1. Considering \(k\neq 0\), what is the relative degree n^∗ of the plant?
The relative degree is defined as the difference between the degree of the denominator’s polynomial order and the numerator’s polynomial order. Since \(\lambda\) is given to be of degree \(n-1\), the numerator’s polynomial of \(G_{nm}\) has degree \(n-1\) while the denominator’s polynomial has degree \(n\). Therefore the relative degree of the transfer function \(G_{nm}\) equals one.
2. What is the (full, i. e., non-minimal) order of \(G_{nm}\) , assuming non-zero parameters?
The order of a transfer function is denoted by the highest power of the denominator’s polynomial. The highest power of the denominator of \(G_{nm}\) is n. Therefore the order of \(G_{nm}\) is n.
3. Is F controllable?
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.
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 \(a_0 \neq 0\), controllability and observability is ensured