How do I find the state space representation of a Linear Fraction Transformation (LFT)?

Question

I am having a problem with solving this question:

As you can see, I have the filtering problem and I need to find a state space representation. I know how to get the state space of a SISO transfer function. But here, I am not sure how to do it. Any advice?

Answer 1

After a lot of searching, I think I found the solution.

The following relations ship should be used: $$ G(s)=C(sI-A)^{-1}B+D $$ and \begin{align} \dot{x}=Ax+Bu \\ y=Cx+Du \end{align}

First we find the system matrix: $$ \frac{CB}{sI-A}=\frac{1}{s+2} $$ so A=-2

Next, we find the input matrices: we can see from the diagram, that only $w$ has affect on the system. This means $b_w=1$ and $b_u=0$.

Lastly, we find the output and feed-through matrices: $$z=\frac{C_z}{s+2}=\frac{1}{s+2}$$ and also $$y=\frac{C_y+D_w(s+2)}{s+2} = \frac{s-1}{s+2}$$ from the above we have that $C_z=1$, $C_y=-3$ and $D_w=1$

Answer 2

Always start with the signal definitions and it follows from there.

$$\begin{align*} y &= H(s) w\\ z &= G(s) w - u\\ u &=Ky \end{align*} $$

Now, write the top two as

$$ \begin{pmatrix}z\\y\end{pmatrix} = \begin{pmatrix}G(s) &-1\\ H(s)&0\end{pmatrix} = \begin{pmatrix}w\\u\end{pmatrix} $$ This is your generalized plant candidate. If it is also stabilizable, and detectable you are done. You can also convert it to state representation.