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Closed Form Time Response of an Infinite Tree of Mechanical Components Described by an Irrational Transfer Function

Guel-Cortez, A. J.; Sen, M.; Goodwine, B. (July 2019, Proceedings of the ... American Control Conference)

In this work we determine the time-domain dynamics of a complex mechanical network of integer-order components, e.g., springs and dampers, with an overall transfer function described by implicitly defined operators. This type of transfer functions can be used to describe very large scale dynamics of robot formations, multi-agent systems or viscoelastic phenomena. Such large-scale integrated systems are becoming increasingly important in modern engineering systems, and an accurate model of their dynamics is very important to achieve their control. We give a time domain representation for the dynamics of the system by using a complex variable analysis to find its impulse response. Furthermore, we validate how our infinite order model can be used to describe dynamics of finite order networks, which can be useful as a model reduction method.

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This work presents a simple procedure for designing fractional $$PD^\mu$$ controllers for a type of implicit operators, which have recently been studied to describe large-scale systems. The methodology developed proposes a geometrical approach that allows characterizing the parameter-space of the $$PD^\mu$$ controller into stable and unstable regions. Several numerical examples illustrate the effectiveness of the proposed results.

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