It is the project managers role to produce the PID and pass it on to the project board for authorisation. In reality the stakeholders, users and business analysts will need to be involved in producing much of the documentation.
The PID is a constantly evolving document and remains important throughout the project life cycle. The PID contains many documents/sections including the project plan, exception plans, risks/issues and therefore is updated throughout the project. It remains a reference point to who is doing "what, when, how, why".
Spend time keeping the PID updated and authorised.