Robust Compensation for Dynamic Friction in Positioning Control of Hard Disk Drives

Copyright © IFAC Motion Control, Grenoble, France, 1998

ROBUST COMPENSATION FOR DYNAMIC FRICTION IN POSITIONING CONTROL OF HARD DISK DRIVES

Y. L. Lou', T. H. Lee', S. S. Ge', C. H. Huang" and T. S. Low"

'Department a/Electrical Engineering, "Data Storage Institute National University a/Singapore, 10 Kent Ridge Crescent, Singapore 119260

Abstract: Friction in actuators of hard disk drives lowers the system gain in low frequency range, which prevents the system positioning accuracy from further improvement. The complexity and multiform of friction make it difficult to obtain a complete friction model. A nonrnodel-based robust friction compensation method is introduced, in which the gain decrease in the low frequency domain is compensated by the infmite gain in a bang-bang compensator. The introduced method is simply integrated into the proximate time-optimal servomechanism controller widely used in the current hard disk drives. Simulation results based on a commercial FUJITSU hard disk are presented Copyright © 1998 IFAC Keywords: Disks, Servomotor actuators, Friction, Bang-bang control.

1. INTRODUCTION

•

Owing to the rapidly increasing demands for high capacity and performance of hard disk drives, servo engineers are required to develop more advanced control strategies. It is expected that the position accuracy of the hard disk drives will reach 25,000 track per inch (positioning accuracy higher than 1 /lm) at the end of this century (Chew, 1996). For the system with such a high accuracy requirement, some nonlinearities usually neglected or simplified in the control system design must be taken into account and reconsidered; friction is one of these nonlinearities. The problem of pivot friction in actuators of hard disk drives has been observed widely by manufactures, and has recently received more attention due to the new challenge for hard disk drives (Abramovitch, et al., 1994; Eddy and Messner, 1995; Wang, et al. , 1994). The foregoing works on friction in hard disk drives are summarized as follows :

•

•

Friction is a function of both displacement and velocity of the system. There are hysteresis loops in terms of friction with respect to displacement and friction with respect to velocity. Due to the complexity of friction, it is difficulty to obtain a complete friction model.

In view of the difficulty in obtaining a complete friction model, a nonmode-based robust friction compensation method is introduced in this paper. In Section 2, friction in positioning control of hard disk drives is discussed. The introduced friction compensation method is explained in Section 3.

Ill] l ill 11111 11I

100

10'

10 2

10 3

10 4 Hz

i-

•

Friction lowers system gain in low frequency range as shown in Fig. 1.

Fig. I. Decrease of system gain in the frequency range due to friction 309

low

amplifier saturation when I Ye I > y, to accelerate or decelerate the target seeking, while the control changes to linear control when IYe I :$ y"

Simulation analysis is presented in Section 4, followed by conclusion in Section 5. 2. FRlCTION IN POSITIONING CONTROL OF HARD DISK DRlVES The positioning control of the read/write head of a hard disk drive actuator involves two functions: target seeking and target tracking. In the target seeking, the head should be forced to move to the target track as quickly as possible, while in the target tracking, the head should be positioned precisely and robustly at the target track.

To ensure the continuity of the system state in the switching of the control modes, the position gain kJ and velocity gain k2 have the relationship (2)

2.1. Proximate time-optimal servomechanism

To satisfy both the requirements for target seeking and target tracking, model switching control strategies, e.g. Proximate Time-Optimal Servomechanism (PTOS), are widely employed in hard disk drive industry (Franklin, et aI., 1990). In the PTOS controller, a nonlinear control function makes the current amplifier saturated when the position error is larger than a predefined threshold to accelerate or decelerate the target seeking, while the control changes to linear control after the position error is smaller than the threshold to eliminate the amplifier chatting between the positive and negative maximum outputs.

