Frequency Response Data-based Resonant
Filter Design considering Phase Stabilizationand Stroke Limitation applied toDual-Stage Actuator Hard Disk Drives
Masahiro Mae ∗ Wataru Ohnishi ∗ Hiroshi Fujimoto ∗∗Department of Electrical Engineering and Information Systems,
Graduate School of Engineering, The University of Tokyo, Japan.
(e-mail: mmae@ieee.org)
Abstract: Disturbance rejection of a Hard Disk Drive (HDD) enables a large amount of data
storage in a recent information society. The aim is to design a feedback controller which rejects
disturbances at multiple frequencies in HDDs. The disturbance rejection is achieved using
resonant filters which have a large peak at disturbance frequencies. The developed approachenables the convex optimization of resonant filters with phase stabilization and stroke limitation
using frequency response data of a controlled system. The disturbance rejection performance ofthe optimized resonant filters is validated in a dual-stage actuator HDD benchmark problem.
Keywords: disturbance rejection, positioning control, data-based design, convex optimization,
frequency response, hard disk drive
1. INTRODUCTION

Disk

Increasing demand for storage capacity of data servers
in a recent information society leads to the importanceof the track-following performance in a Hard Disk Drive(HDD) (Atsumi, 2016). To improve the track-following
performance, the feedback controller should be designed
to reject disturbances in HDDs.

VCM

Model-based approaches are traditionally developed suchas using adaptive control (Ohno and Hara, 2006; PérezArancibia et al., 2010; Shahsavari et al., 2015; Sun et al.,2017), resonant filter (Atsumi et al., 2007), repetitive
control (Fujimoto, 2009; Chen and Tomizuka, 2014),
Youla–Kucera parameterization (Chen and Tomizuka,2016), disturbance observer (Zheng et al., 2017), and coupling controller (Yabui et al., 2020). They basically need amodeling process of a controlled system that makes it difficult to consider model variations of mass-produced HDDs.
These approaches also need a heuristic tuning process.
Data-based approaches are also developed to recover disadvantages of model-based approaches such as using H∞
and H2 robust control (Bashash and Shariat, 2019; Potu
Surya Prakash and Horowitz, 2022). These approaches
only consider the gain stabilization and it can result in aconservative controller design. Hardware constraints suchas maximum strokes also should be considered for actual
implementation.
Although important contributions have been made to
design feedback controllers for disturbance rejection in
HDDs, phase stabilization and stroke limitation are notconsidered in the optimization of feedback controllers. In
this paper, the developed approach enables the structured
multiple resonant filter design considering phase stabilization and stroke limitation.

Spindle Motor
Position Data
User Data
SectorMagnetic HeadPZT Actuator
PZT
Controller
VCM
Controller

e

− r
+

Fig. 1. Hard disk drive with a dual-stage actuator.
The main contributions of this paper are as follows.
(1) Resonant filters for a dual-input single-output system
are designed by iterative convex optimization.
(2) Model variations of HDDs are directly dealt withmultiple sets of frequency response data.
(3) Phase stabilization and stroke limitation are considered in optimization calculation.
2. PROBLEM FORMULATION
Fig. 1 illustrates the basic schematic of a dual-stage
actuator HDD. This HDD consists of a Voice Coil Motor(VCM) and a PieZoelecTric (PZT) actuator. The objectiveof this benchmark problem is to minimize the tracking
error of the magnetic head (Atsumi, 2022). The trackfollowing performance is evaluated in 3σ(yc ) the worst
case of three times of standard deviation value of thecontinuous-time magnetic head position in steady state
response for 1 s. The track pitch is Tp = 52.7 nm andsampling time is Ts = 1/(7200/60)/420 ≃ 1.9841 × 10−6 s.

