Hello I am a junior student majoring in mechanical engineering, taking a course on classical control theory. While learning transfer function and block diagram of a closed loop system, I found a concept of feedback quite intuitive at first glance; however, I am having a hard time understanding the role of the feedback loop for now. The motive for its introduction in the system was acceptable, but the actual form it takes in the block diagram makes me wonder why it is called so. Please refer to the link: https://en.wikipedia.org/wiki/Closed-loop_transfer_function.
The feedback transfer function, H(s), seems to be named so in order to indicate that it was meant to mimic the idea of "feeding something back to the beginning". In reality, however, it literally does nothing more than merely adding some part in denominator. The transfer function of the entire system has changed from G(s) to G(s)/(1+G(s)H(s)) as it adds on the closed loop at the bottom, but what's the point of making it just a bit more complicated? Why don't we just incorporate the loop into the whole system and again define its new transfer function, G'(s) = G(x)/(1+G(s)H(s))?
In the same context, when we see controllers in the block diagram (https://en.wikipedia.org/wiki/PID_controller), we naturally anticipate it to "manipulate" the system to achieve a desired outcome. In my opinion, however, once we've finished any process of fine-tuning, it seems possible to incorporate it into a whole system as is described above since the controller block does not vary at all.
I think I am doubting the representational power of the block diagram or misunderstanding how to extract the concept of feedback and control out of the scheme. Please help me with this.
Thank you.