Im wondering about the difference of an I-state-space controller and an (simple) PI-Controller?
As I know, you get the advantage to place any dynamic with the state-space-control (if the manipulating variable is without limit).. but what else can be pointed to use an I-state-space controller?
Take the second order system
$$ G(s) = \frac{1}{s^2} \tag{1} $$
and a PI controller
$$ C(s) = k_p + \frac{k_i}{s} $$
The closed loop transfer function is
$$ G_{cl}(s) = \frac{k_p s + k_i}{s^3 + k_ps + k_i} $$
Can this system be stable? We know that a necessary condition is that the coefficients of the denominator polynomial of $G_{cl}(s)$ are positive. However, the $s^2$ coefficient is zero.
Conclusion: The second order system $(1)$ cannot be stabilized with PI control.
Now lets look at a state feedback controller with integral action. The transfer function $(1)$ in state space is
$$ \begin{align} \dot{x} &= A x + Bu \\ y &= C x \end{align} $$
with
$$ A = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, B = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, C = \begin{bmatrix} 0 & 1 \end{bmatrix}, x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$
The integral controller is
$$ \begin{align} \dot{x}_i &= e = w - y \\ u_i &= k_i x_i \end{align} $$
You can put that together:
$$ \begin{align} \dot{\xi} &= \begin{bmatrix} A & 0 \\ -C & 0 \end{bmatrix} \xi + \begin{bmatrix} B \\ 0 \end{bmatrix} u + \begin{bmatrix} 0 \\ 1 \end{bmatrix} w \\ y &= \begin{bmatrix} C & 0 \end{bmatrix} \xi \\ \xi &= \begin{bmatrix} x_1 \\ x_2 \\ x_i \end{bmatrix} \end{align} $$
and design a control law with $u_x = \begin{bmatrix} k_1 & k_2\end{bmatrix} x$ and $u_i = k_i x_i$ so
$$ u = -u_x - u_i = -K_a \xi = -\begin{bmatrix} k_1 & k_2 & k_i \end{bmatrix}\xi $$
This will give you a closed loop transfer function:
$$ G_{cl} = \frac{-k_i}{s^3 + k_1 s^2 + k_2 s - k_i} $$
This will be stable as long as you take $k_1, k_2 > 0$, $k_i < 0$ and $k_1 k_2 + k_i > 0$.
Conclusion: The second order system $(1)$ can be stabilized with state feedback + integral action.
Example: Take $k_1 = 6$, $k_2 = 11$, $k_i = -6$ for the closed loop transfer function
$$ G_{cl} = \frac{6}{(s + 1)(s + 2)(s + 3)} $$
and to place the closed loop eigenvalues at $-1, -2, -3$. This would be impossible using PI control.
With full state feedback, not only can we stabilize the system, we can also put its eigenvalues wherever we want.