Signals and systems - Biomedical Engineering Desk Reference - page 69

Solution: Inertia is one property of mass that is defined as
an integral relationship between the mechanical variables
force ( F ) and velocity ( u ):

involving integral and differential operations, just as in an
analog model. In fact, it is a straightforward task to
convert from an analog model to a systems model, but
not vice versa. It requires some mathematical tricks to
structure calculus operations into a format that involves
simply algebraic multiplication.
Occasionally, the operation performed by an element
is simple, such as a scaling of the input; that is, the output
is the same as the input, but multiplied by a constant
gain. In such cases, the equation for G would be a simple
constant defining the multiplying gain. Under static or
steady-state conditions when the inputs have a constant
value, and all internal signals have also settled to a con-
stant value, the element equations, the G 's, can usually
be reduced to constants. If all the element functions are
constants, the model can be solved, that is, the value of
the output and all internal signals determined, using
algebra. A steady-state solution of a systems model is
given in Example 2.3.4 below.
One of the earliest physiological systems models, the
pupil light reflex, is shown in Figure 2.3-9 , and includes
two processes. The pupil light reflex is the response of
the iris to changes in light intensity falling on the retina.
Increases or decreases in ambient light cause the muscles
of the iris to change the size of the pupil in an effort to
keep light falling on the retina constant. (This system was
one of the first to be studied using engineering tools.) The
two-component system receives light as the input and
produces a movement of the iris muscles that changes
pupil area, the aperture in the visual optics. The first box
represents all of the neural processing associated with this
reflex, including the light receptors in the eye. It gener-
ates a neural control signal, which is sent to the second
box. The second box represents the iris musculature, in-
cluding its geometric configuration. The input to this
second box is the neural control signal from the first box
and the output is pupil area.
The systems model shown in Figure 2.3-9 demon-
strates the strengths and the weaknesses of systems

F ¼ mðdv=dtÞ

a modification of Newton's equation, F ¼ ma. To find the
velocity of the mass, solve for v in the above equation by
time integrating both sides of the equation:
ð Fdt ¼ mv ;

ð Fdt ¼ 1
2

ð 4 dt

v ¼ 1
m

v ¼ 4

2 t ¼ 2 ð 5 Þ¼ 10 cm = sec

2.3.3.3 Systems analysis and systems
models

Systems models usually represent whole processes using
so-called black box components. Each element of a sys-
tems model consists only of an input-output relationship
defined by an equation and represented by a geometric
shape, usually a rectangle. No effort is made to determine
what is actually inside the box; hence, the term black box.
The modeler pays no heed to what is the inside the box,
only its overall input-output (or stimulus/response)
characteristics. A typical element in a systems model is
shown graphically as a box or sometimes as a circle when
an arithmetic process is involved ( Figure 2.3-8 ). The
inputs and outputs of all elements are signals with a well-
defined direction of flow or influence. These signals and
their direction of influence are shown by lines and arrows
connecting the system elements ( Figure 2.3-8 ).
The letter G in the right-hand element of Figure 2.3-8
represents the mathematical operation that converts the
input signal into an output signal, usually expressed as
a ratio of output to input:

G ¼ Output
Input

; Output ¼ InputðGÞ [Eq. 2.3.6]

For many elements, the mathematical operation defined
by the letter G in Figure 2.3-8 can be quite complex,

Figure 2.3-9 A systems model of the pupil light reflex. Light falling
on the retina stimulates a neural controller that generates a neural
signal that is sent to the iris muscles, the plant or effector
apparatus. The system involves feedback because as the pupil
(the hole in the iris) reduces in size, it reduces the light falling on the
retina. This is considered a negative feedback system because
a positive increase in the response (in this case a reduction in
pupil size) leads to a decrease in the stimulus (i.e., the light
falling on the retina).

Figure 2.3-8 Typical elements in a system model. The left-hand
element is an 'adder whose output signal is the sum of the two
inputs x 1 and x 2 . The right-hand element is a general element that
takes the input (x 1 þ x 2 in this case) and operates on it with the
mathematical operation G to produce the output signal.