Biomedical Engineering Reference
In-Depth Information
6.2.6
Feedback Control for Attachment Point Motion
The sensitivity of the control force to the mismatch between the actual and
nominal impact velocities, which is characteristic of the open-loop control
of the point of attachment of the spring to the base, can be substantially
reduced by replacing the open-loop control by a feedback control using the
following relation:
X
=−
α(u
+
1
),
(6.20)
where
u
is the current control force (the spring tension) and
α
is the feed-
back gain, which should be taken to be reasonably large. To simulate the
behavior of the system under such a control law, one should add Eq. (6.20)
to Eq. (6.5) and substitute the expression of Eq. (6.14) for
u
. This results
in the system of equations
X
x
¨
+
k(x
−
X)
=
0
,
=
α (k(x
−
X)
−
1
) .
(6.21)
This system should be subjected to the initial conditions
x(
0
)
=
0
,
x(
0
)
=
v
0
,
(
0
)
=
0
.
(6.22)
The first two conditions coincide with those of Eq. (6.6) and the third con-
dition indicates that the spring has not been prestrained before the impact.
Solve the initial-value problem of Eqs. (6.21) and (6.22) and then calculate
the control force using Eq. (6.14) to obtain
e
λ
1
t
e
λ
2
t
,
k(v
0
α)
λ
1
−
λ
2
+
=−
−
=−
−
u(t)
k (x(t)
X(t))
(6.23)
where
λ
1
and
λ
2
are the characteristic numbers of the system of Eq. (6.21)
defined by
1
,
1
.
(6.24)
1
1
kα
2
4
kα
2
kα
2
4
kα
2
λ
1
=−
−
−
2
=−
+
−
For large
α
, the control law of Eq. (6.23) can be approximated by
e
−
kαt
e
−
t/α
.
u(t)
≈
−
(6.25)
The right-hand side of this relation involves two decaying exponentials
with substantially different time constants. The characteristic time of decay
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