It is composed of the seventeen Heirs of each House. There is likely oversight from the Lyorn and/or Issola as regards protocol. It may be that if the Heir is unable to attend, a Deputy can be assigned to stand in for the Heir at the council.
It is up to the Council of Princes to determine when the Cycle has changed, and to confirm each new Emperor. They also determine the Imperial Allotment, which is presumably the portion of the Imperial budget which is granted to the Emperor in his own self.
This is a subset of the Council of Deputies.