I have a SIMO system. $\xi$ is the input and $Y$ is the output. The state space model is given by
\begin{align}
\dot{X} &= AX + B\xi\\
y&=r-Cx
\end{align}
$A$ is $5 \times 5$ matrix. $B$ is $5 \times 1$. The controller $K$ is a state feedback controller such that $A – BK$ is Hurwitz. The schematic overview is shown in the figure. How can I find out the open loop or closed loop bandwidth for this system?
Edit: By bandwidth I mean the frequency at which the open loop transfer function crosses the $0$dB line.
Edit2: Maybe the question wasn't very clear. After I design the feedback control law, my system is
\begin{equation}
\dot{X} = (A – BK)X
\end{equation}
Now the transfer function is essentially from reference $r$ to a chosen output $y$. But given $A-BK$, how do I find the transfer function? Note that the feedback is a state feedback and hence is very difficult to write as a transfer function.
Best Answer
Given a state space model of the following form,
$$ \dot{x} = A\,x + B\,u, \tag{1} $$
$$ y = C\,x + D\,u. \tag{2} $$
The openloop transfer function of this system can be found by taking the Laplace transform and assuming all initial conditions to be zero (such that $\mathcal{L}\{\dot{x}(t)\}$ can just be written as $s\,X(s)$). Doing this for equation $(1)$ yields,
$$ s\,X(s) = A\,X(s) + B\,U(s), \tag{3} $$
which can be rewritten as,
$$ X(s) = (s\,I - A)^{-1} B\,U(s). \tag{4} $$
Substituting this into equation $(2)$ and defining the openloop transfer function $G(s)$ as the ratio between output ($Y(s)$) and input ($U(s)$) yields,
$$ G(s) = C\,(s\,I - A)^{-1} B + D. \tag{5} $$
In a normal block diagram representation the controller has as an input $r-y$, with $r$ the reference value you would like to have for $y$, and an output $u$, which would be the input to $G(s)$. For now $r$ can be set to zero, so the controller can be defined as the transfer function from $-y$ to $u$.
For an observer based controller ($L$ and $K$ such that $A-B\,K$ and $A-L\,C$ are Hurwitz) for a state space model we can write the following dynamics,
$$ u = -K\,\hat{x}, \tag{6} $$
$$ \dot{x} = A\,x - B\,K\,\hat{x}, \tag{7} $$
$$ \dot{\hat{x}} = A\,\hat{x} + B\,u + L(y - C\,\hat{x} - D\,u) = (A - B\,K - L\,C + L\,D\,K) \hat{x} + L\,y. \tag{8} $$
Similar to equations $(1)$, $(2)$ and $(5)$, the transfer function of the controller $C(s)$, defined as the ratio of $U(s)$ and $-Y(s)$, can be found to be,
$$ C(s) = K\,(s\,I - A + B\,K + L\,C - L\,D\,K)^{-1} L. \tag{9} $$
If you want to find the total openloop transfer function from "$-y$" to "$y$" you have to keep in mind that in general $G(s)$ and $C(s)$ are matrices of transfer functions, so the order of multiplication matters. Namely you first multiply the error ($r-y$) with the controller and then the plant, the openloop transfer function can be written as $G(s)\,C(s)$. The closedloop transfer function can then be found with,
$$ \frac{Y(s)}{R(s)} = (I + G(s)\,C(s))^{-1} G(s)\,C(s). \tag{10} $$
It can also be found directly using equations $(2)$ and $(6)$, and the closedloop state space model dynamics,
$$ \begin{bmatrix} \dot{x} \\ \dot{\hat{x}} \end{bmatrix} = \begin{bmatrix} A & -B\,K \\ L\,C & A - B\,K - L\,C \end{bmatrix} \begin{bmatrix} x \\ \hat{x} \end{bmatrix} + \begin{bmatrix} 0 \\ -L \end{bmatrix} r, \tag{11} $$
$$ \frac{Y(s)}{R(s)} = \begin{bmatrix} C & -D\,K \end{bmatrix} \begin{bmatrix} s\,I - A & B\,K \\ -L\,C & s\,I - A + B\,K + L\,C \end{bmatrix}^{-1} \begin{bmatrix} 0 \\ -L \end{bmatrix}. \tag{12} $$