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

*

*$A\in \mathbb{R}^{m\times m}$ - state dynamics

*$B\in \mathbb{R}^{m\times n}$ - control dynamics

*$C\in \mathbb{R}^{n\times m}$ - 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
$$\begin{array}{l}
y^{\prime\prime} = (A-k_pBC)y^\prime -k_iBCy + k_i B\bar{w}
\end{array}$$
This can be written as the first order system
$$
\begin{bmatrix}
y\\z
\end{bmatrix}^\prime =
\begin{bmatrix}
0 & I\\
-k_iBC & A-k_pBC
\end{bmatrix}
\begin{bmatrix}
y\\
z
\end{bmatrix}
+
\begin{bmatrix}
0\\
k_i B\bar{w}
\end{bmatrix}
$$
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.
 A: Hint.
As a LTI is easily Laplace transformable, the problem can be stated as
$$
\cases{
\left(sI-A\right)Y= B U\\
U = \left(k_p I+\frac{k_i I}{s}\right)E\\
E = W_r - W
}
$$
and putting all together
$$
E = W_r - C Y = W_r - \frac 1s C\left(sI-A\right)^{-1}B\left(s k_p I+ k_i I\right)E
$$
and calling
$$
G = I+\frac 1s C\left(sI-A\right)^{-1}B\left(s k_p I+ k_i I\right)
$$
we have
$$
E = G^{-1}W_r
$$
and the error dynamics depend on the zeros from $\det(G)$
as an example, considering
$$
A = \left(
\begin{array}{cc}
 1 & 2 \\
 -3 & 4 \\
\end{array}
\right),\ \ B = \left(
\begin{array}{c}
 1 \\
 0 \\
\end{array}
\right), \ \ C = \left(
\begin{array}{cc}
 1 & 1 \\
\end{array}
\right)
$$
we have
$$
\det(G) = \frac{2 k_i s-14 k_i+2 k_p s^2-14 k_p s+s^3-5 s^2+10 s}{s \left(s^2-5 s+10\right)}
$$
and the finite zeros are given by
$$
(2s-14) k_i+(2s^2-14s) k_p+s^3-5 s^2+10 s = 0
$$
Those zeros now are continuously dependent on $k_p,k_i$ and should be located inside the left complex plane to attain stability. This can be handled using the Routh-Hurwitz criterion. Additionally, to have asymptotic null error we should verify also that once stable, the error dynamics should obey
$$
\lim_{s\to 0}s E =0
$$
