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RAGEON deleted file Stability and Convergence Proof.tex
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\chapter{Investigation of a Non-minimal Plant Structure}
\section{System analysis}
relative degree 1
order of $G_nm = n$
A system can only lose its controlability or observability 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)}\underbrace{\begin{bmatrix}1\\s\\\vdots\\s^{n-2}\end{bmatrix}}_{\xi}
\\\lambda(s) &= a_{n-1}s^{n-1} + a_{n-2}s^{n-2} + ... + a_1s+a_0
\end{align}
Therefore we have to show that the transfer function $F(s)$ is coprime to guarantee controllability and observability. Since $\xi$ only contains potencies of $s$ 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. Hence $a_0 \neq 0$ and controllability and observability is ensured
