Initialization Properties of the KF Algorithm

Where:

But a really nice property of the KF algorithm allows us to do so:

So we can just choose $P_0$ large enough such that $x_0$ belongs to the $3\sigma$ confidence interval, then the algorithm will adjust both $\hat{x}(t)$ and $P(t)$ every times it iterate.

But even if we choose $P_0$ too small, such that $x_0$ is not inside the $3\sigma$ confidence interval, the KF will still converge to the “optimal” solution MSE if the state of the problem is not Gaussian, LSME if it is Gaussian, tho is better to choose $P_0$ bigger, the KF will converge sooner.

NOTE: This is not the case in non-linear systems, where if $P_0$ is too small or too large the algorithm might not converge, the linearization might fail.