外文文獻翻譯譯稿與原文

1、外文文獻翻譯譯稿1 卡爾曼濾波的一個典型實例是從一組有限的，包含噪聲的，通過對物體位置的觀察序列（可能有偏差）預(yù)測出物體的位置的坐標及速度。在很多工程應(yīng)用（如雷達、計算機視覺）中都可以找到它的身影。同時，卡爾曼濾波也是控制理論以及控制系統(tǒng)工程中的一個重要課題。例如，對于雷達來說，人們感興趣的是其能夠跟蹤目標。但目標的位置、速度、加速度的測量值往往在任何時候都有噪聲?？柭鼮V波利用目標的動態(tài)信息，設(shè)法去掉噪聲的影響，得到一個關(guān)于目標位置的好的估計。這個估計可以是對當前目標位置的估計（濾波），也可以是對于將來位置的估計（預(yù)測），也可以是對過去位置的估計（插值或平滑）。命名編輯這種濾波方法以它的發(fā)明

2、者魯?shù)婪?E.卡爾曼（Rudolph E. Kalman）命名，但是根據(jù)文獻可知實際上Peter Swerling在更早之前就提出了一種類似的算法。斯坦利。施密特（Stanley Schmidt）首次實現(xiàn)了卡爾曼濾波器?？柭贜ASA埃姆斯研究中心訪問時，發(fā)現(xiàn)他的方法對于解決阿波羅計劃的軌道預(yù)測很有用，后來阿波羅飛船的導(dǎo)航電腦便使用了這種濾波器。關(guān)于這種濾波器的論文由Swerling（1958）、Kalman (1960)與Kalman and Bucy（1961）發(fā)表。目前，卡爾曼濾波已經(jīng)有很多不同的實現(xiàn)?？柭畛跆岢龅男问浆F(xiàn)在一般稱為簡單卡爾曼濾波器。除此以外，還有施密特擴展濾波器、信

3、息濾波器以及很多Bierman, Thornton開發(fā)的平方根濾波器的變種。也許最常見的卡爾曼濾波器是鎖相環(huán)，它在收音機、計算機和幾乎任何視頻或通訊設(shè)備中廣泛存在。以下的討論需要線性代數(shù)以及概率論的一般知識。 卡爾曼濾波建立在線性代數(shù)和隱馬爾可夫模型（hidden Markov model）上。其基本動態(tài)系統(tǒng)可以用一個馬爾可夫鏈表示，該馬爾可夫鏈建立在一個被高斯噪聲（即正態(tài)分布的噪聲）干擾的線性算子上的。系統(tǒng)的狀態(tài)可以用一個元素為實數(shù)的向量表示。隨著離散時間的每一個增加，這個線性算子就會作用在當前狀態(tài)上，產(chǎn)生一個新的狀態(tài)，并也會帶入一些噪聲，同時系統(tǒng)的一些已知的控制器的控制信息也會被加入。同時

4、，另一個受噪聲干擾的線性算子產(chǎn)生出這些隱含狀態(tài)的可見輸出。 卡爾曼濾波是一種遞歸的估計，即只要獲知上一時刻狀態(tài)的估計值以及當前狀態(tài)的觀測值就可以計算出當前狀態(tài)的估計值，因此不需要記錄觀測或者估計的歷史信息?？柭鼮V波器與大多數(shù)濾波器不同之處，在于它是一種純粹的時域濾波器，它不需要像低通濾波器等頻域濾波器那樣，需要在頻域設(shè)計再轉(zhuǎn)換到時域?qū)崿F(xiàn)?？柭鼮V波器的狀態(tài)由以下兩個變量表示：，在時刻k的狀態(tài)的估計；，誤差相關(guān)矩陣，度量估計值的精確程度?？柭鼮V波器的操作包括兩個階段：預(yù)測與更新。在預(yù)測階段，濾波器使用上一狀態(tài)的估計，做出對當前狀態(tài)的估計。在更新階段，濾波器利用對當前狀態(tài)的觀測值優(yōu)化在預(yù)測階段