Gp
Frp

Cdp

Im

Fmp

Hm

Pcp

ycp

S

yp
d

Gv
r=0

e

+ −

Frv

Cdv

Im

Fmv

Hm

Pcv

S

+

+

+

+

y

Fig. 2. Block diagram of a hard disk drive with a dual-stage actuator.
-150
40
0
-40
102

-200
103

104

0

-250

-180
-360
102

103

104

0
-40
103

104

0
-180
-360
102

103

104

Fig. 4. Amplitude spectrum of output disturbances.

40

102

-300
102

103

104

problem, the external disturbances consist of repeatablerun-out dRRO , rotational vibration df , and fan-induced
vibration dp . These three disturbances can be combined
to one output disturbance d equivalently that is given bydkc (jωkf ) = Pcv,kc (jωkf )df (jωkf ) + dp (jωkf ) − dRRO (jωkf ). (1)The amplitude spectrum of the output disturbance d in 9
cases is shown in Fig. 4.
In this paper, the reference signal is r = 0 in all time andthe objective is designing the resonant filters Frp and Frv
for each actuator in addition to given open-loop controlled
system Gp and Gv to minimize the worst case of trackfollowing error against to the output disturbance d.

Fig. 3. Frequency responses of given open-loop systems.
Top: PZT actuator Gp . Bottom: VCM Gv .

3. CONVEX OPTIMIZATION OF MULTIPLE
RESONANT FILTERS UNDER CONSTRAINTS

Because of the hardware constraints, the output stroke of
the PZT actuator must be smaller than 50 nm.

In this section, the optimization method of multiple resonant filters for a dual-stage actuator HDD is formulated.
The structure of the designed resonant filters is presented.
The optimization problem that directly uses frequency
response data is formulated with conditions of robust
stability, robust performance, and hardware constraints.
The optimization problem is solved by iterative convex
optimization with sequential linearization.

Fig. 2 shows the block diagram of the dual-stage actuator
HDD with continuous-time control systems Pc , discretetime feedback controllers Cd , discrete-time multirate filters
Fm . The subscripts p and v denote a PZT actuator and
VCM, respectively. The continuous-time control system
has 9 cases of model variations and the subscript kc =1, . . . , 9 denotes the index of the 9 cases. m ∈ N is thenumber of multirate. Im is interpolator for m times upsampling. Hm is multirate zero-order-hold in m times upsample. S is sampler.
The frequency response data of the given open-loop controlled systems Gp and Gv in Fig. 2 that consists of thecontrolled systems and pre-designed feedback controllers(Atsumi, 2022) is shown in Fig. 3. In this benchmark

3.1 Structure of designed resonant filters
To improve the track-following performance, disturbances
can be rejected by the resonant filters (Atsumi et al., 2007)
that have the same resonance frequency as the disturbancefrequencies because of the internal model principle. In thispaper, multiple resonant filters are designed in disturbancefrequencies to improve the track-following performance.

Fr (s)Imψ2ρ(1,2) s + ρ(1,1) s
2
s2 + 2ζr,1 ωr,1 s + ωr,1κ..
.
ρ(nr ,2) s2 + ρ(nr ,1) s2
2
s + 2ζr,nr ωr,nr s + ωr,n
r

+
ws

+
+

−1

+

G(jωr ) G(jωr )Fr (jωr )0
Re

Fig. 5. Block diagram of resonant filters.
The block diagram of the designed resonant filters withmultiple resonance frequencies is shown in Fig. 5. The
resonant filter in each actuator is defined as
Fr,ku (jωkf , ρku )nrXρku ,(kr ,2) (jωkf )2 + ρku ,(kr ,1) (jωkf )=1+
2(jωkf )2 + 2ζr,kr ωr,kr (jωkf ) + ωr,k
r
kr =1


1
(jωkf )




2


T  (jωkf )2 + 2ζr,1 ωr,1 (jωkf ) + ωr,1


1
2


(jωkf ) ρku ,(1,1)    (jω )2 + 2ζ ω (jω ) + ω 2 ρ
kfr,1 r,1kf
 ku ,(1,2)  
r,1 

 
=
..
..


 
.
.


