I am looking at a good tutorial on PID control, and I am a little confused about how control works. My question is really whether a proportional controller adds energy to the system to obtain some desired output, because from the before and after plots could not be generated without the introduction of a lot more energy to the system?
So assume I have a simple spring-mass system, meaning a 2nd order ODE. The transfer function after taking the Laplace transform looks like:
$$ \frac{X(x)}{F(s)} = \frac{1}{s^2 + 10s + 20} $$
Now if I use Matlab to simulate this system, I can use a unit step function, as the input. The unit step should just mean that the signal is set to $U(s) = 1$, so the error is the difference between the output and the input $Y(s) - U(S)$. Here is a plot from the tutorial.
Now, the authors of the tutorial introduce proportional gain in a feedback controller. Now the new transfer function is
$$ \frac{X(x)}{R(s)} = \frac{K_p}{s^2 + 10s + (20 + K_p)} $$
They set the gain coefficient $K_p = 300$. This is all very standard looking stuff for the transfer function. However, now the step response is so much stronger. My question is kinda how did the response ramp up so much?
So that is my basic question: if the step response is just a constant value of $u(s) = 1$, then how is the system going from an amplitude of 0.05 to almost 1.3 at one point, based upon feedback control? Like even with feedback that seems like a huge change in the system just by adjusting some constant input. I think I am probably misinterpreting the physics or something, so hopefully someone can set me straight. The math is easy to solve, but understanding whether the answer is sensible is what I am looking for.