Hardware Reference
In-Depth Information
Substituting equation 3.72 into equation 3.73 yields
⎡
⎣
λ
1
λ
q−1
⎤
⎦
⎡
⎣
a
i,q
⎤
⎦
···
λ
1
1
1
λ
2
λ
q−1
a
i,q−1
.
a
i,0
···
λ
2
1
2
.
λ
q
m+n
λ
q−1
m+n
,
··· λ
m+n
1
⎡
⎣
⎤
⎦
−N
a
(λ
1
)
−1
N
p
(λ
1
)d
m
(λ
1
)
−N
a
(λ
2
)
−1
N
p
(λ
2
)d
m
(λ
2
)
.
−N
a
(λ
m+n
)
−1
N
p
(λ
m+n
)d
m
(λ
m+n
)
=
.
(3.74)
If q + l = m + n
d
+1andifD does not have repeated roots, then the
first term in the above equation is a non-singular square matrix, and n
i
and n
can be easily calculated by solving equation (3.74). Furthermore, d has been
specified by equation (3.70). Hence we have the I
V
=
d
.
Example: Let the actuator transfer function be G
p
(s)=
1×10
4
s
2
,whichis
controlled by a PD controller
G
c
(s)=
0.014s +1
0.00014s +1
.
(3.75)
Following the above equations,
s
3
+ 714.3s
2
s
3
+ 714.3s
2
+10
5
s +7.143 × 10
6
,
S =
100000s +7.143 × 10
6
s
3
+ 714.3s
2
+10
5
s +7.143 × 10
6
.
T =
(3.76)
The poles of the closed-loop system are:−848.97,−75.516 + 77.95i,−75.516−
77.95i.
We wish to inject the IVC signal from the reference signal point r.Letthe
transfer function from the initial position value to plant output considering I
v
:
y =(S + I
v
T )y(0).
(3.77)
Solution 1:
Following the equations given above, we have N
a
=(s
2
(s+714.2857). The
roots of N are λ
i
= 100 × [−5.5802,−0.7813 + 0.8183i,−0.7813−0.8183i]. d
m
is to be 20 times faster than these poles which can be determined. Substituting
into equation 3.74 we have a
i
=9.7843×10
−6
, 6.6852×10
−3
, 3.1529×10
−2
, 0.
Hence
5.591 × 10
5
s
3
+3.82 × 10
8
s
2
+1.802 × 10
9
s
s
4
+1.304 × 10
4
s
3
+2.625 × 10
7
s
2
+5.895 × 10
10
s +4.082 × 10
12
.
(3.78)
I
v
=
n
d
=
