Biomedical Engineering Reference
In-Depth Information
u(t)
of Eq. (4.55) for the extremum shows that
the first local extremum occurs at the time instant
t
∗
=
T
Analysis of the function
+
τ
∗
,where
τ
∗
satisfies the relations
c
2
4
ω
2
if
c
2
4
ωc
+
sinh
ωτ
∗
=
4
ω
2
,
cosh
ωτ
∗
=
4
−
k >
0
,
c
2
−
c
2
−
4
ω
2
if
c
2
4
c
(4.58)
τ
∗
=
4
−
k
=
0
,
c
2
4
ω
2
if
c
2
4
ωc
−
sin
ωτ
∗
=
4
ω
2
,
cos
ωτ
∗
=
4
−
k >
0
.
c
2
c
2
4
ω
2
+
+
A detailed derivation of these equations is chosen to be omitted. Substitute
the expressions of Eq. (4.58) into the relations of Eqs. (4.55) and (4.56) to
obtain
exp
−
2
cτ
∗
.
1
|¯
u(t
∗
)
|=|¯
u(T )
|
(4.59)
|¯
|
|¯
|
It is apparent that
u(t
∗
)
<U
since
u(T )
<U
. This completes the proof
of Proposition 4.1.
Construction of Optimal Control Force for Problem 4.1
Use
Eq. (4.18) to construct the optimal control force
F
0
(t)
for Problem 4.1.
Calculate the function
y
0
(t)
. To that end, differentiate the expression of Eq.
(4.42) with reference to Eq. (4.44) to obtain
¨
⎧
⎨
exp
c
t
U
c
k
˜
x
0
(t)
−
−
for 0
≤
t
≤
T,
y
0
(t)
˙
=
(4.60)
⎩
0
for
t > T,
exp
c
t
⎧
⎨
Uk
c
2
k
˜
x
0
(t)
+
−
for 0
<t<T,
y
0
(t)
¨
=
(4.61)
⎩
0
for
t > T.
x
0
(t)
is the solution of Problem 4.4. Therefore, in accor-
dance with Eqs. (4.28), (4.29), (4.36), and (4.37), this function satisfies the
differential equation
The function
˜
x
0
=−
U
+
v(t),
0
≤
t
≤
T
(4.62)
and the boundary conditions
˜
˜
x
0
(
0
)
˜
=
0
,
x
0
(
0
)
=
0
,
x
0
(T )
=
0
.
(4.63)
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