Biomedical Engineering Reference
In-Depth Information
With reference to Eq. (3.16) it follows that
ψ
increases monotonically on
the interval 0
<t<τ
from zero to some positive value and then decreases
monotonically without limit. Therefore, at some time instant
t
∗
> τ
the
derivative
ψ
vanishes and the function
ψ(t)
attains a maximum. From
Eq. (3.18) it follows that max
t
x(t)
≥
max
t
ψ(t)
and, hence, max
t
x(t)
≥
ψ(t
∗
)
. Using the last inequality, the relation max
t
|
x(t)
|
>
max
t
x(t)
,and
the definition of Eq. (3.5) for the criterion
J
1
, we obtain
J
1
≥
ψ(t
∗
).
(3.21)
A control force that is identically equal to
t
∗
and is arbitrarily defined on the interval
t > t
∗
, provided that the inequalities
|
−
U
on the interval 0
≤
t
≤
U
hold for
t > t
∗
, ensures the lower bound
of Eq. (3.21) for the performance index
J
1
. Hence, this control force
characterizes the optimal response
u
0
(t)
of the isolator to the shock pulse
corresponding to the limiting performance, and the absolute minimum
of
x(t)
|≤
ψ(t
∗
)
and
|
u(t)
|≤
the
peak
magnitude
of
the
relative
displacement
of
the
object
is
given by
J
1
(u
0
)
=
ψ(t
∗
).
(3.22)
The corresponding peak magnitude of the load transmitted to the object,
J
2
(u
0
)
=
U,
(3.23)
takes on its upper bound and, therefore, Problems 3.1 and 3.2 are reciprocal
to one another.
If the magnitude of the acceleration of the base due to the shock pulse
does not exceed the maximum magnitude of the absolute acceleration
allowed for the object during all time, which corresponds to
|
v(t)
|≤
U
0, then the optimal control is
u
0
(t)
for
t
v(t)
. For this control, the
object moves as if it were rigidly attached to the base and, accordingly,
J
1
(u
0
)
≥
≡−
=
0.
Rectangular Pulse
The rectangular pulse
⎧
⎨
a
if 0
≤
t<τ,
=
v(t)
(3.24)
⎩
a
>
U,
0 f
t
≥
τ,
belongs to the class of disturbances characterized by Eq. (3.16). For this
disturbance, Problem 3.1, stated by Eq. (3.8), and Problem 3.2, stated by
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