Downgrade rather than close, in almost every version of this. A product change to a no fee card in the same family usually keeps the account and its opening date intact, so your history is untouched, and it keeps a line of credit open which helps your overall utilisation.
Before you do it, two checks. First, move or use any points sitting on the card, because points can be forfeited when a premium card is closed or changed and the rules differ by issuer. Second, ring retention and ask plainly whether there is an offer to keep it, worst case they say no and you downgrade the same call.