Biomedical Engineering Reference
In-Depth Information
Solution: Generate an algebraic equation based on the
configuration of the system and the fact that the output
of each process is just the input multiplied by the asso-
ciated gain term as stated in Eq. 2.3.6.
By definition : G ¼ Out=S 1; H ¼ S 3 =Out ;
and
S 1 ¼ In S 3
where G and H are constants
Hence S 1 ¼ Out=G ;
S 3 ¼ OutðHÞ
Since S 1 ¼ In S 3 substituting in the above :
Out=G ¼ In OutðHÞ
The plot of maximum output values (i.e., variable
output) is a straight line indicating a linear relationship
between the amplitude of output and input signal.
Whatever the process really is, it appears to be linear.
Rearranging : Out ¼ InðGÞOutðGHÞ ;
Outð 1 þ GHÞ¼InðGÞ
Out
In
G
1 þ GH
¼
[Eq. 2.3.7]
The solution is the classic feedback equation. Since the
two elements, G and H, could be represented by simple
gain constants, algebra alone can be used to work out the
input-output equations. What of more complicated sit-
uations where the model is not in steady state and/or the
processes must be defined using differential and integral
operations?
2.3.3.4 Systems and analog analysis:
summary
The basic differences and relative strengths and weak-
nesses of systems analysis versus analog analysis have al-
ready been described. Systems analysis only tries to
represent the behavior of a process whereas analog
analysis makes some effort to mimic the way in which
the process produces that behavior. This is done in analog
models by representing the process using elements that
are, to some degree, analogous to those in the actual
process. Analogous elements have the same general be-
havior as the physiological elements they represent;
hence, analog models usually represent the system at
a lower level, and in greater detail, than do systems
models. However, not all analog models offer this detail.
This can be seen in the windkessel cardiovascular model
of Figure 2.3-5 . The single capacitor, Cp, represents the
combined elastic behavior, or springlike characteristics,
of the entire arterial tree. Analog models often provide
better representation of secondary features such as
energy use, which is usually similar between analog ele-
ments and the actual components they represent.
Figuring out exactly what the process is solely by testing
it with external signals can be a major challenge. The field
of system identification deals with approaches to obtain
a mathematical representation for an unknown system by
probing it with external signals. A few examples and
problems later in this text show some of the techniques
used to evaluate linear systems in this manner. Comparing
the input and output sinusoids it looks like process_x
contains a derivative and a multiplying factor that increases
signal amplitude, but more input-output combinations
would have to be evaluated to confirm this guess.
Example 2.3.4: Find the overall input-output relation-
ship for the systems model below. Assume that the
system is in steady-state condition so that all the signals
have constant values and the two elements, represented
by the equations G and H, are simply gain constants.
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