A NOVEL DESIGN OF SHORT-SEEKING CONTROL FOR DUAL-ACTUATOR HARD DISK DRIVES

Copyright (c) 2005 IFAC. All rights reserved 16th Triennial World Congress, Prague, Czech Republic

A NOVEL DESIGN OF SHORT-SEEKING CONTROL FOR DUAL-AC HARD DISK DRIVES

TUATOR

Li Yang and Masayoshi Tomizuka Department of Mechanical Engineering University of California at Berkeley, Berkeley, Cal ifornia 94720, U.S.A. [email protected] , [email protected]

Abstract: This paper discusses short seeking contro l for a dual-actuator hard disk drive consisting of a coarse actuator (voice coil motor, VCM) and a fine actuator (piezoelectric actuator, PZT). A novel design method is proposed b ased on the dual-stage track following control and initial value adjustment (IVA ). This method, which adjusts the initial values of the track-following controllers f or short seeking, makes the use of the feed-forward controller and reference trajectory un necessary. The proposed design also takes into consideration the saturation property of the fine actuator by using a two-stage IVA scheme: (1) only VCM loop is utilized in the fi rst stage (beyond PZT stroke limit), and IVA is designed to yield a high-speed seeking; (2) PZT loop is switched on when the read/write head approaches the target track in the second stage (within the PZT stroke limit), and IVA is used to produce a fast and smoot h response. Simulation results verify the effectiveness of the proposed method. Copyright © 2005 IFAC Keywords: dual actuator, short seeking, actuator sa

1. INTRODUCTION

turation, hard disk drives (HDDs)

The dual-stage actuator can improve both trackfollowing and track-seeking performance. Many The hard disk drive industry continues to strive fo research r efforts have been devoted to the design of increased storage densities and reduced data access the track following servo using dual-stage actuator s. times. This demands performance improvements of Several designs for dual stage track-following serv os the head positioning system in terms of fast track have been proposed, such as a parallel type (Ding, et seeking and precise track following of the target al., 2000; Semba,et al., 1999), a decoupled type track. To meet these requirements, the servo (Mori, et al., 1991), a master-slave typeKoganezawa ( , bandwidth of the head positioning system must be et al., 1999), and a u-synthesis MIMO type increased to reject a wider range of disturbances s uch (Hernandez, et al., 1999). However, short seeking as disk flutter vibrations, spindle run-out, windag performance e, of dual-stage servos has not been and external vibration. The servo bandwidth, discussed much. It has only been evaluated based on however, is mainly limited by the actuator’s step responses of a closed-loop track-following mechanical resonance modes. Dual-stage actuation feedback system, or alternatively, achieved by a tw ohas been recognized as a powerful candidate to degree-of-freedom (TDOF) servo structure, which expand the servo bandwidth. The dual-actuator disk requires the design and implementation of both the file system consists of two actuators: coarse and f feed-forward ine controller and reference trajectory actuators. The coarse actuator is of low bandwidth (Kobayashi and Horowitz, 2001; Ding, et al., 2004). with a large operating range and it is used for coa rse positioning; the voice coil motor (VCM) is the most In this paper, we propose a novel method for dual popular coarse actuator. The fine actuator is of hi stage gh short seeking control based on dual-stage tra ck bandwidth with a small operating range and it is us following ed control and initial value adjustment. Thi s for fine positioning; the piezoelectric transducer method, which utilizes the same controllers as in (PZT) is a popular fine actuator. track following, tunes the initial values of track-

109

and following controllers for short seeking. By tuning the achieve basic performance. It is designed to be phase-lag-lead compensator. The PZT controller is initial values of the feedback controller, the desi red transient characteristics in short-seeking can be designed to achieve overall performance and is a obtained without the use of feed-forward controller phase-lag compensator with two notch filters. and reference trajectory, which implies no real tim e computation for feed-forward control is needed. As the fine PZT actuator has a stroke limit of only a few micrometers, the design for the short seeking also take into consideration the saturation properties o f the PZT actuator. When the short-seek distance is over PZT stroke limit, a two-stage IVA scheme is designed in such a way that only VCM loop is utilized in the first stage (beyond PZT stroke limi t), and the PZT loop are switched on when the read/write head approaches the target track in the second stage (within the PZT stroke limit). The fi rst stage IVA is designed to yield a high-speed seeking , Frequency (rad/sec) Frequency (rad/sec) and the second stage IVA is used to produce a fast Fig. 2 Frequency responses of VCM and PZT plants and smooth response as the output approaches the command input. 20

