Yes, you can test control loops -- you can even turn it into a unit test. At least in theory.
The problem is that to do the test you need either a working, physical system or a good model. So if you're making a shutdown valve for a chemical plant, you need a physical build of that control valve. Even on that scale, you're talking about something that could potentially fill an engineering lab, be quite noisy and have a considerable amount of stored pneumatic or hydraulic energy. It's possible, but not exactly practical.
The alternative is to model the system, but now the question changes: how can you be certain that your model is accurate and models all the variables? Say your valve is slower when it's cold and you don't model that -- now you have a false positive result ("it works" -- but nobody realised that "temperature" was a dependent variable).
So you take the middle ground - you can have the test jig for a week, so you record the inputs and outputs for a week under varying software conditions. But those recordings are only valid for that specific timing -- if you change the software and change the timing (maybe you move the trim motor slower), you get a model change and a false positive or negative.
It's certainly possible, but it's only possible with a good sized team, and supportive management who realise that the test is absolutely necessary.