Welch & Bishop , An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960
Welch & Bishop , An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960
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发表时间：
1994
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通讯作者：
G. Welch;G. Bishop
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作者：
G. Welch;G. Bishop
1960年，R.E.卡尔曼发表了他著名的论文描述了递归解决离散数据线性滤波问题。从那时起，在很大程度上由于数字计算的进步，卡尔曼滤波器一直是广泛研究和应用的主题，特别是在自主或辅助导航领域。卡尔曼滤波器是一组数学方程，它提供了最小二乘法的有效计算（递归）解。该滤波器在几个方面非常强大：它支持对过去、现在甚至未来状态的估计，即使在建模系统的精确性质未知时也可以这样做。本文的目的是提供一个实用的介绍离散卡尔曼滤波器。本介绍包括基本离散卡尔曼滤波器的描述和一些讨论，扩展卡尔曼滤波器的推导，描述和一些讨论，以及一个相对简单（有形）的例子，带有真实的数字和结果。1. welch@cs.unc.edu，http://www.cs.unc.edu/~welch 2. gb@cs.unc.edu http://www.cs.unc.edu/~gb Welch & Bishop，An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960，R.E.卡尔曼发表了他著名的论文描述了递归解决discretedata线性滤波问题[卡尔曼60]。从那时起，在很大程度上由于数字计算的进步，卡尔曼滤波器一直是广泛研究和应用的主题，特别是在自主或辅助导航领域。关于卡尔曼滤波器的一般思想的非常“友好”的介绍可以在[Maybeck 79]的第1章中找到，而更完整的介绍性讨论可以在[Sorenson 70]中找到，其中还包含一些有趣的历史叙述。更广泛的参考文献包括[Gelb 74; Grewal 93; Maybeck 79; Lewis 86; Brown 92; Jacobs 93]。卡尔曼滤波器解决的一般问题是试图估计离散时间受控过程的状态，该过程由线性随机差分方程（1. 1）控制，测量值为。(1.2)随机变量和分别表示过程噪声和测量噪声。假设它们（彼此）独立，为白色，且具有正态概率分布（式1.3）。(1.4)在实践中，过程噪声协方差矩阵和测量噪声协方差矩阵可能会随着每个时间步长或测量而变化，但这里我们假设它们是恒定的。在没有驱动函数或过程噪声的情况下，差分方程（1.1）中的矩阵将前一时间步的状态与当前步的状态相关联。请注意，在实践中可能会随着每个时间步而变化，但在这里我们假设它是常数。矩阵B将可选控制输入与状态x相关联。测量方程（1.2）中的矩阵将状态与测量zk联系起来。在实践中可能会随着每个时间步长或测量而变化，但在这里我们假设它是恒定的。滤波器的计算起源我们定义（注意“超级减”）为在步骤k处的先验状态估计（给定步骤k之前的过程的知识），并且为在步骤k处的后验状态估计（给定测量）。然后我们可以将先验和后验估计误差定义为xR ∈ xk Axk 1 -布克1 - + + = z R ∈ zk H xk vk + = wk vk p w（）N 0 Q，（）n p v（）N 0 R，（）n Q R n n × Ak 1 - k An l × u R ∈ m n × Hx Hk R ∈ x Hk R n ∈ zk ek xk x Hk，和-ek xk x k。- Welch & Bishop，An Introduction to the Kalman Filter 3先验估计误差协方差为，（1.5），后验估计误差协方差为。(1.6)在推导卡尔曼滤波器的方程时，我们开始的目标是找到一个方程，该方程将后验状态估计计算为先验估计和实际测量与测量预测之间的加权差的线性组合，如下面的（1.7）所示。在下面的“过滤器的可能起源”中给出了（1.7）的一些理由。(1.7)式（1.7）中的差称为测量新息或残差。残差反映了预测测量值与实际测量值之间的差异。残差为零意味着两者完全一致。（1.7）中的矩阵K被选择为使后验误差协方差（1.6）最小化的增益或混合因子。这种最小化可以通过以下方式来实现：首先将（1.7）代入上述定义，代入（1.6），执行指定的期望，取结果的迹对K的导数，将该结果设为零，然后求解K。有关更多详细信息，请参见[Maybeck 79; Brown 92; Jacobs 93]。使公式1.6最小化的K的一种形式为：
In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) solution of the least-squares method. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results. 1. welch@cs.unc.edu, http://www.cs.unc.edu/~welch 2. gb@cs.unc.edu, http://www.cs.unc.edu/~gb Welch & Bishop, An Introduction to the Kalman Filter 2 1 The Discrete Kalman Filter In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discretedata linear filtering problem [Kalman60]. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. A very “friendly” introduction to the general idea of the Kalman filter can be found in Chapter 1 of [Maybeck79], while a more complete introductory discussion can be found in [Sorenson70], which also contains some interesting historical narrative. More extensive references include [Gelb74; Grewal93; Maybeck79; Lewis86; Brown92; Jacobs93]. The Process to be Estimated The Kalman filter addresses the general problem of trying to estimate the state of a discrete-time controlled process that is governed by the linear stochastic difference equation , (1.1) with a measurement that is . (1.2) The random variables and represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions , (1.3) . (1.4) In practice, the process noise covariance and measurement noise covariance matrices might change with each time step or measurement, however here we assume they are constant. The matrix in the difference equation (1.1) relates the state at the previous time step to the state at the current step , in the absence of either a driving function or process noise. Note that in practice might change with each time step, but here we assume it is constant. The matrix B relates the optional control input to the state x. The matrix in the measurement equation (1.2) relates the state to the measurement zk. In practice might change with each time step or measurement, but here we assume it is constant. The Computational Origins of the Filter We define (note the “super minus”) to be our a priori state estimate at step k given knowledge of the process prior to step k, and to be our a posteriori state estimate at step k given measurement . We can then define a priori and a posteriori estimate errors as x R ∈ xk Axk 1 – Buk wk 1 – + + = z R ∈ zk H xk vk + = wk vk p w ( ) N 0 Q , ( ) ∼ p v ( ) N 0 R , ( ) ∼ Q R n n × A k 1 – k A n l × u R ∈ m n × H H x̂k R ∈ x̂k R n ∈ zk ek xk x̂k , and – ≡ ek xk x̂k. – ≡ UNC-Chapel Hill, TR 95-041, February 8, 2001 Welch & Bishop, An Introduction to the Kalman Filter 3 The a priori estimate error covariance is then , (1.5) and the a posteriori estimate error covariance is . (1.6) In deriving the equations for the Kalman filter, we begin with the goal of finding an equation that computes an a posteriori state estimate as a linear combination of an a priori estimate and a weighted difference between an actual measurement and a measurement prediction as shown below in (1.7). Some justification for (1.7) is given in “The Probabilistic Origins of the Filter” found below. (1.7) The difference in (1.7) is called the measurement innovation, or the residual. The residual reflects the discrepancy between the predicted measurement and the actual measurement . A residual of zero means that the two are in complete agreement. The matrix K in (1.7) is chosen to be the gain or blending factor that minimizes the a posteriori error covariance (1.6). This minimization can be accomplished by first substituting (1.7) into the above definition for , substituting that into (1.6), performing the indicated expectations, taking the derivative of the trace of the result with respect to K, setting that result equal to zero, and then solving for K. For more details see [Maybeck79; Brown92; Jacobs93]. One form of the resulting K that minimizes (1.6) is given by1