I know that I already asked this kind of question on the website, but meanwhile I have much more knowledge about the subject and ready to describe my real problem with enough background information I hope.
Assume we are given a system
$$\dot x= Ax + Bu $$ $$y = Cx + Du $$ where $A,B,C,D$ are matrices, and $x, u$ and $y$ are vectors with appropriate sizes
I wrote an article about calculating the poles and zeros of such a system.
First of all we can calculate the transfer function of the system, which is
$$H(s) = C(sI – A)^{-1}B$$
Then I put $H(s)$ in the Smith-McMillan form. All elements are rationals of the form: $\frac{\epsilon_i(s)}{\psi_i(s)}$.
$$ SM_H(s) =
\left( \begin{array}{cccc|ccc}
\frac{\epsilon_1(s)}{\psi_1(s)} & 0 & \ldots & 0 & 0 & \ldots & 0 \\
0 & \frac{\epsilon_2(s)}{\psi_2(s)} & 0 & \vdots & 0 & \ldots & 0 \\
\vdots & & \ddots & 0 & 0 & \ldots & 0 \\
0 & \ldots & 0 & \frac{\epsilon_r(s)}{\psi_r(s)} & 0 & \ldots & 0 \\ \hline
0 & 0 & 0 & 0 & 0 & \ldots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array} \right) $$
Then, the definitions of poles and zeros. We have that the poles of a transfer function matrix are the roorts of the polynomial $p_H(s)$: $$ p_H(s) = \psi_1(s)\psi_2(s)\ldots\psi_r(s)$$
The following theorem is presented: The controllable eigenvalues are the poles of $H(s)$.
$$ \{ \textrm{Poles of } H(s) \} \subseteq \{ \textrm{Eigenvalues of } A \}$$
And then, for the several types of zeros of a given system, we defined a system matrix P(s):
$$P(s) = \left[
\begin{array}{cc}
sI-A & B \\
-C & D \\
\end{array} \right]
$$
where The $\textbf{zeros of the system $\{A,B,C,D\}$}$ are the roots of the zero polynomial $z_P(s)$ of the system. $z_P(s)$ is the monic greatest common divisor of all nonzero minors of order $r = \text{Rank }P(s)$.
And so on and so on. For example we defined invariant zeros, input-decoupling zeros, output-decoupling zeros, input/output decoupling zeros and finally the zeros of the transfer function $H(s)$. They had all specific properties, like the zeros of a transfer function correspond with
So, that was a lot of information I think. But now I come to the point: I have just 2 questions for you about this theory.
What is the whole motivation of this theory? Why are we calculation the poles and zeros of systems? I know that eigenvalues are important because the eigenvalues give us information about stability, controllability, stabilizeability etc. But can you guys give me about 3 reasons to calculate poles and all those different types of zeros?
What exactly is a transfer function and why is it important? I know that it has something to do with relation between input and output of all the indivual states, and that the Laplace Transformation is somehow involved (Which I never learned, but if you can give me a introductionary website/pdf about Laplace let me know!)
Best Answer
These are very general questions that any textbook on control theory would address quite extensively.
However, here are some good (albeit very brief) treatments that address your two main questions:
I don't see how these will help you if you still don't know/understand transform methods, so also look at that reference.