Seems I should be looking for a function of the inputs that is "surprisingly" approximately constant

I think in most situations where you don't have internal observations of the various actors, it's more likely that outputs will be constant than a function of the inputs. That is, a control system adjusts the relationship between an input and an output, often by counteracting it completely--thus we would see the absence of a relationship that we would normally expect to see. (But if we don't know what we would normally expect, then we have trouble.)

Anyone have a link to a good paper on this?

I'm leaning pretty heavily on a single professor/concept for this answer, but there's a phrase called "Milton Friedman's Thermostat," perhaps best explained here (which also has a few links for going further down the trail):

If a house has a good thermostat, we should observe a strong negative correlation between the amount of oil burned in the furnace (M), and the outside temperature (V). But we should observe no correlation between the amount of oil burned in the furnace (M) and the inside temperature (P). And we should observe no correlation between the outside temperature (V) and the inside temperature (P).

An econometrician, observing the data, concludes that the amount of oil burned had no effect on the inside temperature. Neither did the outside temperature. The only effect of burning oil seemed to be that it reduced the outside temperature. An increase in M will cause a decline in V, and have no effect on P.

A second econometrician, observing the same data, concludes that causality runs in the opposite direction. The only effect of an increase in outside temperature is to reduce the amount of oil burned. An increase in V will cause a decline in M, and have no effect on P.

But both agree that M and V are irrelevant for P. They switch off the furnace, and stop wasting their money on oil.

They also give another example with a driver adjusting how much to press the gas pedal based on hills here, along with a few ideas on how to discover the underlying relationships.

I feel like it's worth mentioning the general project of discovering causality (my review of Pearl, Eliezer's treatment), but that seems like it's going in the reverse direction. If a controller is deleting correlations from your sense data, that makes discovering causality harder, and it seems difficult to say "aha, causality is harder to discover than normal, therefore there are controllers!", but that might actually be effective.