IEEE Systems, Man and Cybernetics Magazine - July 2018 - 26

Connections Between Cybernetics
and Fractional Calculus
McCulloch et al. [18] describe how the anatomy and neurophysiology of the retina preprocess visual scenes and
do not simply send the brain snapshots of the world. Such
an insight was crucial to developing linear models of signal transmission from the mesh of cells in the retina
through the optic nerve to the brain. Cybernetics grew out
of such early attempts to model neural processes using
integer-order feedback control. We cannot help but wonder what added features of the visual field might be
encoded in this model if fractional-order space and time
derivatives were employed.
Cybernetics emerged in the 1960s from this biophysical
foundation, and early neural physiologists applied it to
model hand and eye movements and the stimulus-duration
curve for the firing of the action potential, so it is somewhat
surprising that a more formal partnership with fractional
calculus was not established, given the propinquity of time
and place. To elaborate on this point, consider the following
examples taken from the fields of communication and control, key aspects in the Wiener definition of cybernetics.
Communication in Man and Machine
Communication is the exchange of information, and Shannon [19] linked information to changes in entropy. Of
course, the information exchange (or interchange) does
not necessarily have a digital form but can also be
described in an analog form. Whenever we interact, we
deal with information exchange, and the omnipresent dissipation necessarily leads to measurable changes in entropy that provide a measure of the information exchanged
during the interaction. In dynamic processes, this information is usually used for control purposes as feedback.
This relationship is most obvious, of course, in contemporary digital control systems, where the data obtained
during an interaction between the controller and the controlled system (plant) are sampled and fed back for adjustments of the control actions by the controller. Since the
data are digitized, Shannon entropy provides the measure
of the information. Wiener, on the other hand, considered
information as analog and continuous.
In mathematical modeling, differential equations are a
basic tool, and the process of finding a numerical solution
is also an example of the cybernetic approach. To obtain
numerical solutions of fractional-order differential equations and perform numerical simulations, discretization is
used, and we necessarily arrive at a feedback loop. For
simplicity, consider the case of a fractional-order differential equation whose only independent variable is time. The
function value at the next discretized time step is computed based on the values of all the previous time steps, and
then this is added to the previous steps for the next iteration. The weights of the function values at the discretization steps always change with each step. This is a
mathematical manifestation of what we observe in other
26

IEEE SyStEmS, man, & CybErnEtICS magazInE July 2 0 1 8

forms in many fields, such as automation, telecommunications, and neuroscience [20].
In the interaction between humans and real materials,
dissipation and feedback are always present. However,
classical Newtonian time-and, therefore, the classical
integer-order calculus developed by Newton and Leibniz-
by its nature cannot capture both dissipation and feedback, particularly when fractional orders appear in the
system transfer function; here fractional-order calculus
comes to mind as a suitable tool. In studying the interactions between humans and viscoelastic materials (e.g.,
clay, bread dough, and cheese), Scott Blair [21], one of the
founding fathers of rheology, reinvented fractional-order
differentiation and integration while developing psychorheological models to describe compression and relaxation.
He also suggested [22] the classification of viscoelas tic
materials based on invariants that involve fractional powers of time, an approach similar to that embedded in using
the Reynolds number in fluid mechanics, which takes into
account a characteristic length and velocity as well as viscosity and density.
In the field of sociology, Maltz [23] observed similar
complex interactions between humans as units of society
and coined the term psychocybernetics, which takes us
almost all the way back to Ampère's original definition of
cybernetics. Such memory-encoding phenomena are not
restricted to human-material interactions, as demonstrated by the work of Westerlund [24] on relaxation in dielectrics and of Shahsavari and Ulm [25] on the indentation
analysis of viscoelastic solids. The common feature of
these three examples taken from different fields of science
is their approach to time scales: in each case, the time
scale is deemed an intrinsic property of the process, and
this individual time scale is definitely not uniform.
Control in Man and Machine
In Aristotle's Politics, it says [26]:
For if every instrument could accomplish its own
work, obeying or anticipating the will of others, like
the statues of Daedalus, or the tripods of Hephaestus,
which, says the poet, 'of their own accord entered the
assembly of the Gods'; if, in like manner, the shuttle
would weave and the plectrum touch the lyre without
a hand to guide them, chief workmen would not want
servants, nor masters slaves.
Aristotle distinguishes between what he calls instruments of action and instruments of production. Since, for
Aristotle, life is action, the instruments of action will be
those responsible for reproducing life, while the instruments of production will be those whose aim is to produce something apart from its own operation. Here, we
observe not only the objective of automation but also its
two main subfields: the automation of production, which
is the field now called industrial automation, and the
automation of actions, which constitutes the subfield of
the automata. Fundamental to these two fields is what



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