Biomedical Engineering Reference
In-Depth Information
4.1.5
Constant-Force Control versus Optimal Control
The expression of Eq. (4.72) obtained in Section 4.1.3 for the optimal
control force
F
0
(t)
for Problem 4.1 and Fig. 4.6 show that even if the
optimal control
u
0
(t)
for Problem 4.2 is constant on the interval 0
T
,
the optimal control
F
0
(t)
is not constant on this interval and, moreover,
involves impulse components. The near-optimal control
F
0
(t)
constructed
for Problem 4.1 according to the procedure of Section 4.1.4 also contains
impulsive components. At the same time, the optimal control for Problem
4.3 (for the rigid-body model) could seem to be a near-optimal control for
Problem 4.1 if the stiffness of the spring between bodies 1 and 2 is high.
This control does not contain impulse components and for a number of
typical disturbances is a constant-force control. However, this conjecture is
superficial and the situation is more complicated.
Consider a system in which the parameters and the external disturbance
coincide with the respective characteristics of Example 4.1. The optimal
control
F
0
in Problem 4.3 for the rigid model subject to an instantaneous
shock
v(t)
≤
t
≤
=
Vδ(t)
can be taken in the form
⎧
⎨
m
1
+
m
2
−
P
if 0
≤
t
≤
T,
m
1
F
0
(t)
=
(4.99)
m
1
V
P
⎩
0
if
t > T,
T
=
.
To obtain this relation, use the control of Eq. (3.40) as an optimal control
for Problem 4.4 for an instantaneous shock
v
Vδ(t)
. The variables
u
and
U
in Problem 4.4 are related to the variables
F
and
P
in Problem
4.3 by the expressions
u
=
P/m
1
of Eq. (4.27).
Applying these relations to the optimal control found for Problem 4.4 leads
to Eq. (4.99)
The corresponding law of motion of the rigid model of the object relative
to the base is represented as
=
F/(m
1
+
m
2
)
and
U
=
⎧
⎨
Pt
2
2
m
1
Vt
−
for
0
≤
t
≤
T,
x
0
(t)
˜
=
(4.100)
⎩
m
1
V
2
2
P
for
t > T.
J
1
(F )
, defined in (4.22), is given by
The minimum value of the criterion
m
1
V
2
2
P
J
1
( F
0
)
=
.
(4.101)
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