Du C., Xie L. Modeling and Control of Vibration in Mechanical Systems

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34 Modeling and Control of Vibration in Mechanical Systems

−4 −3 −2 −1 0 1 2 3 4

−2

−1.5

−1

−0.5

0.5

1.5

x 10

−3

Displacement (V, 0.5 µm/V)

Control signal u (V)

FIGURE 2.13

Control signal u versus displacement x.

FIGURE 2.14

VCM actuator modeling with friction nonlinearity model F (x).

Modeling of Disk Drive System and Its Vibration 35

In what follows, we shall ﬁnd F (x) to model the relationship between f and x.

The above mentioned operator b ased method to approximate hysteresis will be ap-

plied to model the hysteresis of f and x as shown in Figure 2.6.

in (2.21 ) can be cho sen as the amplitud e of x. Since with the chosen peak-to-

peak values of x = 0.5 , 1 and 3V, the r

are respectively

= 0.25; r

= 0 .5; r

= 1 .5. (2.32 )

With m = 3, n = 3 and the initial valu es π

= 0, ξ

= 0, after 2000 iteratio ns, a

minimum error e is achieved and the optimal parameters are obtained as

a = 9.0024,

= −1.5783, w

= 0.1667, w

= −0.0655,

= −0.1431, w

= −7.3341, w

= 0.4403,

which gives the minimal kT

∞

= 0.08.

With these parameters, f

= F (x(k)) can be calculated fro m (2.20). The time

traces of f

and f are compared in Figure 2.15. It is observed that the time trace

from the model (2.20) can closely track f, and the error f −f

is small. The modeled

hysteresis from x to f

is drawn and compared with the measured one i n Figure 2.6.

It is seen that the modeled hysteresis and the measured one are close to each other.

If th e creep term i n the model (2.20) is removed, i.e., w

= 0, j = 1, ···, m, the

modeling accuracy decreases. Indeed, in this case, the minimal kT

∞

= 0.1293 >

0.08.

Figur es 2.16 and 2.17 show that the quality of the agreement in the frequency

domain between 70 Hz and 150 Hz decreases with lower excitation amplitude espe-

cially f or the 0.5V case.

Note that the operator model f

= F (x(k)) describes the hysteretic characteristics

in Figure 2.6 as a mapping between the actuator position x and the friction force

f. It turns out that the model makes it possible to approximate hysteretic transfer

characteristics withou t modeling the underlying physics. This is different from the

friction models such as the preload and two-slope model i n [40][42]. Compared to

the mod el in [40] [42], an advantage of our operato r based model is that the frequency

response of the actuator can also ﬁt well to the measured one, as shown in Figures

2.16 and 2.17 .

For comparison, we also apply the preload and two slope model [40] for the hys-

teresis, as shown in Fi gure 2.18. The preload model for velocity v and the two-slope

model for position x are given by

= k

v + k

sgn(v), (2.33)

and



x, |x| ≤ s

x + (k

−k

, |x| > s

36 Modeling and Control of Vibration in Mechanical Systems

Using the data with the amplitude of x(t) of 0.25V, we obtain that

= 1.85e − 4, k

= 0.48, k

= 0.0145, k

= 0.0014, s

= 0.009.

Using this model, a comparison with the measured data in the time domain is

shown i n Figure 2.19 for the amplitude of x(t) of 0.5 and 1 V, respectively. It can

be o bserved that the plant input u versus position x ﬁts reasonably well to the mea-

sured results in the time domain. However, in the frequency domain, the magnitude

response f or the case of the amplitude of x(t) of 0.5V, plotted as the dott ed curve in

Figur e 2.17 devi at es signiﬁcantly from the measurement results.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

−2.5

−2

−1.5

−1

−0.5

0.5

1.5

2.5

x 10

−3

Time(sec)

Friction and error

From measurement (f)

From modeling(f

)

Error (f−f

)

FIGURE 2.15

Measured and modeled friction and error.

Modeling of Disk Drive System and Its Vibration 37

FIGURE 2.16

Actuator frequency response for sinusoidal reference with amplitude of 1 and 3 V,

respectively.

FIGURE 2.17

Actuator f requency response for sinusoid al reference with amplitude of 0.5 V.

38 Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.18

Preload and two-slope model for friction modeling.

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

−1

−0.8

−0.6

−0.4

−0.2

0.2

0.4

0.6

0.8

x 10

−3

Displacement (µm)

Input voltage u

Measured

Modeled

0.5V

FIGURE 2.19

Plant input voltage u versus displacement x.

Modeling of Disk Drive System and Its Vibration 39

2.4 Vibration modeling

Vibration in disk drives causes the deviation of the R/W head po sitioning from the de-

sired track center. It is the combin at ion of the repeatable r unout which is synchronous

with the spin dle revolut ion and the nonrepeatable runout. Figure 2.20 shows a sim-

pliﬁed block-diagram of disk drive servo loop. y is the position of the R/W head and

e is the position error signal. The si gnal d

represents all the torque disturb ances

to the syst em. Such disturbances include any torque due to air-turbulence force to

the actuator, the suspension and the slider. The effects of the t orque disturbances are

dominant at frequencies that are r elat ively low when compared to the servo band-

width. The sign al d

represents d isturbances that are due to non-repeatabl e motions

of the disk and motor, suspension and slid er vibrations, which dir ectly add to the

relative p osition of the R/W head and the servo track. The noise si gnal n includes

media and head sensin g noises and also represents the effects of the PES demodu-

lation noise which includes actual electrical noise and A/D quantization noise. The

noise signal n is thus reasonably modeled as a broad-band white noise.

2.4.1 Spectrum-based vibration modeling

FIGURE 2.20

Closed-loop control system disturbances d

, d

and noise n.

