外文文献翻译译稿和原文

外文文献翻译译稿1

卡尔曼滤波的一个典型实例是从一组有限的，包含噪声的，通过对物体位置的观察序列（可能有偏差）预测出物体的位置的坐标及速度。在很多工程应用（如雷达、计算机视觉）中都可以找到它的身影。同时，卡尔曼滤波也是控制理论以及控制系统工程中的一个重要课题。

例如，对于雷达来说，人们感兴趣的是其能够跟踪目标。但目标的位置、速度、加速度的测量值往往在任何时候都有噪声。卡尔曼滤波利用目标的动态信息，设法去掉噪声的影响，得到一个关于目标位置的好的估计。这个估计可以是对当前目标位置的估计（滤波），也可以是对于将来位置的估计（预测），也可以是对过去位置的估计（插值或平滑）。 命名[编辑]

这种滤波方法以它的发明者鲁道夫.E.卡尔曼（Rudolph E. Kalman）命名，但是根据文献可知实际上Peter Swerling在更早之前就提出了一种类似的算法。

斯坦利。施密特（Stanley Schmidt）首次实现了卡尔曼滤波器。卡尔曼在NASA埃姆斯研究中心访问时，发现他的方法对于解决阿波罗计划的轨道预测很有用，后来阿波罗飞船的导航电脑便使用了这种滤波器。关于这种滤波器的论文由Swerling（1958）、Kalman (1960)与Kalman and Bucy（1961）发表。

目前，卡尔曼滤波已经有很多不同的实现。卡尔曼最初提出的形式现在一般称为简单卡尔曼滤波器。除此以外，还有施密特扩展滤波器、信息滤波器以及很多Bierman, Thornton开发的平方根滤波器的变种。也许最常见的卡尔曼滤波器是锁相环，它在收音机、计算机和几乎任何视频或通讯设备中广泛存在。

以下的讨论需要线性代数以及概率论的一般知识。

卡尔曼滤波建立在线性代数和隐马尔可夫模型（hidden Markov model）上。其基本动态系统可以用一个马尔可夫链表示，该马尔可夫链建立在一个被高斯噪声（即正态分布的噪声）干扰的线性算子上的。系统的状态可以用一个元素为实数的向量表示。随着离散时间的每一个增加，这个线性算子就会作用在当前状态上，产生一个新的状态，并也会带入一些噪声，同时系统的一些已知的控制器的控制信息也会被加入。同时，另一个受噪声干扰的线性算子产生出这些隐含状态的可见输出。

，在时刻k的状态的估计；

，误差相关矩阵，度量估计值的精确程度。

卡尔曼滤波器的操作包括两个阶段：预测与更新。在预测阶段，滤波器使用上一状态的估计，做出对当前状态的估计。在更新阶段，

滤波器利用对当前状态的观测值优化在预测阶段获得的预测值，以获得一个更精确的新估计值。

预测

(预测状态

)

(预测估计协方差矩阵）

更新

首先要算出以下三个量：

(测量余量，

measurement residual)

(测量余量协方差)

(最优卡尔曼增益） 然后用它们来更新滤波器变量x与P：

(更新的状态估计)

(更新的协方差估计）

使用上述公式计算

一些 仅在最优卡尔曼增益的时候有效。使用其他增益的话，公式要复杂

不变量（Invariant）

如果模型准确，而且与

的值准确的反映了最初状态的分布，那么以下不变量就保持不变：所有估计的误差均值为零

请注意，其中 实例

考虑在无摩擦的、无限长的直轨道上的一辆车。该车最初停在位置0

处，但时不时受到随机的冲击。我们每隔Δt秒即测量车的位置，但是这个测量是非精确的；我们想建立一个关于其位置以及速度的模型。我们来看如何推导出这个模型以及如何从这个模型得到卡尔曼滤波器。

