RRO Compensation of Hard Disk Drives with RPTC Considering Correlation of Adjacent Tracks

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1 SICE Annual Conference 28 Aug. 2-22, 28, Univ. of Elector-Communications, Japan Compensation of Hard Disk Drives with RPTC Considering Correlation of Adjacent Tracks Hiroaki Nishina 1 and Hiroshi Fujimoto 2 Department of Electrical and Computer Engineering, Yokohama National University, Yokohama, Japan 1 (Tel : ; nishina@hfl.dnj.ynu.ac.jp) 2 (Tel : ; hfuji@ynu.ac.jp) Abstract: This paper presents an improvement method of repetitive controller based on perfect tracking control (PTC) in order to reject variance repeatable runout () of hard disk drive. Authors group proposed the repetitive PTC (RPTC) with switching mechanism. RPTC is constructed by using periodic signal generator (PSG) and PTC. The PSG works to make feedforward signal from the periodic disturbance and the PTC generates control input to cancel the periodic error in steady-state. However, we have not considered the difference of between tracks. This paper proposes a re-learning scheme considering correlation of adjacent tracks. Finally, the advantages of RPTC using proposed scheme are demonstrated through simulations and experimented at HDD equipment. Keywords: multirate control, repetitive control, hard disk drive, adjacent track 1. INTRODUCTION Head-positioning controllers of hard disk drives (HDDs) are generally composed of two control modes; the track-seeking mode and the track-following mode. The feedforward performance is important in the trackseeking mode, and the disturbance rejection performance is important in the track-following mode. Digital twodegree-of-freedom controllers generally have two samplers for the reference signal r(t) and the output, and one holder on the input u(t) as shown in Fig.1. Therefore, there exist three time periods T r, T y, and T u which represent the period of r(t),, and u(t), respectively. The input period T u is generally decided by the speed of the actuator, D/A converter, or the calculation on the CPU. Moreover, the output period T y is also determined by the speed of the sensor or the A/D converter. In the case of HDDs, the position error is detected by the discrete servo signals embedded in the disks. Therefore, the output sampling period T y is decided by the number of these signals and the rotation frequency of the spindle motor. However, it is possible to set the control period T u shorter than T y because of the recent development CPU. Thus, the controller can be regarded as the multirate control system which have the hardware restriction of T u <T y. Then, many multirate controllers have been proposed both for track-seeking and track-following modes [1][2][3]. Repetitive control is a widely used technique to reject periodic disturbances or to track a periodic reference signal[4]. Although this control scheme has excellent performance for low order disturbance modes, it cannot reject relatively higher frequency modes. Authers group proposed repetitive controller based on perfect tracking control (PTC) in order to reject high-order repeatable runout () with periodic signal generator (PSG) [5][6]. There are two approaches to design repetitive controller. One is the feedback approach of the repetitive PTC which has internal model of periodic disturbance [7]. The other is the feedforward approach with switching mechanism such that the high-order can be rejected without any sacrifice of the closed-loop characteristics. In both approaches, multirate feedforward control is utilized to overcome the unstable zero problem of discrete-time plant. PSG works to make the compensate signal from the periodic disturbance of the following track.to generate the compensate signal, PSG needs to average position error signal (PES) at the each sector to reject non repeatable runout (). However, the of one track is different from that of another track. When the track is changed, the compensatione signal generated on one track would be degrade the following performance on the other track. However it is not realistic to wait several rotations of disk to make the averaged signal at every track. In this paper, we propose a re-learning scheme considering correlation of adjacent tracks. The correlation means that the periodic disturbance has similarity in near adjacent tracks. After the compensation of RPTC in the current track, PSG learns the residual error to generate the compensate signal for adjacent tracks. Because the proposed re-learning scheme does not need to stop feedforward compensation, the following performance is not worsen. 