In model predictive control for a system
$$ \begin{cases} x_m(k+1)=A_m x_m(k)+B_m u(k)+\xi_k\\ y_m (k)=C_m x_m(k)+\eta(k) \end{cases} $$
where
$u$ is the manipulated variable (dimension
$y$ is the process output
$x$ is the state variable
$\xi$ is the disturbance
$\eta$ is the measurement noise.
According to this (page 4) and many other references , an augmented system is defined to relate the output to the input rates:
$$ \begin{cases} \Delta x_m(k)= x_m(k)-x_m(k-1)\\ \bar y(k)=C_m x_m(k) \\ x(k)=\begin{bmatrix}\Delta x_m(k) & \bar y(k)\end{bmatrix} ^T\\ \Delta \xi(k)= \xi(k)-\xi(k-1) =\text{zero mean white noise}\\ \end{cases} $$
Then a new input/output relationship with different matrices A, B and C are formed as follows
$$ \begin{cases} x(k+1)= Ax(k)+B\Delta u(k)+\bar \xi(k)\\ y(k)=Cx(k)+\eta(k) \end{cases} $$
$$ A=\begin{bmatrix} A_m & 0 \\ C_m A_m & I \end{bmatrix}\\ B=\begin{bmatrix} B_m \\ C_m B_m \end{bmatrix}\\ C=\begin{bmatrix} 0 & I \end{bmatrix}\\ $$
Now the relationship is based on $\Delta u(k)$ as an input.
Let's forget all noise and disturbances. I am wondering why we use $\Delta x_m$?
We could choose a different design. We could assume that there is an integrator at the beginning of the plant and then use $\Delta u(k)$ as an input, integrate it and feed it to the plant. So the augmented state would become: $$ x(k)=\begin{bmatrix}x_m(k) & u(k)\end{bmatrix} ^T\\ $$ where $u(k)$ is injected to the state.
The relationship would be:
$$ \begin{cases} x(k+1)=A x(k)+B \Delta u(k)+\xi_k\\ y_m (k)=C x_m(k)+\eta(k) \end{cases} $$
where
$$ A=\begin{bmatrix} A_m & B_m \\ 0 & I \end{bmatrix}\\ B=\begin{bmatrix} 0 \\ I \end{bmatrix}\\ C=\begin{bmatrix} C_m & 0 \end{bmatrix}\\ $$
Isn't this model simpler, more readable and with more meaningful states? In addition, in the new design, if one penalizes the actuation amplitude, the cost function does not require an additional weighting matrix to penalize $u$. The matrix $Q$ can perform that.