Kalman滤波算法的原理可以参考： 卡尔曼滤波理解

python中filterpy库中实现了各种滤波算法， 其中就包括了kalman滤波算法。
具体实现代码： https://github.com/rlabbe/filterpy/blob/master/filterpy/kalman/kalman_filter.py

本文针对该代码进行详细解读分析。

1 初始化及参数设置

需要设定的几个参数可参考： 详解多目标跟踪(MOT)算法中的Kalman滤波

• 状态变量的维度： dim_x
• 观测变量的维度：dim_z
• 状态变量：x， 初始化为大小为dim_x， 全0的列向量
• 观测变量：z， 初始化为大小为dim_z， 全0的列向量
• 状态转移矩阵： F, 初始化为维度为dim_x， 值为1的对角阵
• 状态变量协方差矩阵： P, 初始化为维度为dim_x， 值为1的对角阵
• 处理噪声写房产矩阵： Q, 初始化为维度为dim_x， 值为1的对角阵
• 测量矩阵：H,初始化为维度为（dim_z, dim_x）的全0矩阵
• 测量误差协方差矩阵： R, 初始化为维度为dim_z， 值为1的对角阵

 def __init__(self, dim_x, dim_z, dim_u=0):
        if dim_x < 1:
            raise ValueError('dim_x must be 1 or greater')
        if dim_z < 1:
            raise ValueError('dim_z must be 1 or greater')
        if dim_u < 0:
            raise ValueError('dim_u must be 0 or greater')

        self.dim_x = dim_x
        self.dim_z = dim_z
        self.dim_u = dim_u

        self.x = zeros((dim_x, 1))        # state
        self.P = eye(dim_x)               # uncertainty covariance
        self.Q = eye(dim_x)               # process uncertainty
        self.B = None                     # control transition matrix
        self.F = eye(dim_x)               # state transition matrix
        self.H = zeros((dim_z, dim_x))    # measurement function
        self.R = eye(dim_z)               # measurement uncertainty
        self._alpha_sq = 1.               # fading memory control
        self.M = np.zeros((dim_x, dim_z)) # process-measurement cross correlation
        self.z = np.array([[None]*self.dim_z]).T

2 predict

def predict(self, u=None, B=None, F=None, Q=None):
        """
        Predict next state (prior) using the Kalman filter state propagation
        equations.

        Parameters
        ----------

        u : np.array, default 0
            Optional control vector.

        B : np.array(dim_x, dim_u), or None
            Optional control transition matrix; a value of None
            will cause the filter to use `self.B`.

        F : np.array(dim_x, dim_x), or None
            Optional state transition matrix; a value of None
            will cause the filter to use `self.F`.

        Q : np.array(dim_x, dim_x), scalar, or None
            Optional process noise matrix; a value of None will cause the
            filter to use `self.Q`.
        """

        if B is None:
            B = self.B
        if F is None:
            F = self.F
        if Q is None:
            Q = self.Q
        elif isscalar(Q):
            Q = eye(self.dim_x) * Q


        # x = Fx + Bu
        if B is not None and u is not None:
            self.x = dot(F, self.x) + dot(B, u)
        else:
            self.x = dot(F, self.x)

        # P = FPF' + Q
        self.P = self._alpha_sq * dot(dot(F, self.P), F.T) + Q

        # save prior
        self.x_prior = self.x.copy()
        self.P_prior = self.P.copy()

predict的过程比较简单， 就是根据上一次的状态变量， 估计当前的状态变量值， 同时更新状态变量的协方差。

3 update

  def update(self, z, R=None, H=None):
        """
        Add a new measurement (z) to the Kalman filter.

        If z is None, nothing is computed. However, x_post and P_post are
        updated with the prior (x_prior, P_prior), and self.z is set to None.

        Parameters
        ----------
        z : (dim_z, 1): array_like
            measurement for this update. z can be a scalar if dim_z is 1,
            otherwise it must be convertible to a column vector.

            If you pass in a value of H, z must be a column vector the
            of the correct size.

        R : np.array, scalar, or None
            Optionally provide R to override the measurement noise for this
            one call, otherwise  self.R will be used.

        H : np.array, or None
            Optionally provide H to override the measurement function for this
            one call, otherwise self.H will be used.
        """

        # set to None to force recompute
        self._log_likelihood = None
        self._likelihood = None
        self._mahalanobis = None

        if z is None:
            self.z = np.array([[None]*self.dim_z]).T
            self.x_post = self.x.copy()
            self.P_post = self.P.copy()
            self.y = zeros((self.dim_z, 1))
            return

        if R is None:
            R = self.R
        elif isscalar(R):
            R = eye(self.dim_z) * R

        if H is None:
            z = reshape_z(z, self.dim_z, self.x.ndim)
            H = self.H

        # y = z - Hx
        # error (residual) between measurement and prediction
        self.y = z - dot(H, self.x)

        # common subexpression for speed
        PHT = dot(self.P, H.T)

        # S = HPH' + R
        # project system uncertainty into measurement space
        self.S = dot(H, PHT) + R
        self.SI = self.inv(self.S)
        # K = PH'inv(S)
        # map system uncertainty into kalman gain
        self.K = dot(PHT, self.SI)

        # x = x + Ky
        # predict new x with residual scaled by the kalman gain
        self.x = self.x + dot(self.K, self.y)

        # P = (I-KH)P(I-KH)' + KRK'
        # This is more numerically stable
        # and works for non-optimal K vs the equation
        # P = (I-KH)P usually seen in the literature.

        I_KH = self._I - dot(self.K, H)
        self.P = dot(dot(I_KH, self.P), I_KH.T) + dot(dot(self.K, R), self.K.T)

        # save measurement and posterior state
        self.z = deepcopy(z)
        self.x_post = self.x.copy()
        self.P_post = self.P.copy()

update的过程需要提供观测变量z。

1 首先计算卡尔曼增益。
2 计算当前观测值与通过观测矩阵得到的值之间的误差， 这个误差值乘上卡尔曼增益， 再加上predict过程得到的先验状态估计结果， 得到当前的卡尔曼滤波估计结果。
3 更新状态变量协方差矩阵。