Biomedical Engineering Reference
In-Depth Information
Eq. (3.9), have simple analytical solutions. Substitute
v(t)
of Eq. (3.24) into
the expressions of Eqs. (3.19) and (3.20) for the functions
ψ(t)
and
ψ(t)
to obtain
⎧
⎨
⎩
1
U ) t
2
2
(a
−
if 0
≤
t
≤
τ,
ψ(t)
=
(3.25)
a
tτ
2
τ
2
−
1
1
2
Ut
2
if
t
>
τ,
−
⎧
⎨
(
a
−
U
)
t
if 0
≤
t
≤
τ,
ψ(t)
=
(3.26)
⎩
aτ
−
Ut
if
t > τ.
ψ(t)
From the condition
=
0, calculate
aτ
U
t
∗
=
(3.27)
and then substitute the resulting
t
∗
for
t
into Eq. (3.25) to obtain
a(a
−
U)
2
U
τ
2
.
ψ(t
∗
)
=
(3.28)
In accordance with Eq. (3.22),
a(a
−
U)
J
1
(u
0
)
=
τ
2
,
(3.29)
2
U
and an optimal control providing the absolute minimum for the criterion
J
1
is defined as
⎧
⎨
aτ
U
,
−
U
if 0
≤
t
≤
u
0
(t)
=
(3.30)
if
t >
aτ
⎩
0
U
.
The product
aτ
measures the change in the absolute velocity of the base
due to the shock pulse. Therefore,
aτ
=
V
, and Eqs. (3.29) and (3.30) can
be represented as
1
V
τ
,
V
2
2
U
U
J
1
(u
0
)
=
−
(3.31)
⎧
⎨
V
U
,
−
U
if 0
≤
t
≤
u
0
(t)
=
(3.32)
⎩
if
t >
V
0
U
.
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