2. REPETITIVE PERFECT TRACKING CONTROL(RPTC) 2.1 PTC [5] In this paper, it is assumed that the control input can be changed N times during the sampling period of output signal T y. For simplification, the input multiplicity N is set to be equal with the order of nominal plant n since N n is the necessary condition of perfect tracking. But, by using the formulation of [5], this assumption can be relaxed to deal with more general system with N n. Consider the continuous-time nth-order plant described by ẋ(t) =A c x(t)+b c u(t), =c c x(t). (1)

2 Fig. 1 Multirate hold. The discrete-time state equation discretized by the shorter period T u becomes x[k +1]=A s x[k]+b s u[k], (2) where x[k] =x(kt u ) and A s := e AcTu, b s := Tu e Acτ b c dτ. (3) By calculating the state transition from t=it y =kt u to t=(i +1)T y =(k + n)t u in Fig.1, the discrete-time plant P [z] can be represented by x[i +1]=Ax[i]+Bu[i],y[i] =cx[i], (4) where x[i] =x(it y ), z := e sty and multirate input vector u is defined in the lifting form as u[i] := [u 1 [i],,u n [i]] T = [u(kt u ),,u((k + n 1)T u )] T, (5) and the coefficients are given by A = A n s, B =[A n 1 s b s, A n 2 s b s,, A s b s, b s ], (6) c = c c. From (4), the transfer function from x[i +1] R n to the multirate input u[i] R n can be derived as u[i] = B 1 ( I z 1 A ) x[i +1] (7) [ ] O A = x[i +1]. (8) B 1 B 1 From the definition in (6), the nonsingularity of matrix B is assured for a controllable plant. Moreover, all poles of the transfer function (7) are zero from (8). Hence, (7) is a stable inverse system. Then, if the control input is calculated by (9) as shown in Fig.2, perfect tracking is guaranteed at sampling points for the nominal system because (9) is the exact inverse plant. u [i] =B 1 (I z 1 A)r[i] (9) Here, r[i](x d [i + 1]) is previewed desired trajectory of plant state. The nominal output can be calculated as y [i] =cx d [i] =z 1 cr[i]. (1) When the tracking error y[i] y [i] is caused by disturbance or modeling error, it can be attenuated by the robust feedback controller C 2 [z] Fig. 2 Repetitive perfect tracking controller. Fig. 3 Periodic signal generator for 2nd order system. 2.2 RPTC [6] The RPTC is based on perfect tracking control with periodic signal generator (PSG). Because the PSG can be constructed by the series of memories z 1, the computation cost is very low. First, the PTC is designed using multirate feedforward control as minor-loop system to obtain the ideal command response. The measured output y[i] is assumed to have the output disturbance d[i] as y[i] =p[i] d[i] :=cx[i] d[i], (11) where p[i] is the plant output. In this section, the disturbance is assumed to be repetitive signal with period T d. When the tracking error y[i] y [i] is caused by unmodeled disturbance or modeling error, it can be attenuated by the robust feedback controller C 2 [z] as shown in Fig.2. Second, the periodic signal generator is designed to generate desired trajectory r[i]. Because perfect tracking (x[i] =x d [i] or x[i] =z 1 r[i]) is assured, the minorloop nominal system is expressed as y[i] =z 1 r[i] d s [i], r[i] :=cr[i], (12) where d 2 [i] := (1 P [z]c 2 [z]) 1 d[i] and P [z] is the single-rate plant with T y if the minor-loop feedback controller C 2 [z] is a single-rate system. In the RPTC, two schemes can be considered: the feedback and feedforward approaches. In this paper, the feedforward approach is utilized, which has switching shecme. The PSG is can be designed as the outer-loop controller by r[i] = z z N y[i], (13) d 1 where the integer N d is defined as T d /T y. From (12) and (13), the total closed-loop system is represented by r[i] = zn d 1 z N d s [i]. (14) d Therefore, the repetitive disturbance which is modeled as d[i] =(z N d 1) 1 is completely rejected at every

