control//controllability and observability//functional observability

Functional observability is the property that lets an observer estimate a chosen function of a system's state (a few target variables, or a combination of them) from the sensors available, without being able to reconstruct the whole state. It is what matters when a system is too large to observe completely and the question only needs a part of it: the load on a few lines of a power grid, the prevalence of an infection, an intended movement among the activity of thousands of neurons.

Functional observability is the property that lets an observer estimate a chosen function of a system's state (a few target variables, or a combination of them) from the sensors available, without being able to reconstruct the whole state. It is what matters when a system is too large to observe completely and the question only needs a part of it: the load on a few lines of a power grid, the prevalence of an infection, an intended movement among the activity of thousands of neurons.

Classical observability asks whether every state variable can be recovered from the outputs. Functional observability asks the narrower question for a target z=Fxz = Fxz=Fx, and the narrower question can have a yes where the full one has a no, so it needs fewer sensors and a smaller observer: the estimator only has to track the part of the state the target depends on. Montanari and colleagues gave a graph-based test and algorithms that find the minimal set of sensors and the observer of minimum order for a target in large networks, and applied them to detecting attacks on power grids from few phase measurements and to estimating an epidemic's prevalence under limited testing (Functional observability and target state estimation in large-scale networks, PNAS, 2022).

The static core of the idea is one condition: two states that give the same measurements must give the same target value. If two states with different targets look identical to the sensors, no estimator separates them, whatever its design.

With noise the condition becomes a matter of degree. What decides is how much the measurement distributions of the two cases overlap, how much data there is and what an error costs, so a target that is observable in principle can still be out of reach under a given protocol.

It turns the design question around, from how many sensors make the system observable to which sensors make this target observable, which is how state estimation scales to systems with thousands of variables.