Loop-Shaping Method with Unstable Poles for
Magnetic-Head Positioning Control in Hard Disk
Drive
Takenori Atsumi

Shota Yabui

Department of Mechanical Engineering
Chiba Institute of Technology
Narashino, Japantakenori.atsumi@p.chibakoudai.jp

Department of Mechanical Systems Engineering
Tokyo City University
Tokyo, Japanyabuis@tcu.ac.jp

Atsushi Okuyama

Masahiro Mae

Department of Precision Engineering
Tokai University
Hiratsuka, Japanokuyama@tokai-u.jp

Department of Electrical Engineering and Information Systems
The University of Tokyo
Tokyo, Japan
mmae@ieee.org

Abstract—In this paper, we present a loop-shaping method thatincorporates unstable poles for the magnetic-head positioning
system in a hard disk drive (HDD). The proposed method involves
introducing unstable poles instead of the conventional stable poles
for the resonance filter used in loop shaping. In our approach,the resonant filter is designed to realize a circle that includes
the coordinate [−1, 0j] on the Nyquist plot of the open-loop
characteristics in the control system. Validation results using
the HDD benchmark problem demonstrate that our proposed
methods effectively reject disturbances in the high-frequency
range while maintaining robust performance.
Index Terms—Resonant Filter, Unstable Pole, Loop Shaping,
Hard Disk Drives, Positioning Control

I. I NTRODUCTION
The future of cloud services relies on the expansion of
hard disk drive (HDD) capacity, as data demands within cloudservices are growing rapidly. To address this challenge, we aim
to enhance the precision of magnetic-head positioning control
systems, which will ultimately reduce the bit size required fordata storage on a disk.
In magnetic-head positioning control systems, a significant
obstacle to improving positioning accuracy is the presence of
mechanical oscillations in the high-frequency range. Overcoming this disturbance proves complex because the oscillation
frequency often exceeds the servo bandwidth of the control
system or the resonant frequencies of the controlled components.
To tackle this issue, we have proposed loop-shaping methods that incorporate resonant filters. These methods allow
us to effectively suppress disturbances occurring beyond theservo bandwidth and the mechanical resonant frequencies
A part of this work is supported by JSPS KAKENHI Grant Number
JP22K04170.

[1], [2]. In this paper, we present a loop-shaping methodthat introduces unstable poles for the resonant filters. We
validate this approach using the HDD benchmark problem,
demonstrating its applicability to the latest HDDs.
II. L OOP - SHAPING METHOD WITH RESONANT FILTER
The loop-shaping is widely used for improving the positioning accuracy for stable control systems [3]–[11]. In this paper,
we employ resonant filters to improve the stable control system
based on the loop-shaping scheme.
A. Principle of resonant filter (conventional method)
In this study, the frequency response of G[z] at ω [rad/s]
is expressed as G[ejωTs ], where Ts represents the samplingtime of G[z]. We have made the assumption that a similar
representation holds for the frequency response of an unstablesystem.
A control system design with the resonant filter enables us
to add a phase-stable circle in a vector locus of the openloop characteristics of the control system so that the gainof sensitivity function |S[z]| decreases at the disturbance’s
frequency [1]. In this method, a resonant mode R0 is given as
κr
R0 (s) = 2,(1)
s + 2ζr ωr s + ωr 2
where κr is the modal gain, ζr is the damping ratio, and ωr is
the resonance frequency. In order to decrease the gain of thesensitivity function, we employ a resonant filter R[z] which is
discretized Rc (s):Rc (s) = R0 (s)Cr (s),(2)

where Cr is a phase compensator.
Fig. 1 shows the principle of the resonant filter. Here,[ar , br j] means a coordinate of the open-loop characteristic

Im

Im

R

r+

Re

(a) Vector locus of resonant filter

e

L

Re

yd + +(b) Block diagram

Fig. 1. Principle of resonant filter
Fig. 3. Vector locus of a resonant filter with unstable poles.
r+

Rs

e

+

+

L

yd
Disk

Fig. 2. Block diagram with resonant filter.

Spindle motor

Base

PZT actuator

L at a target frequency of the resonant filter (the resonant
frequency of the resonant filter). The resonant filter is designed
so as to add a circle that makes the vector locus away fromthe [−1, 0j] like Fig. 1 (a). To do so, the phase of the resonant
filter should be∠R[ejωr Ts ] = θr ,
(3)
where


θr = arctanbrar + 1

.

