A process for controlling an exhaust system can comprise flowing exhaust gas from the engine past a first oxygen sensor, through a NO.sub.x adsorber, past a second oxygen sensor, through a catalyst and past a third oxygen sensor, wherein the first oxygen sensor, the second oxygen sensor, and the third oxygen sensor, are in operable communication with an electronic control module, and using a switching delay between the first oxygen sensor and the second oxygen sensor to determine a NO.sub.x value, wherein the NO.sub.x value is selected from the group consisting of a NO.sub.x regeneration time, a stored NO.sub.x amount, a NO.sub.x storage efficiency, and combinations comprising at least one of the foregoing NO.sub.x values. A desulfurization process can be initiated when the NO.sub.x value is less than or equal to a first selected value. During the desulfurization process, when the third oxygen sensor signals a condition rich of stoichiometry, oxygen can be provided to the catalyst.