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.