3 sampling point in steady-state. In (12), there exists redundancy to decide r[i] from the PSG output r[i] since we have freedom to select the state variabel x. In order to make the multirate input smooth, it should be given as the derivative from x = [p, ṗ, p, ]. Fig.3 shows one example of the 2nd order plant with x = [p, ṗ], in which the velocity command is generated by ṗ d [i] = (p d [i +1] p d [i 1])/2T y. In the steady-state, the switch turns on dureing one disturbance period T d and PSG store the output. Here, the PTC generates control input u [i] to cancel the periodic steady error. Thus, the plant output p[i] perfectly tracks the periodic disturbance d[i] and the tracking error becomes zero at every sampling point (y[i] =). Since the switch turns on just N d sampling time, the N d memories work as complete feedforward compensator. RPTC without the switching action suppresses harmonic components as well as conventional method based on the internal model principle. 2.3 Averaging runout considered adjacent tracks The periodic disturbance in the HDDs appears as the synchronization with the spindle rotation speed. This is caused by the eccentric disk, the torque ripple of the spindle motor and each sector position noise. Moreover, when the servo information is written out of synchronization of the axis of the disk rotation, relative position error that synchronizes with the disk rotation is generated. The periodic disturbance of each track is different. Especially, the of inner disk is completely different from that of outer disk. The PSG needs to memorize the periodic disturbances of each track. Therefore, when using the periodic disturbance in the previous track, the compensation signal would be different from the periodic disturbance of that track. Here, this paper proposes re-learning scheme with the weighting factor. At the first, it is assumed that PTC compensation is not applied. From (12) and r[i] =, position error signal y[i] is calculated as y[i] = d s [i], where i means the sector number. It is assumed that y [i] is calculated with the average of y[i] at the each sector, where the suffix of y [i] is incremented at every re-learning. It is assumed that the average times is N. The first compensation signal r [i] is calculated as r [i] =zy [i] =zs[z]d [i], and d [i] is the of the start track. At the second, we consider that the head position move to another track. The next position error signal y 1 [i] is calculate as y 1 [i] = z 1 r 1 [i] S[z](d [i]+ d 1 [i]) = z 1 zs[z]d [i] S[z](d [i]+ d 1 [i]) = S[z] d 1 [i], (15) where d 1 [i] =d [i] + d 1 [i] and d 1 [i] is the variation of track runout from d [i]. Finally, we consider to generate the next compensation signal. The d 1 [i] includes not only signal but also signal which should be eliminated to make compensation signal. We consider the averaging times of several position error signal. Considering y [i] is calculated with the average of N times and y 1 [i] is one period, r 2 [i] is calculated as ( ) N r 2 [i] = zs[z] N +1 d 1 [i]+ N +1 d 1[i] = zs[z](d [i]+ 1 d1 [i]) N ( = z y [i]+ 1 ) y 1 [i] N (16) (16) means that the residual position error is added to r 1 [i] with weight 1/(N +1). (16) is a simple method to re-learn the residual position error. By generalizing (16), the control law r 2 [i] can be rewritten as r 2 [i] =r 1 [1] + zαy 1 [i], (17) where α is a free parameter to give higher weight to new data. The weighting factor α should be chosen as 1 N +1 α N N +1. (18) Furthermore, by using Q filter [4], can be further reduced. When cut-off frequency of Q filter is chosen as the cut-off frequency of the feedback controller, is not influenced so much. 3. APPLICATIONS TO REFECTION IN HDD 3.1 Control system design and simulations In the track-following mode of HDD, two kinds of disturbance is injected at the plant output should be considered; repeatable runout () and non-repeatable runout (). While is synchronous with the disk rotation, is not synchronous. Although there are many techniques to reject the in low frequency region, the high frequency is hard to reject by conventional technologies. However, the effect of high-order cannot be neglected since the required servo accuracy is getting drastically severe. Therefore, this