1. An engine control system for a vehicle, comprising: a fuel control module that transitions fueling of an engine from fuel rich to fuel lean; a catalyst fault detection module that diagnoses whether a fault is present in an exhaust catalyst based on a response of an oxygen sensor to the transition, wherein the oxygen sensor is located one of upstream and downstream from the exhaust catalyst; a prediction module that generates a predicted operating parameter of the engine based on a model of the engine and a set of possible target values determined based on an engine torque request; a cost module that determines a cost for the set of possible target values based on comparisons of the predicted operating parameter with predetermined minimum and maximum values; a constraint module that, before the transition, selectively adjusts at least one of the predetermined minimum and maximum values for the fault diagnosis; a selection module that, based on the cost, selects the set of possible target values from a group including the set of possible target values and N other sets of possible target values determined based on the engine torque request, wherein N is an integer greater than zero, and that sets target values based on the selected set of possible target values; and an actuator module that controls an engine actuator based on a first one of the target values.