Kalman Gain

Designed to minimize the mean-square error between $x-\hat{x}$ $L(t)=P(t)C^T{R}_v^{-1}$

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We want to minimize the trace of $\dot{P}$.

$$\begin{array}{rl}\frac{\partial tr(\dot{P})}{\partial L}& =0\\ \frac{\partial tr(\dot{P})}{\partial L}& =-2P{C}^T+2L{R}_v=0\end{array}$$

Which gives us

$$L(t)=P(t)C^T{R}_v^{-1}$$

This is known as the Kalman Gain.

We are certain that this is a minimum since

$$\frac{{\partial }^{2}tr(\dot{P})}{\partial L^2}=2R_v\succ 0$$

$R_v$ is a Positive-Definite Matrix