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UNKNOWN MULTIPLE NARROW-BAND DISTURBANCE REJECTION IN HARD DISK

DRIVES–AN ADAPTIVE NOTCH FILTER AND PERFECT DISTURBANCE

OBSERVER APPROACH

Xu Chen

Department of Mechanical Engineering

University of California, Berkeley

Berkeley, California 94720

Email: maxchen@me.berkeley.edu

Masayoshi Tomizuka

Department of Mechanical Engineering

University of California, Berkeley

Berkeley, California 94720

Email: tomizuka@me.berkeley.edu

ABSTRACT

In this paper, an adaptive control scheme is developed to re-

ject unknown multiple narrow-band disturbances in a hard disk

drive. An adaptive notch ﬁlter is developed to efﬁciently esti-

mate the frequencies of the disturbance. Based on the correctly

estimated parameters, a disturbance observer with a newly de-

signed multiple band-pass ﬁlter is constructed to achieve asymp-

totic perfect rejection of the disturbance. Evaluation of the con-

trol scheme is performed on a benchmark problem for HDD track

following.

1 INTRODUCTION

The track following control in a hard disk drive (HDD) sys-

tem aims at precisely regulating the read/write head on the data

track and reducing the Track Mis-Registration (TMR). One im-

portant source of TMR is the vibration/ﬂutter of disks at their

resonant frequencies. This behavior is driven by the turbulent

air ﬂow within the hard disk assembly, and creates the so-called

non-repeatable multiple narrow-band disturbances [1–3], whose

energy is highly concentrated at several unknown frequencies.

These disturbances differ from track to track and disk to disk

in nature, and may appear at frequencies higher than the open

loop servo bandwidth [1,2]. The above characteristics of narrow-

band disturbances lead to the difﬁculty of rejecting them using

traditional feedback control methods. With the rapid growth in

HDD’s storage density, compensation of the non-repeatable mul-

tiple narrow-band disturbances is becoming more and more an

important issue.

Improving the mechanical design can help to reduce the

narrow-band disturbances but will signiﬁcantly increase the

product cost. Investigation of this problem in the community of

systems and controls has therefore been popular. For example,

Zheng and Tomizuka [4–6] suggested direct and indirect adap-

tive disturbance observer (DOB) schemes to estimate and cancel

the disturbance; Kim et al. [7] proposed a parallel add-on peak

ﬁlter to shape the open loop frequency response; Landau [8] used

Youla-Kucera parametrization to achieve adaptive narrow-band

disturbance rejection on an active suspension. These previous

methods have been focusing on rejecting disturbance with one

narrow-band. Yet, the ﬁeld of multiple narrow-band disturbance

rejection was seldom entered before.

This paper focuses on developing an adaptive control

scheme that compensates any number of unknown narrow-band

disturbances. The main techniques proposed are the minimal pa-

rameter adaptive notch ﬁlter (ANF) for frequency identiﬁcation

and the disturbance observer (DOB) for disturbance rejection.

The minimal parameter ANF was developed in the signal pro-

cessing community, rooted back to [9], reﬁned ﬁrst in [10] and

then in [11]. It has the advantage of high computational efﬁ-

ciency, good numerical robustness, and fast convergence. This

paper proposes a computationally simpliﬁed version of [11], to

ﬁt the algorithm for our controller structure. The accurately

identiﬁed frequencies are then applied to construct a new type

of DOB, which can asymptotically achieve perfect narrow-band

disturbance rejection. Advantages of the proposed algorithm are:

it inherits the excellent properties of the minimal parameter ANF

in the frequency identiﬁcation; it is an add-on compensator and

can be simply integrated to the existing baseline controller struc-

ture; it rejects the narrow-band disturbance with minor inﬂuence

to the system sensitivity function at other frequencies; it can

achieve asymptotic perfect disturbance rejection.

The remainder of this paper is organized as follows. Sec-

tion 2 formally deﬁnes the problem. The proposed DOB struc-

ture, and the design of a multiple narrow band-pass Q-ﬁlter are

presented in Section 3. Section 4 provides the adaptive ANF al-

gorithm for Q-ﬁlter parameter identiﬁcation. The efﬁcacy of the

proposed method is provided in Section 5, with an example of

rejecting two narrow-band disturbances. Section 6 concludes the

paper.