(4)

To realize the circle, we should add a phase-stable resonant
filter R in parallel connection to L like Fig. 1 (b). However,this control system is not able to reject the disturbance because
the output signal of the resonant filter is added to the control
variable after adding the disturbance. Therefore, to reject the
disturbance, the resonant filter Rs is added in series connection
to L like Fig. 2. Rs [z] can be given as the discretized Rsc (s):Rsc (s) =R0 (s)Cr (s)
,
L̃(s)

VCM

HSA

Fig. 4. Hard disk drive.

achieved by unstable poles trace circular paths on the Nyquist
plot. However, their direction is counterclockwise in vector
trajectory.
According to the Nyquist stability criterion, the control
system is stable when the circle realized by the resonant
filter with unstable poles includes the coordinate [−1, 0j] on
the Nyquist plot of the open-loop characteristics. Therefore,
in order to decrease the gain of the sensitivity function, theresonant filter with the unstable pole is designed so as to add
a circle which makes the vector locus like Fig. 3. Therefore,
in the case of the unstable poles, the phase of the resonant
filter at ωr should be∠Rs [ejωr Ts ] = θru − ∠L[ejωr Ts ],(5)where,

where L̃ is approximated L around ωr (the resonant frequencyof R). Note that L̃ does not include unstable zeros. The phaseof the resonant filter at ωr should bejωr Ts∠Rs [e] = θr − ∠L[ejωr Ts].

(6)

In this paper, as discretization method for the resonant filters,
we employ the Bilinear (Tustin) method with frequency prewarping at ωr .
To reject multiple disturbances, we can employ multiple
resonant filters as follows,
Rs [z] =Md
X

Rsm [z, m],

(7)

m=1

where Md is the number of target disturbances, and Rsm isthe resonant filter for each disturbance.
B. Resonant filter with unstable poles (proposed method)
In the resonant filter scheme, we are able to employ
unstable poles instead of stable poles. Similar to resonancecharacteristics using stable poles, resonance characteristics

Magnetic head
θru = arctanbrar + 1

(8)


+ π.

(9)

Note that the resonant filter with unstable poles must have alarge gain to contain the coordinates [−1, 0j] inside the circle
on the Nyquist plot of the open-loop characteristics.
III. M AGNETIC - HEAD POSITIONING SYSTEM IN HDD
A. Control system
Fig. 4 shows the internal components of the HDD with its
cover removed. The HDD includes a VCM (Voice Coil Motor),
PZT (Lead Zirconate Titanate) actuators, an HSA (Head-Stack
Assembly), magnetic heads, disks, and a spindle motor. The
VCM moves the HSA radially across the disks, while the
PZT actuators fine-tune the position of the magnetic heads
with limited stroke. Fig. 5 shows the block diagram of themagnetic-head positioning control system used in this study.
It consists of feedback controllers Cdv and Cdp for the VCM
and the PZT actuator, respectively, an interpolator Ip , multirate filters Fmv and Fmp for the VCM and the PZT actuator,
respectively, a multi-rate zero-order hold Hm , a sampler S,

dRRO

Fmp

m

Cdv

Ip

Fmv

m

df
+

Pcp

+

Pcv

ycp

Gain [dB]

e

Ip

dp
+

+

+

+

yc

yd

+

Fig. 5. Block diagram of magnetic-head positioning system.
TABLE IVARIATIONS OF PLANT PARAMETERS IN THE BENCHMARK PROBLEM .
ωpp
+6%
0%
−6%
+6%
0%
−6%
+6%
0%
−6%

ζpv
−20%
0%
+20%
−20%
0%
+20%
−20%
0%
+20%

ζpp
−20%
0%
+20%
−20%
0%
+20%
−20%
0%
+20%

Kpp
0%
0%
0%
+5%
+5%
+5%
−5%
−5%
−5%

and the plant models Pcv and Pcp for the VCM and the PZT
actuator, respectively. The interpolator Ip , which includes an
up sampler and an averaging filter with the multi-rate number
2, generates the reference signal for the PZT actuator from the
VCM output. In this study, the sampling time of the control
system (Ts ) is 1.98 [µs].

