Design of a repetitive controller: an application to the track-following servo system of optical disk drives T.-Y. Doh, J.R. Ryoo and M.J. Chung Abstract: In an optical disk drive servo system, disturbances with significant periodic components cause tracking errors of a periodic nature. As an effective control scheme for improving periodic disturbance attenuation performance, repetitive control has been applied successfully to the trackfollowing servo system of optical disk drives. The increase in disk rotational speed to achieve a better data throughput leads to the increase in the frequency of periodic disturbance, which needs a high loop gain in a wider control bandwidth. However, this requirement is not easily accomplished because plant uncertainty hinders selecting the bandwidth of a filter in the repetitive controller. The problem of add-on type repetitive controller design for a track-following servo with norm-bounded uncertainties in optical disk drives with high rotational speed is examined. Using the Lyapunov functional for time-delay systems, a sufficient condition for robust stability of the repetitive control system is derived in terms of an algebraic Ricatti inequality or a linear matrix inequality (LMI). On the basis of the derived condition, it is shown that the repetitive controller design problem can be reformulated as an optimisation problem with an LMI constraint on the free parameter. The validity of the proposed method is verified through experiments using a DVD-ROM drive.
1
Introduction
For data read-out in an optical disk drive, the laser spot should be positioned precisely on the centre of a data track in the presence of disturbances. Disturbances acting on the track-following servo are mainly caused by inevitable eccentric rotation of disk and hence are inherently of a periodic nature having the same frequency components as the disk rotational frequency and its integer multiples. Therefore to obtain a good tracking performance, it is essential to attenuate such periodic disturbances because the data track on the disk keeps moving inwards and outwards by the disturbances during the rotation of the disk [1–3]. As the disk rotational speed increases for improving data transfer rate in optical disk drives, so does the frequency of the periodic disturbance, which leads to a high loop gain in a wider control bandwidth in the trackfollowing servo. In the track-following controller design, there are three major specifications, namely disturbance attenuation, robust stability and noise rejection, which conflict with each other. For example, a high loop gain for disturbance attenuation with a restriction on control bandwidth usually # IEE, 2006 IEE Proceedings online no. 20045217 doi:10.1049/ip-cta:20045217 Paper first received 25th October 2004 and in revised form 16th June 2005 T.-Y. Doh is with the Department of Control and Instrumentation Engineering, Hanbat National University, San 16-1, Duckmyung-dong, Yuseon-gu, Daejeon, 305-719, Korea J.R. Ryoo is with the Department of Control and Instrumentation Engineering, Seoul National University of Technology, 172 Gongneung 2-dong, Nowon-gu, Seoul, 139 – 743, Korea M.J Chung is with the Division of Electrical Engineering, Department of Electrical Engineering and Computer Science, Korea Advanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Korea E-mail:
[email protected] IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