2 PROBLEM DEFINITION

Figure 1 shows the baseline feedback control loop for HDD

track following and the proposed compensation scheme to re-

ject the multiple narrow-band disturbances. The full-order plant

model Gpz−1contains the dynamics of the HDD servo sys-

tem including the power ampliﬁer, the voice-coil motor, and the

actuator mechanics. Throughout the paper we use the well for-

mulated open-source HDD benchmark simulation package [12]

as a demonstration tool. The dashed line in Fig. 2 shows

the frequency response of Gpz−1, which is a fourteenth-order

transfer function with several high frequency resonances. The

low order nominal plant model z−mGnz−1matches the low-

frequency dynamics of Gpz−1up to about 2000 Hz, as shown

in Fig. 2. The baseline feedback controller CFB z−1is a com-

monly used PID controller cascaded with three notch ﬁlters. Fig-

ure 3 shows the frequency response of the open loop system

CFB z−1Gpz−1, which has a gain margin of 5.45 dB, a phase

margin of 38.2 deg, and an open loop servo bandwidth of 1.19

kHz.

The signals r(k),d(k),u(k), and n(k), are respectively the

reference, the input disturbance, the control input, and the out-

put disturbance. PES is the position error signal. The multi-

ple narrow-band disturbances of interest are assumed to be con-

tained in d(k), and have unknown frequencies between 300 Hz

and 2000 Hz [1, 2].

1

( )

Q z

−

m

z

−

d

(k

)

n(k

)

P

E

S

C

F

B

(z

¡

1

)

c

(k

)

r

=

0

z

(k

)

1 1

( )

n

G z

− −

1

( )

P

G z

−

^

d

(k

)

u(k

)

Figure 1. Structure of the proposed scheme for multiple narrow-band

disturbance rejection.

We observe from Fig. 3, that the general baseline open loop

system has a decreasing high gain in the low frequency region,

-20

0

20

40

60

80

100

Magnitude (dB)

101102103104

-1440

-1080

-720

-360

0

Phase (deg)

Bode Diagram

Frequency (Hz)

Gp

z-mGn

Figure 2. Frequency responses of the full order plant and its nominal

model.

-50

0

50

Magnitude (dB)

10

1

10

2

10

3

10

4

-720

-360

0

360

720

Phase (deg)

Bode plot: Baseline open loop response

Frequency (Hz)

Figure 3. Frequency response of the open loop system with the baseline

controller CFB z−1.

which indicates that the closed loop system suppresses low fre-

quency disturbance but has less and less attenuation capacity as

the frequency increases. When special attenuation is needed at

several narrow frequency regions (say around 500 Hz and 1000

Hz), the baseline controller would fail to maintain the desired

servo performance. In this paper, we aim at providing a solution

to this important problem.

3 DOB FOR MULTIPLE NARROW-BAND DISTUR-

BANCE REJECTION

The proposed control scheme for multiple narrow-band dis-

turbance rejection is summarized in the dash-dotted box in Fig.

1. The add-on compensator is a disturbance observer (DOB) [13]

with a specially designed adaptive Q-ﬁlter Qz−1. As the name

of DOB suggests, the multiple narrow-band disturbance is ’ob-

served’ by the controller, and fed back to cancel itself. To see

this point, notice ﬁrst that the signal ˆ

d(k)is expressed by, in the

operator notation,

ˆ

d(k) = Gn(z−1)−1[Gp(z−1)(u(k) + d(k))+ n(k)] −z−mu(k)

(1)

A stable inverse model Gnz−1−1is needed in the above

signal processing. If Gnz−1has minimal phase, its inverse can

directly be assigned, if not, stable inversion techniques such as

the ZPET method [14] should be applied.

Noting that Gnz−1−1=z−mz−mGnz−1−1, we can

transform Eq. (1) to

ˆ

d(k)

=z−mnz−mGnz−1−1Gpz−1u(k)−u(k)o

+z−mz−mGnz−1−1Gpz−1d(k)

+Gnz−1−1n(k).(2)

Notice that z−mGnz−1≈Gpz−1under 2000 Hz, which

gives hz−mGnz−1−1Gpz−1iu(k)−u(k)≈0. If in addi-

tion the output disturbance n(k)is small, then Eq. (2) becomes

ˆ

d(k)≈z−md(k) = d(k−m).(3)

The above equation implies that ˆ

d(k)is a good estimate of

d(k), i.e., the disturbance is well ’observed’ by DOB.

