Output feedback sliding mode control for the flying
height of a pickup head in near-field optical disk
drives
W.C. Wu and T.S. Liu
Abstract:An output sliding mode control method to produce a stable flying height for a pickup head in a near-field optical disk drive is presented. When the optical disks have a large amplitude vibration that makes a stable flying height difficult to attain, a piezoelectric bender is used to complement an air bearing at the head/disk interface. Control simulation for an identified model using measured vibration data of an optical disk is performed to demonstrate the robustness of the proposed method.
1 Introduction
Optical disks are popular due to their large data storage capacity, and as such they find extensive use in audio and video media. The need for a large data storage capacity and high quality recording media has resulted in high data storage density requirements for optical disks [1, 2]. However, due to light wave diffraction in a far-field optical environment, the optical pickup device cannot further reduce the light spot size to access a smaller data track width and thereby accomplish a higher data recording density. The application of near-field optics to avoid the diffraction limitation requires the near-field optical disk drives to adopt the structure of a magnetic disk drive but with the magnetic pickup being replaced by a near-field optical pickup head. This allows the disk drive to maintain the space between the pickup head and disk within the near-field focusing length. Thus, the recording density can be increased by means of an improvement in the optical resolution. Additionally, optical disks are generally made out of a plastic-based material due to its low cost. Hence, the optical disk vibration magnitude will be larger than that in an aluminum-substrate hard disk. Therefore, the passive air bearing flying height mechanism barely attains the focusing performance requirement in the presence of severe disk vibration. In order to circumvent the problem and achieve a satisfactory performance in flying height control, we attach a multi-layer piezoelectric bender (PZT) to the suspension arm to serve as an active flying height controller. Similar designs have been widely used in hard disk drives to achieve head-disk spacing control and fine track following control [3, 4].
We aim to apply an output feedback sliding mode controller for flying height control by using a PZT bender. The sliding mode control method is popular due to its
properties of robustness and insensitivity to matched disturbance and model uncertainty [5, 6]. Conventional sliding mode control is based on a state-space design, where information on the system states is needed when construct-ing the slidconstruct-ing surface and controller. However, system states are not measurable in the current study; hence, an output feedback sliding mode control method that solely uses the output error is developed. An output feedback sliding mode controller is developed first to validate the performance of the flying height control. The first and second derivatives of the flying height error are needed in the controller. However, this may result in difficulties in practice due to measurement noise. Therefore, a high-order sliding mode control method[7]with robust differentiators [8] is used to reduce the number of required output derivatives and to avoid the influence of measurement noise. In addition, a high-order sliding mode controller helps to reduce chattering with appropriate order design [7, 9]. Simulations on an identified model using measured vibration data of an optical disk will be presented to demonstrate the controller effectiveness.
2 Flying height control mechanism
To enable the active flying height control capability, a modified flying head with a PZT actuator is fixed to a suspension arm, as shown inFig. 1. The PZT serves as the flying height actuator for the flying head in a near-field optical disk drive. This approach has been used as a fine actuator in high density data storage disk drives, since its response performance is sufficient to achieve a fast and precise movement. With a laser Doppler vibrometer in the system identification, a PZT transfer function that relates the control voltage U in volts to the PZT bending displacement Y in nanometres in the vertical direction is:
YðsÞ UðsÞ¼ ð3:188 10 15 ðs2þ 19:85s þ 1:096 107ÞÞ. ððs þ 2:545 104Þðs2þ 141:2s þ 4:797 106Þ ðs2þ 534:9s þ 1:068 108ÞÞ ð1Þ
As a result, a Bode diagram and an open-loop step response are shown in Figs. 2 and 3, respectively. Considering the
qIEE, 2003
IEE Proceedings online no. 20030885 doi: 10.1049/ip-cta:20030885
The authors are with Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan, Republic of China Paper first received 17th January 2003 and in revised form 8th July 2003
open-loop step response shown inFig. 3, there is an obvious oscillation in the transient response which results in a long settling time. Hence, a closed-loop controller is required to eliminate the flying height oscillation and obtain a fast and stable response.
3 Output feedback sliding mode control
The flying head is required to undergo a desired displace-ment, qd; in the vertical direction in order to maintain a constant flying height between the vibratory optical disk and the optical flying head. The trajectory of the desired displacement qdgenerally will be the optical disk vibration waveform plus a constant flying height. In view of uncertainties and disturbances in the current system, a sliding mode control method is applied.
