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.
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
$$ \lim_{s\to 0}s E =0 $$