results in a reduced stability margin. For this reason, the controller design procedure requires numerous trial and errors and the order of the designed controller becomes too high to be applied to a practical system. Hence, it is difficult for only a conventional controller like a lead-lag controller to achieve the requirements in high-speed rotational optical disk drives. As an effective add-on controller for enhancing periodic disturbance attenuation capability, repetitive controllers for disk drive servo systems have been investigated in [4–9]. Tomizuka et al. [4] and Chew and Tomizuka [5] analysed repetitive control in the discrete-time domain and introduced two filters for repetitive control. Do¨tsch et al. [6] and Steinbuch [9] proposed useful repetitive control schemes for those cases where an optical disk rotates at constant linear velocity and the disk rotational period cannot be measured directly by an auxiliary signal. Moon et al. [7] designed a repetitive controller for an optical disk drive using a graphical design technique based on the frequency domain analysis of linear interval system. Onuki and Ishioka [8] compensated the repeatable runout in hard disk drives by switching filter parameters in the repetitive controller. It is well known that the performance of repetitive control depends on the cutoff frequency of a low-pass filter included in the repetitive controller, that is, the q-filter proposed by Hara et al. [10] and others [4, 5]. For a repetitive controller to be effective, the bandwidth of the q-filter needs to be sufficiently higher than the disturbance frequency. In practice, the bandwidth of the q-filter is not sufficiently extended because of the plant uncertainty and system stability. Moreover, existing approaches dealing with plant uncertainty have focused on the analysis on the robustness of repetitive control system [11–14]. For these reasons, a new method is required to select the cutoff frequency of q-filter, not only guaranteeing system stability but also improving performance in consideration of plant uncertainties. 323
Fig. 1 Dual-stage actuator for track-following servo
This paper proposes a repetitive controller design method for the track-following servo with norm-bounded uncertainties of optical disk drives with high rotational speed. The repetitive controller is installed as an add-on module in the existing track-following servo wherein the feedback compensator is designed independently of the repetitive controller. The compensated plant and the repetitive controller are represented as linear time-invariant systems in the state space. On the basis of the Lyapunov functional for time-delay systems, a simple criterion is proposed, which ensures the asymptotic stability of the closed-loop system inclusive of the repetitive controller and is given in the form of an algebraic Ricatti inequality (ARI) or a linear matrix inequality (LMI). The repetitive controller design problem is shown to be equivalent to an optimisation problem over the free parameter of the low-pass filter subject to an LMI constraint. With improving tracking performance, the designed repetitive controller preserves system stability. The overall track-following servo system including the repetitive controller is implemented using a 32-bit floating point digital signal processor (DSP). The feasibility of the proposed design method is shown by experimental results. The remainder of this paper is organised as follows. Section 2 gives a brief introduction to the track-following servo system of an optical disk drive and the repetitive controller as an add-on module. In Section 3, a stability criterion for the repetitive control system is proposed. It is shown that the repetitive controller is obtained by solving an optimisation problem. In Section 4, digital implementation of the servo system is described and some experimental results are presented. Some concluding remarks are given in Section 5. In the following, smax[.] denotes the maximum singular value. For a symmetric matrix X ¼ XT, X . 0 (,0) means that X is positive (negative) definite. The sizes of the identity matrix I and the zero matrix 0 should be inferred from the context, except special cases.