The PZT plant model in (1) can be rewritten in state-variable form as:
_x x¼ Ax þ Bu þ dðx; tÞ ¼ a1 a2 a3 a4 a5 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 x1 x2 x3 x4 x5 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 þ b1 0 0 0 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 uþ d1 d2 d3 d4 d5 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 y¼ Hx ¼ 0 0 h½ 3 h4 h5x ð2Þ
where yðtÞ 2 R is the flying height, uðtÞ 2 R is the control input voltage, and the system state vector xðtÞ 2 R5: Addi-tionally, the disturbance vector dðx; tÞ 2 R5 caused by system uncertainty and external noise is bounded by:
kdðx; tÞk ðx; tÞ ð3Þ
and ðx; tÞ is a known boundary function.
In near-field optical disk drives, the only feedback information about the flying head generally will be either the displacement or the velocity. Not all system states are available and we assume that only the output displacement error can be obtained. As a consequence, the control design method using system states can not be used and thus an output feedback sliding mode control will be developed.
The flying height error is defined as:
e¼ y qd ð4Þ
Since the relative degree of the system is three according to (1), the twice differentiation term €ee is incorporated in the sliding surface design. Hence, a sliding surface s(e) is prescribed in terms of error terms:
sðeÞ ¼ Ce ¼ ½ 1 L G €ee _ee e 2 4 3 5 ¼ €ee þ L_ee þ Ge ð5Þ where coefficients L and G come from the sliding mode conjugate poles p1¼ a þ bj and p2¼ a bj as:
C¼ ½ 1 L G ¼ ½ 1 2a a2þ b2þ 2ab ð6Þ As long as a sliding mode of the system trajectory exists on a sliding surface sðeÞ ¼ 0; a continuous control ueq called ‘equivalent control’[5] and [6] can replace the undefined discontinuous control on the discontinuity boundary to make the system trajectory continuous along the surface sðeÞ ¼ 0 [6]. Since the system trajectory in the sliding mode is continuous, the sliding function sðeÞ represented in (5) must be differentiable. This means that the time derivative of sðeÞ in (5) is equal to zero; i.e.:
_ssðeÞju¼ueq ¼ e
ð3Þþ L€ee þ G_ee ¼ 0 ð7Þ Assuming the desired displacement qd is a constant and taking the triple differentiation of (4), leads to:
eð3Þ¼ yð3Þ qð3Þd ¼ yð3Þ ð8Þ Substituting (8) into (7) yields:
_ssðeÞju¼ueq ¼ y ð3Þ
þ L€ee þ G_ee ¼ 0 ð9Þ
Further, substituting the state equation from (2) into (9) yields: frequency, rad/s phase, deg magnitude, dB –50 0 50 100 frequency, rad/s a b 102 103 104 105 106 102 103 104 105 106 –270 –225 –180 –135 –90 –45 0
Fig. 2 Bode diagram of open loop system
a Magnitude as a function of frequency b Phase as a function of frequency
spindle motor optical disk PZT pivot arm VCM h f r (t ) v (t )
Fig. 1 Flying head with an embedded PZT actuator
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 time, s amplitude, nm
_ssðeÞju¼ueq¼ d3 dt3ðh3x3þ h4x4þ h5x5Þ þ L€ee þ G_ee ¼ d 2 dt2ðh3x2þ h4x3þ h5x4þ h3d3þ h4d4þ h5d5Þ þ L€ee þ G_ee ¼ d dtðh3x1þ h4x2þ h5x3þ h3d2þ h4d3þ h5d4 þ h3d3þ h4d4þ h5d5Þ þ L€ee þ G_ee ¼ h3ða1x1þ a2x2þ a3x3þ a4x4þ a5x5þ u þ d1Þ þ h4x1þ h5x2þ h4d2þ h5d3þ h3_dd2 þ ðh4þ h3Þ_dd3þ ðh5þ h4Þ_dd4þ h5_dd5þ L€ee þ G_ee ¼ 0 ð10Þ It follows from (10) that the equivalent control ueq can be written as: u¼ ueq ¼ a1x1þ a2x2þ a3x3þ a4x4þ a5x5þ h4 h3 x1þ h5 h3 x2 1 h3 df L h3 €ee G h3 _ee ð11Þ where df ¼ h3d1þ h4d2þ h5d3þ h3_dd2þ ðh4þ h3Þ_dd3þ ðh5þ h4Þ_dd4þ h5_dd5; and hence, it depends on the disturb-ance terms.