5、獲得的預(yù)測值，以獲得一個更精確的新估計值。預(yù)測(預(yù)測狀態(tài))(預(yù)測估計協(xié)方差矩陣）更新首先要算出以下三個量：(測量余量，measurement residual)(測量余量協(xié)方差)(最優(yōu)卡爾曼增益）然后用它們來更新濾波器變量x與P：(更新的狀態(tài)估計)(更新的協(xié)方差估計）使用上述公式計算僅在最優(yōu)卡爾曼增益的時候有效。使用其他增益的話，公式要復(fù)雜一些不變量（Invariant）如果模型準確，而且與的值準確的反映了最初狀態(tài)的分布，那么以下不變量就保持不變：所有估計的誤差均值為零且協(xié)方差矩陣準確的反映了估計的協(xié)方差：請注意，其中表示的期望值, 。 實例考慮在無摩擦的、無限長的直軌道上的一輛車。

6、該車最初停在位置0處，但時不時受到隨機的沖擊。我們每隔t秒即測量車的位置，但是這個測量是非精確的；我們想建立一個關(guān)于其位置以及速度的模型。我們來看如何推導(dǎo)出這個模型以及如何從這個模型得到卡爾曼濾波器。因為車上無動力，所以我們可以忽略掉Bk和uk。由于F、H、R和Q是常數(shù)，所以時間下標可以去掉。車的位置以及速度（或者更加一般的，一個粒子的運動狀態(tài)）可以被線性狀態(tài)空間描述如下：其中是速度，也就是位置對于時間的導(dǎo)數(shù)。我們假設(shè)在（k  1）時刻與k時刻之間，車受到ak的加速度，其符合均值為0，標準差為a的正態(tài)分布。根據(jù)牛頓運動定律，我們可以推出其中且我們可以發(fā)現(xiàn)（因為a是一個標量）

7、。在每一時刻，我們對其位置進行測量，測量受到噪聲干擾。我們假設(shè)噪聲服從正態(tài)分布，均值為0，標準差為z。其中且如果我們知道足夠精確的車最初的位置，那么我們可以初始化并且，我們告訴濾波器我們知道確切的初始位置，我們給出一個協(xié)方差矩陣：如果我們不確切的知道最初的位置與速度，那么協(xié)方差矩陣可以初始化為一個對角線元素是B的矩陣，B取一個合適的比較大的數(shù)。此時，與使用模型中已有信息相比，濾波器更傾向于使用初次測量值的信息。§推§推導(dǎo)后驗協(xié)方差矩陣按照上邊的定義，我們從誤差協(xié)方差開始推導(dǎo)如下：代入再代入 與整理誤差向量，得因為測量誤差vk與其他項是非相關(guān)的，因此有利用協(xié)方差矩陣

8、的性質(zhì)，此式可以寫作使用不變量Pk|k-1以及Rk的定義這一項可以寫作 ： 這一公式對于任何卡爾曼增益Kk都成立。如果Kk是最優(yōu)卡爾曼增益，則可以進一步簡化，請見下文。§最優(yōu)卡爾曼增益的推導(dǎo)卡爾曼濾波器是一個最小均方誤差估計器，后驗狀態(tài)誤差估計（英文：a posteriori state estimate）是我們最小化這個矢量幅度平方的期望值，這等同于最小化后驗估計協(xié)方差矩陣Pk|k的跡（trace）。將上面方程中的項展開、抵消，得到：當矩陣導(dǎo)數(shù)是0的時候得到Pk|k的跡（trace）的最小值：此處須用到一個常用的式子，如下： 從這個方程解出卡爾曼增益Kk：這個增