The position error threshold is defmed as (3)

In the design of a PTOS, after selecting the factor a c (0, 1), the gain kJ is chosen according to the system performance requirement, and the gain k2 is determined by (2). Then function flye) can be determined by (1). 2.2. Friction in hard disk drives

Friction, which depends on many factors such as temperature, asperity of the contacted surfaces, and situation of lubrication, has two different presentations: pre-sliding friction and sliding friction (Armstrong-Helouvry, et aI., 1994). In the pre-sliding stage, which is usually in the range of less than 10-5 m and dominated by the elasticity of the contacting asperity of surfaces, friction depends on both system position and velocity. Nonlinear dynamics such as hysterisis of friction to the displacement and friction to velocity have been observed by many researchers (Abramovitch, et al., 1994; Armstrong-Helouvry, et al., 1994; Eddy and Messner, 1995; Wang, et aI., 1994). In the sliding stage, friction is dominated by the lubrication of the contacting surfaces and has the function of system damping. Friction in sliding stage

Fig. 2 shows a block diagram of a PTOS control of a hard disk drive. In Fig. 2, y, y*, y and Ye are the position signal, reference position, observed position and position error respectively; v and v are velocity signal and observed velocity respectively. The amplifier has the saturation limits of ± f m • The actuator, which is in fact a Voice Coil Motor (VeO), has a torque constant k/) inertia equivalent gain k", and position measurement gain Icy. Friction F(v, y) is a nonlinear function of both system velocity and system position. Since only the position is measurable in a hard disk drive, a state observer is used to construct the necessary state variables for the controller. The nonlinear control functionj(ye) makes the current

y

Fig. 2.

Proximate Time-Optimal Servomechanism (PIOS) control of a hard disk drive. 310

can usually be represented by various single variable functions of velocity. For a hard disk drive with the positioning accuracy requirement higher than micrometers, the friction dynamics in the pre-sliding stage cannot be neglected in the control system design. Friction introduces steady state error, tracking lag, and limit cycles in a servo system. However, for a hard disk drive, which is mainly a regulation system, one of the most important tasks is to decrease the system steady state error to improve the positioning accuracy.

Fig. 3.

In the last stage of a PTOS when IYe I ~ y, andj(ye) = kIY/ k2' the controller turns into a linear PD controller. The system dynamics as shown in Fig. 2 is described by

bounded within the first and the third quadrants as shown in Fig. 3, which can be described mathematically as F(V'Y)::; Fo+ + blv, { F(v,y) ~ Fo- + blv,

(4)

At the steady state, v = v = 0 and state error is then given by

Ye =

v> 0, v

(5)

(6)

3. ROBUST FRlCTION COMPENSATION

I I

A robust friction compensation approach is proposed for static friction (Southward, et at. , 1991). It is further extended to dynamic friction through some modifications in this paper.

which cannot be completely removed because the system gain is limited due to the limited system bandwidth consideration. The most commonly used techniques to solve this problem are (Franklin, et at., 1990): (1) the integration control, and (2) observer-based compensation. The common assumption for both solutions is that the disturbance is a constant bias such that the derivative of the disturbance respect to time is zero, which is the basic assumption for the design of the observer. However, it is well known that an integration control in the positioning system with friction leads to limit cycles (Lee, et at., 1997; Annstrong and Amin 1996), and the assumption that the disturbance does not change with time is unrealistic for the friction in the micrometer level, where the friction dynamics cannot be neglected.

3.1. Robust Friction Compensation The robust friction compensation scheme is shown in Fig. 4, where Fc(c) is an additional friction compensator. The control torque to the VOC is now

The friction compensator is a function of a composite error c defmed as (8)

where

However, since friction dissipates energy, it is

/30 and /31

are positive factors . In the last stage

y

A

V

y

Fig. 4.

< o.

where Fo+ = F(Y,v)lv = 0+ and Fo- = F(y,v)lv = 0-. Therefore, the steady state error due to friction is also bounded.

v = 0, the steady

F(v,y)lv=o kk

Boundedness of friction.

Nonmodel-based robust friction compensation. 311

of the PTOS when I Ye I ~ Y/, flye) = klY.lk2' and the dynamics of the state observer is ignored, e.g. v = v and y = y, the system dynamics becomes

In the following, it is to be shown that the closed system (9) with the control of (1}-{3), (7) and (12)(13), is stable with the origin as a globally asymptotically stable equilibrium point. A continuous positive candidate is constructed as,

where kc > O. The gains in PTOS controller and the coefficients of the composite error should has the relationship

j

kc = (k2ktkv

+.,(1) 12,

function

The time derivative of V(e) along the system trajectory (9) is

(10)

f30 1Pt = (k2kt kv -.,(1) / 2,

Lyapunov

where L1 = (k2ktkv)2 - 4k 1k/(..Js" and it should be ensured that L1 ~ O. Without the friction compensator, the system has the steady state error

When e > e+ or e < [ , the compensator is inactive and the system controller is only the PTOS controller. From (9), the PTOS controller drives the system position error approaching to the steady state error. From (13), since a positive 5 is added in the compensator, before the system error arrives at the steady state error, the system enters the region where the compensator is active.

