Why do we want to know the poles and zeros of a linear system?

Question

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.

1. 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?

2. 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!)

Answer 1

The main focus of control theory is the study of feedback systems. From an electrical engineering perspective, I'll use the example of an electronic amplifier, but the general idea can be applied without loss of generality to any feedback system in engineering, from a thermostat temperature control system to a mechanical governor on a steam engine.

It can be shown that by using feedback in an amplifier, certain nice properties result. The input impedance, output impedance and gain of the amplifier can be controlled independently of device parameters. Feedback also greatly reduces distortion. In essence, you can take an amplifier with a large gain, but uncontrolled parameters, and stabilize its parameters by trading off gain for stabilization using feedback. The feedback system must be carefully designed, however, as if the feedback signal ever becomes "in phase" with the input you will have built an oscillator, not an amplifier!

In the frequency response plot of a transfer function, a pole is where the frequency response "breaks" downwards at -6 dB per octave. Similarly, a zero is where the frequency breaks upwards. A pole is also where the phase response of the system has shifted by -45 degrees, and vice versa for a zero. Both will eventually contribute 90 degrees of phase shift to the system. Electronic amplifiers contain "natural" poles due to the nature of the active devices, so it's easy to see that any sufficiently complicated amplifier containing three or more poles will almost certainly oscillate if feedback from output to input is applied as the phase shifts through 180 degrees.

The poles of a feedback system allow one to tell almost at a glance if a system is stable; if there are poles in the right hand complex plane the system will oscillate (why?). I know that in feedback network design, at least for analog electronic amplifiers, zeros are often placed in the feedback network transfer function to cancel the natural poles of the amplifier, allowing the amplifier to retain a good frequency response and yet remain stable.

Answer 2

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:

• poles/zeros
• transfer functions

I don't see how these will help you if you still don't know/understand transform methods, so also look at that reference.