Biomedical Engineering Reference
In-Depth Information
performance characteristics of the proposed design or prototype with those
of the ideal isolator, an engineer can assess the prospects for improving
the design. Therefore, it is advisable that a design effort be preceded by a
limiting performance analysis of the respective class of systems.
A specific design of an isolation system is characterized by the definition
of the corresponding control force
u
as a function of the relative displace-
ment of the object (
x
), its relative velocity (
x
), and time (
t
). In accordance
with Eqs. (3.3) and (3.4), the relative motion of the object in response to a
prescribed shock pulse
v(t)
is governed by the differential equation
x
¨
=
u(x,
x,t)
˙
+
v(t)
(3.13)
subject to the initial conditions
x(
0
)
=
0
,
x(
0
)
˙
=
0
.
(3.14)
The solution of this initial-value problem
x(t)
is a function of time. This
motion generates a time history of the control force
u
[
t
]
=
u(x(t), x(t),t).
(3.15)
Equation (3.13) with the control
u(x, x,t)
replaced by the time history
u
[
t
]
of Eq. (3.15) subject to the initial conditions of Eq. (3.14) will have the
same solution
x(t)
. Therefore, when searching for the optimal control in
Problems 3.1 or 3.2 among the functions depending only on time, all pos-
sible response characteristics are taken into account. Thus, for the limiting
performance analysis, it suffices to solve the corresponding optimal control
problem in the class of open-loop controls depending only on time.
3.2.2 Shock Pulses with One Excursion beyond the Upper
Bound Allowed for the Control Force: Constant-Force
Deceleration
Suppose that the shock pulse
v(t)
has the form shown in Fig. 3.3. The
magnitude of the pulse is greater than the upper bound
U
allowed for the
control force
u
on the interval from the onset of the pulse until an instant
τ
and lies within the bounds of
|
v
|≤
U
for
t
≥
τ
,thatis,
v(t) > U
if 0
≤
t<τ,
(3.16)
|
v(t)
|≤
U
if
t
≥
τ.
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