卡尔曼滤波Kalman filter原理

应用：运动，信号系统

本质：自回归（auto regression），线性算子（Linear operator）, 离散， 随机系统

该系统由如下方程表示（Controlled by a Linear stochastic difference equation）:

(大写为矩阵，小写为一维变量）

真实过程为线性：

 \(w_{k-1}" title="x_k = Ax_{k-1} + B u_{k-1} + w_{k-1} \)

 

discrete time, linear, dynamic, state space, vector difference equation

State: smallest vector to summarize the past of the system.

Prediction: in the absence of the noise.

State equation:

\(  x(k+1) = F(k) x(k) +G(k)u(k) + v(k)

 \)

\( x(k) \) is the \(n_x\) dimensional state vector

\( v(k) \) is the white noise with covariance \( Q(k)\)

Measurement equation:

\( z(k) = x(k) + w(k)

\)

\( w(k) \) is the white noise with covariance \( R(k)\)

 

 

 

 测量系统也为线性：

$z_k = H x_k +v_k.$

$w_{k-1}$与$v_k$为噪音。