因为车上无动力，所以我们可以忽略掉Bk和uk。由于F、H、R和Q是常数，所以时间下标可以去掉。

车的位置以及速度（或者更加一般的，一个粒子的运动状态）可以被线性状态空间描述如下： 表示的期望值, 。

其中是速度，也就是位置对于时间的导数。

我们假设在（k 1）时刻与k时刻之间，车受到ak的加速度，其符合均值为0，标准差为σa的正态分布。根据牛顿运动定律，我们可以推出

其中

且

我们可以发现

（因为σa是一个标量）。

在每一时刻，我们对其位置进行测量，测量受到噪声干扰。我们假设噪声服从正态分布，均值为0，标准差为σz。

其中

且

如果我们知道足够精确的车最初的位置，那么我们可以初始化

并且，我们告诉滤波器我们知道确切的初始位置，我们给出一个协方差矩阵：

如果我们不确切的知道最初的位置与速度，那么协方差矩阵可以初始化为一个对角线元素是B的矩阵，B取一个合适的比较大的数。

此时，与使用模型中已有信息相比，滤波器更倾向于使用初次测量值的信息。

推

推导后验协方差矩阵

按照上边的定义，我们从误差协方差开始推导如下：

代入

再代入

与

整理误差向量，得

因为测量误差vk与其他项是非相关的，因此有

利用协方差矩阵的性质，此式可以写作

使用不变量Pk|k-1以及Rk的定义这一项可以写作 ：

这一公式对于任何

卡尔曼增益Kk都成立。如果Kk是最优卡尔曼增益，则可以进一步简化，请见下文。 最优卡尔曼增益的推导

卡尔曼滤波器是一个最小均方误差估计器，后验状态误差估计（英文：a posteriori state estimate）是

我们最小化这个矢量幅度平方的期望值，，这等同于最小化后验估计协方差矩阵Pk|k的迹（trace）。将上面方程中的项展开、抵消，得到：

当矩阵导数是0的时候得到Pk|k的迹（trace）的最小值：

此处须用到一个常用的式子，如下：

从这个方程解出卡尔曼增益Kk：

这个增益称为最优卡尔曼增益，在使用时得到最小均方误差。

后验误差协方差公式的化简

在卡尔曼增益等于上面导出的最优值时，计算后验协方差的公式可以进行简化。在卡尔曼增益公式两侧的右边都乘以SkKkT得到

根据上面后验误差协方差展开公式，

最后两项可以抵消，得到

.

这个公式的计算比较简单，所以实际中总是使用这个公式，但是需注意这公式仅在使用最优卡尔曼增益的时候它才成立。如果算术精度总是很低而导致数值稳定性出现问题，或者特意使用非最优卡尔曼增益，那么就不能使用这个简化；必须使用上面导出的后验误差协方差公式。

自适应滤波器是能够根据输入信号自动调整性能进行数字信号处理的数字滤波器。作为对比，非自适应滤波器有静态的滤波器系数，这些静态系数一起组成传

递函数。

对于一些应用来说，由于事先并不知道所需要进行操作的参数，例如一些噪声信号的特性，所以要求使用自适应的系数进行处理。在这种情况下，通常使用自适应滤波器，自适应滤波器使用反馈来调整滤波器系数以及频率响应。

总的来说，自适应的过程涉及到将代价函数用于确定如何更改滤波器系数从而减小下一次迭代过程成本的算法。价值函数是滤波器最佳性能的判断准则，比如减小输入信号中的噪声成分的能力。

随着数字信号处理器性能的增强，自适应滤波器的应用越来越常见，时至今日它们已经广泛地用于手机以及其它通信设备、数码录像机和数码照相机以及医疗监测设备中

假设医院正在监测一个患者的心脏跳动，即心电图，这个信号受到 50 Hz （许多国家供电所用频率）噪声的干扰

剔除这个噪声的方法之一就是使用 50Hz 的陷波滤波器（en:notch filter）对信号进行滤波。但是，由于医院的电力供应会有少许波动，所以我们假设真正的电力供应可能会在 47Hz 到 53Hz 之间波动。为了剔除 47 到 53Hz 之间的频率的静态滤波器将会大幅度地降低心电图的质量，这是因为在这个阻带之内很有可能就有心脏跳动的频率分量。

为了避免这种可能的信息丢失，可以使用自适应滤波器。自适应滤波器将患者的信号与电力供应信号直接作为输入信号，动态地跟踪噪声波动的频率。这样的自适应滤波器通常阻带宽度更小，这就意味着这种情况下用于医疗诊断的输出信号就更加准确。