0

0

M a g n it u d e ( d B )

M a g n it u d e (d B )

-20

-40

-60

-80

-100

-20

-40

-60

-80

-180

0

-270

-45

P h a s e (d e g )

P h a s e (d e g )

-120

-360

-450

-540

10

a

-90

-135

4

10

5

10

6

-180

10

4

10

5

10

6

The remainder of this paper is organized as follows . Section 2 describes the structure of a dual-actuato r HDD and the track-following controller design. Section 3 presents the proposed dual-stage short Fig 3 Tracking following control system for dualseeking control with IVA. Both one-track seek actuator HDDs design and short-span seeking (over PZT stroke limit) design are discussed along with the simulati on results. Conclusions are given in Section 4. 3. DUAL STAGE SHORT SEEKING CONTROL WITH INITIAL VALUE ADJUSTMENT In this section, we will discuss short seeking cont rol with IVA for the dual-actuator HDDs. We first explain one-track seeking design and then discuss t he short-span seeking (over PZT stroke limit) design. Fig. 1 shows the HDD with VCM as a first actuator and a PZT as a secondary actuator. PZT stroke limit is – 1 . 0 um . The actual HDD was modeled through 3.1 One-Track Seeking Design frequency response test and Fig. 2 shows the frequency responses of VCM and PZT plant models. Notice that the VCM pant model has double integrators with three resonance modes and PZT pant model has three resonances at high frequencies .

2. DUAL ACTUATOR SYSTEM AND TRACK FOLLOWING CONTROLLER DESIGN

Read/write

head

PZT

Fig. 4

1-track seek with IVA for dual-actuator HDD

Fig. 4 shows a schematic diagram of 1-track seeking control with IVA for dual-actuator HDDs. The proposed method has the following features: Disk (1) The overall structure during seeking is a oneVCM degree-of-freedom structure. (2) Reference input is set to be a step signal, i.e., Fig. 1 Dual-actuator Hard Disk Drive r(k)=r for k ‡ 0 , where r is the seeking distance. Output y is the head position, which can be In this study, the parallel track-following servo measured. The feedback controllers are the same controller is used to achieve high-accuracy track as the dual-stage track following controllers. following (Ding, et al., 2000). The servo structure is (3) Nonzero initial values are set to both controllers shown in Fig.3.PVCM and PPZT represent the plant for 1-track seeking. models of VCM and PZT, respectively. C1 and C2 Notice that the adjustment of the initial v are, respectively, controllers for the VCM and PZT of the controller improves the transient response actuators. The VCM controller is designed to without changing closed-loop characteristics such a s stabilize the VCM loop for preventing damage to stability and sensitivity. Since we already have th e disk drive systems when the PZT is not activated, track following controllers, finding the initial va lues

110

s

alues

of the controllers becomes the main issue. In the The resulting closed loop system (7) is following sub-section 3.1.1, the closed loop system asymptotically stable because the track following depicted in Fig. 4 is represented in time domain. controller has been designed to stabilize the syste m. Then in sub-section3.1.2, IVA of the controllers is The error dynamics of the closed loop system is: formulated as an optimization problem by finding th e e x ( k + 1) = Ae x ( k ) optimal initial states of the controllers to minimi ze -1 (9) the performance index based on the tracking error. where ex k = x k( )- x ¥( ) x( (¥),) = ( I - A) Br Design of performance index for short-seeking is discussed in sub-section3.1.3. Simulation results are 3.1.2 Initial Value Adjustment of the Controllers given in Sub-section3.1.4. Now the initial value adjustment of the controller be formulated as the following optimization problem 3.1.1 Close loop system in time domain Problem: Consider the control system descried by Assume that the dynamic models of VCM plant and equation (7). Find the initial value of the contro xc(0) that minimizes performance index J: PZT plant in continuous time are ¥ T J= ex k Q e(x )k for ( ),Q > 0 xv (t ) = A01 xv (t ) + B01u v (t ) (1) k =0 y v (t ) = C 01 x v (t ) where ex (k) is as defined in (9). and (2) x p (t ) = A02 x p (t ) + B02u p (t ) Theorem 1: The initial value of the controller xc(0) y p (t ) = C02 x p (t ) that minimizes performance index J is: where xv ˛ ´ nv , u v and y v are the state vector, control xc (=0)K1 r + K2 xk (0) n T input and output of VCM, and x p ˛ ´ p , u p , y p are where (12) = K - 1P[T P I I-A]-(1 B) , K = - P- 1 P 1

