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Copy file name to clipboardExpand all lines: lectures/cross_product_trick.md
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# Eliminating Cross Products
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# 消除交叉项
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## Overview
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## 概述
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This lecture describes formulas for eliminating
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本讲座描述了消除以下内容的公式：
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*cross products between states and control in linear-quadratic dynamic programming problems
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*线性二次动态规划问题中状态和控制之间的交叉项
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*covariances between state and measurement noises in Kalman filtering problems
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*卡尔曼滤波问题中状态噪声和测量噪声之间的协方差
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对于线性二次动态规划问题，主要思路包括以下步骤：
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For a linear-quadratic dynamic programming problem, the idea involves these steps
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* transform states and controls in a way that leads to an equivalent problem with no cross-products between transformed states and controls
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* solve the transformed problem using standard formulas for problems with no cross-products between states and controls presented in this lecture {doc}`Linear Control: Foundations <lqcontrol>`
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* transform the optimal decision rule for the altered problem into the optimal decision rule for the original problem with cross-products between states and controls
That the omitted matrix $H=0$ indicates that there are no cross products between states and controls
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in the equivalent problem.
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省略的矩阵 $H=0$ 表示在等价问题中没有状态和控制之间的交叉项。
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The matrices $(A^*, B, R^*, Q) $ defining the equivalent problem and the value function, policy function matrices $P, F^*$ that solve it are related to the matrices $(A, B, R, Q, H)$ defining the original problem and the value function, policy function matrices $P, F$ that solve the original problem by
The **duality** that prevails between a linear-quadratic optimal control and a Kalman filtering problem means that there is an analogous transformation that allows us to transform a Kalman filtering problem
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with non-zero covariance matrix between between shocks to states and shocks to measurements to an equivalent Kalman filtering problem with zero covariance between shocks to states and measurments.
Thus, $x_t$ is $m \times 1$ and $z_t$ is $k \times 1$.
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The Kalman filtering formulas are
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因此，$x_t$ 是 $m \times 1$ 且 $z_t$ 是 $k \times 1$。
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卡尔曼滤波公式为：
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\begin{align*}
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K(\Sigma_t) & = (A \Sigma_t D' + BF')(D \Sigma_t D' + FF')^{-1}, \\
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\Sigma_{t+1}& = A \Sigma_t A' + BB' - (A \Sigma_t D' + BF')(D \Sigma_t D' + FF')^{-1} (D \Sigma_t A' + FB').
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\end{align*} (eq:Kalman102)
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Define tranformed matrices
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定义转换后的矩阵：
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\begin{align*}
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A^* & = A - BF' (FF')^{-1} D, \\
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B^* {B^*}' & = BB' - BF' (FF')^{-1} FB'.
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\end{align*}
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### Algorithm
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### 算法
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A consequence of formulas {eq}`eq:Kalman102} is that we can use the following algorithm to solve Kalman filtering problems that involve non zero covariances between state and signal noises.
The Kalman gain for the original problem **with non-zero covariance** between shocks to states and measurements is then
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原始问题（具有**非零协方差**的状态和测量噪声）的卡尔曼增益为：
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$$
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K = K^* + BF' (FF')^{-1},
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$$
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The state reconstruction covariance matrix $\Sigma$ for the original problem equals the state reconstrution covariance matrix for the transformed problem.
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原始问题的状态重构协方差矩阵 $\Sigma$ 等于转换后问题的状态重构协方差矩阵。
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## Duality table
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Here is a handy table to remember how the Kalman filter and dynamic program are related.
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