2
System description
Fig. 1 depicts a track-following servo mechanism with a compound actuator composed of a small fine actuator mounted on top of a large coarse actuator. The role of the 324
coarse actuator in track-following mode is simply to move the fine actuator slowly over the operation range. In this paper, therefore, only the fine actuator is considered in the design of the track-following servo system. A typical track-following servo system is shown in Fig. 2 where P(s) is the fine actuator and C(s) is the feedback compensator. The track position on which the laser spot should be located during track-following control is represented as an external input d(t). Fig. 3 shows a typical pattern of the tracking error when only the focus controller is activated. Because of the radial runout shown in Fig. 3 owing to the eccentric rotation of disk, d(t) is a radially oscillating signal with the frequency of disk rotation. Every track produces one cycle of sinusoidal signal as the track passes inwards and outwards from the laser beam spot or vice versa. Therefore the tracking error signal e(t) has a shape of a sinusoid shown in Fig. 3. Because the phase and amplitude of d(t) are not known, d(t) is regarded as an unknown periodic disturbance. In addition, because of inherent limitations of opto-mechanical structure, d(t) cannot be measured and only the available value is an amplified signal of e(t) by the photo detector gain KPD . Therefore a track-following servo system is a representative error feedback control system, and controller design is formulated as a disturbance attenuation problem to find a controller reducing the magnitude of the tracking error below an acceptable level. The fine actuator, which is a voice-coil motor driven by a current amplifier, determines the radial position of the laser spot by moving the objective lens to which a spring and a damper are attached. Therefore a second-order mass – spring –damper model is usually adopted to represent the fine actuator. Because of unavoidable modelling errors
Fig. 2 Track-following servo system IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
Fig. 3 Radial runout and tracking error signal that originate from the eccentric rotation of disk
caused by changes in the operating environment or allowable error margins of components, suppose that, in consideration of parameter uncertainty, the tracking actuator is described as a second-order linear interval transfer function as follows [7, 15]
PðsÞ ¼
s2
þ
þ ½b 2 ; b2 þ þ ½a1 ; a1 s þ ½a 2 ; a2
ð1Þ
þ The value of KPD is in [K2 PD, K PD] because the components of the amplifier also have allowable error margins. Information about the extraneous disturbance, such as maximally injectable magnitude and bandwidth, is specified for the controller design [1, 16]. On the basis of the data, the required loop gain and bandwidth for sufficient disturbance rejection and the minimum stability margin for stable trackfollowing pull-in are calculated. Considering the specifications, the feedback compensator C(s) is designed. As various design methods for C(s) have been proposed in previous works [7, 15, 17], C(s) is assumed to be given without loss of generality. It is noted that as the eccentricity and the rotational speed of disk increase, so do the magnitude and the frequency of disturbance, which leads to the high loop gains and an extended control bandwidth. However, it is very difficult for only C(s) to fulfil this requirement because the control bandwidth should be limited to reject measurement noise in the