Sliding mode control has to enable a system to move towards and stay on a sliding surface; i.e. satisfy the reaching and sliding condition[6]:
s_ss < kjsj ð12Þ
where k is a positive constant, so that the system trajectory reaches the sliding surface in a finite time. Therefore, based on (11), for output feedback sliding mode control we define the control input u as:
u¼ a1x1þ a2x2þ a3x3þ a4x4þ a5x5þ h4 h3x1 þh5 h3 x2þ 1 h3 df maxþk sgnðsÞ L h3 €ee G h3 _ee ¼ ðj f ðxÞ þ kdfjmaxþ kÞsgn sð Þ kL€ee kG_ee
¼ Q sgnðsÞ kL€ee kG_ee ð13Þ where sgnðsÞ ¼ þ1 if s > 0 0 if s¼ 0 1 if s < 0 8 < : ð14Þ fðxÞ ¼ a1x1þ a2x2þ a3x3þ a4x4þ a5x5þ h4 h3x1þ h5 h3x2; 1 > k¼ 1 h3 > 0 Q¼ ðj f ðxÞ þ kdfjmaxþ kÞ > 0 ð15Þ Proof: Substituting (10) and (13) into the reaching and sliding condition in (12) yields
s_ss¼ s½h3ða1x1þ a2x2þ a3x3þ a4x4þ a5x5þ u þ h1d1Þ þ h4x1þ h5x2þ h4d1þ h5d2þ h3_dd1þ h4_dd2þ h3_dd2 þ h5_dd3þ h4_dd3þ h5_dd4þ L€eeþ G _ee ¼ s½h3fðxÞ þ dfþ h3uþ L€ee þ G_ee ¼ s h3fðxÞ þ df h3 fðxÞ þ 1 h3 df maxþk sgnðsÞ ¼ h3 fðxÞ þ 1 h3 df s h3 fðxÞ þ 1 h3 df maxjsj h3kjsj ð16Þ However: fðxÞ þ 1 h3 df s< fðxÞ þ 1 h3 df maxjsj
and from (15), one has h3> 1; hence, (16) yields s_ss < kjsj: This proves the reaching and sliding condition in (12) for the proposed output feedback
sliding mode controller. A
Since the control law given in (13) is discontinuous across the sliding surface, it gives rise to chattering in a trajectory tracking process. Therefore, a saturation function satðsÞ with a sliding layer " is adopted to replace the switching function sgnðsÞ defined in (14), i.e.:
satðsÞ ¼ sgnðsÞs jsj > " " jsj " (
ð17Þ The controller in (13) can thus be written as:
u¼ Q satðsÞ kL€ee kG_ee ð18Þ
The closed-loop block diagram with output feedback sliding mode control is shown in Fig. 4. Since the maximum boundary of the system states function fðxÞ and disturbances ðx; tÞ is usually unknown or inaccurate. The design parameter Q in (18) can be adjusted based on an energy function expressed by:
EðeÞ ¼ Z
e2dt ð19Þ
in order to minimise the control error.
4 Output feedback high-order sliding mode
control
The sliding surface in (5) adopts e¨ and e˙ in order to construct an output feedback sliding mode controller. Since only flying height error can be measured in practice, the first and second differentiation of e are used instead of direct measurements of e˙ and e¨. Hence, a high-order sliding mode controller with robust differentiators is presented for the flying height control. Using high-order sliding mode reduces the number of output derivatives required and the robust differentiator yields exact differentiation with finite-time convergence. Q plant qd e s u y + − + + + − − + sat(s) G L L dtd dt d G k + e e
4.1
Third-order sliding mode controller
Considering the system in (1) of relative degree three, a third-order sliding mode controller design is adopted. The control purpose is to eliminate the flying height error, hence, the sliding surface is defined as:¼ e ¼ y qd ð20Þ
With the sliding surface, a general form of relay type third-order sliding mode control for a relative degree three system can be written as[7]: u¼ A3sgn €þ 2 j _j3þ 2 1=6 sgn_þ jj2=3sgn h i ð21Þ where A3 is a positive constant. Accordingly, to use (21) only knowledge of the relative degree of the system is needed in advance while the exact plant model is not required. Thus, the information needed is simply the current value of . In (21), the required time derivatives of can be evaluated by various differentiators [10] and [11]. To modify (21), define sliding functions as[12]:
S0¼ ð22Þ
S1¼ _SS0þ A1jS0j 2=3
sgnðS0Þ ð23Þ
where A1 is a positive constant. Using (22) and (23), then (21) becomes an output feedback third-order sliding mode controller expressed by:
u¼ A3sgnf_SS1þ A2jS1j 1=6 sgn½_SS0þ A1jS0j 2=3 sgnðS0Þg ð24Þ where A2 is a positive constant. Further, (24) can be rewritten as:
u¼ A3sgnðS2Þ ð25Þ
where
S2¼ _SS1þ A2jS1j1=6sgnðS1Þ ð26Þ
4.2
Robust differentiator
