卡尔曼滤波推导

##概述
推导卡尔曼滤波算法中的卡尔曼增益表达式
##模型
状态方程：
$$x_{k+1}={\Phi }_{k+1}{x}_k+w_k$$
测量方程：
$$z_k=H_k{x}_k+v_k$$
并假定：
E[x0]=μ0xE[wk]=0 ∀kE[vk]=0 ∀kcov{ wk,wj}=Qkδkjcov{ vk,vj}=Rkδkjcov{ x0,x0}=P0cov{ wk,vj}=0 ∀kcov{ x0,wk}=0 ∀kcov{ x0,vj}=0 ∀k \begin{aligned} \\ & E[x_0] = \mu_0^x\\ & E[w_k] =0\ \forall k \\ & E[v_k] =0\ \forall k \\ & cov\{w_k,w_j\}=Q_k\delta_{kj}\\ & cov\{v_k,v_j\}=R_k\delta_{kj}\\ & cov\{x_0,x_0\}=P_0\\ & cov\{w_k,v_j\}=0 \ \forall k\\ & cov\{x_0,w_k\}=0 \ \forall k\\ & cov\{x_0,v_j\}=0 \ \forall k\\ \end{aligned} \\ ​E[x0​]=μ0x​E[wk​]=0 ∀kE[vk​]=0 ∀kcov{ wk​,wj​}=Qk​δkj​cov{ vk​,vj​}=Rk​δkj​cov{ x0​,x0​}=P0​cov{ wk​,vj​}=0 ∀kcov{ x0​,wk​}=0 ∀kcov{ x0​,vj​}=0 ∀k​
##推导
根据上述条件将推导以下方程：

1. 状态推断
   $${\hat{x}}_{k+1}^{-}={\Phi }_{k+1}{\hat{x}}_k$$
2. 协方差推断
   $$P_{k+1}^{-}={\Phi }_{k+1}{P}_k{\Phi }_{k+1}^T+Q_k$$
3. 卡尔曼增益计算
   $$K_{k+1}=P_{k+1}^{-}{H}_{k+1}^T(H_{k+1}{P}_{k+1}^{-}{H}_{k+1}^T+R_{k+1}{)}^{-1}$$
4. 状态更新
   $${\hat{x}}_{k+1}={\hat{x}}_{k+1}^{-}+K_{k+1}(z_{k+1}-H_{k+1}{\hat{x}}_{k+1}^{-})$$
5. 协方差更新
   $$P_{k+1}=P_{k+1}^{-}-K_{k+1}{H}_{k+1}{P}_{k+1}^{-}$$
   令：在k+1步可以得到一个k步的无偏差估计，误差项${\tilde{x}}_k={\hat{x}}_k-$