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# Linear Estimation

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

### 1.3.2 Algorithm

#### Innovation

Innovation defines the new information of the measurement. It is the difference between the actual measurement $Y_{k}$ and the predicted measurement $H\hat{X}_{k|k-1}$. It is denoted by $Z_{k}$ and expressed as, $$Z_{k}=Y_{k}-H_{k}\hat{X}_{k|k-1}.$$

#### Aposteriori estimation (Measurement update)

The algorithm starts with some initial guess. Aposteriori estimate $\hat{X}_{k|k}$, at step $k$: $$\hat{X}_{k|k}=\hat{X}_{k|k-1}+K_{k}Z_{k},$$ where expression of Kalman gain matrix (derived earlier) is: $$K_{k}=P_{k|k-1}H_{k}^{T}[H_{k}P_{k|k-1}H_{k}^{T}+R_{k}]^{-1}.$$ Posterior error covariance matrix $P_{k|k}$, could be evaluated as, $$P_{k|k}=[I-K_{k}H_{k}]P_{k|k-1}.$$

#### Prediction (Time update)

Prior estimation of the states could be obtained with the help of process equation: $$\hat{X}_{k+1|k}=\phi_{k}\hat{X}_{k|k}.$$ Prior error covariance matrix $P_{k+1|k}$ is calculated by: $$P_{k+1|k}=\phi_{k}P_{k|k}\phi_{k}^{T}+Q_{k}.$$ From the above discussion, we see that the Kalman filter algorithm described here is recursive in nature. The figure on the top describes it.