paper applies the proposed multirate repetitive controllers FF- RPTC. The plant is a 2.5-in HDD with 12 nm track pitch. The sampling period of this drive is T y =43.5μsand the control input is changed N =2times during this period. In the design of controller, the nominal plant P n (s) is modeled as double integrator system as shown P n (s) = kp ms 2 e Ls, (19) [ 1 e T ys ] P n [z] = Z P n (s), (2) s y [k] = P n [z]u [k], (21) where e Ls (L = 3μs) is the time-delay in order to reduce the modeling error due to the calculation delay and power amplifier. When time-delay is considered, PSG need to fix the memory with ahead same samples of the time-delay of plant. The the high-order detail model P (s)

4 Table 1 Parameters of resonance modes. l f l [Hz] ζ l A l S[z] T[z] (a) Nominal model (b) Detail model Fig. 5 1st mode distrubance. Gain[dB] Freq[Hz] Fig. 4 Sensitivity functions S[z], T [z] that is considered rigid mode and lth resonance modes as ( ) P (s) = k p 1 l m s 2 + A i s 2 +2ζ i=1 i ω i s + ωi 2 e Ls = k p b k s k 1 + b k 1 s k b 2 s + b 1 m s k + a k s k 1 e Ls. (22) + + a 2 s + a 1 The parameter of the resonance mode shows Table.1. The rotation frequency of spindle motor is 9 Hz and the number of sector is N d = 256. The minor loop FB controller C 2 [z] is designed by the PID controller with 1 khz crossover frequency. When the resonance mode is considered, notch filter is used to suppress ω n =2π52 rad/sec resonance mode. The notch filter is given as F (s) = s2 +2ζ n ω n s + ωn 2 s 2 +2ζ d ω n s + ωn 2, (23) where ζ n =.1,ζ d =.1. The notch filter was discretized by prewarp tustin transformation, in which the sampling time is T u = T y /2. To remove the noise caused by resonance mode or to reduce effect, Q filter is used on the output of PSG in Fig.2. The Q filter is given as [ ] z + γ + z 1 Nq r f [k] = r[k], (24) γ +2 where r[k] is output of PSG, r f [k] is output of Q filter. This filter is a low-pass filter with zero phase-delay, and the N q sample ahead value is needed as (24). Moreover, the smaller γ 2 and the high-order N q have bigger roll-off order while the disturbance rejection performance becomes poorer. In the detail model simulation,chosen parameter as γ =2, N q =8and then cut-off frequency becomes 2kHz. Fig.4 shows the sensitivity S[z] and complementary sensitivity T [z] of the closed-loop. The injected disturbance signal is calculated form the approximate inverse of sensitivity function and position error signal (PES) obtained from experiments. The simulations use the only signal which is extracted from experimental data by the averaging operation of total PES (a) Nominal model (b) Detail model Fig. 6 1st 4th mode distrubance Fig. 7 FFT of the Fig.6(a) Fig. 8 FFT of the Fig.6(b). Fig.5(a) Fig.8 shows the simulation results. Fig.5(a) and Fig.5(b) show the time response for the nominal model and the detail model. The single mode disturbance of 9 Hz sinusoidal signal is added with 1 track amplitude to output disturbance. The time origin (t =)is at the instance that the switch turns on to start compensation. And, in the Fig.5(a) the PTC works form t = T d to perfectly tracks the with zero error With the detail model P (s) using Q filter, the compensation signal became poorer than nominal model. Therefore, there exist a little residual vibration. In Fig.6(a) and Fig.6(b) from the 1st to the 4th mode disturbance of 9 Hz is added as output disturbance. Fig.7 and Fig.8 show the FFT of the Fig.6(a) and Fig.6(b) respepectively. Because the resonace modes and Q filter, there are modeling error. However, residual signal is decreased by feedback controler C 2 [z]. Next, the proposed method is simulated with the nominal model. In this simulation, we considered that the track or head change as the amplitude change of runout. The track seeking is not simulated. The averaging times N is 5 and the weighting factor α is.4. In Fig.9 and Fig.1 show the simulation results with the rariation of disturbance. The disturbance is the 1st 4th mode of spindle frequency and change the amplitude of the each 2nd 4th mode at t =. As shown in Fig.9, RPTC tracks the with zero

