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

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?

• This looks more like a physics problem, than a maths problem. From your diagram you should be able to use the theory in your notes to write the conditions that $z,y$ need to satisfy (as linear equations in $x,w,u$. It is not clear what $x$ is, since it does not even appear in the diagram but again you would know, and would need to write a linear equations that is its derivative. Is this what you have tried? – AnyAD Jun 12 at 1:10

\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.

• I tried it your way but I do not know how to go from the second form that you present, to the state space form. Could you explain that part as well or is there any material you could maybe link me to? – Dimitris Pantelis Jun 18 at 21:06

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 $D_w=1$