In reality, the inﬂuence of n(k)and other noise can not

be omitted. To compensate only the narrow-band disturbance

without inﬂuencing other components in d(k), it is proposed to

pass ˆ

d(k)through a multiple band-pass ﬁlter Qz−1, whose

pass-bands are sharply located at the narrow-band frequencies.

The compensation signal c(k)formed by ﬁltering ˆ

d(k)through

Qz−1, therefore contains only the multiple narrow-band dis-

turbance. Adding the negative of c(k)to the control input, we

achieve the compensation.

3.1 Q-ﬁlter Design for Perfect Disturbance Rejection

Assume that the multiple narrow-band frequencies are at

Ω1,...,Ωn(in Hz). Estimation of these frequencies will be dis-

cussed in the next section. In this section, we present the struc-

tural design of Qz−1, and assume Ωi’s are known.

To have sharp pass-bands at Ωi’s, poles of Qz−1need to

be located close to these frequencies. The second constraint on

Qz−1comes from Eq. (3), that the estimated disturbance ˆ

d(k)

is a m-step delayed version of the true signal d(k). This translates

to the need on Qz−1to be able to provide a m-step forward

action to ˆ

d(k).

Suppose the basis Q-ﬁlter for single narrow-band distur-

bance rejection has the form Q0z−1=NQz−1/DQz−1,

where NQz−1and DQz−1are polynomials of z−1. Us-

ing the results in the last subsection, that ˆ

d(k)≈d(k−m)and

c(k) = Qz−1ˆ

d(k), we have

d(k)−c(k)≈1−z−mQz−1d(k)

=DQz−1−z−mNQz−1

DQ(z−1)d(k).(4)

When d(k)is a narrow-band sinusoidal signal with fre-

quency Ωo(in Hz) and sampling time Ts(in sec), by direct ex-

pansion, one can show that 1−2cos(ωo)z−1+z−2d(k) = 01,

where ωo=2πΩoTs. Therefore, if DQz−1is stable in Eq. (4),

and

DQ(z−1)−z−mNQ(z−1) = J(z−1)1−2cos(ωo)z−1+z−2,

(5)

where Jz−1is a polynomial of z−1, we can achieve asymptotic

perfect disturbance rejection, i.e.,

lim

k→∞(d(k)−c(k)) = 0.(6)

In our nominal model, m=1. Assign Jz−1=1, and let

DQz−1be given by

DQz−1=1−2αcos(ω0)z−1+α2z−2,(7)

where the shaping coefﬁcient αis a real number smaller than but

close to 1, such that Qz−1is stable and has two poles αe±jω0

at the narrow-band frequency.

Substituting Eq. (7) to Eq. (5), and applying the method of

undetermined coefﬁcients, we get

NQz−1=2(1−α)cos(ω0) + α2−1z−1.(8)

Introduce now the shorthand notation θ0=cos(ω0). The

ideal Q-ﬁlter for single narrow-band disturbance rejection is thus

given by

Q0z−1=2(1−α)θ0+α2−1z−1

1−2αθ0z−1+α2z−2.(9)

When d(k)contains nnarrow-band components, the above

equation can be extended to

Qz−1=

n

∑

i=1

2(1−αi)θi+α2

i−1z−1

1−2αiθiz−1+α2

iz−2,(10)

11/1−2cos(ωo)z−1+z−2is also known as the internal model of the nar-

row band disturbance d(k).

Figure 4 shows the frequency response of the above

Qz−1with α1=α2=0.998, Ts=3.788 ×10−5sec, θ1=

cos(2π×500Ts), and θ2=cos(2π×1200Ts). Notice that at the

central frequencies, the magnitude and the phase of Qz−1are

1 (0 dB) and 0 deg, respectively. Therefore, passing a broad band

disturbance ˆ

d(k)through Qz−1, one ﬁlters out other compo-

nents and gets only the multiple narrow-band signals at 500 Hz

and 1200 Hz.

