Biomedical Engineering Reference
In-Depth Information
u
ff
(
t
)
Q
(
q
)
+
f
(
t
)
f
d
(
t
)
u
(
t
)
K
P
+
e
(
t
)
u
fb
(
t
)
−
Figure 16.12
Two D.O.F. adaptive control.
As shown in Figure 16.12,
P
is an unknown transfer function from the control input
u
(
t
) to the
control output
f
(
t
),
K
, and
Q(
u
)
are the feedback and feedforward controllers, respectively,
with the adjustable parameter
u
.
P
is assumed unknown and is stable and has stable inverse.
The objective here is not only limited to the convergence of the control error
e
(
t
)
!
0asin
the previous adaptive control or learning control researches, but also to realize the inverse of
P
by
Q(
u
)
. It is clear that, once
Q
(
u
)
!
P
1
, then the control system can track any other types of desired
signals without any loop delay. The only condition here is to perform an adaptation process with
respect to a desired input that satisfies a PE condition that will be explained later in the theorem.
Firstly, let us describe the state space equation of the unknown
P
1
as
d
h
1
(
t
)
dt
¼
F
h
1
(
t
)
þ
gf
d
(
t
)
(16
:
24)
d
h
2
(
t
)
dt
¼
F
h
2
(
t
)
þ
gu
0
(
t
)
(16
:
25)
u
0
(
t
)
¼
c
0
h
1
(
t
)
þ
d
0
h
2
(
t
)
þ
k
0
f
d
(
t
)
¼ u
0
h
(
t
)
(16
:
26)
k
0
T
,
h
(
t
)
¼
[
h
1
(
t
)
h
2
(
t
)
f
d
(
t
)
T
u
0
¼
[
c
0
d
0
Here,
F
is any stable matrix and
g
is any vector with (
F
,
g
) being controllable. In Equation (16.26),
c
n
]
T
c
0
¼
[
c
1
c
2
d
n
]
T
d
0
¼
[
d
1
d
2
and
k
0
are unknown parameters to be estimated. The parameterization of (16.24)-(16.26) can yield
an arbitrary transfer function from
f
d
to
u
0
(
t
). If the matrices
F
and
g
are represented in the
controllable canonical form
2
4
3
5
010
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
T
F
¼
.
.
.
.
,
g
¼
0
½
01
0
0
01
f
1
f
2
f
2
then the transfer function from
f
d
to
u
0
can be calculated as
T
f
d
,
u
0
¼
P
1
(
s
)
¼
k
0
þ
c
0
(
sI
F
)
1
g
¼
k
0
s
n
þ
(
f
n
k
0
þ
c
n
)
s
n
1
þþ
(
f
1
k
0
þ
c
1
)
s
n
þ
(
f
n
d
n
)
s
n
1
þþ
(
f
1
d
1
)
(16
:
27)
1
d
0
(
sI
F
)
1
g
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