22 12

2

and where

Av

v

n

˛ ´ , Bp ˛ ´ p , Cp ˛ ´

Ap

1· n p

ex (0) = x(0) - x(¥ ) =

u p (k ) = C pc x pc (k ) + Dpce(k )

xvc ˛ ´

nvc

, Avc

x pc ˛ ´

n pc

, Apc

2 nvc

˛ ´, Bvc

n2pc

˛ ´, Bpc

˛nvc ´ ,Cvc ˛ ´1· nvc, Dvc ˛ ´ npc

(11)

(13)

where

Suppose VCM controller and PZT controller C1(z) , C2 (z) can be expressed in time domain as follows: xvc(k + 1) = Avcxvc (k) + Bvce(k) (5) uv (k) = Cvcxvc (k) + Dvce(k) and x pc k( + 1)= Apcx pc k( + ) Bpce k( ) where

(10)

J = ex (0T) P ex (0)

˛ ´nv , Cv ˛ ´ 1· nv ,

n2p

ller

(3) SinceA is asymptotically stable, (13) has a Proof: unique positive definite solution P>0, and J can be transformed to: (4)

x p (k + 1) = Ap x p (k ) + Bpu p (k ) y p (k) = C p x p (k) 2 ˛n´v , B

can .

22 12

the state vector, control input and output of PZT. and P22 ,P12 are obtained by the Lyapunov Eq.: The discrete time equivalents of (1) and (2) with P11 P12 AT PA - P = - Q ,P =: T sampling periodTs can be expressed by P12 P22 xv (k +1) = Av xv (k) + Bvuv (k) yv (k) = Cv xv (k)

1· npc

˛ ´ ,Cpc ˛ ´

So, J = eTk

(P011) ek

xk (0) xc (0)

- x(¥ ) =:

ek (0) ec (0)

(14)

(+ 0) ekT2 (0P)12ec (0)+ ecT (0)P22ec (0)

It is easy to show that ¶ J / ¶ xc (0) = 0 is equivalent to

¶ J / ¶ ec (0) = 0 . Solving ¶ J / ¶ ec (0) =0 , we get -1 T ec (0) = - P22 P12ek (0) From (14), (16) and (9), we conclude that J is minimized for xc (=0)K1 r + K2 xk (0) , where K1, K2 are as defined in (12).

(6)

, Dpc ˛ ´ .

Noting that xk (0) may be assumed zero for 1-track Combining the controllers equations (5), (6) and seek, the initial value of the controllers turned out to plants equations (3), (4) and referring to Fig. 4, we be a simple form as: can obtain the closed loop system: -1 T (17) 0 r 1K ,= [ 22 P 12P I]( I-A)-1 B c x ( =) 1K x k( + 1)= Ax k( )+ B r (7) k

y= (Cx ) k (+ )D r

where x k (=) :xv[k T( )xp, k T( )xvc, k T( )xpc, k T( T)] ˛ ´

nv +np +nvc+npc

3.1.3 Design of Performance Index Function

Matrices A, B, C, D can be computed from the system matrices in (3)~(6). Notice that the states of the closed loop systemx(k ) can be separated into two parts: the states of two plants xk(k)) ( and the states of two controllers xc((k)): xk (k ) :=

xv k( ) x p (k )

, xc (k ) :=

(8)

xvc k( ) x pc (k )

.

The performance indexJ plays an important role in determining the initial value of the controller and thus the short seeking performance. Design I- considering tracking error at sampling points It is straightforward to evaluate the tracking error at sampling points and consider smoothness of the

111

(16)

output and control input in the performance index as: J=

where Ju =

¥

¥

2

k=0

¥

J1 = e y t( 2) dt =

(18) 0

2

[ey (k) + q1e y (k) ] + q2 J u

=

ey(k):=[ey(k +1)- ey(k)]/Ts

e y k( )= y k( -) y ¥( ),

[

T

]

U k - U ¥ [ (U)k - (U )] ¥ [ ( U) k =( u()], u, p (k) v) k )(

k=0

T

(19)

=

and q1 , q2 are weighting factors.