Case 4
Case 5
Case 6

Case 7
Case 8104 Case 9

Frequency [Hz]

-90
-180
-270
-360
103

104

Frequency [Hz](a) Pcv
30

Gain [dB]

ωpv
+4%
0%
−4%
+4%
0%
−4%
+4%
0%
−4%

Case 1Case 2
Case 3020
10Case 1Case 2
Case 3

Case 4
Case 5
Case 6

Case 7
Case 8
Case 90-10103Phase [deg.]

Case
1
2
3
4
5
6
7
8
9

0
-20
-40
-60
-80
103

Phase [deg.]r =0 +

Cdp

104

Frequency [Hz]

180
90
0
-90
-180
103

104

Frequency [Hz](b) Pcp
Fig. 6. Frequency responses of mechanical characteristics in the benchmark
problem.

B. Benchmark problem
A technical committee of HDD servo researchers fromprominent universities and a Japanese HDD manufacturer devised an open-source HDD benchmark problem [12]. Experts
in this field presented their works using the HDD benchmark
problem [2], [13]–[21].
1) Controlled object [22]: The ambient temperature ofmost HDDs ranges from 5 to 60 deg. Therefore, thisbenchmark problem provides two boundary samples: a lowtemperature (LT) model and a high-temperature (HT) model.
The nominal plant is a room-temperature (RT) model.
The LT condition increases the mechanical resonance frequencies and decreases the damping ratios of both the VCM
and the PZT actuators compared to the RT condition. The
VCM mechanical resonance frequency (ωpv ) increases by 4%,
the PZT actuator mechanical resonance frequency (ωpp ) by
6%, and the damping ratios of both the VCM (ζpv ) and the
PZT actuators (ζpp ) decrease by 20%.
The HT condition has the opposite effect: it decreases themechanical resonance frequencies and increases the damping
ratios of both the VCM and the PZT actuators compared to
the RT condition. The VCM mechanical resonance frequency
decreases by 4%, the PZT actuator mechanical resonancefrequency by 6%, and the damping ratios of both the VCM
and the PZT actuators increase by 20%.
The magnetic-head positioning control system also requires
robustness against the gain variation of the PZT actuators.
Therefore, this benchmark problem uses three parameters
for the PZT actuator gain (Kpp ) variation: −5%, 0%, and+5%. As a result, this benchmark problem has nine cases

for the controlled object. Table I shows the variations ofplant parameters from the nominal model. Fig. 6 shows thefrequency responses of Pcv and Pcp .
2) Disturbance: This benchmark problem considers three
disturbances for the magnetic-head positioning control system
as follows:
RV (Rotational Vibration): This is df in Fig. 5. It is
the rotational acceleration caused by the operating HDDs
next to each other, which degrades the magnetic-head
positioning system.
• Fan-induced vibration: This is dp in Fig. 5. It is the large
air vibration (wind noise) produced by the cooling fansin the data center storage box, which affects the HDDs
through an airborne path.
• RRO (Repeatable RunOut): This is dRRO in Fig. 5. It is theresidual error after the RRO compensation by the learning
process in the production lines of the latest HDDs. Thisbenchmark problem uses the dRRO as measurement noise(random signal).
•

3) Control system with pre-set controller: This benchmark
problem provides a pre-set controller as an example. Fig. 7
shows the frequency responses of the control system characteristics with the pre-set controllers: (a) Bode plot of openloop, (b) Nyquist plot of open-loop, and (c) Gain Bode plot of
sensitivity function. Fig. 8 shows the simulation results of yc
with the pre-set controllers: (a) time domain and (b) amplitude
spectrum.

Gain [dB]

Case 1Case 2
Case 3

60
40
20
0
-20

Case 7
Case 8
Case 9Rsp+er =0 +

102

103

dRRO

104

Frequency [Hz]

0

Rsd
+

Rsv

+

+

+

Cdp

Ip

Fmp

Hm

Cdv

Ip

Fmv

Hm

df
+

yd

ycp

Pcp

+

Pcv

dp
+

+

+

+

yc

S

Fig. 9. Block diagram of HDD benchmark problem with the resonant filters.

-90
-180
-270
-360
102

103

104

Frequency [Hz](a) Open-loop (Bode plot).

2

Case 4
Case 5
Case 6

Case 7
Case 8
Case 9Phase [deg.]

Case 1Case 2
Case 3

4

Imaginary axis++

Gain [dB]

Phase [deg.]

