Can a controller have a negative gain?
1) Sure, this inverting amplifier circuit has a negative gain.



Gain = output/input, so a negative gain simply means that the polarity of the signal is inverted between input and output.

2) OH YES in a system with negative feedback, like this:



The sign of the feedback signal (running through (H) ) cannot be changed sign-wise or the feedback would not longer be negative feedback but positive. That results in a completely different system behavior.

Imagine cruise-control in your car, when going up a hill the cruise-control compensates to keep the speed constant. That's negative feedback, the speed becomes less (we're going up a hill) power is increased. In a positive feedback system, power would be decreased so you'd slow down ! That's not what you want.