Biomedical Engineering Reference
In-Depth Information
FIGURE 4-13
Float controller
reservoir for a
heart-lung machine.
In a real system, the input may change in an arbitrary way, but to quantify its behavior
responses are usually determined for impulses, steps, ramps, and sinusoidal inputs. If the
system is linear and time invariant (LTI), the output will be the algebraic sum of any
number of responses to any of these functions.
It is, in fact, possible to construct any of the inputs described from the sum of a
sequence of impulses, and therefore the impulse response provides a common method to
characterize a system in the time domain. However, the manipulation of sums of impulse
responses is cumbersome, so another method of analysis, the Laplace transform, is easier.
Consider the simple example of a float-valve controller that could be used to maintain
the level of blood in an open reservoir in a heart-lung machine, as shown in Figure 4-13.
In this system, the rate at which blood enters the tank is dependent on the difference
between the current blood depth,
h
(m), and the final depth,
h
fin
(m).
A simple first-order differential equation can be written
du
dt
=
k
(
h
fin
−
h
)
(4.62)
The solution of this form of differential equation can be determined by inspection
e
−
kt
h
=
h
fin
(
1
−
)
(4.63)
In the case of a second-order differential equation such as the mass-spring-damper system
discussed earlier, the response is described by the natural frequency of the system and the
damping ratio. Depending on the magnitudes of these parameters, the response to a step
input could be an undamped sinusoid, a damped sinusoid, or an exponential, as shown in
Figure 4-7.
It is often more difficult to fit the response of a system to one of the standard solutions,
particularly if the order is high and there are feedback components. This is another reason
for describing systems in terms of their Laplace transforms, which make them far easier
to manipulate and solve. The process is shown in Figure 4-14.
For a function,
f
(
t
)
, in the time domain, the Laplace transform is
∞
f
(
t
)
e
−
st
dt
F
(
s
)
=
(4.64)
0
Under normal circumstances it is not necessary to evaluate this integral because tables
have been compiled for the most commonly occurring functions, which, in conjunction
with a few rules for handling combinations of transforms, allow for the solution of most
problems encountered in biomechatronics and other disciplines. Some of the rules are
listed in Table 4-5.
