deletions | additions       diff --git a/Stability and Convergence Proof.tex b/Stability and Convergence Proof.tex  index daab8ab..e088ada 100644  --- a/Stability and Convergence Proof.tex  +++ b/Stability and Convergence Proof.tex 
...
order of $G_nm = n$  A transfer function loses system can only lose  its controllability controlability  or observability through zero-pole-cancelation if a zero-pole-cancellation occurs  \cite{lunze2}. Conversely if no cancellation takes place controllability and observability is ensured.  \begin{align}  F(s) &= \frac{1}{\lambda(s)}[1,s,\ldots,s^{n-2}]^T  \\\lambda(s) &= a_{n-1}s^{n-1} + a_{n-2}s^{n-2} + ... + a_1s+a_0  \end{align}  Therefore it is necessary but not sufficient we have to show  that thezeros of the  transfer function $F(s)$ don't cancel out the roots. is coprime to guarantee controllability and observability.  The only possible case in which a zero-pole-cancellation occurs is for $a_0 = 0$. Due to the fact that $\lambda(s)$ is Hurwitz all eigenvalues are less than zero.  F is controllable due to the fact that thus $a_0$ can't be zero.