Realizations of a Special Class of Admittances With Strictly Lower Complexity Than Canonical Forms

This paper investigates the simplified realization problem of a special class of positive-real admittances similar to biquadratic functions but with an extra pole at the origin, which is widely used in the analysis of suspension systems. The results in this paper are motivated by passive mechanical control with the inerter. The concept of strictly lower complexity is first defined, whose indices in this paper are the total number of elements, the number of resistors (dampers), and the number of capacitors (inerters). We then derive a necessary and sufficient condition for this class of admittance to be realized by the networks that are of strictly lower complexity than the canonical realization by the Foster Preamble method. To solve this problem, it is shown that it suffices to consider the following: 1) networks with at most four elements, 2) irreducible five-element resistor-inductor (RL) networks, and 3) irreducible five-element resisitor-inductor-capacitor (RLC) networks. Other cases are shown to be impossible. By finding their corresponding network configurations through a series of constraints and deriving the corresponding realizability conditions, the final condition can be obtained. Finally, the U-V plane and numerical examples are provided to illustrate the theoretical results.