I was supposed to find state space representation and its matrices of this system:
and I have no idea, how to do this. We were told not to transfer the system to time domain, but I can only do state space representation from time domain schemas.
When I tried to solve this, I got matrices
$$A = \left( \begin{array}{ccc@{\ }r}
-a & k \\
-b & -p \\
\end{array} \right)$$
$$B = \left( \begin{array}{ccc@{\ }r}
0 \\
b \\
\end{array} \right)$$
$$C = \left( \begin{array}{ccc@{\ }r}
1 & 0 \\
\end{array} \right)$$
$$D = \left( \begin{array}{ccc@{\ }r}
0
\end{array} \right)$$
I went like:
$$X_2(s) = (U(s)-X_1(s)) \cdot \frac {b}{s+p} $$
$$X_1(s) = X_2(s) \cdot \frac {k}{s+a}$$
That could mean:
$$sX_2(s) + pX_2(s) = bU(s) – bX_1(s) \to \dot x_2(t) = bu(t) – bx_1(t) – px_2(t)$$
$$sX_1(s) + aX_1(s) = kX_2(s) \to \dot x_1(t) = -ax_1(t) + kx_2(t)$$
and output sould be:
$$y(t) = x_1(t)$$
that would lead to matrices I wrote. But I don't know, if I can do that this way, or if that is what was the task, cause we were told not to transfer to time domain, but I can't imagine how to do it without transfer I did.
Best Answer
From what I see what you did is correct.
I also couldn't think of a good way to do this without going to the time domain. Do note that from transfer function to state space, there are infinite possibilities. But some of the realizations are called canonical forms. See this wiki page for Canonical realizations.
Maybe what you can do is to get the complete transfer function for $$\frac{y}{u} = \frac{bk}{s^2 + (p+a)s + bk + pa}$$ first. And then just use that canonical transformation to plug in the numbers into the matrices by observation.