high-frequency region. In order to effectively reject the periodic disturbance in the high-speed optical disk drives, the repetitive controller Crp(s) is added to the existing track-following controller as an add-on module shown in Fig. 4 where q(s) is a lowpass filter to ensure system stability and T is the constant and known period of d(t). Note that Crp(s) is equivalent to the modified repetitive controller with a(s) ¼1 as proposed by Hara et al. [10]. The repetitive control system design problem is compressed into the design of a low-pass filter q(s) which not only preserves system stability of the track-following servo system but also improves the tracking accuracy. In other words, the point of the problem is to maximise the bandwidth of q(s) for smaller tracking error by preserving the system stability.
3 3.1
be represented by x_ p ðtÞ ¼ ðAp þ D Ap Þxp ðtÞ þ Bp uðtÞ yðtÞ ¼ ðCp þ DCp Þxp ðtÞ
ð2Þ
where xp(t) [ Rnp is the state of G(s), u(t) [ R is the output of the repetitive controller and y(t) ¼ 2eT (t) is the electrically amplified error which should be attenuated. The uncertainties will be assumed in the form Hp1 D Ap ð3Þ ¼ FEp DCp Hp2 where Hp1 , Hp2 and Ep are known real matrices which characterise the structure of the uncertainty and F is an unknown matrix function with the property of smax[F] 4 1. Without loss of generality, the low-pass filter q(s) is assumed to be first order, that is qðsÞ ¼
vc s þ vc
ð4Þ
where vc is the cutoff frequency of the low-pass filter. Remark 1: From the fact that the ideal q(s) for perfect tracking is 1, it is possible to intuitively anticipate that a firstorder filter will lead to better performance than other high-order filters, as long as their cutoff frequencies are identical, which is verified in [7]. Moreover, compared with a first-order filter, because high-order filters cause larger phase-lag and are relatively complicated, there are difficulties in implementing them in a digital control system with high sampling frequency. Let us realise q(s)e2sT in Crp(s) of Fig. 4 Yrc ðsÞ vc esT Xc ðsÞ ¼ U ðsÞ s þ vc Xc ðsÞ
ð5Þ
where U(s) and Yrc(s) are Laplace transforms of u(t) and yrc(t) [ R, respectively, and Xc(s) is that of dummy-state
Repetitive control system design Stability criteria
For convenience, let KPDC(s)P(s) be a compensated plant G(s). By transforming it into the state equations, G(s) can IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
Fig. 4 Repetitive track-following servo system 325
variable xc(t) [ R. Then, the numerators and the denominators on both sides of the equation are equated to each other. This results in
Lemma 1 [18]: Let S1 and S2 be real constant matrices of compatible dimensions and M(t) be a real matrix function satisfying MT(t)M(t) I. Then the following inequalities hold
Yrc ðsÞ ¼ vc esT X c ðsÞ U ðsÞ ¼ sX c ðsÞ þ vc X c ðsÞ
1. S1M(t)S2 þ ST2MT(t)ST1 r21S1ST1 þ rST2S2 for any r.0
ð6Þ ð7Þ
By inverse Laplace transformation, these two equations can be written as follows x_ c ðtÞ ¼ vc xc ðtÞ þ uðtÞ
ð8Þ
yrc ðtÞ ¼ vc xc ðt T Þ
ð9Þ
If we eliminate yrc(t) using the relation u(t) ¼ eT (t) þ yrc(t), the repetitive controller Crp(s) can be realised as x_ c ðtÞ ¼ vc xc ðtÞ þ vc xc ðt T Þ þ eT ðtÞ
Lemma 2 [19]: The LMI QðxÞ ST ðxÞ
By introducing the augmented state vector x ¼ [xTp xTc ]T, we can obtain the following closed-loop system to be used to check the stability of the overall system: x_ ðtÞ ¼ ðA þ D AÞxðtÞ þ Ad xðt T Þ
ð11Þ
QðxÞ SðxÞR1 ðxÞST ðxÞ , 0
RðxÞ , 0;
Proof of Theorem 1: 1. Consider a Lyapunov functional V(x(t), t) given by ðt V ðxðtÞ; tÞ ¼ xT ðtÞPxðtÞ þ xT ðuÞQxðuÞ du ð15Þ tT T
A¼
Ap Bp Cp Cp
0 ; vc
Ad ¼
0 0
and the matrix D A is expressed in the form Hp1 Bp Hp2 D A ¼ HFE ¼ F½Ep Hp2
vc Bp vc
