卡尔曼滤波详细推导

╰半夏微凉° 2022-11-28 13:46 140阅读 0赞

卡尔曼滤波（Kalman filtering）是一种利用[线性系统][Link 1]状态方程，通过系统输入输出观测数据，对[系统状态][Link 2]进行最优估计的算法，由于观测数据中包括系统中的噪声和干扰的影响，所以最优估计也可看作是[滤波][Link 3]过程。卡尔曼滤波器的本质是线性最小均方误差估计，而均方误差是协方差矩阵的迹。卡尔曼滤波有好几种公式推导方法，本文从最小二乘估计的方法推导卡尔曼滤波过程。

卡尔曼滤波有以下几个特点：

(1)卡尔曼滤波处理的对象时随机信号

(2)被处理信号无有用和干扰之分，滤波的目的是要估计出所有被处理信号

(3)系统的白噪声激励和量测噪声并不是需要滤除的对象，它们的统计特性正式估计过程中需要利用的信息

所以确切的说卡尔曼滤波是属于最优估计理论，与常规滤波器具有完全不同的概念和含义，就实际运用而言卡尔曼滤波实质上

是由一套数字计算机实现的递推算法，每个递推周期中包含对被估计量的时间更新和量测更新过程。

1、系统的变量定义

![x(k)][x_k]为系统的真值

![z(k)][z_k]为系统的测量值

![k(k)][k_k]为卡尔曼滤波增益

![A][]系统的状态转移矩阵

![B][]为系统的输入增益矩阵

![H][]为系统的量测矩阵

![P(k|k)][P_k_k]为最优估计协方差矩阵

![P(k|k-1)][P_k_k-1]为预测值协方差矩阵

![x(k|k)][x_k_k]为k时刻的最优估计值

![x(k|k-1)][x_k_k-1]为k时刻的预测值，系统的后验概率估计

![x(k-1|k-1)][x_k-1_k-1]为k-1时刻的最优估计值

![u(k)][u_k]为系统的测量噪声，其均值![E(u(k)) = 0][E_u_k_ _ 0]，协方差矩阵为R且服从正态分布的测量噪声

![w(k)][w_k]为系统的过程噪声，其均值![E(w(k)) = 0][E_w_k_ _ 0]，协方差矩阵为Q且服从正态分布的过程噪声

2、系统的状态方程

![x(k) = Ax(k-1) + Bu(k) + w(k)][x_k_ _ Ax_k-1_ _ Bu_k_ _ w_k] ......................................................................(1)

![z(k) = Hx(k) + u(k)][z_k_ _ Hx_k_ _ u_k].......................................................................................(2)

![x(k|k-1) = Ax(k-1|k-1) + Bu(k)][x_k_k-1_ _ Ax_k-1_k-1_ _ Bu_k].........................................................(3)

![x(k|k) = x(k|k-1) + k(k)(z(k) - Hx(k|k-1))][x_k_k_ _ x_k_k-1_ _ k_k_z_k_ - Hx_k_k-1]........................................(4)

公式(1)为系统状态转移方程，(2)为系统的测量方程，(3)为预测方程（后验概率估计），(4)为最佳估计方程

卡尔曼滤波器最终从上述公式求解出最优估计值![x(k|k)][x_k_k]，在求解最优估计值时首先需要推导出公式(4)的卡尔

曼滤波增益![k(k)][k_k]，具体的公式推导见下节

3、卡尔曼滤波推导

遵循误差最小原则，定义误差![e(k) = x(k) - x(k|k)][e_k_ _ x_k_ - x_k_k]求误差最小化，![P(k|k)][P_k_k 1]为误差的协方差矩阵

![P(k|k) = E\[(x(k) - x(k|k))(x(k)-x(k|k))^\{T\}\]][P_k_k_ _ E_x_k_ - x_k_k_x_k_-x_k_k_T]................................................(5)