-60

-40

-20

0

Magnitude (dB)

10

1

10

2

10

3

10

4

-540

-360

-180

0

180

Phase (deg)

Frequence Response of Q-filter

Frequency (Hz)

Figure 4. Frequency response of a Q-ﬁlter with two narrow pass-bands.

The error rejection property of a control system is com-

monly evaluated by its sensitivity function Sz−1, which is the

transfer function from the output disturbance to the feedback er-

ror signal. Figure 5 shows the frequency response of Sz−1

for the proposed overall control structure when Qz−1is ﬁxed.

With the add-on compensation scheme, PES at 500 Hz and 1200

Hz can get greatly attenuated, due to the deep notches in the

magnitude response at the corresponding frequencies, while the

inﬂuence to the sensitivity at other frequencies is neglectable.

Stability of DOB (see [15, 16]) requires the nominal model

z−mGnz−1to have no zeros outside the unit circle and that

Qejω

<1

|∆(ejω)|∀ω,(11)

where ∆(z−1) = Gp(z−1)−z−mGn(z−1)/z−mGn(z−1)repre-

sents the multiplicative model mismatch. Plotting the magnitude

responses of 1/∆z−1and Qz−1in Fig. 6, we see that the

proposed DOB is stable as long as the narrow-band frequency is

less than 3000 Hz.

4 ADAPTIVE NOTCH FILTER FOR FREQUENCY IDEN-

TIFICATION

The proposed Q-ﬁlter design requires knowledge of the fre-

quency information θi=cos(2πΩiTs), which is not priori avail-

able. In this section, we apply an adaptive notch ﬁlter (ANF)

-80

-60

-40

-20

0

20

Magnitude (dB)

10

1

10

2

10

3

10

4

-45

0

45

90

135

180

Phase (deg)

Frequency response: error r ejection function/sensitivity func tion

Frequency (Hz)

w ith DOB

w ithout DOB

Figure 5. Frequency response of the closed loop sensitivity function.

100101102103104

-60

-40

-20

0

20

40

60

80

Magnitude (dB)

Bode Diagram

Frequency (Hz)

1/∆(z)

Q(z)

Figure 6. Magnitude responses of 1/∆z−1and Qz−1.

algorithm to estimate these quantities. As has been discussed

before, ˆ

d(k)contains the multiple narrow-band disturbance as

well as other noise components. A low-pass ﬁlter is thus ﬁrst

constructed to ﬁltered out the components in ˆ

d(k)that are out of

interest. The ﬁltered signal z(k)is ﬁnally a multiple narrow-band

signal with small noise-signal ratio, and can be identiﬁed using

the parameter estimation scheme in this section.

4.1 Theory

The intuition of the ANF algorithm comes from the fact that

energy of the narrow-band disturbance z(k)is highly concen-

trated at several frequencies Ω= [Ω1,...,Ωn]T(in Hz). If one

constructs a notch ﬁlter Hz−1with multiple center frequen-

cies at [Ω1,...,Ωn]T, and passes z(k)through Hz−1, the out-

put eo(k)should have the least energy.

Introduce the normalized frequency ωi=2πΩiTs(in radi-

ans). The transfer function of a qualifying notch ﬁlter [11] is

given by

Hz−1=

n

∏

i=1

Hoωi,z−1,(12)

where

Hoωi,z−1=1−2βz−1cosωi+β2z−2

1−2αz−1cosωi+α2z−2.(13)

Hoωi,z−1has two poles at αe±jωiand two zeros at βe±jωi.

With the shaping coefﬁcients βchosen close to 1, and α<β<

1, the ﬁlter will have a strong attenuation to the input signal at

frequency ωi.

To ﬁnd the unknown ω0

is, an adaptive algorithm is neces-

sary. Considering the numerator and the denominator of Hz−1

as two entire sections2and estimating their coefﬁcients, Neho-

rai developed the algorithm in [9]. Applying the cascaded se-

ries of second-order sections as shown in Eq. (12), and updat-

ing directly ωigave rise to [10], which was later reﬁned in [11].

The above algorithms identify only nparameters, which is the

minimum possible number for nnarrow-band components. The

main advantage of [11] over [9] is that cascaded ﬁlter structures

are numerically more efﬁcient and stable when the ﬁlter order is

high [17].