Since a controlled closed-loop can provid e steady signals, the vibration source

analysis is based on the signal that is collected f rom the closed-loop system. From

Figur e 2.20 ,

e(k) = −P (z)S(z)d

(k) − S(z)d

(k) + S(z)n(k), (2.34)

40 Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.21

Closed-loop control system with di sturbance and noise models.

where P (z) is the transfer function of the discretized plant model P (s) and the sen-

sitivity function or error rejection function is given by

S(z) =

1 + P (z)C(z)

. (2. 35)

Figur e 2.22 shows the sensitivity function S(z) of a closed-loop control system.

Assume that d

, d

, and n are uncorrelated. The power sp ectrum denoted by S

the error signal e is given by

= |P (z)S(z)|

(k)|

+ |S(z)|

(k)|

+ |S(z)|

|n(k)|

. (2.36)

Figur e 2.23 shows the spectrum of NRRO component in the error signal e. Two

humps are obviously observed in the baseline curve. One is in t he fr equ ency range

lower than 100 Hz, th e other one is after 500 Hz. Considering (2.36) and Fig ure

2.22, the second hump is caused by S(z) through |S(z)|

|n(k)|

, and the ﬁrst hump

is due to d

through P (z)S(z) with a hump in a lower frequency range. Hence the

disturbance and noise modeling can be carried out as foll ows.

In Fig ure 2.21, models D

(s), D

(s) and N(s) are used to describe d

(s), d

(s)

and n(s) respectively, and w

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

variance 1. Eq. (2.36) becomes

= |P (z)S(z)|

(z)|

+ |S(z)|

(z)|

+ |S(z)|

|N(z)|

, (2.37)

where D

(z), D

(z) and N (z) are the discrete forms of D

(s), D

(s) and N (s),

respectively. D

(z) and N (z) can be determined by ﬁtting wei ghted versions of

P (z)S(z) and S(z) to the baseline curve of the spectrum and the spik es are consid-

ered as the effect of D

(z). Hence the steps to obtain D

, N and D

are

Modeling of Disk Drive System and Its Vibration 41

FIGURE 2.22

Sensitivity function S(z).

42 Modeling and Control of Vibration in Mechanical Systems

FIGURE 2.23

PES NRRO spectrum.

Step 1). Find S

(j),

(j) =

min

i=1+(j−1)q

(i), j = 1, 2, ···, L/q, (2.38 )

where L is the length of S

and q is as small as possibl e.

Step 2). Compute D

(z),

(z)|

= W

(z)S

/|P (z)S(z)|

, (2. 39)

where W

is a low-pass weighting function used to select S

in low frequency range.

Step 3). Compute N(z),

|N(z)|

= W

/|S(z)|

, (2.40)

where W

is the high-pass weighting function to select S

in h igh frequency range.

Step 4). Till now, the baseline curve S

can be ﬁt wel l by the i dentiﬁed D

and N.

The remaining part o f the spectrum is then regarded as D

. Thus,

|D(z)|

= {S

− [|P (z)S(z)|

(z)|

+ |S(z)|

|N(z)|

]}/|S(z)|

. (2.41 )

Modeling of Disk Drive System and Its Vibration 43

(s), D

(s) and N (s) are then obtained as follows from D

(z), D

(z) and

N(z) using the bilinear approxi mation method [81].

(s) =

1.3916 × 10

−5

(s + 575.8)(s + 575.6)(s

+ 0.04389s + 161.6)

+ 315.5s + 8.17 8 × 10

)(s

+ 315.4s + 8.178 × 10

)

, (2.42)

(s) =

0.701588(s + 1 .271 ×10

)

+ 6.125 × 10

−5

s + 4.373 ×10

)

(s + 708.4)

+ 0.0001317 s + 4.376 × 10

)

(2.43 )

N(s) =

1.1695(s + 1.431 × 10

)(s + 766 .2)(s

+ 8609s + 4.672 × 1 0

)

(s + 4 603)(s + 1538)(s

+ 4451s + 1.507 × 10

)

. (2.44)

With t he disturbance and noise models, the feedback controller C(z) can be opti-

mized to minimize the erro r due to w = [w

]

by using the H

optimal control

method or other advanced cont rol methods.

2.4.2 Adaptive modeling of disturbance

2.4.2.1 Neural network approximation

A neural network usually consists of a large number of simpl e processing elements

called nodes. The nodes are interconnected by weighted links wit h weight param-

eters adjustable. The different arrangement of the nodes and the interconnections

deﬁnes various architectures of neural networks [73][74], which are suitable to dif-

ferent kinds of applicati ons. In control engineering, a multi-layer neural network

is usually used to generate the mapping from input to output since it can approxi-

mate any function under mild assumption with any desired accuracy. The function

approximation is deﬁned as follows.

Deﬁnition 2.1 [75] If f(x) : R

→ R

is a continuous vector function deﬁned

on a compact set Ω, and any y(W, x) : R

× R

→ R

is an approximatin g

function that depends continuously on W and x, then, the approximation problem is

to d et ermine the opti mal W denoted by W

⋆

, for some index d such that

d(y(W

⋆

, x), f(x)) ≤ ε, (2.45 )

for an acceptable small ε.

There are a number of neural networks studied for function approximation such as

multi-layer perceptron networks, radial basis function (RBF) networks, and higher

order neural networks. The RBF network is suitable for online non linear adaptive

modeling and control, because it is a linearly parameterized network , has spatiall y

localized learning capability and thus has better memory during learning, and ex-

hibits a fast initial rate of learning convergence.

The th ree-layer neural netw ork shown in Figure 2.24 is a RBF network, where

x ∈ R

, y ∈ R

, and s ∈ R

are respectively the input , the output, and the

activation function vectors, and w

is the second to the third layer interconnection