Integral action in prediction — finding predicted inputs from augmented model

Assume that we have a state space model:

$$x(k+1) = A_m x(k) + B_m u(k)$$

And we implement integral action to the model by creating the agumented state space model:

$$\begin{bmatrix} x(k+1)\\ y(k+1) \end{bmatrix} = \begin{bmatrix} A_m & 0 \\ C_mA_m & 1 \end{bmatrix}\begin{bmatrix} x(k)\\ y(k) \end{bmatrix} + \begin{bmatrix} B_m\\ C_mB_m \end{bmatrix} \begin{bmatrix} u(k) \end{bmatrix} \\ \begin{bmatrix} y(k) \end{bmatrix} = \begin{bmatrix} 0 &1 \end{bmatrix}\begin{bmatrix} x(k)\\ y(k) \end{bmatrix}$$

This will put a pole at $$1$$ on right half plane and the output $$y(k)$$ will increase over time if $$u(k) = c \forall k$$, where $$c$$ is a constant value.

Question:

I have a weak memory that this is the way to include integral action with state feedback for prediction. But I'm am very unsure. Note I'm not talking about Linear Quadratic Integral control.

Can this agumented linear model be used to predict its future values $$u(k)$$ and then the first value $$u(0)$$ can be used in the none-augmented state space model above for tracking $$y(k)$$ after reference $$r(k)$$ in a closed loop system?