The thought of an across-the-board regulation based rent cap never crossed my mind, but that actually could be effective and fair. If there was some kind of easy to understand formula based on the unit, potential landlords would easily be able to calculate whether it makes financial sense instead of simply cutting costs and squeezing as much rent out as possible. There wouldn't be an incentive to kick people out (can't jack the rent) but there would be one to keep it maintained/updated since they'd be competing on everything but price. Honestly, I wouldn't mind if my rent went up a bit if it meant my unit would be properly maintained or I had the freedom to move somewhere similar without doubling my rent.
Edit: you could make it more enticing to the current landlords by easing some renter protections, like making it much easier to remove problem tenants