 

ρ
 
(jωkf )

ku ,(nr ,1)ρku ,(nr ,2) 2 (jωkf )2 + 2ζr,nr ωr,nr (jωkf ) + ωr,n
r 

2


(jωkf )
2
2
(jωkf ) + 2ζr,nr ωr,nr (jωkf ) + ωr,nr= ρT(2)
ku ϕ(jωkf ),where the subscripts that correspond to each actuator isku ∈ {p, v}, the number of designed resonance frequency
is nr ∈ N, the index of the resonance frequency is
kr = 1, . . . , nr , the tuning parameter is ρ = [ρp , ρv ] thatconsists of ρku ∈ R2nr +1 , the resonance angular frequency
is ωr,kr , and the damping coefficient is ζr,kr .
The designed resonant filter consists of the sum of theresonant modes and phase compensators that are definedasκs2 + κψs
F (s) = 2.
(3)
s + 2ζr ωr s + ωr2
Fig. 6 shows the vector locus using a resonant filter andthe coefficients κ and ψ represent the gain and phase ofeach resonant mode.
3.2 Optimization problem formulation
In this paper, robust stability, robust performance, andhardware constraints are considered in frequency response
data. For robust performance, the resonant filters are optimized to minimize the worst case of the error amplitude
spectrum. For hardware constraints, the stroke limitationof the PZT actuator is considered that the maximum
amplitude spectrum of yp is less than the maximum value
with the given feedback controller. For robust stability,
the vector locus with resonant filters must be on the sameside against (−1, j0) and at the outside of the modulus

Fig. 6. Vector locus using a resonant filter with modulus
margin and phase stabilization in Nyquist diagram.
margin. From these conditions, the optimization problem
is formulated as follows.
minimizemax |ekc (jωkf )|
(4a)
ρ

∀k

∀
c , kfsubject to |yp,kc (jωkf )| ≤ yp,max(4b)

ws (jωkf )|Skc (jωkf , ρ)| ≤ 1

(4c)


 π
π
− ≤ ∠ 1 + Lkc (jωkf , ρ) − ∠ 1 + Gkc (jωkf ) ≤ ,
2
2

(4d)

∀ k ,∀ kcfwhere ws is the weighting of the sensitivity function andGkc (jωr ) = Gp,kc (jωkf ) + Gv,kc (jωkf ),(5)
Lkc (jωkf , ρ) = Lp,kc (jωkf , ρp ) + Lv,kc (jωkf , ρv ), (6)Lp,kc (jωkf , ρp ) = Gp,kc (jωkf )Frp (jωkf , ρp ),(7)
Lv,kc (jωkf , ρv ) = Gv,kc (jωkf )Frv (jωkf , ρv ),
(8)
1
.
(9)
Skc (jωkf , ρ) =1 + Lkc (jωkf , ρ)
3.3 Convex optimization using sequential linearization
In (4a), the objective function can be equivalently givenby1minimize max |ekc (jωkf )| ⇔ maximize min∀ k ,∀ k∀ k ,∀ k |e
ρ
ρ
(jω
c
c
f
f
kc
k )|

f

1
⇔ minimize −min
1 + Lkc (jωkf , ρ)
, (10)∀ k ,∀ kρ
c
f dkc (jωkf )where the error frequency response data is given byekc (jωkf ) = Skc (jωkf , ρ)dkc (jωkf ).
(11)
In (4b), the amplitude spectrum of the output of the PZT
actuator is evaluated as
Lp,kc (jωkf , ρp )dkc (jωkf ),(12)|yp,kc (jωkf )| =1 + Lkc (jωkf , ρ)and the maximum value with the given feedback controller
is given by
Gp,kc (jωkf )dkc (jωkf ).
(13)
yp,max = max∀ k ,∀ k
1 + Gkc (jωkf )cf
In (4d), the angle of the vector locus is evaluated by atan2function using the real and imaginary part of the vector
locus. From these analyses, the optimization problem isgiven by (14).
The optimization problem (14) is nonlinear and nonconvex. Using sequential linearization, the optimization
problem can be calculated by the iterative convex optimization and is given by (15).

