2007 IEEE International Conference on Control and Automation ThCl-2Guangzhou, CHINA - May 30 to June 1, 2007

Nonlinear Tracking Control for a Hard Disk DriveDual-Stage Actuator System

Jinchuan Zheng*, Minyue Fu*, Youyi Wangt and Chunling Dut*School of Electrical Engineering and Computer Science

The University of Newcastle, Callaghan, NSW 2308, AustraliaEmail: Jinchuan.Zheng @newcastle.edu.au

tSchool of Electrical & Electronic EngineeringNanyang Technological University, Singapore 639798tA*STAR, Data Storage Institute, Singapore 117608

Abstract-This paper presents a nonlinear tracking controlmethod for a hard disk drive (HDD) dual-stage actuator (DSA)system that consists of a voice coil motor (VCM) actuator anda piezoelectric (PZT) microactuator. Conventional track seekingcontrollers for DSA systems were generally designed to enable theVCM actuator to approach the target track without overshoot.However, we observe that this strategy is unable to achieve theminimal settling time when the target tracks are beyond thePZT actuator stroke limit. To further reduce the settling time,we design the VCM actuator controller to yield a closed-loopsystem with a small damping ratio for a fast rise time and certainallowable overshoot. Then, a composite nonlinear control law isdesigned for the PZT actuator to reduce the overshoot caused bythe VCM actuator as the system output approaches the targettrack. Experimental results show that the proposed dual-stageservo outperforms the conventional dual-stage servo in short-spanseeking and additionally achieves better track following accuracythan the VCM only single-stage servo.

I. INTRODUCTION

The continuous increase of track density and high speeddata access in hard disk drives (HDDs) require the headposition accurately maintained along the track center (trackfollowing) and swiftly moved from one track to another (trackseeking). However, the conventional voice coil motor (VCM)only single-stage servo is hard to provide high performancedue to its mechanical resonance modes, various disturbancesand noise in HDDs. The dual-stage actuators (DSA) areintroduced to overcome the limitation [1], [2]. In DSA servosystems, the VCM actuator is used as the primary stage toprovide long track seeking but with poor accuracy and slowresponse time while the secondary stage such as a piezoelectric(PZT) microactuator [3] is used to provide higher precisionand faster response but with a stroke limit. By combiningthe DSA system with properly designed controllers, the twoactuators are complementary to each other and the defectsof one actuator can be compensated by the merits of theother one. Therefore, the DSA system can provide both largedisplacement, high positioning accuracy and fast track seeking.

It is a challenging task to design the DSA controllers toyield an optimal performance because of the specific char-acteristics: 1) The DSA system is a dual-input single-output(DISO) system, which means that for a given desired trajectory,alternative inputs to the two actuators are not unique. Thus, aproper control strategy is required for control allocation. 2) Thesecondary actuator typically has a very limited travel range,which results in the actuator saturation problem. A variety of

Fig. 1. A hard disk drive dual-stage actuator with a piezoelectric microactu-ator (Throughout this paper, we will refer to the combined VCM and E-blockassembly as VCM actuator and the combined PZT and suspension assemblyas PZT actuator).

approaches have been reported to deal with the DSA controlproblems. For example, control design for track followingand settling can be found in [4]-[6]. The secondary actuatorsaturation problem was explicitly taken into account duringthe control design [7], [8]. In [9], a decoupled track-seekingcontroller using a three-step design approach is developedto enable high-speed one-track seeking and short-span track-seeking for a dual-stage servo system. The control design forthe PZT actuator by minimizing the destructive interferenceis proposed in [10] to attain desired time and frequencyresponses.

In this paper, we study a dual-stage HDD with a push-pull PZT microactuator as shown in Fig. 1. It consists of aVCM actuator as the primary stage and a PZT actuator as thesecondary stage. The PZT is located between the suspensionand the E-block, which is moved by the VCM. The twoactuators are respectively driven through a PZT amplifier and aVCM driver. The VCM driver has a voltage input limit of +3.5V. The PZT actuator has a stroke limit of +0.5 ,um and thePZT amplifier has a voltage input limit of +1.5 V. The headposition is measured using a laser Doppler vibrometer (LDV)in our experiment setup. We assume that the coupling effectsbetween the two actuators are negligible and then measure thefrequency responses of the VCM and PZT actuator separately,which are shown with the dashed lines in Fig. 2.

