Kalman prediction step ($m_t^{-}=A_t{m}_{t-1}+B_t{u}_{t-1},P_t^{-}=A_t{P}_{t-1}{A}_t^{\top }+Q_t$)

The Kalman prediction step advances the previous posterior mean and posterior covariance matrix through the state transition:The minus superscript marks information before observing $y_t$. The transpose and matrix multiplication propagate the old uncertainty; $Q_t$ adds the uncertainty of the new process noise. A known control input changes the mean but contributes no additional variance. If the control input itself is uncertain, that uncertainty needs its own model.