Biomedical Engineering Reference
In-Depth Information
Now substitute
u
=˜
u
0
(the optimal control for Problem 4.4) into Eq. (4.82)
to obtain
U
k
.
J
1
(
˜
≤
˜
+
J
1
(
u
0
)
u
0
)
(4.86)
Since
u
0
is the optimal control for Problem 4.2, it follows that
J
1
(u
0
)
≤
J
1
(
u
0
).
˜
(4.87)
The inequalities of Eqs. (4.86) and (4.87) imply
U
k
.
J
1
(
J
1
(u
0
)
−
u
0
)
˜
≤
(4.88)
The
relations
of
Eqs.
(4.85)
and
(4.88)
give
the
desired
estimate
of
Eq. (4.74).
The inequality of Eq. (4.74) can be represented as
u
0
)
1
u
0
)
1
U
k J
1
(
U
k J
1
(
J
1
(
J
1
(
˜
−
≤
≤
˜
+
J
1
(u
0
)
(4.89)
u
0
)
˜
u
0
)
˜
to provide a two-sided estimate for the optimal value of the performance
index
J
1
in Problem 4.2. The lower and upper bounds are determined on the
basis of the solution of Problem 4.4 for a single-degree-of-freedom system.
The ratio
U
k J
1
(
η
=
(4.90)
u
0
)
˜
characterizes the accuracy of the approximation of the optimal value of the
performance index
J
1
(u
0
)
in Problem 4.2 (and, hence, in Problem 4.1) by
the value of the performance index
J
1
(
˜
u
0
)
, which is optimal for Problem 4.3.
The lower the value of
η
, the higher the accuracy. The quantity
η
measures
the ratio of the static deformation of the two-body model by the force
U
to
the peak magnitude of the displacement of the rigid body in Problem 4.4.
Proposition 4.3.
The absolute error of the approximation of the control
force
u
0
(t)
by the control force
u
0
(t)
in terms of the performance index
˜
satisfies the inequality
2
U
|
J
1
(u
0
)
−
J
1
(
u
0
)
˜
| ≤
k
.
(4.91)
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