V_ ðxðtÞ; tÞ ¼ xT ðtÞ½PðA þ D AÞ þ ðA þ D AÞT P þ QxðtÞ þ xT ðtÞPAd xðt T Þ þ xT ðt T ÞATd PxðtÞ
0
ð12Þ
By realising the repetitive controller as (10), the closedloop system (11) can be realised as a typical time-delay system. The system (11) satisfies the following assumptions. Assumption 1: In the repetitive track-following servo system shown in Fig. 4, the closed-loop system without the repetitive controller is stable for all uncertainties. Assumption 2: Ap 2 BpCp is stable.
xT ðt T ÞQxðt T Þ xT ðtÞ½PðA þ D AÞ þ ðA þ D AÞT P þ Q þ PAd Q1 ATd P xðtÞ
ð16Þ
To remove the uncertainty in (16), we use (3) and (12) and apply lemma 1 (1). Then we obtain PD A þ D AT P ¼ PHFE þ ET FT HT P l1 PHHT P þ lET E
ð17Þ
for some l . 0. Substituting (17) into (16), it becomes
Theorem 1: Subject to assumptions 1 and 2, the repetitive control system shown in Fig. 4 is robustly asymptotically stable independent of period T if one of the following two equivalent conditions holds: 1. There exist matrices 0 , P ¼ P T [ R(npþ1)(npþ1) and 0 , Q ¼ Q T [ R(npþ1)(npþ1) and a scalar l . 0 satisfying the ARI PA þ AT P þ Q þ PAd Q1 ATd P þ l1 PHHT P þ l ET E , 0
(npþ1)(npþ1)
and 0 , Q ¼ where 0 , P ¼ P [ R QT [ R(npþ1)(npþ1) are weighting matrices. Taking the derivative of V(x(t), t) along the solution of (11) yields
where
SðxÞ ,0 RðxÞ
where Q(x) ¼ QT(x), R(x) ¼ RT(x) and S(x) depend affinely on x, is equivalent to
ð10Þ
uðtÞ ¼ vc xc ðt T Þ þ eT ðtÞ
ð13Þ
2. There exist matrices 0 , P ¼ P T [ R(npþ1)(npþ1) and 0 , Q ¼ QT [ R(npþ1)(npþ1) and a scalar l.0 satisfying the LMI 2 3 PA þ AT P þ Q PAd PH lET 6 ATd P Q 0 0 7 6 7,0 ð14Þ T 4 H P 0 lI 0 5 lE 0 0 lI In the proof of theorem 1, the following lemmas will be used. 326
2. ST1 S2 þ ST2S1 m22ST1 S1 þ m2ST2S2 for any m . 0
V_ ðxðtÞ; tÞ xT ðtÞ½PA þ AT P þ Q þ PAd Q1 ATd P þ l1 PHHT P þ lET E xðtÞ
ð18Þ
If V˙(x(t), t) , 0 when x(t) = 0, then x(t) ! 0 as t ! 1 and the repetitive control system is globally asymptotically stable independent of T for all admissible uncertainties satisfying (3). From (18), this stability condition is guaranteed if ARI (13) holds. 2. Applying lemma 2, it is easy to show that ARI (13) is equivalent to LMI (14). A Theorem 1, which is based on stability conditions for general linear time-delay systems with norm-bounded uncertainties [18, 20], is a result obtained by qualifying the conditions for repetitive control systems. 3.2
Repetitive controller design
Now, we find out the maximum cutoff frequency vc of the low-pass filter to ensure system stability of the trackfollowing control system, using the result of theorem ˆ c and 1. For convenience, we represent vc as the sum of v IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
dvc , that is vc ¼ v^ c þ dvc
ð19Þ
where vˆc is a roughly estimated value from the nominal plant and dvc is an unknown value which we should find. Then, A and Ad are affinely dependent on the free parameter dvc and are represented as the following form A ¼ A0 þ A1 dvc
ð20Þ
Ad ¼ Ad0 þ Ad1 dvc
ð21Þ
of q-filter in the repetitive controller both to improve tracking accuracy and to ensure the above stability condition. Remark 2: The stability condition of theorem 2 is a little more conservative than that of theorem 1 because lemma 1 (2) is applied to eliminate non-linearities arising in terms like dvcPA1 and dvcAT1 P. In compensation for the conservatism, however, theorem 2 provides a simple method to find vc .
where 0 Ap Bp Cp 0 ; A1 ¼ A0 ¼ Cp v^ c 0 0 v^ c Bp 0 Bp Ad0 ¼ ; Ad1 ¼ 0 v^ c 0 I
0 I
Remark 3: The final objective of problem (23) is to minimise g satisfying the LMI condition (22). However, there are non-linear terms like m21P and 2 gQ in LMI condition (22). Thus, to solve linear convex optimisation problem (23), Q and m should be fixed.