把误差![e(k) = x(k) - x(k|k)][e_k_ _ x_k_ - x_k_k]中的最优估计值![x(k|k)][x_k_k]用公式(4)替代

![e(k) = x(k) - \\left \\\{ x(k|k-1) + k(k)(z(k) - Hx(k|k-1)) \\right \\\}][e_k_ _ x_k_ - _left _ x_k_k-1_ _ k_k_z_k_ - Hx_k_k-1_ _right].............................(6)

上述误差方程中的![z(k)][z_k]用公式(2) ![z(k) = Hx(k) + u(k)][z_k_ _ Hx_k_ _ u_k] 代替得出：

![e(k) = x(k) - \\left \\\{ x(k|k-1) + k(k)(Hx(k) + u(k) - Hx(k|k-1)) \\right \\\}][e_k_ _ x_k_ - _left _ x_k_k-1_ _ k_k_Hx_k_ _ u_k_ - Hx_k_k-1_ _right]

![= \[x(k) - k(k)Hx(k)) -(x(k|k-1) - k(k)Hx(k|k-1)\] - k(k)u(k)][_x_k_ - k_k_Hx_k_ -_x_k_k-1_ - k_k_Hx_k_k-1_ - k_k_u_k]

![= (I - k(k)H)x(k) - (I - k(k)H)x(k|k-1) - k(k)u(k)][_I - k_k_H_x_k_ - _I - k_k_H_x_k_k-1_ - k_k_u_k]

![= \[I - k(k)H\]\[x(k) - x(k|k-1)\] - k(k)u(k)][_I - k_k_H_x_k_ - x_k_k-1_ - k_k_u_k]................................................(7)

![e(k)e(k)^\{T\} = \[I - k(k)H\]\[x(k) - x(k|k-1)\]\[x(k) - x(k|k-1)\]^\{T\}\[I - k(k)H\]^\{T\}][e_k_e_k_T_ _ _I - k_k_H_x_k_ - x_k_k-1_x_k_ - x_k_k-1_T_I - k_k_H_T]

![\- \[I - k(k)H\]\[x(k) - x(k|k-1)\]u(k)^\{T\}k(k)^\{T\}][- _I - k_k_H_x_k_ - x_k_k-1_u_k_T_k_k_T]

![gif.latex][]![\- k(k)u(k)\[I - k(k)H\]\[x(k) - x(k|k-1)\] + k(k)u(k)u(k)^\{T\}k(k)^\{T\}][- k_k_u_k_I - k_k_H_x_k_ - x_k_k-1_ _ k_k_u_k_u_k_T_k_k_T]

由于![\[x(k) - x(k|k-1)\]][x_k_ - x_k_k-1]与![u(k)][u_k]无关，并且![E\[u(k)\] = 0][E_u_k_ _ 0 1]

![E\[\[x(k) - x(k|k-1)\]\[x(k) - x(k|k-1)\]^\{T\}\] = P(k|k-1)][E_x_k_ - x_k_k-1_x_k_ - x_k_k-1_T_ _ P_k_k-1]  所以![P(k|k)][P_k_k]协方差矩阵如下：

![P(k|k) = E\[e(k)e(k)^\{T\}\]= \[I - k(k)H\]P(k|k-1)\[I - k(k)H\]^\{T\} + k(k)Rk(k)^\{T\}][P_k_k_ _ E_e_k_e_k_T_ _I - k_k_H_P_k_k-1_I - k_k_H_T_ _ k_k_Rk_k_T]...........(8)

综上所示我们得到了公式(8)，我们需对公式(8)求![k(k)][k_k]的导数，使之![P(k|k)][P_k_k]最小。

我们需将公式(8)展开如下所示：

![gif.latex][] ![P(k|k) = P(k|k-1) - (k(k)HP(k|k-1))^\{T\} - k(k)HP(k|k-1) \\\\+ k(k)\[HP(k|k-1)H + R\]k(k)^\{T\}][P_k_k_ _ P_k_k-1_ - _k_k_HP_k_k-1_T_ - k_k_HP_k_k-1_ _ k_k_HP_k_k-1_H _ R_k_k_T]....................(9)