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1-4244-0818-0/07/$20.00 (¢ 2007 IEEE

-~~~~~~~~~~~~~~~~~~240 f g'o ' 2 s2 X

ro4~~~~~~~~-3

10' X0 10-;1=- 10

(a) VCM actuator (b) PZT actuator

Fig. 2. Frequency responses of a hard disk drive DSA system (dashed line:measured response; solid line: measured response with resonance compen-sator; dotted line: simulated response with resonance compensator).

By far, most of the work on the DSA track seeking controlto follow a step command input is based on the strategythat the VCM actuator control loop is designed to have littleovershoot, and the PZT actuator control loop is designed tofollow the position error of the VCM actuator [9]-[1 1]. Underthis conventional strategy, the total settling time can be reducedby the time that it takes for the PZT actuator to reach itsstroke limit. However, we observe that when the target track isbeyond the PZT actuator travel range, this strategy is unable tominimize the total settling time because the PZT actuator canmake little contributions due to its very limited travel range.To further reduce the settling time under this circumstance, wepropose that the VCM actuator controller can be designed toyield a closed-loop system with a small damping ratio for a fastrise time allowing a certain level of overshoot, and then as theVCM actuator approaches the target track the PZT actuatorcontrol loop is used to reduce the overshoot caused by theVCM actuator. In this way, the total settling time is muchless than that of the conventional control provided that theovershoot is within the PZT actuator stroke limit.

To perform the aforementioned control strategy, we willuse the DSA control structure in Fig. 3, which consists ofresonance compensators, a state estimator and a nonlineartracking controller. The design of these control blocks willbe presented in Section II. Section III shows the experimentalresults. Conclusions are given in Section IV.

II. NONLINEAR TRACKING CONTROL DESIGN

Our objective here is to design the controllers in Fig. 3such that the two actuators cooperate to enable head positiony to track a step command input of amplitude Yr rapidlywithout exhibiting a large overshoot. In this section, we firstlydescribe the resonance compensation in Section I1-A and stateestimator in Section 1I-B for the DSA system. Then, SectionII-C presents the proximate time-optimal control law to yielda VCM actuator closed-loop system with a small dampingratio so as to achieve a quick rise time. Finally, we developa composite nonlinear control law with a step-by-step designprocedure in Section II-D and E for the PZT actuator, whichcan reduce the overshoot caused by the VCM actuator as thehead position approaches the target track.

Resonancecompensator DSA plant

r~~------

Fig. 3. Block diagram of a complete DSA control system.

A. Resonance Compensation

The resonances of the B-block and the suspension exerts

adverse effects on the tracking performance. Here, we use a

compensator with cascaded notch filters to actively damp the

resonances. The transfer function of the resonance compen-

sator is given by

F n s2 +2:iS+ at)2Fv,p lls2+2(2w2'i+ 1i <2i

ii 12+22a)s L)(1)

where wi denotes the resonance frequency, (Ii and (2i are thedamping ratios chosen to notch the resonance peak. We haveimplemented an Fv to damp the main resonance modes at 4.2,6, and 9 kHz in the VCM actuator, and an Fp to damp the reso-nance modes at 6.8, 8.8, and 11.1 kHz in the PZT actuator. Thesolid lines in Fig. 2 show the measured frequency responsesof the VCM and PZT actuator after resonance compensation.It can be seen that the main resonances of the actuators arelargely damped. From the measured response with resonancecompensation, we observe that the compensated VCM actuatorand PZT actuator can be reliably approximated as a puredouble integrator and a second-order model, respectively in thefrequency range of interest. Furthermore, similar to [11], weplace two saturation blocks before the resonance compensatorsand set the saturation levels U-1 and U12 sufficiently small suchthat the control signals uv and up never exceed the voltagelimits of the VCM driver and PZT amplifier, respectively. Inthis way, we can treat the DSA with resonance compensationas a linear DISO system, which is represented in a state-spaceform as follows:

, ,

2: 2

Y

Alx + Blsat(ul), x:(O)°A2X2 + B2sat(U2), X2 (O)Yv + YP CIXI + C2X2

-0= 0

where the state x1 [Yv yV]T, X2 [YP YP]T,

A I O1 [B

0] C= [ 1 0 ],

(2)

(3)

[ 1 0 ] (4)

and the saturation function sat(ui) (i 1, 2) is defined as

sat(ui) sgn(ui)min{ui, luil}

where uii is the saturation level of the ith control input.