(11)

In the region where the compensator is active, the time derivative of V( e) is

The friction compensator is designed according to the steady state error as shown in Fig. 5.

(12) (16)

otherwise,

0,

From (12) and (13), ri (e) is negative in the region of

o < e ~ e+ or e- ~ e < o. Thus, l&Cl)1 ~ 0 as t ~ 00 . +

Furthermore, since (8) can be viewed as a stable firstorder differential equation in Ye(t) with &Cl) as an input, it follows that IYe(I)1~ 0 as t ~ 00 . Therefore, the origin is a globally asymptotically stable equilibrium point.

+

e _ =e~ +5 ,

1

(13)

e =e, -5 ,

where e/ and e,- are the positive and negative steady state errors respectively, and 5 > 0 is a design parameter.

3.2. Remarks 1. When the system error is within the set of e C (e-, e), any positive value of 5 is enough to guarantee that the feedback force will always exceed the static friction force level. In this region, the non linear proportional feedback force is essentially a bang-bang force . 2. Since the gain of the bang-bang friction compensator is infinitive, it compensates the decrease of the system gain in low frequency range due to friction. 3. Although the first stage of the PTOS control is also a bang-bang control, the bang-bang friction compensator changes its output in a much smaller scale than the first stage of

G-

Fig. 5. Friction compensator. 312

Table 1. Parameters in the simulation investigation

0"0

Values 102

O"J

5x 10-4

0"2

10-3

Parameters

Fe Fs Vs

k"

5xl0-4 7.25xl0-4 10-4 3.852xl04

N/m Nlmls Nlmls N N Nlmls lIKgrn2

Parameters ky k,

Im k, k2

Ice I!JJ.

Values

PTOS. The method here will not lead to current amplifier much energy loss. 4. Over the entire phase plan, system state trajectories are absolutely continuos. There cannot exist a sliding mode as in the theory of variable structure control. 5. In the design of the friction compensator, kc, Po and PI should firstly be determined from the system position and velocity gains in PTOS controller by (10). Then, after the selection of 8, the compensator can be designed by the steady state error according to (12) and (13). 6. The difference of the proposed method from the one in (Southward, et aI., 1991) is that the friction compensator here is not only a function of position error Ye but also of velocity v in the defmition of the composite error c. so as to ensure the system stability for the dynamic friction.

Values

Parameters

m s A

3000 0.1l34 0.55 2.59xlO 02 15 03 1.7x 10 1

6.67xlO·3 lxl0· S 0.7 10-6 1 0.55 2xlO·s

PJ

t5 a Yo Vo io Zo

m mls

A m

of the hard disk drive seeks tracks between ±5000 tracks, where one track is defined as Yo = 1 J..Iffi. Dynamic hysterisis of friction with respect to position and velocity are presented in Fig. 6. Fig. 7 shows the I xl04

0.2

r--

t

L

O.5

L

y, O.1

Yo

rx

Yo

~$l

\

0

0

-D.5

-D.I

-I

-D.2

2

~/Y:t.

yYto\ JJ

-~

IY 1-1"'" I1 \ I , 11 I 1\ IJ

II \ \

o

1\

-I

IV

V

t v

-2

o

0.05

1-

0.1 s

(a) 1.5

X

lOo)

L

3. NUMERICAL SIMULATION

"-

o

Numerical simulations based on a FUJITSU hard disk are carried out to verifY the introduced method in this paper. The bristle friction model proposed in (Canudes de Wit, et al., 1995) is used, which captures most of the friction behaviors. The model is mathematically represented as

) -I

-I

o

-D.5

0.5

I x 10'

.1:._ Yo

(b) l.5

X

lOo)

(17)

/

o where z(t) describes the average deflection of the bristles, and g(v) is a positive function depending on many factors such as material properties, lubrication and maybe also temperature. The factors 0"0, 0"1 and 0"2 are stiffness, damping and viscous friction coefficients respectively. Vs is the Stribeck velocity, Fe is the Coulomb friction level and Fs is the level of stiction friction. The parameters used in the simulation investigation are given in Table 1.

