You got my DD's dog? Sounds just like him. He comes over to my place if there is no one in their house for long periods.
Poor thing sits and does that same thing - eyes close, head nods, lower and lower and then, just like his grandpa, suddenly jerks it up as if to say I'm not sleepy, no, not me. And the eye lids start to droop again and the cycle repeats.
Little people, eh?