Frequency Response and Transfer Functions of Large Self-similar Networks

TL;DR

This paper develops algorithms to compute the frequency response and transfer functions of large self-similar networks, revealing their dynamics are either integer or fractional order, and demonstrates their approximation capabilities.

Contribution

It introduces novel algorithms for analyzing large self-similar networks' dynamics and characterizes their transfer functions as integer or fractional order systems.

Findings

Finite networks exhibit integer-order dynamics.

Infinite networks exhibit fractional or irrational order dynamics.

The analysis enables approximation of irrational expressions with rational functions.

Abstract

This paper focuses on computing the frequency response and transfer functions for large self-similar networks under different circumstances. Modeling large scale systems is difficult due, typically, to the dimension of the problem, and self-similarity is the characteristic we exploit to make the problem more tractable. For each circumstance, we propose algorithms to obtain both transfer functions and frequency response, and we show that finite networks' dynamics are integer order, while infinite networks are fractional order or irrational. Based on that result, we also show that the effect of varying a network's operating condition to its dynamics can always be isolated, which is then expressed as a multiplicative disturbance acting upon a nominal plant. In addition, we analyze the non-integer-order nature residing in infinite dimensional systems in the context of self-similar networks. Finally, leveraging the main result of this paper, we also illustrate its capability of approximating some irrational expressions by using rational functions.