In the UK, there is Council Housing, this may be a good alternative to rent control because it provides housing at a nominal cost and gives the private landlords some competition. The only problem with this type of offering is it will stick in the craw of the American psyche who abhor the government competing against the private sector.

Rents do get bid up due to things like the minimum wage rising. This is because the power relationship between landlord and tenant is heavily skewed in favour of the landlord. The employee/employer relationship also suffers from the same disease.

Disclosure: I'm a US/UK dual citizen, and a landlord myself.