According to lots of books about predictive control, they recommend to having internal integration inside the model.
For example if we have a state space model:
$$x(k+1) = Ax(k) + Bu(k) \\ y(k) = Cx(k)$$
Then integration will be:
$$x(k+1) = Ax(k) + Bu(k)$$ $$y(k+1) = CAx(k) + y(k) + CBu(k)$$
Where $y(k)$ is the output.
When I try to simulate a first order system $Gs) = \frac{1}{3+s}$ with sample time $h = 0.1$, I got this step answer:
And adding some integration:
As you can see, a pole is placed at $1$. So what's the purpose of having internal integration? Removal of the steady state error?