Hardware Reference

In-Depth Information

bance models and measurement noises, we have [44][45]:

⎨

⎡

⎣
A
a
B
a
C
i
00

⎤

0 A
i
00

00A
o
0

000A
n

⎦

A =

,

⎡

⎣
B
a
D
i
00

⎤

⎦

⎡

⎣
B
a

⎤

⎦

B
i
00

0

0

0

0

B
1
=

,B
2
=

,

B
o

0

(3.57)

⎩

0

0 B
n

C
1
=

[ C
a
0,C
o
C
n
],

D
11
=

[ 0 D
o
,D
n
],

C
2
=

[ C
a

0,C
o
0],

D
21
=

0,

D
22
=

0.

Using LMI toolbox in MATLAB, matrices X, L, Y, F, Q, R, S and J can be

found by solving the matrix inequalities 3.56 to minimize trace(W). Hence an

optimal solution in the form of 3.54 can be obtained.

3.4.3 An Application Example

Let us consider the actuator and disturbance models given below [44]:

2.861 × 10
21

(s
2
+50.27s +1.579 × 10
4
)(s
2
+ 816.8s +1.668 × 10
9
)
,

G
p
=

1.3916
×
10
−5
(s + 575.8)(s + 575.6)(s
2
+0.04389s + 161.6)

(s
2
+ 315.5s +8.178 × 10
4
)(s
2
+ 315.4s +8.178 × 10
4
)

G
I
=

,

1.1695(s +1.431
×
10
4
)(s + 766.2)(s
2
+ 8609s +4.672
×
10
7
)

(s + 4630)(s + 1538)(s
2
+ 4451s +1.507 × 10
7
)

G
N
=

,

0.7016(s +1.271
×
10
4
)
2
(s
2
+6.125
×
10
−5
s +4.373
×
10
8
)

(s + 708.4)
2
(s
2
+0.0001317s +4.376 × 10
8
)

G
O
=

(3.58)

w
i
(i =1, 2, 3) are independent white noises with variance 1.

The optimal control obtained through the LMI approach is

G
c
(z)=
1
×
10
−5
(4.289z
5
−
3.541z
4
−
7.898z
3
+6.485z
2
+3.745z
−
3.073)

z
5
−0.3219z
4
−1.37z
3
+0.2231z
2
+0.4031z +0.06635

.

The frequency responses of the controller and open loop transfer functions are

shown in Figures 3.35 and 3.36.

The plant output disturbance is composed of two components. One is the

repeatable run-out (RRO), which is attributed to disk shift and written in po-

sition error. The RRO is phase-locked to the spindle rotation. The other is

the non-repeatable runout (NRRO), which is not phase-locked to the spindle

rotation, and is attributed to spindle, disk and actuator assembly resonances.