Biomedical Engineering Reference
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the rattlespace [Eq. (3.6)],
J
1
=
max
t
x(t)
−
min
t
x(t),
(3.55)
and the peak magnitude of the force transmitted to the object [Eq. (3.7)],
J
2
=
max
t
|
u(t)
|
.
(3.56)
For the discrete-time model, the maximization or minimization over the
continuous time
t
should be replaced by the maximization or minimiza-
tion with respect to the index of the nodal points of the discretization.
Let
J
1
d
,
J
1
d
,and
J
2
d
denote the discrete-time analogues of the criteria of
Eqs. (3.54) - (3.56). Then, accordingly,
J
1
d
=
max
k
|
x
k
|
,
k
=
0
,
1
,...,N,
(3.57)
J
1
d
=
max
k
x
k
−
min
k
x
k
,
k
=
0
,
1
,...,N,
(3.58)
J
2
d
=
max
k
|
u
k
|
,
k
=
1
,...,N.
(3.59)
3.3.2
Numerical Solution of Problem 3.1
Discrete-Time Statement
In terms of the discrete-time model of
Eq. (3.49) and the performance criteria of Eqs. (3.54) - (3.59), Problem 3.1
is stated as follows: For the system of Eq. (3.49), find an optimal set of
the control variables
u
i
,i
1
,...,N
, that minimize the criterion
J
1
d
(or
J
1
d
) subject to the constraint
J
2
d
=
≤
U
which can be represented as
|
u
i
|≤
U,
i
=
1
,...,N.
(3.60)
Minimization of the Peak Magnitude of the Displacement
The
peak magnitude of the relative displacement of the object,
J
1
d
, can be
defined as the least upper bound for the quantities
|
|
=
x
k
,k
1
,...,N
,
that is,
J
1
d
=
D
0
{
D
0
||
x
k
|≤
D
0
,k
=
0
,
1
,...,N
}
,
min
(3.61)
where
D
0
is an auxiliary variable. The inequality
|
x
k
|≤
D
0
is equivalent
to simultaneous linear inequalities
x
k
≥−
D
0
and
x
k
≤
D
0
. Therefore,
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