In the following theorem, a modified stability condition is proposed, which is represented as an LMI and convex in P and the inverse of dvc . Theorem 2: Subject to assumptions 1 and 2, the repetitive control system shown in Fig. 4 is robustly asymptotically stable independent of period T if, for some weighting matrix Q . 0 and constant m . 0, there exist a matrix P . 0 and scalars g ¼ dv21 c , l satisfying the LMI 2 3 P11 P12 P13 T 4 P12 P22 0 5,0 ð22Þ T P13 0 P33 with the shorthand " # PA0 þ AT0 P þ Q PAd0 P11 :¼ ATd0 P Q " # " PAd1 m1 P m AT1 PH P12 :¼ ; P13 :¼ 0 0 0 0
P22 :¼ diagðg Q; gI; gIÞ;
lET 0
#
P33 :¼ diagðlI; l IÞ
Proof of Theorem 1: By using (19) and (20) and applying lemma 1 (2), (1, 1) element of (14) reduces to PA þ AT P þ Q ¼ PA0 þ AT0 P þ Q þ dvc ðPA1 þ AT1 PÞ PA0 þ AT0 P þ Q þ dvc ðm2 PP þ m2 AT1 A1 Þ for some constant m . 0. (1, 2) and (2, 1) elements of (14) become PAd ¼ PAd0 þ PAd1 dvc AT P ¼ ATd0 P þ dvc ATd1 P by (21), respectively. After applying lemma 2 and some mathematical manipulation like elementary row operation, we obtain (22). A From theorem 2, the repetitive controller design problem can be formulated as a standard problem of linear objective minimisation subject to an LMI constraint as follows: Minimise g ¼ dv1 c over P; l and g satisfying ð22Þ ð23Þ The sum of vˆc and the maximum dvc ¼ g21, which is obtained by solving (23), is the largest cutoff frequency IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
Remark 4: The role of vˆc in the matrix A0 is to guarantee the ˆ c should be larger than feasibility of LMI (22). Therefore v zero. If vˆc is very close to zero or excessively large, there may not exist a feasible solution. In practical situations, if we know the system information in a degree, it is possible ˆ c. to guess the amount of v 4
Experiments
A commercial DVD-ROM drive with a disk rotating at a constant angular velocity (CAV) of 2400 rpm, that is, the DVD 4 speed, was used for experiments where the frequency of periodic disturbance is 40 Hz. The resonance mode and damping ratio of the tracking actuator are approximately 62 Hz and 0.08, respectively. In consideration of +10% parametric uncertainty, the tracking actuator is modelled as PðsÞ ¼
s2
½68:7; 84:0 ðm=VÞ ð24Þ þ ½55:8; 68:2s þ ½138308; 169043
In the experimental setup, the track pitch of the disk is 0.74 mm and the measurable range is +0.37 mm, which corresponds to the amplified tracking error signal, eT (t), of +2.0 V. Therefore KPD is 5.4 106 V/m. Because of the allowable error of components, KPD also has +10% parametric uncertainty. The control objective is to maintain the tracking error within +0.01 mm against various disturbances bounded by a prescribed limit. To this end, a servo bandwidth of about 3 kHz is required, together with a loop gain exceeding 70 dB in the low-frequency region. The existing feedback controller which satisfies these requirements and provides sufficient gain and phase margin is a lead-lag compensator expressed by CðsÞ ¼
3:87ðs þ 1364Þðs þ 9425Þ ðs þ 942Þðs þ 87965Þ
ð25Þ
Thus, by transforming the interval plant KPD P(s)C(s) into the state-space equation, the compensated plant G(s) is represented by (2) with (3). See more details in [15] and [21] – [23] to find how to linearly interconnect a nominal plant and the uncertainty F. The repetitive controller design is accomplished by solving the LMI optimisation problem (23). Let vˆc , m and Q be 251.3 rad/s, 103 and 0.1I, respectively. As a result, the positive definite solution which is solved by LMI 327
control toolbox [24] is 2 4:54 105 6 2:29 103 6 3 P¼6 6 4:26 10 4 0:141 2:30 105
3
2:29 10 8:27 10:4 203 5:59 104
3
4:26 10 10:4 129 414 0:0166
3 0:141 2:30 105 203 5:59 104 7 7 414 0:0166 7 7 5 9146 0:199 5 0:199 4:14 10
l is 0.015 and g is 7.72 1025. Therefore the maximum cutoff frequency vc of q(s) satisfying theorem 2 is vc ¼ v^ c þ dvc ¼ 13204:7 rad=s As shown in Fig. 5, in case that the cutoff frequency of q(s) is selected as 13204.7 rad/s (2101.6 Hz), the openloop gain increases at the fundamental frequency of the periodic disturbance and its integer multiples. However, the wider the bandwidth of q(s) is, the more the phase margin of the system decreases. Hence, although the performance to attenuate the periodic disturbance improves, it is possible that the overall system becomes unstable by small parameter variation. If a cutoff frequency is selected, which is both less than the obtained maximum frequency and much larger than the fundamental frequency of the disturbance, the track-following servo system not only improves the performance of disturbance attenuation but also ensures a sufficient system stability margin. For this purpose, we choose both one-fifth of the maximum frequency and about ten times as large as the frequency of disk rotation, that is, vc ¼ 2640.9 rad/s (420.3 Hz). Fig. 6 shows effects of the designed repetitive controller for the 16 extremely perturbed compensated plants, G(s) when vc is 2640.9 rad/s. Compared with Fig. 5, instead of