9、益稱為最優(yōu)卡爾曼增益，在使用時得到最小均方誤差。§后驗誤差協(xié)方差公式的化簡在卡爾曼增益等于上面導(dǎo)出的最優(yōu)值時，計算后驗協(xié)方差的公式可以進行簡化。在卡爾曼增益公式兩側(cè)的右邊都乘以SkKkT得到根據(jù)上面后驗誤差協(xié)方差展開公式，最后兩項可以抵消，得到.這個公式的計算比較簡單，所以實際中總是使用這個公式，但是需注意這公式僅在使用最優(yōu)卡爾曼增益的時候它才成立。如果算術(shù)精度總是很低而導(dǎo)致數(shù)值穩(wěn)定性出現(xiàn)問題，或者特意使用非最優(yōu)卡爾曼增益，那么就不能使用這個簡化；必須使用上面導(dǎo)出的后驗誤差協(xié)方差公式。自適應(yīng)濾波器是能夠根據(jù)輸入信號自動調(diào)整性能進行數(shù)字信號處理的數(shù)字濾波器。作為對比，非自適應(yīng)濾波器有靜

10、態(tài)的濾波器系數(shù)，這些靜態(tài)系數(shù)一起組成傳遞函數(shù)。對于一些應(yīng)用來說，由于事先并不知道所需要進行操作的參數(shù)，例如一些噪聲信號的特性，所以要求使用自適應(yīng)的系數(shù)進行處理。在這種情況下，通常使用自適應(yīng)濾波器，自適應(yīng)濾波器使用反饋來調(diào)整濾波器系數(shù)以及頻率響應(yīng)?？偟膩碚f，自適應(yīng)的過程涉及到將代價函數(shù)用于確定如何更改濾波器系數(shù)從而減小下一次迭代過程成本的算法。價值函數(shù)是濾波器最佳性能的判斷準則，比如減小輸入信號中的噪聲成分的能力。隨著數(shù)字信號處理器性能的增強，自適應(yīng)濾波器的應(yīng)用越來越常見，時至今日它們已經(jīng)廣泛地用于手機以及其它通信設(shè)備、數(shù)碼錄像機和數(shù)碼照相機以及醫(yī)療監(jiān)測設(shè)備中假設(shè)醫(yī)院正在監(jiān)測一個患者的心臟跳動，

11、即心電圖，這個信號受到 50 Hz （許多國家供電所用頻率）噪聲的干擾剔除這個噪聲的方法之一就是使用 50Hz 的陷波濾波器（en:notch filter）對信號進行濾波。但是，由于醫(yī)院的電力供應(yīng)會有少許波動，所以我們假設(shè)真正的電力供應(yīng)可能會在 47Hz 到 53Hz 之間波動。為了剔除 47 到 53Hz 之間的頻率的靜態(tài)濾波器將會大幅度地降低心電圖的質(zhì)量，這是因為在這個阻帶之內(nèi)很有可能就有心臟跳動的頻率分量。為了避免這種可能的信息丟失，可以使用自適應(yīng)濾波器。自適應(yīng)濾波器將患者的信號與電力供應(yīng)信號直接作為輸入信號，動態(tài)地跟蹤噪聲波動的頻率。這樣的自適應(yīng)濾波器通常阻帶寬度

12、更小，這就意味著這種情況下用于醫(yī)療診斷的輸出信號就更加準確。擴展卡爾曼濾波器在擴展卡爾曼濾波器（Extended Kalman Filter，簡稱EKF）中狀態(tài)轉(zhuǎn)換和觀測模型不需要是狀態(tài)的線性函數(shù)，可替換為（可微的）函數(shù)。函數(shù)f可以用來從過去的估計值中計算預(yù)測的狀態(tài)，相似的，函數(shù)h可以用來以預(yù)測的狀態(tài)計算預(yù)測的測量值。然而f和h不能直接的應(yīng)用在協(xié)方差中，取而代之的是計算偏導(dǎo)矩陣（Jacobian）。在每一步中使用當前的估計狀態(tài)計算Jacobian矩陣，這幾個矩陣可以用在卡爾曼濾波器的方程中。這個過程，實質(zhì)上將非線性的函數(shù)在當前估計值處線性化了。這樣一來，卡爾曼濾波器的等式為：預(yù)測使用Jacob