Case 4
Case 5
Case 6

0
-2
-4

40
20
0
-20
-40
102

103

Frequency [Hz]

104

180
90
Rsv
Rsp
Rsd

0
-90
-180
102

103

Frequency [Hz]104

Fig. 10. Frequency responses of resonant filters using stable poles only.

-6
-6

-4

-2

0

2

4

Real axis

6

8

Rsp : The resonant filter Rsp is intended for the PZT loop
and compensates for disturbances above 4 kHz.
• Rsd : This resonant filter is applied to the open-loop of thecontrol system. It compensates for the gain degradationof the sensitivity function |S[z]| caused by the use of Rsv
or Rsp .
It’s important to note that all controllers, except for the
resonant filters, remain consistent with the pre-set control
system used in the HDD benchmark problem. The resonant
frequencies of Rsv and Rsp were determined based on simulation results of yc using the pre-set controllers shown in Fig. 8(b).
•(b) Open-loop (Nyquist plot).
100Gain [dB]

-10
-20
-30
-40
Case 1Case 2
Case 3

-50
-60
102

Case 4
Case 5
Case 6103

Case 7
Case 8
Case 9104

Frequency [Hz](c) Sensitivity functions.
Fig. 7. Frequency responses of the control system with the pre-set controller
in the benchmark problem. The worst case of ||S||∞ is 6.39 [dB].

-80

10
-90

Amplitude [dB]

Amplitude [nm]

5

0

-5
Case 1Case 2
Case 3-100200Case 4
Case 5
Case 6

400

Case 7
Case 8
Case 9600

Time [ms](a) Time domain

-100
-110
-120
-130
-140

800

1000

-150
101

Case 1Case 2
Case 3

Case 4
Case 5
Case 6102

Case 7
Case 8
Case 9103

104

Frequency [Hz](b) Frequency domain

Fig. 8. Simulation results of yc with the pre-set controller in the benchmark
problem. The worst case of 3σ = 14.54 [% of track width].

IV. C ONTROL SYSTEM DESIGN WITH RESONANT FILTERS
In this study, we employ a control system equipped withthree types of resonant filters: Rsv , Rsp , and Rsd , as depicted
in Fig. 9, for the HDD benchmark problem. Specifically:•

Rsv : This resonant filter is designed for the VCM loop
and compensates for disturbances occurring below 4 kHz.

A. Using resonant filter with stable poles (conventional
method)Fig. 10 shows the frequency responses of resonant filters
Rsv (solid), Rsp (dashed), and Rsd (dot-dashed) using stable
poles only. In this case, resonances created by Rsd have small
peak gains because Rsd compensate for the increased gain of|S[z]| by employing Rsv or Rsp . Fig. 11 shows frequency
responses of the control system ((a) Bode plot of the openloop, (b) Nyquist plot of the open-loop, (c) Gain Bode plot of
the sensitivity function). In this case, the worst case of ||S||∞
is 6.38 [dB].
B. Using resonant filter with unstable poles (proposed
method)The resonant filters with the unstable poles must have large
peak gain in order to contain the coordinates [−1, 0j] inside
the circle on the Nyquist plot of the open-loop characteristics.
Therefore, we employ the unstable poles for Rsv only. Fig. 12shows the frequency responses of resonant filters using unstable poles for Rsv . In this case, Rsp and Rsd do not employ
unstable poles. Fig. 13 shows the frequency responses of thecontrol system. In this case, the worst case of ||S||∞ is 6.38
[dB].

Case 7
Case 8
Case 9

103

102

0
-90
-180
-270
-360
102

103

104

Frequency [Hz]Case 1Case 2
Case 3

60
40
20
0
-20

104

Frequency [Hz]Phase [deg.]

Phase [deg.]

102

Case 4
Case 5
Case 6

Gain [dB]Gain [dB]

Case 1Case 2
Case 3

60
40
20
0
-20

Case 1Case 2
Case 3

104

103

104

0
-180
-270
-360
102

Frequency [Hz](a) Open-loop (Bode plot).
6

Case 7
Case 8
Case 9Case 1Case 2
Case 3
Case 4
Case 5

4

0

2

-2

Case 6
Case 7
Case 8
Case 9

0
-2

-4

-4

-6
-6

-4

-2

0

2

4

Real axis

6

-6

8

(b) Open-loop (Nyquist plot).