卡尔曼滤波器的本质是线性最小均方误差估计，而均方误差是协方差矩阵的迹，所以我们

只需对![P(k|k)][P_k_k]的迹求导![\\frac\{\\partial tr(P(k|k))\}\{\\partial k(k)\}][frac_partial tr_P_k_k_partial k_k]。由于![tr(A^\{T\}) = tr(A)][tr_A_T_ _ tr_A] ，所以![tr(P(k|k-1))][tr_P_k_k-1]如下：

![tr(P(k|k-1)) = tr(P(k|k-1)) - 2tr\[k(k)HP(k|k-1)\] \\\\+ tr\[k(k)\[HP(k|k-1)H + R\]k(k)^\{T\}\]][tr_P_k_k-1_ _ tr_P_k_k-1_ - 2tr_k_k_HP_k_k-1_ _ tr_k_k_HP_k_k-1_H _ R_k_k_T] .................................(10)

对迹求导我们需要知道以下几个矩阵迹求导公式：

![\\frac\{\\partial tr(AB)\}\{\\partial A\} = B^\{T\}][frac_partial tr_AB_partial A_ _ B_T]             若矩阵B是对称阵则![\\frac\{\\partial tr(ABA^\{T\})\}\{\\partial A\} = 2AB][frac_partial tr_ABA_T_partial A_ _ 2AB]  其中tr表示矩阵的迹。

![\\frac\{\\partial tr(P(k|k))\}\{\\partial k(k)\} = - 2P(k|k-1)^\{T\}H^\{T\} + 2k(k)\[HP(k|k-1)H + R\]][frac_partial tr_P_k_k_partial k_k_ _ - 2P_k_k-1_T_H_T_ _ 2k_k_HP_k_k-1_H _ R] ...............................(11)

令![\\frac\{\\partial tr(P(k|k))\}\{\\partial k(k)\} = 0][frac_partial tr_P_k_k_partial k_k_ _ 0] 得出：    ![k(k) = \\frac\{P(k|k-1)H^\{T\}\}\{HP(k|k-1)H + R\}][k_k_ _ _frac_P_k_k-1_H_T_HP_k_k-1_H _ R]  ..........................................(12)

把公式(12)的表达式替换公式(9)最后一项![k(k)\[HP(k|k-1)H + R\]k(k)^\{T\}][k_k_HP_k_k-1_H _ R_k_k_T]的![k(k)][k_k]得到以下式子：

![P(k|k) = P(k|k-1) - (k(k)HP(k|k-1))^\{T\} - k(k)HP(k|k-1) \\\\+ P(k|k-1)H^\{T\}k(k)^\{T\}][P_k_k_ _ P_k_k-1_ - _k_k_HP_k_k-1_T_ - k_k_HP_k_k-1_ _ P_k_k-1_H_T_k_k_T]

![= P(k|k-1) - (k(k)HP(k|k-1))^\{T\} - k(k)HP(k|k-1) \\\\+ (k(k)HP(k|k-1))^\{T\}][P_k_k-1_ - _k_k_HP_k_k-1_T_ - k_k_HP_k_k-1_ _ _k_k_HP_k_k-1_T]

![= P(k|k-1) - k(k)HP(k|k-1) = (I - k(k)H)P(k|k-1)][P_k_k-1_ - k_k_HP_k_k-1_ _ _I - k_k_H_P_k_k-1]..................................(13)

当![k(k)][k_k 1]的取值等于公式(12)时，最优估计值的协方差矩阵![P(k|k)][P_k_k]方差最小。但该公式中有一个

未知的量![P(k|k-1)][P_k_k-1]预测协方差矩阵，接下去我们将就继续推导![P(k|k-1)][P_k_k-1]的表达式。

定义预测误差![\{e\}' = x(k) - x(k|k-1)][e_ _ x_k_ - x_k_k-1]，预测协方差矩阵![P(k|k-1) = E\[\{e\}'\{e^\}'^\{T\}\]][P_k_k-1_ _ E_e_e_T]

