How to represent the nonlinear model as a state space in Unscented Kalman Filter
Rather than write

$$
x_t = Ax_{t-1} + f(u_{t-1},\mathbf{w})
$$

as the state update equation, I'd write:

$$
\xi = \left[ \begin{array}{c}
x_t\\
u_t 
\end{array}
\right] 
$$

and then
$$
\xi_t = g(\xi_{t-1},A,\mathbf{w})
$$

so that
$$
y_t = \left[ 1\ \ \ 0 \right] \xi_t + v_t
$$

Then you could apply the non-additive noise formulation of the EKF to get your filter equations.