Flanker Lydiate was released from his contract with former club Racing Metro prior to the November internationals and joins Cardiff Blues' Sam Warburton in signing one of the new deals.
Under the terms of the Rugby Services Agreement, players returning to Wales are expected to return to their region of origin - Newport Gwent Dragons in Lydiate's case - but the 26-year-old has been allowed to move elsewhere for family reasons.
"I am delighted to have secured a national dual contract and I am looking forward to doing my best for the Ospreys while I aim to win selection for Wales," he said.
"Having a national dual contract gives me the certainty I need for my future and I will now be able to concentrate one hundred per cent on my rugby.
"I enjoyed my time with Racing Metro but this is the right move for me as I want to achieve my full potential as a player based here in Wales."
Lydiate, who has 41 Wales caps, was one of 12 players offered national dual contracts by the Welsh Rugby Union, with further announcements expected shortly.