Simulink Modeling Problem
My guess is that your fancy "tune PID" aimed for irrealistic response characteristics and ended up with those absurd gains for your controller, which should be the cause for also absurd control actions.

Get rid of the derivative term (K_d = 0; they are problematic specially if your noise contain high frequency components) and start by using small gains for K_p and K_i, and increase them as you observe the system response. K_p should be set for getting a reasonably good initial response (but don't overshoot) and K_i should be set for making the steady-state error zero in an acceptable time.