Rent control doesn't "create" any housing. It simply puts a cap on the price of housing. I'm a part-time Real Estate investor, and I would never invest in a city that had a rental cap, since the point of investments is to make money, not start a charity. Public housing has the same problem. You end up putting a bunch of poor people in one building, who statistically end up being associated with crime and drug use. This drives down real estate values in the adjoining neighborhood and makes real-estate less attractive to investors and you wind up creating a slum.
Want to promote more affordable housing? Keep the government far away. The market has its fair share of issues and inefficiencies, but it's still more efficient than affordable housing programs dreamed up by government bureaucrats.