−γminimizeρsubject to γ −∀ k ,∀ k
c
f

(14a)

1
1 + Lkc (jωkf , ρ) ≤ 0
dkc (jωkf )(14b)|dkc (jωkf )|Lp,kc (jωkf , ρp ) − 1 + Lkc (jωkf , ρ) ≤ 0
yp,maxws (jωkf ) − 1 + Lkc (jωkf , ρ) ≤ 0Im(1 + Lkc (jωkf , ρ))Im(1 + Gkc (jωkf ))π∓ atan2± atan2− ≤ 0 when ± Re(1 + Gkc (jωkf )) ≥ 0
Re(1 + Lkc (jωkf , ρ))Re(1 + Gkc (jωkf ))
2




Im(1 + Lkc (jωkf , ρ))Im(1 + Gkc (jωkf ))π± atan2∓ atan2− ≤ 0 when ± Re(1 + Gkc (jωkf )) ≥ 0
Re(1 + Lkc (jωkf , ρ))Re(1 + Gkc (jωkf ))2
minimizeρki|dkc (jωkf )|Reyp,maxws (jωkf ) − Re

± atan2

(14f)


∗
1
(1 + Lkc (jωkf , ρki −1 )) 
dkc (jωkf )1

(1 + Lkc (jωkf , ρki ))  ≤ 0
1
dkc (jωkf )
(1 + Lkc (jωkf , ρki −1 ))dkc (jωkf )!
∗Lp,kc (jωkf , ρp,ki −1 )Lp,kc (jωkf , ρp,ki ) − Re
Lp,kc (jωkf , ρp,ki −1 )∗
1 + Lkc (jωkf , ρki −1 )1 + Lkc (jωkf , ρki )1 + Lkc (jωkf , ρki −1 )Im(1 + Lkc ,ki −1 (jωkf , ρ))Re(1 + Lkc ,ki −1 (jωkf , ρ))


+

(15b)

!
∗

1 + Lkc (jωkf , ρki −1 )1 + Lkc (jωkf , ρki )≤0
1 + Lkc (jωkf , ρki −1 )

(15c)

!
≤0

Re(1 + Lkc ,ki −1 (jωkf , ρ))Im(1 + Lkc ,ki (jωkf , ρ)) − Im(1 + Lkc ,ki −1 (jωkf , ρ))Re(1 + Lkc ,ki (jωkf , ρ))
1 + Lkc ,ki −1 (jωkf , ρ)

(15d)
!


± atan22

Im(1 + Gkc (jωkf ))Re(1 + Gkc (jωkf ))Im(1 + Gkc (jωkf ))Re(1 + Gkc (jωkf ))−when ± Re(1 + Gkc (jωkf )) ≥ 0


(14e)

(15a)



subject to γ − Re 
∀ k ,∀ k
c
f



(14d)

−γ


∓ atan2(14c)

Im(1 + Lkc ,ki −1 (jωkf , ρ))Re(1 + Lkc ,ki −1 (jωkf , ρ))


+

π
≤0
2

(15e)

Re(1 + Lkc ,ki −1 (jωkf , ρ))Im(1 + Lkc ,ki (jωkf , ρ)) − Im(1 + Lkc ,ki −1 (jωkf , ρ))Re(1 + Lkc ,ki (jωkf , ρ))
1 + Lkc ,ki −1 (jωkf , ρ)2when ± Re(1 + Gkc (jωkf )) ≥ 0
4. VERIFICATION OF DISTURBANCE REJECTIONPERFORMANCEIn this section, the verification of disturbance rejection
performance is conducted in a dual-stage actuator HDD
benchmark problem. The result shows that the resonant
filters are optimized with the conditions of robust stability, robust performance, and hardware constraints. The
track-following performance with designed resonant filters
outperforms that without resonant filters.

