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This paper presents a novel synthesis method of parametrically varying output-feedback controllers for discrete-time linear parameter-varying systems in an l1-framework. The plant parameter-dependence may be affine, polynomial or rational. Real-time parameter measurements are taken into account by the controller in order to change its dynamics according to the current plant operating conditions. The proposed scheme guarantees stability and bounds on the l∞-norm (peak-norm) of performance outputs, and can therefore be seen as gain-scheduled l1 -optimal control. Examples demonstrate the functionality of the method.