50.2.3 Kalman filter initial estimates

Source: Schneider 2013 How to not make the EKF fail

Initial state estimate $\mathbf{x}_0$, $\mathbf{P}_0$

• Filter generally not badly affected by wrong initial state $\mathbf{x}_0$, but convergence will be slow if we are way off

• If $\mathbf{P}_0$ too small whereas $\mathbf{x}_0$ is way off

  • the gain K becomes small
  • filter relies on the model more than on the measurements

Thus: important to have a consistent pair $\mathbf{x}_0$, $\mathbf{P}_0$

Possible to use $$ \mathbf{P}_0 = \text{diag} \left( \left( \mathbf{\hat{x}}_0 - \mathbf{x}_0 \right)^\text{T} \left( \mathbf{\hat{x}}_0 - \mathbf{x}_0 \right) \right) $$ using $$ \mathbf{\hat{x}}_0 - \mathbf{x}_0 = 0.5 \left( \mathbf{x}_u - \mathbf{x}_l \right) $$