What is the general form of a transfer function

Question

I repeatedly see two representations of the general transfer function in the literature. The first is the following which is factorization of the numerator and denominator polynomials:

$$\textbf{G}(s) = k \frac{ (s - a_{1})(s - a_{2})(s - a_{3})\ldots }{ (s - b_{1})(s - b_{2})(s - b_{3})(s - b_{4})\ldots }$$

The alternative, below, eludes me and I'll checked many sources but none seem to justify it. The difference is a lowly $s$ term in the denominator raised to a power. Is this general form related the open-loop transfer function?

$$\textbf{G}(s) = k \frac{ (s - a_{1})(s - a_{2})(s - a_{3})\ldots }{s^m (s - b_{1})(s - b_{2})(s - b_{3})(s - b_{4})\ldots }$$

My question is what is the difference, if any in these two forms?

Answer 1

You may write the second equation on the first form by using $b_i = 0$ corresponding to those poles appearing at $s = 0$. Hence, the only difference is that in the second form you know that there are $m$ poles at $s=0$ while in the first form they may still be there, but one will have to check the values of $b_i$ to determine if they are there.

Note that a system with poles at $s=0$ is a marginally stable system and that the poles are typically introduced by integrators, which have a transfer function $1/s$.

Answer 2

The first representation of the Transfer Function can have poles at zero,this is not represented explicitly in that equation.The second form of the transfer function representation, takes into account,the poles located at z=0.The presence of a pole at zero causes instability.