Passive Mechanical Realizations of Bicubic Impedances with No More Than Five Elements for Inerter-Based Control Design

Abstract This paper mainly investigates the passive realization problems of bicubic (third-order) impedances as damper-spring-inerter networks consisting of no more than five elements. First, the special case where a bicubic impedance contains a pole or a zero on the imaginary axis or at infinity is discussed. Then, assuming that there is no pole or zero on the imaginary axis or at infinity, the realizations of bicubic impedances as five-element networks are investigated. Necessary and sufficient conditions for the realizability as five-element series-parallel networks and as five-element non-series-parallel networks are derived, respectively, where 22 series-parallel configurations and 11 non-series-parallel configurations are presented to cover the conditions. Finally, two numerical examples together with positive-real controller designs for a quarter-car suspension system are presented for illustrations. The results of this paper can contribute to the synthesis of low-complexity passive mechanical (or electrical) networks, which are motivated by the synthesis and design of inerter-based vibration control systems.