A fuel control module transitions engine fueling from rich to lean. A catalyst fault detection module diagnoses whether a fault is present in an exhaust catalyst based on a response of an oxygen sensor to the transition. A prediction module generates a prediction based on a model and a set of possible target values. A cost module determines a cost for the set of possible target values based on comparisons of the prediction with minimum and maximums. Before the transition, a constraint module selectively adjusts at least one of the minimum and maximums for the fault diagnosis. Based on the cost, a selection module selects the set of possible target values from a group of sets of possible target values and sets target values based on the selected set of possible target values. An actuator module controls an engine actuator based on a first one of the target values.