Page 690 - The Mechatronics Handbook

Identification of Ignorance

The final role of modeling is to help us identify our ignorance. This takes two forms. First, the construction
of the model often requires us to name parameters whose numerical value is unknown. This leads us to
estimate the possible range of such parameters or to design experiments to determine the parameter’s value.
Identifying missing details is the first step in determining them. In this way models help us identify our
ignorance of details of the problem or our proposed solution. Secondly, the predictions we make with the
help of our models are testable. The adequacy of a model to describe system behavior can be determined by
comparing actual system behavior with predictions based on that model. Differences between actual system
performance and a model’s predictions point to errors in the model that may be significant. By testing the
validity of our models we identify aspects of the real system that are inadequately reproduced in our model
or artifacts of the model that do not exist in the real system. These discrepancies may point to an inappropriate
assumption or to a fundamental misunderstanding of the physical system. In either case, the source of these
discrepancies should be understood in order to determine if our model is adequate for its intended purpose.

22.3 Modeling of Systems and Signals

In the area of systems and signals we typically deal with the modeling of system dynamics and information
flows through the system and its environment. As can be seen from the discussion above, this is but a
small part of the modeling needed for system design. However, there are a number of decisions that must
be made in selecting a model for the behavior of a dynamic system. These include the choice between
analytic and numerical modeling, the use of distributed or lumped parameters, the approach used to
model random factors, and the choice between linear and nonlinear models. These issues are considered
in the paragraphs below.

Analytical vs. Numerical Models
In this area both analytical and numerical tools are available to support our efforts, so we have a number
of decisions to make regarding the level of complexity of the models we choose to use for various tasks.
Analytical tools provide insights into overall system behaviors and can allow us to make comparisons
between dissimilar systems based on a similarity in their models. These analogous systems often prove
useful in the concept generation phase of design because they allow us to bring to bear solutions to
diverse problems to the solution of the problem at hand. However, as described above, these analogies
usually break down upon closer inspection. In the case of analogies in dynamics, they are typically based
on systems described by equivalent differential equations (e.g., electronic and mechanical oscillators).
However, these linear differential equations are but simplifications that ignore nonlinearities and time-
dependent behavior that may greatly affect system behavior. Therefore, when applying analogies one
must always be careful to explicitly state assumptions made in the modeling process.
Numerical tools, on the other hand, do not yield the same insights of analytical tools, but provide
other and more detailed insights into real system behavior and performance. Typically these tools become
most useful later in the design process, both for tradeoff analysis and for detailed design work. The
availability of these tools opens many doors to modeling options that have in the past been closed and
allows us to consider options that have historically been considered unwieldy. For example, numerical
solution of nonlinear differential equations allows modeling of systems that had previously been
approached only with linear approximations.

Partial vs. Ordinary Differential Equations
One modeling choice that must be made is whether we must model the dynamics of the system using

ordinary or partial differential equations. Partial differential equations must be used whenever values of
interest vary spatially as well as temporally. For example, if we are considering the design of a robot arm,
we are free to use ordinary differential equations if we consider the arm to be rigid, partial differential

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