扩展卡尔曼滤波器

在扩展卡尔曼滤波器（Extended Kalman Filter，简称EKF）中状态转换和观测模型不需要是状态的线性函数，可替换为（可微的）函数。

函数f可以用来从过去的估计值中计算预测的状态，相似的，函数h可以用来以

预测的状态计算预测的测量值。然而f和h不能直接的应用在协方差中，取而代之的是计算偏导矩阵（Jacobian）。

在每一步中使用当前的估计状态计算Jacobian矩阵，这几个矩阵可以用在卡尔曼滤波器的方程中。这个过程，实质上将非线性的函数在当前估计值处线性化了。 这样一来，卡尔曼滤波器的等式为：

预测

使用Jacobians矩阵更新模型

更新

预测

如同扩展卡尔曼滤波器（EKF）一样, UKF的预测过程可以独立于UKF的更新过程之外，与一个线性的（或者确实是扩展卡尔曼滤波器的）更新过程合并来使用；或者，UKF的预测过程与更新过程在上述中地位互换亦可。

外文文献翻译原文1 Kalman filtering, also known as linear quadratic estimation (LQE), is

an algorithm that uses a series of measurements observed over time,

containing noise (random variations) and other inaccuracies, and produces estimates

of unknown variables that tend to be more precise than those based on a single measurement alone. More formally, the Kalman filter operates recursively on streams of noisy input data to produce a statistically optimal estimate of the

underlying system state. The filter is named after Rudolf (Rudy) E. Kálmán, one of the primary developers of its theory.

The Kalman filter has numerous applications in technology. A common application is for guidance, navigation and control of vehicles, particularly aircraft and spacecraft. Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics. Kalman filters also are one of the main topics in the field of Robotic motion planning and control, and sometimes included in Trajectory optimization.

The algorithm works in a two-step process. In the prediction step, the Kalman filter

produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some amount of error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with higher certainty. Because of the algorithm's recursive nature, it can run in real time using only the present input

measurements and the previously calculated state and its uncertainty matrix; no additional past information is required.

It is a common misconception that the Kalman filter assumes that all error terms and measurements are Gaussian distributed. Kalman's original paper derived the filter using orthogonal projection theory to show that the covariance is minimized, and this result does not require any assumption, e.g., that the errors are Gaussian.[1] He then showed that the filter yields the exact conditional probability estimate in the special case that all errors are Gaussian-distributed.

Extensions and generalizations to the method have also been developed, such as

the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The underlying model is a Bayesian model similar to a hidden Markov model but where the state space of the latent variables is continuous and where all latent and observed variables have Gaussian distributions.

The Kalman filters are based on linear dynamic systems discretized in the time domain. They are modelled on a Markov chain built on linear operators perturbed by errors that may include Gaussian noise. The state of the system is represented as a vector of real numbers. At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from the controls on the system if they are known. Then, another linear operator mixed with more noise generates the observed outputs from the true ("hidden") state. The Kalman

filter may be regarded as analogous to the hidden Markov model, with the key difference that the hidden state variables take values in a continuous space (as opposed to a discrete state space as in the hidden Markov model). The Kalman filter is a estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows, the notation represents the estimate of at time n given observations up to, and including at time m ≤ n

. The state of the filter is represented by two variables: , the state estimate at time k given observations up to and

including at time k;

, the a posteriori error covariance matrix (a measure of the

estimated of the state estimate).

The formula for the updated estimate and covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the section.

Invariants

If the model is accurate, and the values for and accurately reflect the distribution of the initial state values, then the following invariants are preserved: (all estimates have a mean error of zero)

is the expected value of , and covariance matrices accurately reflect the where

covariance of estimates

Example application, technical[edit]

Consider a truck on frictionless, straight rails. Initially the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of the truck every Δt seconds, but these measurements are imprecise; we want to maintain a model of where the truck is and what its velocity is. We show here how we

derive the model from which we create our Kalman filter. Since are constant, their time indices are dropped.

The position and velocity of the truck are described by the linear state space

where is the velocity, that is, the derivative of position with respect to time.

We assume that between the (k 1) and k timestep uncontrolled forces cause a constant acceleration of ak that is normally distributed, with mean 0 and standard deviationσa.

