[TeMPOraL]

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.

[DougBTX]

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?

[barrkel]

Replace the system with one that chooses A only; replace it with one that chooses B only; analyze distributions; analyze original system.

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.