Implicit and fractional-derivative operators in infinite networks of integer-order components

Complex engineering systems may be considered to be composed of a large number of simple components connected to each other in the form of a network. It is shown that, for some network configurations, the equivalent dynamic behavior of the system is governed by an implicit integro-differential operator even though the individual components themselves satisfy equations of integer order. The networks considered here are large trees and ladders with potential-driven flows and integer-order components in the branches. It has been known that in special cases the equivalent operator for the overall system in the time domain is a fractional-order derivative. In general, however, the operator is implicit without a known time-domain representation such as a fractional derivative would have, and can only be defined as a solution to an operator equation. These implicit operators, which are a generalization of commonly known fractional-order derivatives, should play an important role in the analysis and modeling of complex systems. This paper illustrates the manner in which they naturally arise in the modeling of integer-order networked systems.