I'm reading a book labeled "Predictive Control System Design and Implementation Using MATLAB®, by. Liuping Wang." It's an easy book for basic understanding in linear Model Predictive Control.
Anyway! According to the book, the book is learning out Predictive Control with integration included in a augmented state space model:
$$ \begin{bmatrix} \Delta x_m(k+1) \\ y(k+1) \end{bmatrix} = \begin{bmatrix} A_m & 0_m^T\\ C_mA_m & 1 \end{bmatrix}\begin{bmatrix} \Delta x_m(k) \\ y(k) \end{bmatrix} + \begin{bmatrix} B_m\\ C_m B_m \end{bmatrix} \Delta u(k) $$
$$y(k) = \begin{bmatrix} 0_m & 1 \end{bmatrix}\begin{bmatrix} \Delta x_m(k) \\ y(k) \end{bmatrix} $$
Where $$\Delta x_m(k+1) = x_m(k+1) - x_m(k)$$ $$\Delta u(k) = u(k) - u(k-1)$$ $$\Delta x_m(k) = x_m(k) - x_m(k-1)$$
Question:
How can this be integration? Why cant the augmented model be:
$$ \begin{bmatrix} x_m(k+1) \\ y(k+1) \end{bmatrix} = \begin{bmatrix} A_m & 0_m^T\\ C_mA_m & 1 \end{bmatrix}\begin{bmatrix} x_m(k) \\ y(k) \end{bmatrix} + \begin{bmatrix} B_m\\ C_m B_m \end{bmatrix} u(k) $$
$$y(k) = \begin{bmatrix} 0_m & 1 \end{bmatrix}\begin{bmatrix} x_m(k) \\ y(k) \end{bmatrix} $$
Why is $\Delta$ important?
You start of with the dynamics
$$ \left\{ \begin{align} x(k+1) &= A\,x(k) + B\,u(k) \\ y(k) &= C\,x(k) \end{align}\right. \tag{1} $$
so
$$ y(k+1) = C\,x(k+1) = C\,A\,x(k) + C\,B\,u(k) \tag{2} $$
but your augmented model would give
$$ y(k+1) = C\,A\,x(k) + C\,B\,u(k) + \color{red}{y(k)} \tag{3} $$
which is incorrect (at least inconsistent with the dynamics you started with).
By instead using $\Delta x(k) = x(k) - x(k-1)$ and $\Delta u(k) = u(k) - u(k-1)$, so $x(k) = \Delta x(k) + x(k-1)$ and $u(k) = \Delta u(k) + u(k-1)$, then equation $(2)$ can also be written as
$$ y(k+1) = C\,A\,\left(\Delta x(k) + x(k-1)\right) + C\,B\,\left(\Delta u(k) + u(k-1)\right). \tag{4} $$
When using the definition of $y(k)$ and $x(k)$ then the terms of equation $(4)$ containing $x(k-1)$ and $u(k-1)$ can be rewritten as
$$ C\,A\,x(k-1) + C\,B\,u(k-1) = C\,x(k) = y(k) \tag{5} $$
therefore
$$ y(k+1) = y(k) + C\,A\,\Delta x(k) + C\,B\,\Delta u(k). \tag{6} $$
One could of course also augment the model by using equation $(2)$, which gives
$$ \begin{bmatrix} x(k+1) \\ y(k+1) \end{bmatrix} = \begin{bmatrix} A & 0 \\ C\,A & 0 \end{bmatrix} \begin{bmatrix} x(k) \\ y(k) \end{bmatrix} + \begin{bmatrix} B \\ C\,B \end{bmatrix} u(k) \tag{7} $$
but in this case there is not really any direct integration in $y$, since normally for integration you have something of the following form $y(k+1) = y(k) + \Delta y(k)$, for example in forward Euler you have $\Delta y(k) = \Delta t\,\dot{y}(k)$. This is not the case in equation $(7)$, but is in equation $(6)$.