(5)

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A2 0 1. B2 0

, C2a, a2 b2

The DSA model parameters in (3)-(5) are identified to matchthe measured responses with resonance compensator, whichare given by

b= 1.7 x 108, a -109, a2 -3.1 x 104 (6)b2 4.3 x108, Ui 3 V, U2 1.25 V. (7)

Fig. 2 shows that the identified models (dotted lines) can matchthe measured ones (solid lines) precisely in the frequency rangeof interest. Thus, from now on we will take the DSA systemin (2) as the plant model for further control design.

B. State Estimator Design

Since the head position y is the only measurable signalfor feedback control, we need to estimate the VCM and PZTactuator states by using the identified DSA model in (2)-(7).The state estimator is given by

i= ADXC + BDsat(u) + L(y -CDS) (8)

where x [Yv Yv YR p1 U =[UI 2]T,

AD= [~iO A BD [ O B2]CD [C1 C2].

The state estimator gain L can be calculated using the poleplacement method [12] by selecting the well-damped estimatorpoles as two to six times faster than the DSA servo bandwidth.Remark 1: According to the separation principle [12], the

complete DSA control system in Fig. 3 is stable if thecontrol laws ui and U2 that are obtained assuming actual statefeedback and the state estimator are both stable. Thus, wecan design the control laws and the estimator separately yetused together. For the simplicity of design, we will present theVCM and PZT actuator control law ui and U2 in Section II-Cand D assuming that the true state x is available for feedback.However, the estimated states x will be used in final controllerimplementation.

C. VCM Actuator Control Design

The role of the VCM actuator is to provide large seekinglength beyond the PZT stroke limit. Thus, time optimal controlis critical to move the head position quickly from one track toanother. The proximate time-optimal servomechanism (PTOS)is a practical near time-optimal controller that can accom-modate plant uncertainty and measurement noise. Hence, weapply the PTOS control law [13] to the VCM actuator E1 in(2) and the controller is independent of the PZT control loop.The PTOS control law is given by

input remains continuous as well, we have the followingconstraints

2k1 Uib1k2 "'Yi k

(12)

The PTOS control law introduces a linear region close to thesetpoint to reduce the control chatter. In the region IeCI < yi,the control is linear and thus the gain K [k1 k21 can bedesigned by any linear control techniques. For instance, usingthe pole placement method [12] we obtain a parameterizedstate feedback gain K as follows

K b [42w2 47aw(1,] (13)

where (I and W) (Hz) respectively represent the dampingratio and undamped natural frequency of the closed-loopsystem CI(sI -A1 + BIK)- Bl, whose poles are placedat27wwi g1+ 1 2.

In conventional DSA control systems, the VCM actuatorcontroller is generally designed to have little overshoot suchas by choosing a large damping ratio in (13). However, in ourproposed control a small damping ratio is chosen for a fast risetime and the resultant overshoot is within the PZT stroke limit,which can be reduced by the PZT actuator under a compositenonlinear control law as will be given in Section II-D.

D. PZT Actuator Control DesignThe goal of the control design for the PZT actuator E2 in (2)

is to enable the PZT actuator to reduce the overshoot caused bythe VCM actuator. We have the following step-by-step designprocedure.

Step 1: Design a linear feedback control law

U2L FX2 (14)

where F [fi f2] is chosen such that the PZT actuatorcontrol system as given by

Xc2 A2X2 + B2sat(FX2) (15)

is globally asymptotically stable (GAS) and the correspondingclosed-loop system in the absence of input saturation C2(sI-A2 -B2F)-1B2 has a larger damping ratio and a higherundamped natural frequency than those of the VCM actuatorcontrol loop. To do this, we choose

F =-B P (16)

where P pT > 0 is the solution of the following Lyapunovequation

ATP+ PA2 -Q (17)

UI sat [k2(f(eI) - V)]

f(el) =l el for leiI >yl

sgn(e1)(V u--l~cl-f)for jell > ye1 Yr -Yv

9 for a given Q QT > 0. Note that the solution of P exists(9) since A2 is Hurwitz. To involve the closed-loop propertiesI f,, explicitly with the control law, we defineU10)

(18)(1 1)

where sat[.] is with the saturation level of Ui1, a~is referredto as the acceleration discount factor, k, and k2 are constantgains, and Yl represents the size of a linear region. To make thefunctions f (el) and f' (el) continuous such that the control

where q, and q2 are tuning parameters. Substituting (18) into(17) yields P, which gives the feedback gain (16) as follows

F [2 [a2q, alq2- ql.2ala2 (19)

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Q q, 0

, q, > 0, q2 > 00 q2

Moreover, the resulting poles of the closed-loop systemC2 (sI-A2-B2F)- 1B2 with (19) if complex conjugate havethe undamped natural frequency and damping ratio as follows:

1 b2W4)2 - q- ai,

b2q -b2alq2 -2aia22=lb2

4aIa2 V j-1a aI

Thus, we can easily achieve the desired W)2 and (2 by choosinga proper pair of q, and q2.