=

Lemma 1: The performance indexJ in (18) can be transformed into the standard quadratic form: ¥

J =

k =0

e x k( T) Q e x ( k )

Q = C C + q1Cd T Cd + q2 Cu T Cu , A- I Cd = C Ts

, Cu =

dv } { y[ k( )(v )- y ¥( )]

k =0 0

Ts

¥

{ eTk ( k )(v )Qcc ek ( k )(v ) dv}

k =0 0

T

¥ k =0 ¥

T s ˆT ˆ {eTx (k ) S ( e A v ( H T Qcc H )e A v dv) Sex (k )} 0

T {eTx (k ) S Qˆ c Sex k( )}=

k =0

where

T Q1 = S Qˆ c S

,

¥

ex k( T) Q1ex k( )

k =0

Ts

ˆT ˆ T Qˆc = e A v (HT QccH)e A vdv , Q cc = Cˆ 0 Cˆ 0 0

Thus, we conclude that the performanceJ index in (23) can also be transformed into the standard

T

where

Ts

¥

- DvcCv - Dvc , Cp Cvc, ,O

(20)

- DpcCv - D , pcCp ,O,Cpc

quadratic form as:J =

¥ k =0

ex k( T) Q e x (k ) .

The proof is omitted due to the space limitation.

3.1.4 One-track Seeking Simulations

Design II- considering the inter-sampling behaviors

We used the same plant models (as in fig. 2) and track-following controllers as discussed in Section 2, and assumed a density of 25.4 kTPI (one-track pitch is 1 um) for the simulations. Fig. 5 shows the response of the output without initial value adjustment of the controllers for 1-track seeking. The dark line shows the head position, the light line and dotted line are VCM and PZT position respectively. Simulation results of 1-track seeking using proposed method are shown in Fig. 6. From Fig. 6, we can see that a fast and smooth response in short seeking can be achieved by IVA of the controllers. The design parameters q1, q2 are set as follows after some trial (23) and errors: q1=0.09, q2=0.005. And initial values of the controllers are calculated by (17).

Design method I only evaluates the tracking error at the sampling instants. However, the plant is a continuous-time system, and it is natural to evaluate the continuous-time signals directly. Especially in dual-stage servos, inter-sampling ripple may take place if the second actuator has mechanical resonance modes at high frequencies beyond sampling frequency, which can be excited by the control input. To incorporate the inter-sampling error, we modifyJ in (18) as follows: ¥

J = [ e y (t ) 2 + q1e y (t ) 2 ]dt + q 2 J u 0

where e y (t ) = y (t ) - y (¥ ), Ju is as defined in (19).

To evaluate (23), the pieces of the continuousIt could be noted from Fig. 6 that the dual-stage time statexk(t) andek(t) are introduced as: seeking control greatly improves the responses. It x k k( )(v )= x k kT ( s+v) takes about 0.3 ms to move the head to the desired (24) ,v˛ [T0s, ) ek k( )(v )= ek kT ( s+v) track so that the head can read or write data on the T data track. The PZT actuator first moves towards the where e x (t ) = x(t ) - x(¥ ) = ek (t ) T , ec (t ) T target track, and then returns to its stroke center to cancel out the movement of the VCM actuator after Lemma 2: ek(k)(v) is a function of error-vector ex (k): the head is on track. We can see PZT is within its ˆ (25) limit. Fig 6 (c) compares the performance of e k ( k )( v ) = He A v S e x ( k ) , v ˛ [0T, s ] stroke Design I and Design II, which confirms that interwhere sampling ripple is decreased using Design II. A B A B ˆ A 01 O , Aˆ 01 = 01 01 , Aˆ 02 = 02 02 Aˆ = O

AˆO02

S=

Sv Sp

Sv =

]

O

O

I nv , O · O n p · nv , I n p , O n p · ( nvc + n pc )

,H =

O

I nv[

Cu

I nv ,O nv · ( n p + 2 ) 0 ,..., 0 ,1,0

2

Position(track)

[

,O], O

IO, n[p ,O]

