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This paper presents a procedure for the synthesis of positive real controllers based on matrix inequalities. Problems with H2 and H∞ cost are considered and the resulting bilinear matrix inequality problems are solved using local, iterative algorithms. The procedure is applied to the synthesis of passive suspensions for the optimization of certain performance measures for a quarter-car model. The characterization of the positive real constraint using matrix inequalities and the use of a new mechanical element called the inerter, permits the optimization over the entire class of positive real admittances and the realization of the resulting admittance using passive elements. The optimization results are compared with previous results obtained using optimization over fixed-structure admittances. The proposed method can reproduce the previous results and achieve better results in certain cases. Results of the experimental testing of a mechanical network involving an inerter are presented.