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A robust stability test for a class of constrained cross-directional controllers is found. Under special circumstances, the stability test is executed on a mode-by-mode basis and greatly simplified to a frequency-domain criterion. The test is also exploited to develop tuning algorithms. The control system involves a quadratic program embedded within an internal model control antiwindup structure and achieves optimal steady-state performance when the plant is known. Both the nonlinearity in the controller and the plant uncertainty satisfy certain integral quadratic inequalities. This allows us to obtain conditions for robust stability that can be expressed as linear matrix inequalities via the Kalman-Yakubovich-Popov lemma.