Head position VCM position PZT position

1.5

1

0.5 0

,Sp =

On p · n v ,In p , On p · 2

-0.5

0 ,..., 00, 1

y k( )( v = ) yv k( )( v + ) y p k( )( v )= C01xv (k)(v) + C02xp (k)(v) ˆ =[ C C ], = ˆC 0 kx k ( v)( ),C 0 01 02 y ( k )( v ) - y ( ¥ ) = Cˆ 0 e k ( k )( v )

and using Lemma 3 , we obtain

1

2

3

4

5

6

Time(s)

7

8 x 10

Fig. 5

The proof is omitted due to the space limitation. Noting that

0

-3

1-track seek without IVA dual-actuator for HDD

From Eq. (17), we can see the initial values of the controllers are scaled with seeking distance r, so are the responses of both actuators. As seeking distance becomes longer, the PZT response will exceed its stroke limit as the upper figure of Fig 7, where r=3. If (26)the saturation property is considered, the performance is degraded as the lower figure of Fig.

112

Position(track)

7. In the next section, we will discuss the short-span(2) Reference input is set to be a step signal for the seeking when the head moves over the PZT whole short-span seeking, i.e.,r(k)=r for k ‡ 0 , actuator’s stroke limit. where r is the seeking distance. The feedback controllers are the same as the dual-stage track 2 following controllers. VCM position PZT position 1.5 (3) Nonzero initial values are set to the Head position corresponding controller (controllers) at the start 1 instant of each stage to achieve desired short 0.5 seek performance. 0 For simplicity, we will explain in two-stage design Time(s) -0.5 procedure. 0 1 2 3 4 5 6 7 8 (a)

x 10

-3

Position(track)

2 VCM position PZT position Head position

1.5 1

0.5 0

-0.5

0

1

2

3

4

5

(b)

6

Time(s) 8

7 x 10

-3

Fig.8 Dual-stage Short-seek control considering PZT actuator saturation

with

IVA

Position(track)

1.06 1.04 1.02

1

0.98

Time(s)

0.96

0.5

1

1.5

2

2.5

(c)

3

3.5

4

(a)

x 10 -3

Fig. 6 1-track seek with IVA for dual-actuator HDDs:(a) Design I; (b) Design II; (c) Head position around target track, dark line:Design II; light line:Design I.

(b) Stage I

5 VCM position PZT position Head position

Position(track)

4

3 2

1 0

-1

0

1

2

3

4

5

6

Time(s) 8

7 x 10

-3

(c) Stage II

5 VCM position PZT position Head position

Position(track)

4

Fig.9 Dual-stage short-seek with two-stage IVA

3 2

1

3.2.2 Stage-I Design ( e > r0 )

0 -1

0

1

2

3

4

5

6

Time(s) 8

7 x 10

-3

Fig. 7 3-um seek using IVA (Response Over PZT Stroke Limit) 3.2 Short-span Seeking (Over PZT Stroke Limit)

In the first stage we use the servo system shown as Fig. 9 (b), which involves only VCM loop. The initial value ofC1 can be obtained following the same procedure as Section 3.1 by first obtaining the closed loop system depicted in Fig.9 (b) in time domain as: x k( + 1)= Ax k( )+ B r k y= (Cx ) k (+ )D r

3.2.1 Overall Control Structure for Short Seeking

where Fig. 8 and 9 show the schematic diagrams of short BD seeking control with IVA for dual-actuator HDDs, x (k) := xv(k) , A = Av BvCvc - BvDvc [C O] , B = v vc v xvc(k) where the seeking distance is over PZT stroke limit. 0 Avc Bvc Bvc The proposed method has the following features: C = [C v O ] , D = O (1) The overall short-span seeking is divided into C1 can be obtained two stages, as referred to Fig. 9: in the firstThe initial value of controller stage, where the head is beyond the PZT using Theorem 1 as: xvc( 0) = K1 r operation range, only VCM actuator and its where K1=[P22-1P12T , I](I-A)-1B, andP22 ,P12 are controller is used; in the second stage, where the obtained by solving the Lyapunov equation: head is approaching the target track (within the T P11 P12 , A is as defined in (28) PZT operation range), the PZT actuator and its A PA- P = - Q ,P =: P12T P22 controller are switched on.

113

(27)

(28)

The objective of IVA in the Stage I is to get a rapid response, so we design the performance index only based on the tracking error ignoring the smoothness ¥ terms as: J = e y (k ) 2 , and soQ=CTC by Lemma 1, k =0

where C is defined in (28).