To implement the above controller in (24) we require information on the successive derivatives _SS0and _SS1: Hence, a robust differentiator is adopted to evaluate exact derivative values. A general form of arbitrary order robust differ-entiator can be applied as[13]:
_zz0¼ k0jz0 f ðtÞj n=ðnþ1Þ sgnðz0 f ðtÞÞ þ z1 . . . _zzi¼ kijz0 f ðtÞj ðn iÞ=nþ1 sgnðz0 f ðtÞÞ þ ziþ1 i¼ 1; . . . ; n 1 _zzn¼ knsgnðz0 f ðtÞÞ ð27Þ
where z0; z1; . . . ; zn are estimations of input fðtÞ; _ffðtÞ; . . . ; fðnÞðtÞ and k0; . . . ; kn are positive constants. Using (27), the control input in (24) can be rewritten as:
u¼ A3sgn 1þ A2j 0j 1=6 sgn m1þ A1jm0j 2=3 sgnðm0Þ h i n o ð28Þ with the robust differentiators being written as:
_m m0¼ kk0jm0 S0j 1=2 sgnðm0 S0Þ þ m1 _m m1¼ kk1sgnðm0 S0Þ ð29Þ and _ 0 ¼ ~kk0j 0 S1j1=2sgnð 0 S1Þ þ 1 _ 1 1 ¼ ~kk1sgnð 0 S1Þ ð30Þ
where the value of the sliding function S1was evaluated by: S1 ¼ m1þ A1jm0j
2=3
sgnðm0Þ ð31Þ
4.3
Fourth-order sliding mode control
In order to eliminate chattering, it was proved by[7] that introducing successive time derivatives u; _uu; . . . ; uðr k 1Þas new auxiliary variables and uðr kÞ as a new control input achieves different modifications of each rth order sliding mode controller with a system relative degree of k¼ 1; 2; . . . ; r: The resulting control input is a ðr k 1Þ smooth function of time when k < r; a Lipschitz function when k¼ r 1; and a bounded infinite-frequency switch-ing function when k¼ r: Accordingly, in order to generate a control input that is smoother than an infinite-frequency switching one, a fourth-order sliding mode control is designed by modifying the original system in (2) as:
_x
x¼ Ax þ Bu y¼ Hx _u
u¼ t ð32Þ
where the actual control input u of the original system is treated as a new auxiliary variable of the higher relative degree system in (32) and t is the new control input of the fourth-order sliding mode control written as:
t¼ A4sgn _SS 2þ A3jS2j 1=12 þ sgn_SS1þ A2jS1j 1=6 _ sgn½_SS0þ A1jS0j 2=3 sgnðS0Þ ¼ A4sgnðS3Þ ð33Þ where S0¼ S1¼ _SS0þ A1jS0j 2=3 sgnðS0Þ S2¼ _SS1þ A2jS1j 1=6sgnðS 1Þ S3¼ _SS2þ A3jS2j1=12sgnðS2Þ ð34Þ The robust differentiators in (27) can be applied here in a manner similar to Section 4.2 that deals with the third-order sliding mode controller.
5 Flying height control simulation
5.1
Output feedback sliding mode control
With the plant model in (1), Figs. 5a and 5b compare simulation results of the output feedback sliding mode controller with a switching function sgnðsÞ in (13) and a saturation function satðsÞ in (18). The 1000 nm step responses and sliding surfaces are shown inFigs. 5a and 5b, respectively. It is demonstrated that the saturation function indeed eliminates chattering. Figures 6a and 6b respectively compare 10 nm step responses and control inputs with and without white noise in the control input. The mean and variance of the white noise are 0.0475 and 3.2526, respectively. As a result, the flying height is not affected, although the control input is accompanied by white noise.
5.2
Third-order sliding mode control with
robust differentiators
The use of robust differentiators i.e. (29) and (30) in the third-order sliding mode controller, (28), leads to the step response and sliding functions, depicted in Fig. 7a and Figs. 7b to 7drespectively. It can be found that the sliding functions, as defined in (20), (22), (23) and (26), indeed converge to zero. The tracking capability does not degrade at all when using robust differentiators for derivative values. In order to verify the robustness of the present controller with robust differentiators, a white noise is introduced into the measured flying height error signal. The mean and variance of the white noise are 0.0001 and 0.1001, respectively. Figure 8 compares the step responses of the third-order sliding mode controller with and without the robust differentiator. As a result, the controller perform-ances with the presented differentiator does not degrade at all, whereas the other one becomes unstable.
