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In many books I've seen the generalized transfer function expressed using:
\$\displaystyle G(s) = K \frac{(s + z_1)(s + z_2)\ldots}{s^n (s + p_1) (s + p_2)\ldots} \$
What I don't understand is where the lonely \$s^n\$ term in the denominator comes from? I've searched high and low but I've not found any explanation for this term.
The s^n term is a pure integrator and the function may be integrated multiple times and that's where the n comes from.
There's no reason that n cannot be negative. Pure integrators are more common than pure differentiators in control systems, so that's probably why it sits in the denominator.
Real systems are causal. That means that the order of the denominator is higher or equal compared to the nominator. Or in other words, the number of poles is equal or higher than the number of zeros. So if you are to model a system, then the transfer function will be of the form you have shown.
Note that for a controller this is not necessarily true as Chu said. Then you can have more zeros than poles