When the r/w head approaches r=1, PZT actuator is switched on, and the response is such that the PZT actuator first moves to the target track, and then returns to its stroke center to cancel out the movement of the VCM actuator after the head is on track. We can see PZT is within its operation range during the whole short-span seeking .

3.2.3 Stage-II Design ( e £ r0 ) 4. CONCLUSION The PZT loop is activated at the instance e=r0, and in stage-II, we use the servo system in Fig. 9 (c), whichThis paper has proposed a short seeking control is the same as that of 1-track seeking, except that themethod based on track-following controllers and initial value of the plantsxk(0) is not zero here. initial value adjustment for dual-actuator HDDs. IVA Nonzero initial values are set to the controllers at theimproves transient characteristics from the viewpoint switching instance. The initial value here refers to of minimizing the performance index function. By the state at the switching instance. The initial values incorporating the inter-sampling error and the of C1and C2 can be obtained following the same smoothness of output and control input in the procedure as 1-track seeking by first obtaining the performance index, initial value adjustment of the closed loop system in Fig.9 (c) as (7), and thentrack-following controllers can achieve desired short finding the optimal initial values of the controller seeking performance. To avoid micro-actuator using Theorem 1. Notice that the initial values of thesaturation, a two-stage IVA scheme is designed to controllers are determined by (11) instead of (17), provide a high-speed movement in the first stage due to the nonzero initial values of the plants. Theusing only VCM loop, and fast and smooth objective of IVA inStage II is to get a fast and positioning in the second stage using both actuators. smooth response as the output approaches the target, Simulations for both 1-track seeking (within PZT and performance index is designed using Design stroke limit) and short-span seeking (over PZT stroke method I as (18), or using Design method II as (23) limit) are performed. Simulation results confirmed that an excellent short-seeking performance can be for enhanced inter-sampling performance . achieved without saturating the fine actuator using the proposed method. 3.2.4 Short-span Seeking Simulations Now, we evaluated3-um seeking responses, which REFERENCES are over the PZT actuator stroke limit. Simulation results of 3um seeking using proposed method are Ding, J., M. Federico and M. Tomizuka, “Short shown in Fig. 10. Here r= 3 and we setr0 to be 2. We Seeking Control with Minimum Jerk can see that a fast and smooth response in short Trajectories for Dual Actuator Hard Disk Drive seeking can be achieved. Compared with Fig.10 (a), Systems,” Proc. Amer. Control Conf., 2004 Fig.10 (b) shows that inter-sampling performance is Ding, J., M. Tomizuka, and H. Numasato, “Design improved usingDesign II. and robustness analysis of dual stage servo system”, Proc. American Control Conference, Chicago, IL, USA, June 2000, pp. 2605-2609. Hernandez, D., S. Park, R. Horowitz, and A. K. Packard, “Dual-stage track-following servo design for hard disk drives,” in Proc. Amer. Control Conf., 1999, pp. 4116–4121. Kobayashi, M. and R. Horowitz, “Track Seek (a) Control for Hard Disk Drive Dual-Stage Servo 4 Systems”, Trans. on Magnetics, Vol. 37, No. 2, 3 March 2001. 2 Koganezawa, S., Y. Uematsu, and T. Yamada, VCM position PZT position 1 “Dual-stage actuator system for magnetic disk Head position 0 drives using a shear mode piezoelectric Time(s) -1 microactuator,” IEEE Trans. on Magnetics, 0 1 2 3 4 5 6 7 8 vol.35, no.2, Mar. 1999. x 10 (b) Mori, K. et al., “A dual-stage magnetic disk drive Fig. 10 3-um seeking by two-stage IVA considering actuator using a piezoelectric drive for a high track density,” IEEE Trans. Magn., vol. 27, no. PZT saturation: (a) Design I; (b) Design II; 6, pp. 5298–5300, 1991. Fig. 10 also indicates how the two actuators Semba, T. et al., “Dual-stage servo controller for contribute to the responses during the short-span HDD using MEMS microactuator,”IEEE Trans. seeking. At first, only VCM actuator is activated. Magn., vol.35, no.5, pp.2271–73, 1999.

Position(um)

4

3

2

VCM position PZT position Head position

1

0

-1

0

1

2

3

4

5

6

7

8

Time(s)

-3

Position(um)

x 10

-3

114