Earlier quoted context omitted.
If a control system's output is uncorrelated with what it is controlling, then it's not really controlling it. Also, to see a clear linear correlation, you might need to know what you're looking for. A controller output may look uncorrelated with the target system state, until you realize it's a I controller, so you need to look for correlation between controller's output and the integral of the system's state.
Let’s say there is a system that controls whether a second system receives input from random source A or random source B, and there is some difference in the distribution of values generated from each source. Starting without knowing the relationship between the two systems, is there a good general approach that could pick out what it is doing?
Or replace A with known non-random source, and then B, and analyze.
I'm not sure what the argument is though. If something has an effect, there will be a correlation. Measuring the correlation may require modifying the system. We've known that measurement unavoidably affects the system under measurement for some time now. It's just a matter of degree.