# State space representation in s-domain

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.

• Can you show us your calculations in addition to the result you get? – user17592 Apr 14 '13 at 12:12

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.