We have a linear system with observation as follows:
$x(t+1)=Ax(t)+Bu(t)+w(t)$
$y(t)=Cx(t)+z(t)$
for given information $y(0\sim t)$, we can construct the dynamics of $\hat{x}(t)$ using the Kálmán filter equations.
What if we are given information $y(0\sim t-N)$ where $N$ is a constant?
It might be challenging to think about this in the continuous time case because your update equations will probably be a time-delay ODE. If you permit yourself to think about this in discrete time, then one possibility is to augment the state to include $x[k], x[k-1], \cdots, x[k-t], u[k-1], u[k-2], \cdots, u[k-T]$. Depending on how you organize this new super state, the dynamics matrix ($A$ )is a block matrix with mostly off-diagonal blocks corresponding to your original $A$ and $B$. The observation matrix is such that you only get information about $x[k-t]$.
With that, you reduced the original time-delay problem into a regular discrete-time LTI filtering problem.
