Doubt with the conversion from output PID signal to PWM on inverted pendulum control problem
i found that some of that papers say that the PID's output is the PWM
  signal


PWM represents an analogue signal just as the output of an ADC can represent an analogue signal. So, having a PID output that is purely analogue should be the same as having the same analogue signal represented by PWM. The whole point of PWM is that the operating frequency of PWM is high enough so that the recipient analogue/mechanical system does not see it as anything other than the original analogue signal. 

In other words the receipient mechanical system is so slow at responding compared to the PWM switching frequency that it doesn't respond to the up/down nature of PWM but rather it just averages things out.

Go research class D audio amplifiers if you want a comparison.