This paper studies controller design for feedback systems in the presence of asymmetrically bounded signals, using a case study. An asymmetric objective functional is used to consider the asymmetrically bounded signals, which makes possible to derive a linear programming problem. Solving this LP makes possible to design controllers that minimize certain outputs, fulfilling at the same time hard constraints on certain signals. The method is presented by application to a hydrogen reformer, a system in petrochemical plants that produces hydrogen from hydrocarbons: a mixed sensitivity problem is stated and solved, with an additional constraint given by the asymmetric limitations on the magnitude and rate of the control signal, and the asymmetricity in the disturbances.