4.1 Conditions
The frequency response data is used from 100 Hz to
Nyquist frequency Fs /2 = 1/2Ts = 25.2 kHz. The frequency response data is arranged at linearly even intervals
in every 1 Hz and the number of data points is nf = 25101.
Nyquist diagram, sensitivity function, amplitude spectrumof e, and amplitude spectrum of yp without resonant filters
are shown in Fig. 7, Fig. 8, Fig. 9, and Fig. 10. From Fig. 9,
the resonant filters are designed at eight frequencies withvertical black dotted lines, and the dumping coefficients of
all resonant filters are set to ζr = 0.05. For the initialcondition, all tuning parameters are set to ρ = 0 and
Frp = Frv = 1. In the robust stability condition, themodulus margin is set to 1/ws = 6 dB.

!

∓ atan2

π
− ≤0
2

(15f)
4.2 Optimization results
The optimization of the resonant filter design is conducted
by YALMIP (Lofberg, 2004) and Mosek (Mosek, 2021)
until the improvement of the objective function from theprevious iteration becomes less than 0.1 %. Nyquist diagram, sensitivity function, amplitude spectrum of e, andamplitude spectrum of yp with resonant filters are shownin Fig. 11, Fig. 12, Fig. 13, and Fig. 14. The optimization
result shows that the resonant filters are designed withconditions of robust stability, robust performance, andhardware constraints in 9 cases of the controlled system.
4.3 Simulation results
The time domain simulation is conducted in a dualstage actuator HDD benchmark problem without and withresonant filters in 9 cases of the controlled system. Fig. 15
shows the track-following performance. It shows that the
tracking errors with resonant filters are smaller than those
without resonant filters in all 9 cases. Fig. 16 shows the
maximum stroke in a PZT actuator. It shows that the
maximum stroke values with resonant filters are almost thesame as those without resonant filters and both controlled
systems satisfy the stroke limitation. As a result, thecontrolled system with resonant filters outperforms thatwithout resonant filters.

200

4

200

4

100

2

100

2

0

0

0

0

-100

-2

-100

-2

-200
-200-100 0 100 200

-4
-4

-200
-200-100 0 100 200

-4
-4

-2

0

2

4

Fig. 7. Nyquist diagram without resonant filters.

10

0

0

-10

-10

-20

-20

-30

-30

-40

-40
103

104

Fig. 8. Sensitivity function without resonant filters.

-50
102

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

103

104

Fig. 9. Amplitude spectrum of e without resonant filters.

0
102

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

103

104

Fig. 10. Amplitude spectrum of yp without resonant filters.

4

103

104

103

104

Fig. 13. Amplitude spectrum of e with resonant filters.

1

0
102

2

Fig. 12. Sensitivity function with resonant filters.

0.5

0
102

0

Fig. 11. Nyquist diagram with resonant filters.

10

-50
102

-2

0
102

103

104

Fig. 14. Amplitude spectrum of yp with resonant filters.

20

15

10

5

0
1

2

3

4

5

6

7

8

9

Fig. 15. Simulation results of track-following performance
3σ(yc ). × and ⃝ denote without and with optimized
resonant filters.
50
40
30
20
10
0
1

2

3

4

5

6

7

8

9

Fig. 16. Simulation results of maximum stroke in a PZT
actuator max|ycp |. × and ⃝ denote without and with
optimized resonant filters.
5. CONCLUSION
In this paper, the design method of optimal resonant
filters is developed to improve track-following performancein a dual-stage actuator HDD. The resonant filters withstructured parameterization are optimized by iterative
convex optimization directly using the frequency response
data of the controlled system. Robust stability, robust
performance, and hardware constraints are considered forthe optimization calculation in 9 cases of the controlled
system. The disturbance rejection performance of the optimized resonant filters is validated in a dual-stage actuator
HDD benchmark problem. Ongoing researches focus on
the usage of both frequency domain and time domain data,
designing other feedback controllers simultaneously, andthe optimal design of multirate filters.
REFERENCES
Atsumi, T. (2016). Emerging Technology for HeadPositioning System in HDDs. IEEJ Journal of Industry
Applications, 5(2), 117–122. doi:10.1541/ieejjia.5.117.
Atsumi, T. (2022). Magnetic-head positioning control
system in HDDs. URL https://jp.mathworks.com/
matlabcentral/fileexchange/111515.

