What are the conditions on a linear time invariant system for a PI controller to converge to a specified set point?

What are the conditions on a linear time invariant system for a PI controller to converge to a specified set point? Specifically, given the system:

$$begin{array}{l}
y^prime = Ay + Bu \
w = Cy
end{array}$$

where

• $Ain mathbb{R}^{mtimes m}$ – state dynamics
• $Bin mathbb{R}^{mtimes n}$ – control dynamics
• $Cin mathbb{R}^{ntimes m}$ – observation dynamics
• $y in mathbb{R}^m$ – state
• $u in mathbb{R}^n$ – control
• $w in mathbb{R}^n$ – observations

I’d like to drive the system $y$ to an observable point $bar{w}$ using a PI controller, but don’t understand the conditions necessary on $A$, $B$, and $C$ to make this possible. From what I can tell, we should have:

$$
u = k_p (bar{w}-w) + k_i int_0^t (bar{w}-w)
$$

which gives

$$begin{array}{l}
y^prime = Ay + B(k_p (bar{w}-w) + k_i int_0^t (bar{w}-w)) \
w = Cy
end{array}$$

or

$$begin{array}{l}
y^prime = Ay + B(k_p (bar{w}-Cy) + k_i int_0^t (bar{w}-Cy))
end{array}$$

which, after taking the derivative and regrouping terms gives

$$begin{array}{l}
y^{primeprime} = (A-k_pBC)y^prime -k_iBCy + k_i Bbar{w}
end{array}$$

This can be written as the first order system

$$
begin{bmatrix}
y\z
end{bmatrix}^prime =
begin{bmatrix}
0 & I\
-k_iBC & A-k_pBC
end{bmatrix}
begin{bmatrix}
y\
z
end{bmatrix}
+
begin{bmatrix}
0\
k_i Bbar{w}
end{bmatrix}
$$

It seems like everything works fine if $BC$ is invertible, but I’d like to know if there’s a better condition. If $n < m$, then $BC$ is less than full rank and it seems like this means it’s not always possible to find a control, but I’m not sure what the theory states or the words to search for in order to better determine this.