Most modern cats run with postcatalyst sensor feedback as well but with a switching sensor in the front. Depending on the formulation the catalyst may want to have a lean bias or a rich bias but when you add the postcatalyst sensor that can be adjusted automatically so as the catalyst and oxygen sensor ages the actual lambda that the catalyst sees does not change away from best conversion. I do my ECM software using lambda as a reference (it makes the math easy when the fuel composition changes), I end up calibrating the front O2 sensor for a target lambda (not voltage like the 1980's and 1990's GM software seems to use) and usually the target is actually slightly rich for better NOx conversion at the expense of CO - a typical target for me is about 0.988 to 0.998 lambda depending on the catalyst. The problem with a front-sensor system is that the front sensor has a tolerance of about 0.02 lambda with age (for a Bosch LSF4.2, anyways) and if you can keep tighter than that you can improve conversion efficiency a lot - I have seen NOx emissions on a ETC (european transient cycle) drop an order of magnitude (from 4 g/kWh to 0.4 g/kWh) just by turning on the postcatalyst sensor. The tolerance of that same oxygen sensor after the catalyst is about 0.0005 lambda. The front switching tends to store and release oxygen in the ceria and zirconia OSC components and after the reactions in the catalyst the exhaust gas is pretty empty of oxygen which makes the postcat O2 sensor very sensitive. A typical target for gasoline is a postcatalyst voltage of between 0.48 and 0.62 volts depending on the NOx requirement. When running with a UEGO most (but not all) catalysts want some dithering of air-fuel ratio; there do exist a few catalysts without much oxygen storage capabilities that want a perfectly stoichiometric mixture. A catalyst with a monster-sized oxygen storage capability is quite difficult to control due to time delays, there are ways of doing it but those are mostly trade secrets.
Another odd duck is the non-PGM catalysts (base metal cats) that use copper and other materials like that. The feedback algorithm is totally martian on those catalysts but they must be used with extremely low-sulphur fuels.
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