0

-10

-10

Gain [dB]

-20
-30
-40

-60
102

-4

-2

0

2

4

Real axis

6

8

10

0

-50

-6

(b) Open-loop (Nyquist plot).

10

Gain [dB]

103

-90

Imaginary axis

Imaginary axis

2

Case 4
Case 5
Case 6

Case 7
Case 8
Case 9Frequency [Hz](a) Open-loop (Bode plot).
4

Case 4
Case 5
Case 6

Case 1Case 2
Case 3

Case 4
Case 5
Case 6103

Frequency [Hz]

-20
-30
-40

Case 7
Case 8
Case 9Case 1Case 2
Case 3

-50
-60
102

104

Case 4
Case 5
Case 6103

Case 7
Case 8
Case 9104

Frequency [Hz](c) Sensitivity functions.

(c) Sensitivity functions.

Fig. 11. Frequency responses of the control system with resonant filters using
stable poles only. The worst case of ||S||∞ is 6.38 [dB].

Fig. 13. Frequency responses of the control system with resonant filters using
unstable poles for Rsv . The worst case of ||S||∞ is 6.38 [dB].

20
10

-90

-20
103

Frequency [Hz]

104

180
90
Rsv
Rsp
Rsd

0
-90
-180
102

103

Frequency [Hz]104

Fig. 12. Frequency responses of resonant filters using unstable poles for Rsv .

V. VALIDATION
In order to validate the proposed method, we conducted thesimulations with the HDD benchmark problem. Fig. 14 shows
the simulation results with results of yc with resonant filters
using stable poles only. Fig. 15 shows the simulation resultswith results of yc with resonant filters using unstable poles

5

Amplitude [dB]

-40
102

Phase [deg.]-80

0
Amplitude [nm]Gain [dB]

40

0

-5

-10
0

Case 1Case 2
Case 3

Case 4
Case 5
Case 6

Case 7
Case 8
Case 9

200

400

600

Time [ms](a) Time domain

-100
-110
-120
-130
-140

800

1000

-150
101

Case 1Case 2
Case 3

Case 4
Case 5
Case 6102

Case 7
Case 8
Case 9103

104

Frequency [Hz](b) Frequency domain

Fig. 14. Simulation results of yc with resonant filters using stable poles only.
The worst case of 3σ = 7.31 [% of track width].

for Rsv . In both figures, (a) indicates the time domain characteristics, (b) indicates the frequency domain characteristics(amplitude spectrum).
In Fig. 16 (a), we present a comparison of the 3σ values
for yc between the results with the loop-shaping method withresonant filters and the result with the pre-set controller. The

Displacement [% of track width]5

Amplitude [dB]

Amplitude [nm]

-90

0

-5

-10
0

Case 1Case 2
Case 3

Case 4
Case 5
Case 6

Case 7
Case 8
Case 9

200

400

600

-100
-110
-120
-130
Case 1Case 2
Case 3

-140

800

1000

-150
101

Case 4
Case 5
Case 6102

Case 7
Case 8
Case 9103

104

Frequency [Hz]Time [ms](a) Time domain(b) Frequency domain

Fig. 15. Simulation results of yc with resonant filters using unstable poles
for Rsv . The worst case of 3σ = 7.27 [% of track width].

results clearly demonstrate that loop-shaping methods with thestable poles only improve the positioning accuracy without
losing the stability margins. Moreover, employing the resonant
filters with the unstable poles for Rsv enables us to improve
the positioning accuracy as much as the resonant filters withthe stable poles.
In Fig. 16 (b), we present a comparison of the maximum
values for ycp between the results with the loop-shaping
method with resonant filters and the result with the pre-set
controller. This figure shows that our proposed method enables
us to improve the positioning accuracy without increasing thestrokes of the PZT actuators.
VI. C ONCLUSION
In order to compensate for the disturbances beyond theservo bandwidth for the magnetic-head positioning system
in HDDs, we have proposed the loop-shaping method thatincorporates unstable poles. The proposed method involves
introducing unstable poles instead of the conventional stable
poles for the resonance filter used in loop shaping whichenables us to decrease the gain of the sensitivity functions
at disturbance frequencies. Validation results using the HDD
benchmark problem demonstrate that our proposed methods
effectively reject disturbances in the high-frequency range
while maintaining robust performance without increasing thestrokes of the PZT actuators.
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Stable pole
Unstable pole (Rsv)
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2
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