Figure 1: Frequency response function of a model with uncertain parameters.
Figure 2: Fuzzy-arithmetical analysis of dynamical systems with uncertain model parameters.
A common problem in the numerical simulation of real-world systems is
the fact that
exact values for the parameters of the models can exhibit a high level
of uncertainty. This
non-determinism in numerical models may arise as a consequence of
motivating some categorization of uncertainties. Although other
classifications are possible in almost the same manner, the following categorization proves to be
well-suited in this context:
aleatory uncertainties, such as natural
variability or scatter, on the one side, and on the other side, epistemic uncertainties,
which arise from
an absence of information, rare data, vagueness in parameter definition,
numerical implementation, or simplification and idealization processes
employed in the
All these conditions manifest as uncertain model
in some situations as uncertain initial or boundary conditions.
Consequently, the results
that are obtained for simulations that only use one specific set of
values as the most
likely ones for the model parameters cannot be considered as
representative of the whole
spectrum of possible model configurations. Furthermore, this fake
exactness provided by
the numerical simulation of models with uncertain but exact-valued
parameters can significantly aspect the comparison between numerical simulations and
Namely, such a comparison may be rated as unsatisfactory if the
results do not well match the experimental ones, even though it might be
satisfactory, if the uncertainties inherent to the models would have
taken into account in the simulation procedure.
While aleatory uncertainties have successfully been taken into account
by the use of probability theory and, in practice, by Monte Carlo simulation, the
additional modeling of
epistemic uncertainties still remains a challenging topic. As a
practical approach to solve
this limitation, a special interdisciplinary methodology to
comprehensive modeling and
analysis of systems has been developed which allows for the inclusion of
uncertainties – in particular of those of epistemic type – from the very beginning of the
This approach is based on fuzzy arithmetic, a special field of fuzzy set
has gained practical relevance after the introduction of the
Transformation Method in
The Transformation Method can be implemented in conjunction with any existing commercial software solution to perform a non-deterministic analysis on problems with uncertain parameters.
As a by-product, the Transformation Method provides special measures that quantify the influence of the uncertainty of each individual input parameter on the overall uncertainty of the output.
In addition to the propagation of uncertainties in a simulation process, it is possible to employ an inverse fuzzy arithmetical approach to validate simulation models as well as to assess the quality of the models.