Step 2: Construct the nonlinear feedback control law

Hence, the PZT actuator closed-loop system with linear feed-back control only (15) is GAS. Furthermore, the PZT actuatorclosed-loop system with the composite nonlinear control law(22) can be expressed as

X2 A2X2 + B2sat(FX2 + LU2N)- (26)

It is obvious that the system (26) satisfies the converging-inputbounded-state (CIBS) property (See [15] for the definition)since A2 is Hurwitz, Isat(.)I <_ U2, and the nonlinear controlinput U2N has

lim U2N(t) 0t oo

(27)

U2N = I(Yr,Y)H YVYr ] (20)

H + (al + b2fl + biki) (a2 + b2f2 + blk2)] (21)

where H is taken to achieve desired closed-loop systemdynamics, which will be clear in Section II-E; and vy(Y, y)is any nonnegative function locally Lipschitz in y, which ischosen to enable the PZT actuator to reduce the overshootcaused by the VCM actuator as the head position y approachesthe target track. The choice of y(Yr, y) will be discussed inSection II-E.

Step 3: Combine the linear and nonlinear feedback controllaws derived in Steps 1 and 2 to form a composite nonlinearcontroller for the PZT actuator

U2 UUIL + U2N

FX2 + Y(Yr, Y)H [YV. Yr1 (22)

With the VCM actuator controller in (9) and the PZTactuator controller as given by (22), we have the followingresults regarding the step response of the DSA closed-loopsystem.Lemma 1: Consider the DSA system in (2) with the VCM

actuator E1 under the PTOS control law (9) and the PZTactuator E2 under the nonlinear control law (22) for anynonnegative function IYr, y) locally Lipschitz in y. Then thecomposite control law will drive the head position y to trackasymptotically any step command input of amplitude Yr.

Proof: The VCM actuator closed-loop system under thePTOS control law can be represented as

z1I Alx + B, sat[k2(f (eIYV)] (23)

where f(el) is defined in (10). It has been proved in [13] thatthe system (23) can track asymptotically any step commandinput of amplitude Yr, i.e.,

lim YV(t) Yr,t Oo

lrm Pv(t) == 0.

Next, we define a Lyapunov function V xT P,P given in (17). Evaluating the derivative of V alctrajectories of the system in (15) yields

V=S2 PX2 + 2 Pz~2xf(AfP + PA2)X2 + 2BTPX2sat(FX2)

2xfQx2 -2FX2sat(FX2)< -x QX2 < 0.

(24)

X2 withong the

(25)

which can be easily deduced from (20) and (24).The proof finishes by observing that the DSA closed-loop

system formed by (23) and (26) has a cascaded structure andit satisfies the conditions of Theorem 1 in [15]. It then followsthat the cascade system formed by (23) and (26) is GAS atthe origin. Thus, for the PZT actuator closed-loop system (26)we have

lim x2(t) 0t oo

(28)

and therefore,

lim y(t) lim [Clxl (t) + C2X2 (t)]tOo tOo

Yr. (29)

Remark 2: Lemma 1 shows that the value of IYr, y) doesnot affect the ability of the overall DSA closed-loop system totrack asymptotically any step command input. However, fromthe perspective of transient performance, a proper choice ofIYr, y) should be carried out to improve the performance ofthe overall DSA closed-loop system. This is the key propertyof the proposed control design.

E. Selecting IYr, y) for Improved Performance

The function y(Yr, y) is used to tune the control law toachieve our objective, a quick step response without a largeovershoot. More specifically, we design the VCM actuatorcontrol loop to have a small damping ratio for a quick risetime and employ the PZT actuator control loop that is designedto be highly damped to reduce the overshoot caused by theVCM actuator as the head position y approaches the targettrack. This control strategy implies that the dynamics of theDSA closed-loop system should be dominated by the VCMactuator control loop when the head position is far away fromthe target track, while dominated by the PZT actuator controlloop when the head position approaches the target track. Thepurpose of the function y(r, y) is to fulfill a smooth transitionfrom the VCM control loop to the PZT control loop.

