a[1 – exp(–ε

)], etc., where a is a positive constant.

One (but by no means the only) natural way of choosing the

random variable X

is to require that this choice should minimize

the average loss or risk

E{L[x

) – X

)]} = E[E{L[x(t

) – X

)]|y(t

), …, y(t)}] (3)

Since the first expectation on the right-hand side of (3) does not

depend on the choice of X

but only on y(t

), ..., y(t), it is clear that

minimizing (3) is equivalent to minimizing

E{L[x

) – X

)]|y(t

), ..., y(t)} (4)

Under just slight additional assumptions, optimal estimates can be

characterized in a simple way.

Theorem 1. Assume that L is of type (2) and that the conditional

distribution function F(ξ) defined by (1) is:

(A) symmetric about the mean

ξ :

F(ξ –

ξ ) = 1 – F( ξ – ξ)

(B) convex for ξ ≤

ξ :

F(λξ

+ (1 – λ)ξ

) ≤ λF(ξ

) + (1 – λ)F(ξ

)

for all ξ

, ξ

≤ ξ and 0 ≤ λ ≤ 1

Then the random variable x

*(t

|t) which minimizes the average

loss (3) is the conditional expectation

*(t

|t) = E[x

)|y(t

), …, y(t)] (5)

Proof: As pointed out recently by Sherman [25], this theorem

follows immediately from a well-known lemma in probability

theory.

Corollary. If the random processes {x

(t)}, {x

(t)}, and {y(t)}

are gaussian, Theorem 1 holds.

Proof: By Theorem 5, (A) (see Appendix), conditional distribu-

tions on a gaussian random process are gaussian. Hence the re-

quirements of Theorem 1 are always satisfied.

In the control system literature, this theorem appears some-

times in a form which is more restrictive in one way and more

general in another way:

Theorem l-a. If L(ε) = ε

, then Theorem 1 is true without as-

sumptions (A) and (B).

Proof: Expand the conditional expectation (4):

E[x

)|y(t

), …, y(t)] – 2X

)E[x

)|y(t

), …, y(t)] + X

)

and differentiate with respect to X

). This is not a completely

rigorous argument; for a simple rigorous proof see Doob [15], pp.

77–78.

Remarks. (a) As far as the author is aware, it is not known what

is the most general class of random processes {x

(t)}, {x

(t)} for

which the conditional distribution function satisfies the re-

quirements of Theorem 1.

(b) Aside from the note of Sherman, Theorem 1 apparently has

never been stated explicitly in the control systems literature. In

fact, one finds many statements to the effect that loss functions of

the general type (2) cannot be conveniently handled mathe-

matically.

valued random variables. In that case, the estimation problem is

stated as: Given a vector-valued random process {x(t)} and ob-

served random variables y(t

), ..., y(t), where y(t) = Mx(t) (M

being a singular matrix; in other words, not all co-ordinates of

x(t) can be observed), find an estimate X(t

) which minimizes the

expected loss E[L(||x(t

) – X(t

)||)], || || being the norm of a

vector.

Theorem 1 remains true in the vector case also, provided we

re- quire that the conditional distribution function of the n co-

ordi- nates of the vector x(t

Pr[x

) ≤ ξ

,…, x

) ≤ ξ

|y(t

), …, y(t)] = F(ξ

, …,ξ

)

be symmetric with respect to the n variables ξ

– ξ

, …, ξ

– ξ

and convex in the region where all of these variables are

negative.

Orthogonal Projections

The explicit calculation of the optimal estimate as a function of

the observed variables is, in general, impossible. There is an

important exception: The processes {x

(t)}, {x

(t)} are gaussian.

On the other hand, if we attempt to get an optimal estimate

under the restriction L(ε) = ε

and the additional requirement that

the estimate be a linear function of the observed random

variables, we get an estimate which is identical with the optimal

estimate in the gaussian case, without the assumption of linearity

or quadratic loss function. This shows that results obtainable by

linear estimation can be bettered by nonlinear estimation only

when (i) the random processes are nongaussian and even then (in

view of Theorem 5, (C)) only (ii) by considering at least third-

order probability distribution functions.

In the special cases just mentioned, the explicit solution of the

estimation problem is most easily understood with the help of a

geometric picture. This is the subject of the present section.

Consider the (real-valued) random variables y(t

), …, y(t). The

set of all linear combinations of these random variables with real

coefficients

∑

iya

)( (6)

forms a vector space (linear manifold) which we denote by

Y(t).

We regard, abstractly, any expression of the form (6) as “point”

or “vector” in

Y(t); this use of the word “vector” should not be

confused, of course, with “vector-valued” random variables, etc.

Since we do not want to fix the value of t (i.e., the total number

of possible observations),

Y(t) should be regarded as a finite-

dimensional subspace of the space of all possible observations.

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