I have this system here:
In this example the state space representation $ \frac{dx}{dt} = Ax + bu $ and the corresponding transition matrix $\Phi(t)$ is asked for.
So to get the state space, I think this method is the fastest one. For that my solution is: $\frac{dx}{dt} = \begin{pmatrix} 0&1&0 \\0&0&1 \\ -6 & -11 & -6\end{pmatrix}x + \begin{pmatrix} 0 \\0 \\1 \end{pmatrix}u$
$y = \begin{pmatrix} -2 &-6 &-2 \end{pmatrix}x + u $
But in this case, getting $\Phi(s) = (sI-A)^{-1}$, is hard work. ;-) And at my exam this is wasting too much time.
So there is another method getting the state space.
There you use the eigenvectors and put them in the diagonal of the A matrix like that:
$A = \begin{pmatrix} -1&0&0 \\0&-2&0 \\ 0 & 0 & -3\end{pmatrix}$ Then you use this equation: $G(s) = c^T(sI-A)^{-1}b+d$ (d = 1 I think)
and use coefficient comparison.
But here is my problem. I don't know how I shall solve this equation.