Consider the DSA system (2) with the control laws in (9)and (22), and assume that the tracking error (yr -y) is smallsuch that the control inputs do not exceed the limits and thecontrol law (9) works within its linear region. Thus, the DSAclosed-loop system can be expressed as

J:

= Ax+ Byr*ly =Cx:

1504

(30)

where

A AlA1BK 0 lA[(j,yr)B2H A2-+B2F '

B -Xy(yr,y)B2H] Ic= [01021.

The DSA closed-loop transfer function from Yr to y can beobtained by

G(s) C(sI -A)-1B

(1-)GI (s) +-G2 (s)

III. EXPERIMENTAL RESULTS AND DISCUSSION

The proposed DSA tracking control has been applied to thedual-stage HDD in Fig. 1. The controller is implemented on areal-time DSP system (dSPACE 1103) with a sample rate of40 kHz. The state estimator gain is calculated as

L 104 x [5.7 51164 1.3 36952]T.

The PTOS controller in (9) for the VCM actuator is obtainedby choosing a), 1200 Hz and thus Yi 9 ,um. We find that(I can be adjusted as

(31)r 0.3, Yr < 0.5,um

(i (Yr) 1.3x(ln(y)+0. 7) y > 0.5 ,umL w'2+((n(y,)+0.7)2 Ywhere (35)

Gi(s) b1k,s2 + b1k2s + b1k,G2(S) - -(a, + b2fl)

S2- (a2 + b2f2)s -(al + b2fl)

(32) such that the resultant overshoot caused by the VCM approx-imately equals to the PZT stroke limit (i.e., 0.5 ,um) when

(33) Yr > 0.5 Am. Hence, the linear gain K is given by

K = [0.33 0.0009(1(yYr)].are the closed-loop transfer function of the VCM and PZTactuator control loop, respectively.

At this point, it is clear that the DSA closed-loop systemdynamics (31) change from the VCM control loop to thePZT control loop when ay increases from 0 to 1. This desiredfeature is due to the proper selection of H in (21). From theperspective of zero placement, when ay changes from 0 to 1the zeros of (31) is moved from the pole locations of the PZTcontrol loop (32) to those of the VCM control loop (33). Sincethe zeros near the poles reduce the effects of the poles on thetotal response, we can use y to tune the system dynamicsfor desired performance. A similar control technique for SISOlinear systems can be found in [14], which however uses thenonlinear feedback law to increase the damping ratio of theclosed-loop system poles to reduce the overshoot.The function IYr, y) can be chosen as a function of the

tracking error, Yr -y. The following shows one choice of -y:

(Yr Y) e- vr-v (34)

where d3 > 0 is a tuning parameter. The function y(Yr, y) in(34) changes from 0 to 1 as y Yr. The parameter d3 can

be adjusted with respect to the amplitude of Yr relative to thePZT actuator travel limit Y2:

1) If Yr < Y2, 3 should be sufficiently small such thatay converges to 1 quickly, which implies that the PZTcontrol loop dominates the DSA closed-loop systemdynamics over the whole control stage.

2) If Yr > Y2, 3 should be large so as to divide the controlstages into two parts. At the initial stage, ay closes to 0,which implies that the VCM control loop dominates theDSA closed-loop system dynamics to achieve a fast risetime. When the position output y approaches the targettrack, ay closes to 1, which implies that the DSA closed-loop system dynamics is dominated by the PZT controlloop that is highly damped. This high damping propertycan in turn imply that the PZT actuator is enabled toreduce the overshoot caused by the VCM actuator.

(36)

For the PZT control design, we choose q= 1.3 and q29 x 10-9, and thus the linear feedback gain F is given by

F =-[0.28 0.00007] (37)

which results in w)2 5300 Hz and (2 0.95. The nonlinearfeedback gain is given by

H -[2.4 0.00015 -0.000035(1(Yr)] (38)

with (i (Yr) in (35). The nonlinear function (34) is chosen as

) e-0.05 y-l-Y(Yr Y) -0.5ly-yr

Yr < 0.5 AtmYr > 0.5 Atm

(39)

In order to compare the proposed control with the conven-tional control where the VCM control loop is generally tunedto have no overshoot, we choose (I 0.9 for any Yr andretain the other tuning parameters, then following the samedesign procedure yields a conventional controller that is usedfor comparison with our proposed controller.