5 .3.2 Position Reference Table 2 ±3σ of pes siganl (track). peak filter RPTC proposed method Fig. 9 RPTC without re-learning Fig. 1 RPTC with re-learning (proposed method). error after the switch turns on. However, when the disturbance changed, residual vibration appeared. On the other hand, the proposed method in Fig.1, the residual error disappeared after the switch turns on again to regenerate the compensation signal. At the first switching action, PSG learned the mainly periodic disturbance. When the disturbance changed, the residual error appears same as conventional method. When the switch turns on in the second time, PSG learned the the residual error and weighted periodic disturbance. Then, the 2nd switch turn off, PTC made the better compensation signal, and the residual error disappeared. Moreover, because RPTC maintains the compensation during the switch is turning on, the following performance is not worser. 4. EXPERIMENTS ON RPTC In the experiments, the RPTC with re-learning (proposed method) is compared with peak filter which attenuates 9 Hz disturbance and RPTC without re-learning. The proposed method and RPTC without re-learning method do not use peak filter. Each feedback controller C 2 [z] is PID controller. C 2 [z] is the same with simulation. Notch filters are used to suppress ω 1 =2π2385 and ω 2 =2π52 rad/sec resonance modes. Because of the high-order resonance modes and the noise, the order of Q filter is set to N q =2. Then the cut-off frequency becomes 1 khz. In Experiment of RPTC, the compensation signal is made from averaging position error signal at the each sector. In the experiment, the track is not changed. However without track changing, we show that the proposed method is effectivity. The average time N is 5 and the weighting factor α is.4. The experimental results are shown Fig.11(a) Fig.13(c). PES, and signals are shown in Fig.11(a),Fig.12(a) and Fig.13(a). signal and signal are given with intentioned off-set to prevent overlap each other. As shown in Fig.11(b), Fig.12(b) and Fig.13(b) which are FFT of signal, under the cut-off frequency of Q filter are suppressed. On the other hand, the area of the drequency band in which the sensitivity function has peak or over cut-off frequency of Q filter are not suppressed. If the cut-off of Q filter are higher than the area of the drequency band in which the sensitivity function, the frequencies of the band deteriorate the PES signal. FFT of signal is shown Fig.11(c), Fig.12(c) and Fig.13(c). The proposed method is little better than the others. Moreover, the proposed method decreases some disturbance mode which could not decrease with RPTC. Table.2 shows the ±3σ for each cases. As shown in Table.2, the proposed method attenuate position error signal well. So, we can show that the proposed method can improve the track following quality. Moreover, we can easily presume that the proposed method also can improve the track following quality when the track is changed. 5. CONCLUSION In this paper, a novel re-learning scheme of repetitive controller based on PTC is proposed. In the conventional RPTC without re-learning scheme, the runout change is attenuated by feedback controller. When the runout is so different from the compensation signal, RPTC needs to learn the runout with long time averaging. In the proposed RPTC with efficient re-learning scheme considering correlation of adjacent tracks, the which has similarity with adjacent tracks can be attenuated with feedforward compensation. Because the proposed method does not need to stop feedforward compensation, the following performance is not worsen. The future works will be the experiments of track change, the expansion of feedback controller bandwidth and the effective rejection of. REFERENCES [1] K. Ohno and R. Horowitz, A multi-rate nonlinear state estimator for hard disk drives,amer. Control Conf., pp (23). [2] M. Hirata, M. Takiguchi and K. Nonami, Trackfollowing control of hard disk drives using multirate sampled-data H control, Conf. Decision Contr., pp (23). [3] L. Yang, and M. Tomizuka, Multi-rate shortseeking control of dualactuator hard disk drives for computation saving, Amer. Control Conf., pp (25). [4] K. K. Chew and M. Tomizuka, Digital contol of

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