I JI f j

It'

I1 ./

-I

-2

o

-I

~­ Vo

2

(c) Fig. 6. Dynamic characteristics of friction. (a) System dynamics when the head seeks between ±5000 tracks. (b) Normalized friction torque to velocity. (c) Normalized friction torque to position.

Simulation results are shown in Fig. 6 and Fig. 7. The traces in Fig. 6 are recorded when the read/write head 313

I x lO'

0.2

z.. 0.5

W vivo l - T

y, O.1 yo

yo

-0 .5

V

~V

0

0

)< .... ~

"f '\

-0.1

"! alao -I

~/ ~~

/

2

4. CONCLUSION

t

A nonmodel-based friction compensation method has been introduced to positioning control of hard disk drives, in which the gain decrease in low frequency range due to friction is compensated by the infinite gain in a bang-bang compensator. The introduced method does not need an exact friction model and is very simply to be integrated into the proximate timeoptimal servomechanism controller, which is widely used in the hard disk drive industry.

.E...

"0

0

v Vo

-I

l

-2

- 0.2

o

0.01

1-

0.02 s

(a) REFERENCE 1)(10 4

0.2

}- ~ 1~y/~o~ ' - / t Jo l

2

-'---'

z.. 0.5

y, O.1

yo

yo 0

0

L

0

-0.5

-0.1

-I

-0.2

vivo

V V

~

:><,..-

"( 1\

t

Abramovitch, D., F. Wang, and G. Franklin (1994). Disk drive pivot nonlinearity mode ling part I: frequency domain. Proceedings of the America Control Conference, Baltimore, Maryland, June 1994, 2600-2603 . Arrnstrong, B. and B. Amin (1996). PID control in the presence of static friction: a comparison of algebaic and describing function analysis. Automatica, 32, 679-692. Arrnstrong-Helouvry, B., P. Dupont, and C. Canudas de Wit (1994). A survey of models, analysis tools and compensation methods for control of machines with friction . Automatica, 30, 10831138. Canudas de Wit, C., H. Olsson, K. J. Astrom, and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Trans. Automat. Contr. , 40, 419--425. Chew, K. K. (1996). Control system challenges to high track density magnetic disk storage. IEEE Trans. on Magnetics, 32,1799-1804. Eddy, K. and W. Messner (1995). Dynamics affecting tracking bias in hard disk drive rotary actuators. Proceedings of the America Control Conference, Seattle, Washington, June 1995, 1055-1060. Franklin, G. F., J. D. Powell and M. L. Workman, Digital Control of Dynamic Systems. second edition, Addision-Wesley Publishing Company, 1990. Lee, T. H., Y. L. Lou, and S. S. Ge (1997). Analysis and design of high precision servo systems with dynamic friction Proceedings of IEEE Singapore International symposium on Control theory and applications, Singapore, June 1997,271-275. Southward, S. C., C. J. Radeliff, and MacCluer (1991). Roubust non linear stick-slip friction compensation. ASME J. of Dynamic Systems, Measurement, and Control, 113,639-645. Wang, F., T . Hust, D. Abramovitch, and G. Franklin (1994). Disk drive pivot nonlinearity modeling part 11: time domain. Proceedings of the America Control Conference, Baltimore, Maryland, June 1994,2604--2607.

a

"0

0

v

vo

r-I

'1 alao I

-2 0

0.01

1 -

0.02 s

Cb) 0.1 y, yo

0

I )( 10- 2 0.5

!>..

'.

0 -0.5 0

0.01

1-

0.02 s

(C)

Fig. 7. Performance improvement by robust dynamic friction compensation. (a) PTOS control of a hard disk drive. (b) Robust compensation of friction in a hard disk drive. (c) Normalized position error and compensator output in robust friction compensation scheme. performance improvement by the proposed robust dynamic friction compensation. The step responses of acceleration, velocity, position and position error without friction compensation are shown in Fig. 7(a). The influence of friction leads to a static error. In Fig. 7(b), with the proposed friction compensation, system positioning accuracy is improved effectively. Fig. 7(c) shows the compensation component comparing with the position error, where chatting in the compensator is observed. However, it should be noted that the chatting in the compensator is very small in the whole control output. 314