# 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.

The poles of a transfer function representation of a LTI system are the same as the eigenvalues of the system matrix of a state space model representation of that same LTI system if both representations are in a minimal form. So the transfer function does not have pole-zero cancelation and the state space model is controllable and observable.

To see why they should be the same you can look at the transformation between state space model to transfer function

$$G(s) = C (s\,I - A)^{-1} B + D.$$

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.