Biomedical Engineering Reference
In-Depth Information
For the following discussion, assume that the object in the rigid-body
model is decelerated with the maximum intensity
U
during a time interval
[0
,T
] and comes to a complete stop at the instant
t
=
T
after the shock
pulse. In this case,
u
0
(t)
≡−
U
for
t
∈
[0
,T
] and Eq. (4.32) gives
1
exp
c
t
,
U
k
k
ξ
0
(t)
=
−
−
0
≤
t
≤
T.
(4.35)
Assumptions
Assumption 1.
The shock pulse
v(t)
has a finite duration
τ
,thatis,
v(t)
≡
0for
t
>
τ
.
Assumption
2.
The
optimal
control
in
the
rigid-body
model
is
a
constant-force control defined by
u
0
(t)
≡−
U,
0
≤
t
≤
T,
T > τ,
(4.36)
where
T
is a time instant at which the object comes to a complete
stop and, hence,
˜
x
0
(T )
=
0
.
(4.37)
Assumption 3.
The maximum of the absolute value of the function
y
0
(t)
˜
over the time interval 0
≤
t
≤
T
satisfies the relation
y
0
(T
∗
)
˜
=
t
∈
[0
,T
]
|˜
max
y
0
(t)
|
,
(4.38)
where
T
∗
is an instant of time at which the maximum occurs. This
implies, in particular, that
y
0
(T
∗
)
˜
≥
0.
Basic Proposition
Proposition 4.1.
The control law
−
U
for 0
≤
t
≤
T,
u
0
(t)
=
(4.39)
u(t)
¯
for
t > T,
where
u(t)
is the solution of the differential equation
¯
¯
c
¯
u
+
u
+
k
u
¯
=
0
(4.40)
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