slight gain decrease in high-frequency region, not only a sufficient phase margin is ensured but also the high open-loop gains are generated at the specified frequencies. The track-following servo including the repetitive controller was digitally implemented on a 32-bit floating point DSP, TMS320C6701, as illustrated in Fig. 7. The program for the track-following servo was executed at a sampling rate of 200 kHz, which is an extremely high sampling rate for control applications but common in commercial DVD-ROM drives. The powerful computing capabilities of the DSP make it possible to implement the digital servo algorithm having such a high sampling rate. The controller designed in continuous-time domain was transformed to a discrete-time controller by the polezero-mapping method, which generates a reasonable result with a sufficiently high sampling frequency compared with control bandwidth. In the experiment, the repetitive controller was turned on at 0.09 s. Fig. 8 shows the tracking error and the control input of the fine actuator P(s) before and after the application of the repetitive controller. Although the results are slightly affected by the measurement noise, the effect of a repetitive controller is evident in the results. After the repetitive controller is turned on, the control input is increased. Therefore, the external disturbance of the disk rotational frequency 40 Hz is almost completely attenuated by the repetitive controller. As shown in Fig. 9, in the steady state, it is confirmed that the tracking error is apparently reduced by the repetitive controller. The repetitive controller enabled track-following servo system to reduce the tracking error to a value below the boundary (+0.01 mm), resulting in more reliable reading/writing of data from/to the optical disk. The improved performance is clearly illustrated by the fast Fourier transform (FFT) results shown in Fig. 10. The repetitive controller reduced the tracking error remarkably at 40 Hz, which is the frequency of disk rotation and its multiples, which leads to
Fig. 5 Open-loop gains and phases for the 16 extremely perturbed compensated plants, G(s), without and with the repetitive controller, Crp(s) when vc ¼ 13204.7 rad/s 328
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
Fig. 6 Open-loop gains and phases for the 16 extremely perturbed compensated plants, G(s), without and with the repetitive controller Crp(s) when vc ¼ 2640.9 rad/s
the improved tracking accuracy. However, because the bandwidth of q(s) is restricted to 420 Hz, high-order harmonics of tracking error is hardly decreased. 5
Concluding remarks
Fig. 7 Experimental environment
This paper considered the problem of repetitive controller design for an uncertain track-following servo in optical disk drives with a high-speed rotation mechanism. For this purpose, we first introduced the track-following servo mechanism in optical disk drives. The closed-loop system including the compensated plant and the repetitive controller was modelled as a linear time-delay system with norm-bounded uncertainties. A sufficient condition for robust stability was obtained, which is represented as an ARI or an LMI. It was shown that the repetitive controller
Fig. 8 Transient response of the track-following servo system with repetitive controller
Fig. 9 Steady-state tracking errors without and with repetitive controller
a Tracking error e(t) b Control input of the fine actuator P(s)
a Tracking error without repetitive controller b Tracking error with repetitive controller
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006
329
Fig. 10 FFT results of tracking errors without (dash line)/with (solid line) repetitive controller
design problem can be reformulated as an optimisation problem to minimise the inverse of cutoff frequency of the low-pass filter q(s). Compared with other methods, the proposed design technique can immediately find the maximum cutoff frequency of q(s) by solving the LMI optimisation problem in spite of parametric uncertainties. Experimental results were presented to validate the effectiveness. 6
References
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