We notice, however, from the control aspect of view, that

the value of ωiis not directly needed (cosωiis the term really

implemented in the controller). To directly estimate ωi, [10]

and [11] needed to calculate trigonometric functions within each

iteration, which can be an expensive task in HDD servo control,

since the microprocessor has limited ability in these computa-

tions. To localize the ANF method for HDD control, we intro-

duce θi=cos(ωi), and modify the algorithm in [11] as follows:

In Eq. (13), let

Aoθi,γz−1=1−2γz−1θi+γ2z−2,γ=α,β,(14)

and introduce the unknown parameter vector θ= [θ1,θ2·· · θn]T,

Eq. (12) becomes

Hθ,z−1=

n

∏

i=1

Hoθi,z−1=

n

∏

i=1

Aoθi,βz−1

Ao(θi,αz−1),(15)

where we used the notation Hθ,z−1to emphasize that this is

a transfer function with unknown parameter θ. The objective of

ANF design is to ﬁnd the best parameter estimate, such that the

following cost function is minimized

Vk=

k

∑

j=1

1

2[eo(j)]2,(16)

2i.e., applying the digital ﬁlter in its direct-form structure.

where e0(k) = Hθ,z−1z(k)is the output error.

The transfer function Hθ,z−1is nonlinear in θ. To ﬁnd

the best estimation, the celebrated Gauss-Newton Recursive Pre-

diction Error Method (RPEM) (chapter 11 in [18]) suggests to

apply the following iterative formulas

ˆ

θ(k) = ˆ

θ(k−1) + F(k−1)ψ(k−1)eo(k)

λ(k) + ψT(k−1)F(k−1)ψ(k−1),(17)

F(k) = 1

λ(k)F(k−1)−F(k−1)ψ(k−1)ψT(k−1)F(k−1)

λ(k) + ψT(k−1)F(k−1)ψ(k−1),

(18)

where ψ(k−1)=[ψ1(k−1),...,ψn(k−1)]T,ψi(k−1) =

−∂eo(k)/∂ˆ

θi(k−1), and λ(k)is the forgetting factor.

The above modiﬁed algorithm has several nice properties:

1. stability of the Gauss-Newton RPEM is guaranteed if

H(ˆ

θ(k),z−1)is stable during the adaptation [18], which can

be easily checked by monitoring if |ˆ

θi(k)|<1, due to our

cascaded construction of Eq. (15).

2. ˆ

θ(k)unbiasedly converges to a local minimum [18].

3. it inherits most of the advantages of [9,11], such as fast con-

vergence, computational efﬁciency, and numerical robust-

ness. Moreover, it does not require computing sine and co-

sine functions.

4.2 Algorithm

Similar to [11], to obtain ﬁrst e0(k) = Hθ,z−1z(k), we

introduce

xj(k) =

j

∏

i=1

H0θi,z−1z(k),(19)

from which we have

xj(k) = 1−2βθiz−1+β2z−2

1−2αθiz−1+α2z−2xj−1(k),(20)

i.e., in the state-space representation

Zi(k+1) = h2αθi−α2

1 0 iZi(k) + h1

0ixi−1(k),(21)

xi(k)=[2(α−β)θi,β2−α2]Zi(k) + xi−1(k).(22)

We can then iteratively get e0(k), with eo(k) = xn(k)and

x0(k) = z(k).

To get ψi(k−1) = −∂eo(k)/∂ˆ

θi(k−1), we notice that

∂eo(k)

∂ˆ

θi(k−1)=∂Hˆ

θi(k−1),z−1z(k)

∂ˆ

θi(k−1)

=∂Hoˆ

θi(k−1),z−1

∂ˆ

θi(k−1)∏

j6=i

Hoˆ

θj(k−1),z−1z(k)

=∂Hoˆ

θi(k−1),z−1

∂ˆ

θi(k−1)H−1

oˆ

θi(k−1),z−1eo(k).(23)

Using Eq. (14) and Eq. (15), we get

∂Hoθi,z−1

∂θi

=

∂Ao(θi,βz−1)

∂θiAoθi,αz−1

A2

o(θi,αz−1)

−Aoθi,βz−1∂Ao(θi,αz−1)

∂θi

A2

o(θi,αz−1),(24)

where ∂Aoθi,γz−1/∂θi=−2γz−1,γ=α,β. Substituting the

above back to Eq. (23), and changing θito its estimated value

ˆ

θi(k−1), we arrive at the following simple formula:

ψi(k−1) = 2[eFi(β,k)−eFi(α,k)],(25)

where eFi(γ,k) = γz−1/Aoˆ

θi(k−1),γz−1eo(k),γ=α,β,

which can again be calculated using a state-space realization

Wi(γ,k+1) = h2γˆ

θi(k)−γ2

1 0 iWi(γ,k) + h1

0ieo(k),(26)

eFi(γ,k) = γ,0Wi(γ,k).(27)

We notice that the above result has a similar structure with

that in [11], but does not require to calculate sine or cosine func-

tions. Analogous to [11], the recursive parameter estimation is

ﬁnally summarized as follows:

Initialization: αo=0.8, αend =0.995, αrate =0.99,3β=

0.9999, Zi(0) = Wi(γ,0) = 0, F(0)≈100/E[eo]2·I,ˆ

θ(0) =

initial guess of the parameters, λ(0) = λ0,λ(∞) = λend ,λrate =

0.99.

Main loop: for k=1,2, . . .

step 1, prediction error computation: for i=1 : n

xi(k) = [2(α−β)ˆ

θi(k−1),β2−α2]Zi(k)+xi−1(k),(28)

with x0(k) = z(k)and eo(k) = xn(k).

3αis designed to increase exponentially from αoto α∞, at the rate of αrate ,

such that the notches get sharper and sharper to better capture the narrow-band

frequencies.

step 2, regressor vector computation: for i=1 : n

eFi(γ,k) = γ,0Wi(γ,k),γ=α,β,(29)

ψi(k−1) = 2(eFi(β,k)−eFi(α,k)).(30)

step 3, parameter update using Eq. (17) and Eq. (18).

step 4, projection of unstable parameters: for i=1 : n, if

|ˆ

θi(k)|>1, ˆ

θi(k) = ˆ

θi(k−1).

step 5, a posteriori prediction error ¯e(k)computation and state

vector update: for i=1 : n

¯xi(k) = 2(α−β)ˆ

θi(k),β2−α2Zi(k) + ¯xi−1(k),

(31)

Zi(k+1) = h2αˆ

θi(k)−α2

1 0 iZi(k) + h1

0i¯xi−1(k),(32)

with ¯xo(k) = z(k)and ¯e(k) = ¯xn(k).

Wi(γ,k+1) = h2γˆ

θi(k)−γ2

1 0 iWi(γ,k) + h1

0i¯e(k),(33)

for γ=α,β.

step 6, forgetting factor and notch ﬁlter shape coefﬁcient update:

replace αby αend −[αend −α]αrate , and

λ(k+1) = λend −[λend −λ(k)]λrate .(34)

Remark: (1) The above algorithm does not involve any sine or

cosine functions, but instead performs one additional simple step

to assure the stability of the ﬁlter. (2) The a posteriori informa-

tion is applied as in [18], to improve the estimation precision.

(3) As long as the initial parameter guesses are not too far away

from the true values, the estimation is unbiased even under the

presence of noise [18].

5 SIMULATION RESULT

The proposed adaptive compensator for multiple narrow-

band disturbance rejection was implemented in the HDD bench-

mark simulation tool [12]. The baseline control system is as

shown in Section 2. All common disturbances in HDD, including

the torque disturbance, the disk ﬂutter disturbance, the repeatable

runout (RRO), and the measurement noise, are added in the sim-

ulation. The sampling time, the spindle rotation speed, and the

track density are respectively 3.788 ×10−5sec, 7200 rpm, and

100k Tracks Per Inch (TPI). The multiple narrow-band distur-

bance in NRRO was modeled as the sum of several sinusoidal

signals [1, 2], and injected at the input to the plant with center

frequencies 500 Hz and 1200 Hz.

Figure 7 shows the time trace of the position error signal

without compensation. It is observed that the peak values of

PES exceeded the standard PES upper bound of 15% Track Pitch

(TP). The dotted line in Fig. 9 presents the spectrum of the PES

without compensation. We can see that the PES had strong en-

ergy components at 500 Hz and 1200 Hz. Without compensa-

tion, the Track Mis-Registration (TMR), deﬁned as 3 times the

standard deviation of the PES, was 21.87% TP.

