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# Linear Estimation

## 1. Kalman Filter Tutorial (Cont'd...)

### 1.3 Kalman Filter

#### 1.3.1 Derivation of Kalman filter

We define a priori estimate error $e_{k|k-1}$ is $$\label{3d} e_{k|k-1}=\hat{X}_{k|k-1}-X_{k},$$ and a posteriori estimate error $e_{k|k}$ is $$\label{3e} e_{k|k}=\hat{X}_{k|k}-X_{k}.$$ Apriori estimate error covariance $P_{k|k-1}$ is defined as $$\label{3f} P_{k|k-1}=E[e_{k|k-1}e_{k|k-1}^{T}],$$ and aposteriori estimate error covariance $P_{k|k}$ is defined as $$\label{3g} P_{k|k}=E[e_{k|k}e_{k|k}^{T}].$$ Now, substituting error expression, apriori error covariance becomes $$\label{3h} P_{k|k-1}=E[(\hat{X}_{k|k-1}-X_{k})(\hat{X}_{k|k-1}-X_{k})^{T}].$$ Now, the expression of the state estimator could be written as: $$\label{3i} \hat{X}_{k|k}=\hat{X}_{k|k-1}+K_{k}(Y_{k}-H_{k}\hat{X}_{k|k-1}),$$ where, $K_{k}$ is the blending factor, which is called as Kalman gain matrix.

Now, to design Kalman filter we have to find $K_{k}$. From the above expression we could write, $$\label{3j} P_{k|k}=E[(\hat{X}_{k|k}-X_{k})(\hat{X}_{k|k}-X_{k})^{T}].$$