Atsumi, T., Okuyama, A., and Kobayashi, M. (2007).
Track-Following Control Using Resonant Filter in Hard
Disk Drives. IEEE/ASME Transactions on Mechatronics, 12(4), 472–479. doi:10.1109/TMECH.2007.901944.
Bashash, S. and Shariat, S. (2019). Performance enhancement of hard disk drives through data-driven control design and population clustering. Precision Engineering, 56(November 2018), 267–279. doi:10.1016/
j.precisioneng.2018.12.007.
Chen, X. and Tomizuka, M. (2014). New Repetitive
Control With Improved Steady-State Performance andAccelerated Transient. IEEE Transactions on Control Systems Technology, 22(2), 664–675. doi:10.1109/
TCST.2013.2253102.
Chen, X. and Tomizuka, M. (2016).
Discrete-Time
Reduced-Complexity Youla Parameterization for DualInput Single-Output Systems. IEEE Transactions on
Control Systems Technology, 24(1), 302–309. doi:10.
1109/TCST.2015.2422796.
Fujimoto, H. (2009). RRO Compensation of Hard Disk
Drives With Multirate Repetitive Perfect Tracking Control. IEEE Transactions on Industrial Electronics,
56(10), 3825–3831. doi:10.1109/TIE.2009.2017558.
Lofberg, J. (2004). YALMIP : a toolbox for modeling andoptimization in MATLAB. In 2004 IEEE International
Conference on Robotics and Automation (IEEE Cat.
No.04CH37508), 284–289. IEEE. doi:10.1109/CACSD.
2004.1393890.
Mosek (2021). MOSEK 9.3. URL https://www.mosek.
com.
Ohno, K. and Hara, T. (2006). Adaptive resonant mode
compensation for hard disk drives. IEEE Transactions
on Industrial Electronics, 53(2), 624–630. doi:10.1109/
TIE.2006.870660.
Pérez-Arancibia, N.O., Tsao, T.C., and Gibson, J.S.
(2010). A new method for synthesizing multiple-period
adaptive–repetitive controllers and its application to thecontrol of hard disk drives. Automatica, 46(7), 1186–
1195. doi:10.1016/j.automatica.2010.04.007.
Potu Surya Prakash, N. and Horowitz, R. (2022). DataDriven Robust Feedback Control Design for MultiActuator Hard Disk Drives. In 2nd IFAC Modeling,
Estimation and Control Conference.
Shahsavari, B., Keikha, E., Fu Zhang, and Horowitz,
R. (2015). Adaptive Repetitive Control Design With
Online Secondary Path Modeling and Application to
Bit-Patterned Media Recording. IEEE Transactions
on Magnetics, 51(4), 1–8. doi:10.1109/TMAG.2014.
2364737.
Sun, L., Jiang, T., and Chen, X. (2017). Adaptive Loop
Shaping for Wideband Disturbances Attenuation in Precision Information Storage Systems. IEEE Transactions
on Magnetics, 53(5), 1–13. doi:10.1109/TMAG.2017.
2654200.
Yabui, S., Atsumi, T., and Inoue, T. (2020). Coupling
Controller Design for MISO System of Head Positioning
Control Systems in HDDs. IEEE Transactions on Magnetics, 56(5), 1–9. doi:10.1109/TMAG.2020.2978156.
Zheng, M., Zhou, S., and Tomizuka, M. (2017). A design
methodology for disturbance observer with application
to precision motion control: An H-infinity based approach. In 2017 American Control Conference (ACC),
3524–3529. IEEE. doi:10.23919/ACC.2017.7963492.