How to represent the nonlinear model as a state space in Unscented Kalman Filter There is an Autoregressive model of order 1 (AR(1)) that is excited by a non-linear signal as the input:
$$x_t = \rho x_{t-1} + u_t \tag{1}$$
The time series $u_t$ is generated from a nonlinear map,
$$u_t = f(u_{t-1},\mathbf{w}) \tag{2}$$ where $f$ is the nonlinear function.
The observations are $$y_t = x_t + v_t \tag{3}$$ where $v_t$ is the measurement noise that is an Additive White Gaussian Noise.

Q1: Can I re-write the model (1) as a state space in the following way:

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

$$y_t =  x_t+ v_t \tag{4}$$

Is the above representation correct? If not then I shall be grateful for the correct technique to represent it.
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.