Passive Controller Realization of a Biquadratic Impedance With Double Poles and Zeros as a Seven-Element Series–Parallel Network for Effective Mechanical Control

This work is concerned with the passive realizability problem for a biquadratic impedance with double poles and zeros as a seven-element series–parallel $RLC$ (damper-spring-inerter) network, which is a series or a parallel connection of two networks: a three-element series–parallel network and a four-element series–parallel network. First, a necessary and sufficient condition is obtained for the realizability of a biquadratic impedance with double poles and zeros as a series–parallel network containing no more than six elements (see Theorem  1 ). Then, the main theorem is presented (see Theorem  2 ), which is derived for three-reactive-element, four-reactive-element, and five-reactive-element settings, respectively (see Lemmas  2 – 4 ). The results can be directly utilized to physically construct a passive mechanism containing dampers, springs, and inerters, which can be regarded as a passive controller for some mechanical control systems.