I am working on my master Thesis and I was going through this Thesis : https://scholarship.rice.edu/bitstream/handle/1911/89289/RICE0327.pdf?sequence=1&isAllowed=y
Here, the author computes the switching curves for minimum time control of some "famous" systems using the concept of Entire Complex Function. I was wondering how to use the same techinique for systems of higher order which are decoupled by mean of state but have the same input. To be more specific, I have the following system :
$M\ddot{x}=u$
$\ddot{y}+2\gamma\dot{y}+\omega^2y=\frac{u}{b}$
where $M,b,\gamma,\omega>0$. How can I compute then the switching curve?
I tried to rewrite the system in state space, thus
$\dot{x}=\begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -\omega^2 & -2\gamma \end{pmatrix}x+\begin{pmatrix} 0\\ \frac{1}{M}\\ 0\\ \frac{1}{b} \end{pmatrix}u$
and then computing the transfer function. Since there are no measurements $y$, the transfer function would simply be :
$G(s)=(sI-A)^{-1}B$
which is a $4$ x $1$ matrix. I thought then that to "cancel" all the poles, the matrix $G(s)$ should simply be losing rank. But it doesn't seem to work since there is a term which can never be zero. How do I proceed?
P.S. As stated by the author himself in the conclusion, this method CAN work for higher order systems.
P.P.S. If you don't want to read the full Thesis, and I can't blame you, just go from page 4 to 7/8.