It's an interesting question. You could definitely create an app that would run on the Ubuntu computers in the background. You could then design the apps to send some sort of data message back to a server app which could track usage. With that you would at least know which systems are no longer running in their original configuration.
But you wouldn't know if the computer: failed, was re-installed, or simply hasn't been turned on.
Alternatively you could try and create some sort of hardware appliance that talked to an app or service on the computer. And then have that appliance send the data. But the problems with that as I see are:
- cost of device/appliance
- difficulty in hiding it
- how to connect it to the internet so it can transmit data