13、ians矩陣更新模型更新預(yù)測如同擴展卡爾曼濾波器（EKF）一樣, UKF的預(yù)測過程可以獨立于UKF的更新過程之外，與一個線性的（或者確實是擴展卡爾曼濾波器的）更新過程合并來使用；或者，UKF的預(yù)測過程與更新過程在上述中地位互換亦可。外文文獻翻譯原文1Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing n

14、oise (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 opt

15、imal 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 co

16、ntrol 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

17、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 (necess

18、arily 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 th

19、e 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

20、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 Gaus

21、sian-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

22、0;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 per

23、turbed 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 t

24、he 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

25、a continuous space (as opposed to a discrete state space as in the hidden Markov model).The Kalman filter is a recursive 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

26、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

27、variables:· , the a posteriori 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 accuracy of the state estimate).The Kalman filter can be written as

28、 a single equation, however it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also know

29、n as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep. In the update phase, the current a priori prediction is combined with current observation information

30、to refine the state estimate. This improved estimate is termed the a posteriori state estimate.Typically, the two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if

31、 an observation is unavailable for some reason, the update may be skipped and multiple prediction steps performed. Likewise, if multiple independent observations are available at the same time, multiple update steps may be performed (typically with different observation matrices Hk).1415§P

32、redictPredicted (a priori) state estimatePredicted (a priori) estimate covariance§UpdateInnovation or measurement residualInnovation (or residual) covarianceOptimal Kalman gainUpdated (a posteriori) state estimateUpdated (a posteriori) estimate covarianceThe formula for the updated estimat

33、e and covariance above is only valid for the optimal Kalman gain. Usage of other gain values require a more complex formula found in the derivations section.InvariantsIf the model is accurate, and the values for  and  accurately reflect the distribution of the initial s

34、tate values, then the following invariants are preserved: (all estimates have a mean error of zero)··where  is the expected value of , and covariance matrices accurately reflect the covariance of estimates···Example application, technicaleditConsider

35、 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 t seconds, but these measurements are imprecise; we want to maintain a model of where the tru

36、ck 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 spacewhere  is the velocity, that is,

37、 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 deviationa. From Newton's laws of motion w

38、e conclude that(note that there is no  term since we have no known control inputs. Instead, we assume that ak is the effect of an unknown input and  applies that effect to the state vector) whereandso thatwhere  andAt each time step, a noisy measurement of the

39、 true position of the truck is made. Let us suppose the measurement noise vk is also normally distributed, with mean 0 and standard deviation z.whereandWe know the initial starting state of the truck with perfect precision, so we initializeand to tell the filter that we know the exact

40、 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

41、the information already in the model.Deriving the a posteriori estimate covariance matrixStarting with our invariant on the error covariance Pk | k as abovesubstitute in the definition of and substitute and and by collecting the error vectors we getSince

42、the measurement error vk is uncorrelated with the other terms, this becomesby the properties of vector covariance this becomeswhich, using our invariant on Pk | k1 and the definition of Rk becomesThis formula (sometimes known as the "Joseph form

43、" 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 derivationThe Kalman filter is a minimum mean-square error estimator. The error in the a

44、 posteriori state estimation isWe 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 posterioriestimate covariance matrix . By expanding out the terms in the equation above and co

45、llecting, 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 thatSolving this for Kk yields the Kalman gain:This gain, which is known as the&

46、#160;optimal Kalman gain, is the one that yields MMSE estimates when used.Simplification of the a posteriori error covariance formulaThe formula used to calculate the a posteriori error covariance can be simplified when the Kalman gain equals the optimal value derived a

47、bove. Multiplying both sides of our Kalman gain formula on the right by SkKkT, it follows thatReferring back to our expanded formula for the a posteriori error covariance,we find the last two terms cancel out, givingThis formula is computationally cheaper and thus nearly always used i

48、n 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 a

49、bove 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 algorit

50、hms, 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 lo

51、op 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 mo

52、dify 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 a

53、s 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) noiseNotch filter method to eliminate noise of this is the use of 50Hz (en:notch filte