5.3
Fourth-order sliding mode control
Figure 9adepicts the step response and Fig. 9bthe actual control input u created by the modified plant model in (32) and the output feedback fourth-order sliding mode con-troller of (33). The derivatives of S0; S1 and S2 are directly calculated. The control input in Fig. 9b shows that high-order sliding mode control can achieve a smooth control input and reduce chattering.
5.4
Flying height control with measured disk
vibration waveform
To validate the proposed controller, we performed a flying height control simulation for the PZT model of (1) using vibration displacement data vðtÞ of a near-field optical disk. The vibration data is measured by using a dual-beam laser Doppler vibrometer for a polycarbonate-substrate-based near-field optical disk at a constant speed of 5400 rpm. The dominant frequency of the disk vibration is 90 Hz synchronous with the spindle motor speed. The applied PZT bender is model PL122.251 made by Physik Instrumente. Its displacement in the bending direction is 0 – 250 mm corresponding to an input voltage of 0 – 60 V.
In order to maintain a constant flying height in the presence of disk vibration, the flying height control can be treated as a trajectory tracking control task dealing with a disk vibration waveform vðtÞ between approximate ^ 60 mm, as shown inFig. 10a, so as to control the flying head height equal to the sum of the disk vibration amplitude and a constant flying height hf; i.e. the focusing depth for a near-field optical disk as depicted in Fig. 1. Hence, the reference input can be expressed as:
rðtÞ ¼ vðtÞ þ hf ð35Þ
Using the third-order sliding mode controller with robust differentiators, the flying height error in tracking the disk 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 200 400 600 800 1000 1200 height, nm 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 200 400 600 800 1000 1200 time, ms a b height, nm 0.5 0.6 0.7 0.8 0.9 1.0 –6 –4 –2 0 2 4 6 time, ms sliding surface ( × 10 11) switching function saturation function Fig. 5
a Step response of the sliding mode controller with switching function and with saturation function
b Sliding surfaces of the sliding mode controller with switching function and with saturation function
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0 2 4 6 8 10 12 time, ms a b height, nm 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 –30 –20 –10 0 10 20 30 40 50 60 70 time, ms control input, V
without white noise with white noise
Fig. 6
a Comparison of 10 nm step response with and without white noise disturbance
vibration waveform is shown in Fig. 10b. The tracking control error reduces from the initial value of 242 nm to lie within ^ 30 nm after 1 ms. The initial large tracking error of up to 93 nm comes from the initial estimation error of the robust differentiator.
6 Conclusions
An output feedback sliding mode flying height controller with a PZT actuator has been developed for flying height control for near-field optical disk drives accompanied by a severe and uncertain disk vibration. In order to reduce the number of required output derivatives and avoid chattering, a high-order sliding mode controller with a robust differentiator has been presented for application in the
0 10 20 30 40 50 60 70 –80 –60 –40 –20 0 20 40 60 time, ms a b height , µ m 0 1 2 3 4 5 6 7 8 9 10 –250 –200 –150 –100 –50 0 50 100 time, ms
flying height err
o
r, nm
Fig. 10
a Measured surface vibration of optical disk b Output error in flying height control
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 500 1000 1500 height, nm 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 –2 0 2 4 time, ms time, ms a b control input, V Fig. 9
a Step response of the fourth-order sliding mode controller b Control input at the fourth-order sliding mode controller
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 –400 –200 0 200 400 600 800 1000 1200 1400 time, ms height, nm
with robust differentiator without robust differentiator
Fig. 8 Step responses under measurement noise of the third-order sliding mode controller with and without robust differentiators 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 200 400 600 800 1000 1200 time, ms a d b c height, nm 0 0.1 0.2 0.3 0.4 0.5 0.6 –1000 –500 0 500 S0 S1 (× 10 7) S2 (× 10 13) 0 0.1 0.2 0.3 0.4 0.5 0.6 –10 –5 0 5 0 0.1 0.2 0.3 0.4 0.5 0.6 –2 –1 0 1 time, ms time, ms time, ms Fig. 7
a Step response of the third-order sliding mode controller with robust differentiators
b Sliding function S0of the third-order sliding mode controller with robust
differentiators
c Sliding function S1of the third-order sliding mode controller with robust
differentiators
d Sliding function S2of the third-order sliding mode controller with robust
flying height control. Simulation results demonstrate the effectiveness of the proposed mechanism and control method.
7 Acknowledgment
This work was supported by ‘Photonics science and technology for tera era’, Center of Excellence, Ministry of Education, Taiwan under contract 89-E-FA06-1-4.
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