Biomedical Engineering Reference
In-Depth Information
For the system governed by the differential equation of (7.6) subjected
to the initial conditions of Eq. (7.7), find an optimal control
u
u
0
(t)
in the class of integrable functions that minimizes the peak displacement
J
1
, provided that the criterion
J
2
does not exceed the prescribed positive
number
H
,thatis,
=
=
{
|
≤
}
J
1
(u
0
)
min
u
J
1
(u)
J
2
(u)
H
.
(7.10)
This problem is characterized by four parameters,
m
,
v
0
,
,and
H
.The
transition to the dimensionless (primed) variables
v
0
H
2
/
3
H
2
/
3
v
8
/
3
H
2
/
3
v
5
/
3
u
m
x
=
t
=
u
=
x,
t,
,
0
0
(7.11)
H
2
/
3
v
5
/
3
H
2
/
3
v
8
/
3
1
H
J
2
=
J
1
=
J
2
=
,
J
1
,
0
0
reduces this number to 1. Use the variables of Eq. (7.11) in Eqs. (7.6) - (7.10)
and omit the primes to obtain the relations of the same form but with
m
=
1,
v
0
=
1. The only free parameter remaining after this change of
variables is the dimensionless parameter
.
When constructing the solution of the problem, the dimensionless vari-
ables will be used. The primes will be omitted, apart from the cases where
the dimensionless variables are considered along with the dimensional ones.
1, and
H
=
7.2.2
Construction of the Solution
To determine the minimum of the criterion
J
1
, it suffices to solve the prob-
lem stated in the previous section on the time interval [0
,T
], where
T
is
the instant (unknown in advance) at which the velocity
x(t)
vanishes for
˙
the first time. For the control defined as
u(t)
≡
0for
t > T
, the point mass
remains indefinitely at the position
x(T )
.
To construct the solution, a number of transformations will be performed.
Integrate Eq. (7.6) subject to the initial condition of Eq. (7.7) to obtain the
expression for the velocity
t
x
˙
=
1
−
v(t),
v(t)
=−
u(τ ) dτ.
(7.12)
0
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