State-Space Representation of Forward and Backward Filters
The backwards filter, as the name suggests filters the input signal with the same filter $(F,G,H,D)$ backwards in time

$$
\begin{align}
x(t-1) &amp;= F\,x(t) + G\,u(t) \\
y(t) &amp;= H\,x(t) + D\,u(t)
\end{align} \tag{1}
$$

But in order to define a filter using a state space model, the update law does have to go forwards in time, which can be done by solving the first equation for $x(t)$

$$
x(t) = F^{-1}\,\left(x(t-1) - G\,u(t)\right) \tag{2}
$$

however $x(t)$ can only a function of $x(t-1)$ and $u(t-1)$. This can be solved by using a new definition for the state $z(t) = x(t) + F^{-1}\,G\,u(t)$, so

$$
z(t) = x(t) + F^{-1}\,G\,u(t) = F^{-1}\,x(t-1) = F^{-1}\,\left(z(t-1) - F^{-1}\,G\,u(t-1)\right) \tag{3}
$$

which can be rewritten in the following more standard form

$$
z(t+1) = F^{-1}\,z(t) - F^{-2}\,G\,u(t). \tag{4}
$$

Substituting the definition of $z(t)$ in to the expression for $y(t)$ gives

$$
y(t) = H\,x(t) + D\,u(t) = H\,\left(z(t) - F^{-1}\,G\,u(t) \right) + D\,u(t) \tag{5}
$$

which can be simplified down to

$$
y(t) = H\,z(t) + \left(D - H\,F^{-1}\,G\right)\,u(t). \tag{6}
$$

Combining equation $(4)$ and $(6)$ gives the following (forwards) state space model, which is equivalent to the (backwards) state space model in equation $(1)$

$$
\begin{align}
z(t+1) &amp;= F^{-1}\,z(t) - F^{-2}\,G\,u(t) \\
y(t) &amp;= H\,z(t) + \left(D - H\,F^{-1}\,G\right)\,u(t)
\end{align} \tag{7}
$$

So the backwards filter based on the forwards filter $(F,G,H,D)$ can be written as $(F^{-1},-F^{-2}\,G,H,D - H\,F^{-1}\,G)$.