I hope this is the right place to ask since I couldn't find a control systems SE.
Lets say we have the transfer function $P(s) = \frac{1}{s(s+15)(s+10)} = \frac{1}{s(s^2+25s+150)}$(completely made up numbers) of some sort of device. This is without any controller or feedback attached.
How should I look at the $\frac{1}{s}$ that's in there? Do I treat it as a separate multiplier or do I handle this as a third order system?
Without the integrator this is a simple 2nd order system with clearly defined relative damping and natural frequency because of $\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$.
Technically I could treat the 1/s as an I controller with the value of $P_i$ value of 1. Would this system even have a steady state error with a simple unity P feedback? Technically it would automatically be a PI controlled system right?
If the questions don't seem clear it's because I don't really know what to ask here.
edit: I managed to read something about types of a transfer function(0, 1 or 2) which only modifies the error you will have. The transfer function shown here would be considered a type 1 because it has 1 pole in the origin(2 means type 2 and 0 means type 0 I guess). Type 1 means that the steady state error will be 0 for a step input, which confirms my speculation about the error. I still don't know if it has any other effect.
So this picture shows the exact same thing in 2 forms. They're are functionally the same. The PID controller is C(s) in this case. Well it's a PI controller with K = 1, without me actually adding anything to the system. I've just added a negative feedback loop.