0 5 10 15 20

-20

-10

0

10

20

Revolution

PES (%TP)

Figure 7. Time trace of the position error signal without compensation.

0 5 10 15 20

-20

-10

0

10

20

Revolution

PES (%TP)

Figure 8. Time trace of the position error signal with compensation.

500 1000 1500 2000 2500

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequency (Hz)

Magnitude

w/ compensation 3σ = 10.25 %TP

w/o compensation 3σ = 21.87 %TP

Figure 9. Spectra of the position error signals with and without the com-

pensator.

With the same baseline controllers, the proposed add-on

compensation scheme was applied to improve servo perfor-

mance. The low-pass ﬁlter in Fig. 1 was designed using MAT-

LAB’s Filter Design Toolbox, to have a cut-off frequency of

2000 Hz.The parameter adaptation was initialized at ˆ

θi(0) =

cos(2πΩo

iTs), where Ωo

1=100 Hz and Ωo

2=1000 Hz, in view

of the fact that the multiple narrow-band disturbance of interest

lies between 300 Hz and 2000 Hz. Figure 10 shows the online

identiﬁcation of the parameters ˆ

θ1and ˆ

θ2. Figure 11 shows the

equivalent online frequency estimation, via the transformation

ˆ

Ω1=cos−1(ˆ

θ1)/(2πTs)and ˆ

Ω2=cos−1(ˆ

θ2)/(2πTs). It is ob-

served that the parameters converged to their true values within

one revolution, i.e., 0.00833 sec.

0 1 2 3 4 5

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Revolution

Estimated parameters

Figure 10. Online parameter estimation of the two narrow-band signals.

0 1 2 3 4 5

0

1000

2000

3000

4000

5000

6000

Revolution

Estimated frequency (Hz)

Figure 11. Equivalent frequency identiﬁcation of the two narrow-band

signals.

Recall the stability condition at the end of section 3, that

ˆ

Ωi(k)should be lower than the DOB stability threshold 3200 Hz.

Correspondingly, ˆ

θi(k)should be larger than cos(2π×3200TS)

for implementation. Once the estimated parameters fell into this

region, the adaptive DOB was constructed to reject the multiple

narrow-band disturbance.

Notice that the proposed Q-ﬁlter is given by

Qz−1=

n

∑

i=1

2(1−αi)θi+α2

i−1z−1

1−2αiθiz−1+α2

iz−2.(35)

In the simulation example, n=2 and α1=α2=0.998. To gen-

erate the compensation signal c(k) = Qz−1ˆ

d(k)at each itera-

tion, Eq. (35) was realized by, in the state space form,

Si(k+1) = 2αθi−α2

1 0 Si(k) + 1

0ˆ

d(k)(36)

ci(k) = (1−α)4αθ2

i−1−α,−2α2θiSi(k) + 2θiˆ

d(k)

(37)

with θireplaced by its latest stable estimate ˆ

θi(k), and c(k) =

Qz−1ˆ

d(k) = ∑n

i=1ci(k).

Figure 8 shows the resulting PES time trace. It is seen that

after a transient response of about 1 revolution, the PES was re-

duced to within 10% TP. In the frequency domain, we observe

from Fig. 9, that the strong energy components at 500 Hz and

1200 Hz were greatly attenuated, while the spectrum of the PES

at other frequencies was almost identical to that without compen-

sation. The TMR was reduced to 10.25% TP, implying a 53.13%

improvement.

6 CONCLUSION

In this paper, an adaptive control scheme was proposed for

rejecting multiple narrow-band disturbances in HDD track fol-

lowing. It consists of an adaptive notch ﬁlter to estimate the

frequencies of the disturbances and a disturbance observer with

a multiple band-pass ﬁlter tuned for the estimated frequencies.

Simulation results on a realistic open-source HDD benchmark

problem showed that the proposed algorithm signiﬁcantly re-

duced the PES and the TMR. The proposed method is suitable

in control systems that demand heavy disturbance attenuation at

several frequencies.

ACKNOWLEDGMENT

This work was supported by the Computer Mechanics Lab-

oratory (CML) in the Department of Mechanical Engineering,

University of California at Berkeley. The authors gratefully ac-

knowledge Dr. Qixing Zheng’s useful discussions.

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