Positive real synthesis using matrix inequalities for mechanical networks: application to vehicle suspension

This paper presents a procedure for the synthesis of positive real controllers based on matrix inequalities. The H₂ and H_∞ problems 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 optimisation of certain performance measures for a quarter-car model. The characterisation of the positive real constraint using matrix inequalities and the use of a new mechanical element called the inerter, permits the optimisation over the entire class of positive real admittances and the realisation of the resulting admittance using passive elements. The optimisation results are compared with previous results obtained using optimisation over fixed-structure admittances. The proposed method can reproduce the previous results and achieve better results in certain cases.