A. Track Seeking

We firstly evaluate the track seeking responses within thePZT stroke limit. Fig. 4 shows the result for Yr 0.5 ,m. Wecan see that the settling times under the proposed control andthe conventional control are almost the same. This is becausewithin the PZT stroke limit the PZT control loop dominates theDSA closed-loop system dynamics whatever the VCM controlloop is tuned. However, when the seeking length is beyond thePZT stroke limit as shown in Fig. 5, the settling time underthe proposed control is significantly reduced compared with theconventional control. The tests of other seeking lengths are alsoconducted and summarized in Table I for easy comparison. It isshown that the proposed control can further reduce the settlingtime by more than 14% for short-span seeking within 2 Atmand 5 ,m. However, the reduction ratio decreases for long-span seeking because the PZT stroke limit is slight relative tolarge seeking lengths.

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TABLE ICOMPARISON OF THE SETTLING TIME IMPROVEMENT

Seeking length Settling Time (ms) Improvement

(Am) Conventional Proposed (%)0.5 0.25 0.25 02 0.38 0.3 213 0.4 0.3 255 0.42 0.36 147 0.41 0.4 2.5

(a) Proposed control (b) Conventional control

Fig. 4. 0.5 ,um seek. LDV gain: 2 ,umlV, Chl: Dual-stage output, y; Ch2:Estimated VCM output, Yv; Ch3: Estimated PZT output, yp; Ch4: PZTcontrol signal, up. Within the PZT stroke limit, the settling time in bothcontrol are similarly 0.25 ms.

(a) Time responses (b) Power spectrum density

Fig. 6. Position error signals (PES).

(b) Conventional control

Fig. 5. 2 ,um seek. LDV gain: 2 ,umNV, Chl: Dual-stage output, y; Ch2:Estimated VCM output, Yv; Ch3: Estimated PZT output, yp; Ch4: PZTcontrol signal, up. The settling time in the conventional control is 0.38 ms,which is reduced to 0.3 ms in the proposed control.

B. Track FollowingFinally, we evaluate the position error signal (PES) of the

proposed DSA control system in Fig. 3 by injecting thedisturbance d and setting Yr 0. The disturbance signals arecollected from a real HDD, which consist of repeatable runout(RRO) and non-repeatable runout (NRRO). The experimentalsteady-state PES (i.e., PES y -d) are shown in Fig. 6.The dual-stage servo decreases the PES 3u value from 167.1nm (VCM only servo) to 143.4 nm, which is a 14%o reductionratio. The power spectrum density (PSD) of the PES indicatesthat the dual-stage servo yields a considerable improvement inthe frequency range from 0 to 2500 Hz. Above 2500 Hz thedual-stage servo leads to some small degradation. This trade-off is due to the waterbed effect in linear control systems.

IV. CONCLUSIONS

We have proposed a nonlinear tracking control for a diskdrive DSA system with a PZT actuator. Distinct from the con-ventional control, the VCM actuator control loop is designedto have a small damping ratio for a fast rise time. The PZTactuator control loop is then enabled by a nonlinear controllaw to reduce the overshoot caused by the VCM actuator asthe system output approaches the target track. Experimentalresults showed that the proposed control can significantlyreduce the settling time in short-span seeking by more than14%o compared with the conventional control. Moreover, theproposed DSA servo system can achieve a 14% reduction ratioof the PES 3u compared to the VCM only servo.

REFERENCES

[1] K. Mori, T. Munemoto, H. Otsuki, Y Yamaguchi, and K. Akagi, "A dual-stage magnetic disk drive actuator using a piezoelectric device for a hightrack density," IEEE Trans. Magn., vol. 27, no. 6, pp. 5298-5300,

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[5] X. Huang, and R. Horowitz, "Robust controller design of a dual-stagedisk drive servo system with an instrumented suspension," IEEE Trans.Magn., vol. 41, no. 8, pp. 2406-2413, Aug. 2005.

[6] H. Numasato and M. Tomizuka, "Settling control and performance of adual-actuator system for hard disk drives," IEEEIASME Trans. Mecha-tron., vol. 8, no. 4, pp. 431-438, Dec. 2003.

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[15] E. Sontag, "Remarks on stabilization and input-to-state stability," inProc. IEEE Conf: Decision and Control, 1989, pp. 1376-1378.

(a) Proposed control

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Sih,gl.-stag. (VCM .,,Iy) s.,. (3-1 67.1 o-, si"gl.-Stag. (VCM ONY) S.,o

0.8E 0.6

0.4

3:.D 0.2

).20 20 .o 60 80 100 0 1000 2000 3000 4000 5000 6000 7000 8000

D.1-stag. s.,. (3-143.4 o-, D-1-stag. s.,.

0.8

0.60

0.4

0.2

0