式子![x(k)][x_k]用公式(1)替代，![x(k|k-1)][x_k_k-1] 用公式(3)替代得出：

![\{e\}' = Ax(k-1) + Bu(k) + w(k) - Ax(k-1|k-1) - Bu(k)][e_ _ Ax_k-1_ _ Bu_k_ _ w_k_ - Ax_k-1_k-1_ - Bu_k]

![= Ax(k-1) - Ax(k-1|k-1) + w(k)][Ax_k-1_ - Ax_k-1_k-1_ _ w_k]

![\{e\}'\{e^\}'^\{T\} = \[A(x(k-1) - x(k-1|k-1)) + w(k)\]\[A(x(k-1) - x(k-1|k-1)) + w(k)\]^\{T\}][e_e_T_ _ _A_x_k-1_ - x_k-1_k-1_ _ w_k_A_x_k-1_ - x_k-1_k-1_ _ w_k_T]

![= A(x(k-1) - x(k-1|k-1))(x(k-1) - x(k-1|k-1))^\{T\}A^\{T\}][A_x_k-1_ - x_k-1_k-1_x_k-1_ - x_k-1_k-1_T_A_T]![gif.latex][]

![\+ A(x(k-1) - x(k-1|k-1))w(k)^\{T\} + w(k)(x(k-1) - x(k-1|k-1))^\{T\}A^\{T\}][A_x_k-1_ - x_k-1_k-1_w_k_T_ _ w_k_x_k-1_ - x_k-1_k-1_T_A_T]

![\+ w(k)w(k)^\{T\}][w_k_w_k_T]

由于![w(k)][w_k]与![(x(k-1) - x(k-1|k-1))][x_k-1_ - x_k-1_k-1]无关，且![E\[w(k)\] = 0][E_w_k_ _ 0 1] 所以：

![\{e\}'\{e^\}'^\{T\} = A(x(k-1) - x(k-1|k-1))(x(k-1) - x(k-1|k-1))^\{T\}A^\{T\} + w(k)w(k)^\{T\}][e_e_T_ _ A_x_k-1_ - x_k-1_k-1_x_k-1_ - x_k-1_k-1_T_A_T_ _ w_k_w_k_T]

![P(k|k-1) = E\[\{e\}'\{e^\}'^\{T\}\] = AP(k-1|k-1)A^\{T\} + Q][P_k_k-1_ _ E_e_e_T_ _ AP_k-1_k-1_A_T_ _ Q]......................................................(14)

综上所示卡尔曼滤波的核心公式如下：

![x(k|k-1) = Ax(k-1|k-1) + Bu(k)][x_k_k-1_ _ Ax_k-1_k-1_ _ Bu_k]       预测方程

![x(k|k) = x(k|k-1) + k(k)(z(k) - Hx(k|k-1))][x_k_k_ _ x_k_k-1_ _ k_k_z_k_ - Hx_k_k-1]       最优估计方程

![k(k) = \\frac\{P(k|k-1)H^\{T\}\}\{HP(k|k-1)H + R\}][k_k_ _ _frac_P_k_k-1_H_T_HP_k_k-1_H _ R]              卡尔曼滤波增益

![P(k|k) = (I - k(k)H)P(k|k-1)][P_k_k_ _ _I - k_k_H_P_k_k-1]         最优估计协方差矩阵递推公式

![P(k|k-1) = AP(k-1|k-1)A^\{T\} + Q][P_k_k-1_ _ AP_k-1_k-1_A_T_ _ Q]       预测协方差矩形递推公式

[Link 1]: https://baike.baidu.com/item/%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F/6814705
[Link 2]: https://baike.baidu.com/item/%E7%B3%BB%E7%BB%9F%E7%8A%B6%E6%80%81/896358
[Link 3]: https://baike.baidu.com/item/%E6%BB%A4%E6%B3%A2/2938301
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