Biomedical Engineering Reference
In-Depth Information
The initial conditions of Eq. (4.41) for the variable
u
follow from the
ξ
at the time instant
T
:
conditions for the variables
ξ
and
ξ(T)
=
ξ
0
(T ),
ξ(T)
=
0
.
(4.53)
The first of these conditions follows from the relation
ξ(t)
=
ξ
0
(t)
,which
is valid for 0
≤
t<T
, and the continuity of the function
ξ(t)
. The second
condition follows from the relation
˜
x(T )
=
0 of Eq. (4.37), the continuity of
the function
x(t)
, and the first relation of Eq. (4.50). To obtain the condition
˙
ξ(T)
ξ(t)
0.
To obtain the first relation of Eq. (4.41), substitute the initial conditions
of Eq. (4.53) for the variables
ξ
and
ξ
into Eq. (4.51) and use the definition
of Eq. (4.52) for the variable
=
0, proceed in the relation
=˙
x(t)
to the limit as
t
→
T
+
u
. To obtain the second relation, differentiate
Eq. (4.51) with respect to time and use Eq. (4.52) to arrive at the relation
¯
¯
k ξ
u
=−
c
u
¯
−
(4.54)
k ξ
0
(T )
.
It remains to prove the inequality of Eq. (4.46) for the control
ξ(T)
and then use the conditions
=
0and
u(T )
¯
=−
u(t)
.The
solution of the initial-value problem of Eqs. (4.40) and (4.41) gives
¯
k ξ
0
(T )
exp
−
T)
(t
1
u(t)
¯
=
2
c(t
−
−
T),
(4.55)
where
⎧
⎨
c
2
ω
sinh
ωt
1
4
c
2
−
cosh
ωt
if
−
k >
0
,
1
1
4
c
2
2
ct
−
1
if
−
k
=
0
,
(t)
=
⎩
k
.
(4.56)
c
2
ω
sin
ωt
1
4
c
2
1
4
c
2
−
cos
ωt
if
−
k<
0
,
ω
=
−
Equation (4.40) coincides in form with the governing equation for a
damped linear oscillator. Due to the energy dissipation, the maximum of
the quantity
occurs either at the initial time instant (
t
=
T
in the case
under consideration) or at the instant of the first local extremum of the
function
|
u
|
u(t)
vanishes.
u(t)
when the derivative
At the time instant
t
=
T
the constraint
|¯
u
|≤
U
holds, since, in accor-
dance with Eqs. (4.41) and (4.35),
|
u(T )
|=
k ξ
0
(T )
=
U
1
exp
c
T
.
k
−
−
(4.57)
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