# Relation between **poles** of $\boldsymbol{G}(s)$ and **eigenvalues** of system matrix $A$.

In page 602 of [1], it states:

It is also possible to relate zeros to the properties of a coprime factorization of a given transfer function matrix. In particular, if we write $$\boldsymbol{G}(s) = \boldsymbol{G_N}(s)[\boldsymbol{G_D}(s)]^{-1}=[\boldsymbol{\bar{G}_D}(s)]^{-1}\boldsymbol{\bar{G}_N}(s)$$ ..., then the zeros of $\boldsymbol{G}(s)$ correspond to those values of $s$ where either $\boldsymbol{G_N}(s)$ or $\boldsymbol{\bar{G}_N}(s)$, or both loose rank. Similarly, we can define the poles of $\boldsymbol{G}(s)$ as those values of $s$ where $\boldsymbol{G_D}(s)$ or $\boldsymbol{\bar{G}_D}(s)$, or both loose rank.

In this definition, from the perspective of a transfer function matrix $\boldsymbol{G}(s)$, the poles are defined as those values of $s$ where $\boldsymbol{G_D}(s)$ or $\boldsymbol{\bar{G}_D}(s)$, or both loose rank. And it also states that the continuous MIMO system is stable iff all poles are on LHP.

However, $\boldsymbol{G}(s)$ can also be transformed into a state-space representation denoted by $S(A,B,C,D)$, or $\dot{x}=Ax+Bu, y=Cx+Du$. From the perspective of the state-space representation, the system $S$ is stable iff all the eigenvalues of $A$ are on LHP.

So, the question is, what is the relationship between poles of $\boldsymbol{G}(s)$ and eigenvalues of $A$ of the system $S$? Are they the same?

[1]: G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control system design. 2001.

$$G(s) = C (s\,I - A)^{-1} B + D.$$
Namely if the state space model is equivalent to the transfer function then when $s$ equals an eigenvalue of $A$ the matrix $s\,I-A$ loses rank. Which should be similar to $\mathbf{\bar{G}_D}(s)$ or $\mathbf{G_D}(s)$ to lose rank.