Minimal seven‐element series‐parallel realizability of a certain positive‐real biquadratic impedance

Abstract

SummaryThe minimal complexity realization of low‐order positive‐real immittances as resistor–inductor–capacitor (RLC) (or damper‐spring‐inerter) networks is an essential unsolved problem in the theory of network synthesis, and the results can be directly applied to the inerter‐based mechanical systems control. This paper solves the realizability problem of a certain class of biquadratic impedances as any seven‐element series‐parallel RLC network, where the impedance contains a repeated zero and pole of multiplicity two. In contrast to the previous results of this problem, the work in this paper considers the general case and removes the restrictions on the network structures and the number of reactive elements. The final results present a necessary and sufficient condition for the realizability of such an impedance as any seven‐element series‐parallel RLC network. A network synthesis example in suspension control design is presented to illustrate the results of this paper and the applications in inerter‐based mechanical control.

Stability Analysis and Design of n-DOF Vibration Systems Containing Both Semi-Active and Passive Mechanical Controllers

“…With the invention of inerters [ 1 , 2 , 3 ], it is possible to systematically realize any passive mechanical system as the physical interconnection of dampers, springs, inerters, etc., which has motivated the recent investigations on the synthesis of passive networks under low-complexity constraints [ 4 , 5 , 6 , 7 , 8 , 9 ]. Passive mechanical networks containing dampers, springs, and inerters (or called damper–spring–inerter networks) have been widely applied as passive mechanical controllers to many vibration control systems, such as seat suspension systems [ 10 ], beam-type vibration systems [ 11 ], vehicle suspension systems [ 12 , 13 , 14 , 15 , 16 ], vibration absorbers [ 17 , 18 ], bridge vibration systems [ 19 ], wind turbine systems [ 20 ], storage tanks [ 21 , 22 ], building vibration systems [ 23 ], etc.…”