# Bode plot of the open loop given the state space - SIMO

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 $$\dot{X} = (A - BK)X$$

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.

• How is $Y$ calculated? Normally in state space as follows, $Y=C\,X+D\,\xi$. Normally for a state space model with an observer based controller you first estimate the full state with $\dot{\hat{X}}=A\hat{X}+B\xi+L(Y-D\xi-C\hat{X})$ and $\xi=-K\hat{X}$, where $L$ and $K$ can be designed by you, such that $A-BK$ and $A-LC$ are Hurwitz matrices. Also do you know if your state space model is minimal (controllable and observable)? Nov 24 '16 at 13:37
• Sorry I have reformulated the question because it was formulated badly...So I have already designed the feedback controller that makes the system Hurwitz. Point is I was to find out the open-loop and closed-loop bandwidth of the system.
– Zero
Nov 24 '16 at 15:25
• Well, the bandwidth gives an estimate of the frequency range for which the control input is effective. Since you have a SIMO system you can use instead of $|G(j\omega_B)|=1$ (which you have used for the SISO system) the condition $\sigma_{\min}(G(j\omega_B))=1$ where $\sigma_{\min}$ denotes the smallest singular value. In fact, in this case, this is equivalent to $\|G(j\omega_B)\|^2=G^H(j\omega_B)G(j\omega)=1$ i.e. $B^T(-j\omega_B-A)^{-T}(j\omega_B-A)^{-1}B=1$ for the open-loop system and similarly for the closed-loop system (replace $A-BK$ instead of $A$).
– RTJ
Nov 27 '16 at 12:11

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,

$$\begin{bmatrix} \dot{x} \\ \dot{\hat{x}} \end{bmatrix} = \begin{bmatrix} A & -B\,K \\ L\,C & A - B\,K - L\,C \end{bmatrix} \begin{bmatrix} x \\ \hat{x} \end{bmatrix} + \begin{bmatrix} 0 \\ -L \end{bmatrix} r, \tag{11}$$

$$\frac{Y(s)}{R(s)} = \begin{bmatrix} C & -D\,K \end{bmatrix} \begin{bmatrix} s\,I - A & B\,K \\ -L\,C & s\,I - A + B\,K + L\,C \end{bmatrix}^{-1} \begin{bmatrix} 0 \\ -L \end{bmatrix}. \tag{12}$$

• I appreciate the answer....But the portion that relates to my problem is when I have the feedback already designed, then my input-output relation is from $r$ to $y$. You are right. But I do not see how I have the transfer function now?
– Zero
Nov 28 '16 at 19:33
• @Zero $G(s)$, $C(s)$ and $\frac{Y(s)}{R(s)}$ are all (matrices of) transfer functions. Nov 28 '16 at 19:58
• Yeah you are right...I just asked the question wrong...I wanted to ask a general question to confirm my method but I think I messed up the question. Still given the question your answer is right. You can always treat the system as multivariable (Postlethwaite). I'll accept the answer given the question :)
– Zero
Nov 29 '16 at 14:48