Yes, I believe so, however if the periodic tenancy is working it might be worth just leaving as-is unless you want to increase the rent (in which case yes, you would have to provide written notice).
The only issue you have if you go ahead with a new AST agreement would be the reluctance from the tenant if they are unsure as to their length of stay and decide to move out altogether, which may be a blessing provided you have an interested party ready to move in and agree to your AST.
A good link to have a gad-at is below:
http://england.shelter.org.uk/get_advice/private_renting
Hope whatever you do works out well notlynx :-)