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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$RLC$</tex-math> </inline-formula> (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 <xref ref-type="theorem" rid="theorem1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</xref> ). Then, the main theorem is presented (see Theorem <xref ref-type="theorem" rid="theorem2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</xref> ), which is derived for three-reactive-element, four-reactive-element, and five-reactive-element settings, respectively (see Lemmas <xref ref-type="lemma" rid="lemma2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</xref> – <xref ref-type="lemma" rid="lemma4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</xref> ). 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.