From Newton's laws of motion we conclude that

(note that there is no term since we have no known control inputs. Instead, we applies that effect to the state

assume that ak is the effect of an unknown input and

vector) where

and

so that

where

and

At each time step, a noisy measurement of the true position of the truck is made. Let us suppose the measurement noise vk is also normally distributed, with mean 0 and standard deviation σz.

where

and

We know the initial starting state of the truck with perfect precision, so we initialize

and to tell the filter that we know the exact position and velocity, we give it a zero

covariance matrix:

If the initial position and velocity are not known perfectly the covariance matrix should be

initialized with a suitably large number, say L, on its diagonal.

The filter will then prefer the information from the first measurements over the information already in the model.

Deriving the a posteriori estimate covariance matrix

Starting with our invariant on the error covariance Pk | k as above

substitute in the definition of

and substitute

and

and by collecting the error vectors we get

Since the measurement error vk is uncorrelated with the other terms, this becomes

by the properties of vector covariance this becomes

which, using our invariant on Pk | k 1

and the definition of Rk becomes

This formula (sometimes known as the "Joseph form" of the covariance update equation) is valid for any value of Kk. It turns out that if Kk is the optimal Kalman gain, this can be simplified further as shown below.

Kalman gain derivation

The Kalman filter is a minimum mean-square error estimator. The error in the a

posteriori state estimation is

We seek to minimize the expected value of the square of the magnitude of this

vector, . This is equivalent to minimizing the trace of the a

. By expanding out the terms in the equation

posterioriestimate covariance matrix

above and collecting, we get:

The trace is minimized when its matrix derivative with respect to the gain matrix is zero.

Using the gradient matrix rules and the symmetry of the matrices involved we find that

Solving this for Kk yields the Kalman gain:

This gain, which is known as the optimal Kalman gain, is the one that yields MMSE estimates when used.

Simplification of the a posteriori error covariance formula

The formula used to calculate the a posteriori error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our

Kalman gain formula on the right by SkKkT, it follows that

Referring back to our expanded formula for the a posteriori error covariance,

we find the last two terms cancel out, giving

This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with numerical stability, or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the a posteriori error covariance formula as derived above must be used.

An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorithms, most adaptive filters are digital filters. Adaptive filters are required for some applications because some parameters of the desired processing operation (for instance, the locations of reflective surfaces in areverberant space) are not known in advance or are changing. The closed loop adaptive filter uses feedback in the form of an error signal to refine its transfer function.

Generally speaking, the closed loop adaptive process involves the use of a cost function, which is a criterion for optimum performance of the filter, to feed an algorithm, which

determines how to modify filter transfer function to minimize the cost on the next iteration. The most common cost function is the mean square of the error signal.

As the power of digital signal processors has increased, adaptive filters have become much more common and are now routinely used in devices such as mobile phones and other communication devices, camcorders and digital cameras, and medical monitoring equipment.

Assuming the hospital is monitoring a patient's heart beating, namely, ECG, the signal is 50 Hz (frequency is used by many countries supply) noise

Notch filter method to eliminate noise of this is the use of 50Hz (en:notch filter) of the signal filtering. However, because of the power supply in hospital. There will be a little fluctuation, sowe assume that the power supply real may fluctuate in the 47Hz to 53Hz. In order to eliminate47 to static filters will greatly reduce the frequency of 53Hz between the ECG quality, this isbecause in the stopband within might well have a frequency component of beating heart.

In order to avoid the possible loss of information, you can use the adaptive filter. The adaptive filter will supply signal and power of patients directly as the input signal, dynamicallytracking noise fluctuation frequency. Adaptive filter this usually stopband width is smaller,which means in this case an output signal for medical diagnosis is more accurate.

Hybrid Kalman filter[edit]

Most physical systems are represented as continuous-time models while discrete-time measurements are frequently taken for state estimation via a digital processor. Therefore,

the system model and measurement model are given by

where

.

Initialize

Predict

The prediction equations are derived from those of continuous-time Kalman filter without update from measurements,

i.e., . The predicted state and covariance are calculated respectively by solving a set of differential equations with the initial value equal to the estimate at the previous step.

Update

The update equations are identical to those of the discrete-time Kalman filter.

外文文献翻译译稿2

外文文献翻译原文2

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