significance of zeros of a transfer function?

Question

In control theory, the poles of a transfer function give information about the stability and behavior of a system.

I'm not sure and can't find anywhere what the significance of the zeros of a transfer function is. What information do they give you?

Every book chapter I've read so far about "Transfer Function Poles and Zeros" tells you what the zeros are but never answers the question of why we care about the zeros at all!

Answer 1

For simplicity suppose we have a strictly proper system with real distinct poles like the following

$$G(s)=\frac{(s-z_1) \dots (s-z_m)}{(s-p_1) \dots (s-p_n)}$$

Then we can factor it like

$$G(s)=\frac{a_1}{s-p_1} + \dots + \frac{a_n}{s-p_n}$$

where $a_i$ is determined by the zeros of the system. Taking inverse Laplace transform we obtain

$$g(t)=a_1 e^{p_1 t} + \dots + a_n e^{p_n t}$$

Hence we can say that zeros of the system determines the contribution of each pole, but not the stability of the system.

Answer 2

The zeros are more fundamental than the poles in the following sense: while poles can be assigned by feedback, the zeros can only be canceled. Therefore, an unstable zero cannot be moved: you have to live with whatever effect it has on the performance of your system, even after closing feedback loops.

And what are those effects? The zeros create a "notch" in the frequency response. The gain at those frequencies is attenuated. Those are the frequencies at which you cannot expect the system to respond. If you need good response at such frequencies, feedback control will not help - you need to redesign the system itself.

Another important property of the zeros can be visualized using the root-locus diagram. For high feedback gains, the closed-loop poles move towards the zeros. Stable zeros pull the roots towards the stable side of the complex plain; unstable zeros, which define non minimum-phase systems, pull the roots to the unstable half-plane, limiting the use of high feedback gains. For minimum-phase systems, some form of high-gain feedback control may be stabilizing (although such methods are often of dubious practical value).