An Indirect Adaptive Approach to Reject Multiple Narrow-Band Disturbances in Hard Disk Drives

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1 An Indirect Adaptive Approach to Reject Multiple NarrowBand Disturbances in Hard Disk Drives Xu Chen Masayoshi Tomiuka Department of Mechanical Engineering, University of California, Berkeley, CA, 9472, USA s: Abstract: This paper presents an indirect adaptive control scheme that rejects unknown multiple narrowband disturbances in hard disk drive systems. The proposed algorithm first finds the model of the disturbance the internal model) and then adaptively estimates its parameters. The design of a bandpass filter with multiple narrow passbands is then presented and used to construct a disturbance observer DOB) for disturbance rejection. The proposed algorithm estimates the minimal amount of parameters, and is computationally simple. Evaluation of the proposed algorithm is performed on a benchmark problem for HDD track following. Keywords: Adaptive control, disturbance rejection, multiple narrowband disturbances, HDD 1. INTRODUCTION In track following control of hard disk drives HDDs), both the repeatable runout RRO) and the nonrepeatable runout NRRO) contribute to Track MisRegistration TMR). RRO is synchronous with the HDD spindle rotation, and can be compensated by customied control algorithms such as adaptive feedforward cancellation or repetitive control Sacks et al. 1995)). NRRO, however, differs from track to track, and can appear at frequencies higher than the servo bandwidth Ehrlich and Curran 1999)). Among the various components in NRRO, disk motion, such as disk fluttering due to turbulent air flow in the hard disk assembly, is the major contributor, and arises as multiple narrowband disturbances 1 Guo and Chen 2); Ehrlich and Curran 1999); McAllister 1996)). With the rapid growth in HDD s storage density, the adverse influence of disk motion on the servo performance is becoming more and more an important issue. Rejection of multiple narrowband disturbances is thus the key to achieve low TMR in track following. Investigations of this important problem have been popular in the field of control theory. The existing solutions have mainly been rooted in rejecting disturbance with one narrowband component. For example, Zheng and Tomiuka 27, 28) suggested direct and indirect adaptive disturbance observer DOB) schemes to estimate and cancel the disturbance; Kim et al. 25) proposed a parallel addon peak filter to shape the open loop frequency response; Landau et al. 25) achieved adaptive narrowband disturbance rejection on an active suspension, based on Youla parametriation. Yet, the problem of multiple narrowband disturbance rejection was seldom examined This work was supported by the Computer Mechanics Laboratory CML) in the Department of Mechanical Engineering, University of California at Berkeley. 1 Disturbances whose energy is concentrated at several frequencies. before. Landau et al. 25) s algorithm can be extended to reject n narrow bands, but requires the estimation of 2n parameters. This paper focuses on developing an adaptive control algorithm that rejects arbitrary number of unknown narrowband disturbances in NRRO. The model of the disturbance, i.e., its internal model, is firstly derived. A new adaptive frequency identification method is then proposed to estimate the parameters of this model, which are then applied to construct a bandpass Qfilter with multiple narrow passbands. Finally, expanding the DOB structure in Zheng and Tomiuka 28) to multiple narrowband disturbance rejection, we form a disturbance observer with the newly designed Qfilter. Advantages of the proposed compensation scheme are: 1) it estimates the minimal number of parameters, which is equal to n, the number of narrowband components; 2) it is stable over a wide range of frequencies, disturbances outside the servo bandwidth can also be compensated; 3) it has fast convergence rate, and is easy to implement. The remainder of this paper is organied as follows. Section 2 formally defines the problem and introduces the proposed solution. Section 3 presents the proposed adaptive frequency identification scheme. The design of DOB with a multiple narrow bandpass Qfilter is shown in Section 4. An example of rejecting two narrowband disturbances is provided in Section 5. Section 6 concludes the paper. 2. THE PROBLEM AND THE PROPOSED SOLUTION Figure 1 shows the proposed block diagram for HDD track following. It reduces to the baseline feedback control loop if we remove the addon compensator inside the dashdotted box. Throughout the paper we use the well formulated opensource HDD benchmark simulation package Hirata

2 27)) as a demonstration tool. The fullorder plant model G p ) contains the dynamics of the HDD servo system including the power amplifier, the voicecoil motor, and the actuator mechanics. The dashed line in Fig. 2 shows the frequency response of G p ), which is a fourteenthorder transfer function with several high frequency resonances. The baseline feedback controller C F B ) is a third order PID controller cascaded with three notch filters. The baseline open loop system has a gain margin of 5.45 db, a phase margin of 38.2 deg, and an open loop servo bandwidth of 1.19 kh. The reference r is ero in track following control. The signals d k), u k), nk), and P ES, are respectively the input disturbance, the control input, the output disturbance, and the position error signal. It is assumed that the multiple narrowband disturbance of interest is contained in d k), and lies between 3 H and 2 H Guo and Chen 2); Ehrlich and Curran 1999)). r P ES C F B 1 ) DOB ck) 1 Q ) IMP based frequency estimation Addon Compensator k) uk) m ^dk) Band pass Filter dk) G 1 ) P 1 1 G ) n Fig. 1. Structure of the proposed control scheme Phase deg) Magnitude db) G p m G n Bode Diagram Frequency H) nk) Fig. 2. Frequency response of G p ) and m G n ) Figure 3 shows the spectrum of the position error signal on one track when the baseline controller is applied. It is observed that several sharp spikes are present due to the multiple narrowband disturbances, which we aim to reject. The proposed solution is to add a compensator as shown in the dashdotted box in Fig. 1. Within the compensator, the loworder nominal plant model m G n ) matches the lowfrequency dynamics of G p ) in the frequency response, as shown in Fig. 2. A stable inverse model G n ) is needed in the design of our proposed compensator. If G n ) has minimum phase, its inverse can directly be assigned, if not, stable inversion techniques such as the ZPET method Tomiuka 1987)) should be applied. Magnitude Frequency H) Fig. 3. PES spectrum with baseline controller The compensation signal c k) is designed, by constructing the DOB, to approximate and cancel the multiple narrowband disturbances. To see this point, notice first that the signal ˆd k) is expressed by, in the operator notation, ˆd k) G n ) [ Gp ) u k) d k)) n k) ] m u k). 1) Since below 2 H, G p ) m G n ), i.e., G n ) Gp ) m, Eq. 1) becomes ˆd k) m d k) G n ) n k). 2) If in addition the output disturbance n k) is small, then the above equation is further simplified to ˆd k) m d k) d k m), 3) which implies that ˆd k) is a good estimate of the disturbance d k). Therefore, the multiple narrowband disturbance is contained in ˆd k). In reality, the influence of n k) can not be ignored. A bandpass filter BP ) is constructed to filter out the signals in ˆd k) that are not of our interest. This is practical since the frequency region of the narrowband disturbances is usually roughly known. The filtered signal k) is finally a multiple narrowband signal 2 with small noisetosignal ratio, and can be applied for the parameter estimation scheme to be presented in Section 3. With the estimated knowledge of the multiple narrowband disturbance, a multiple bandpass filter Q ) can then be constructed. The compensation signal c k) formed by filtering ˆd k) through Q ), therefore contains only the multiple narrowband disturbance. Adding the negative of c k) to the control input, we achieve the compensation. 3. ADAPTIVE DISTURBANCE IDENTIFICATION 3.1 The Internal Model and the Adaptation Algorithm The multiple narrowband disturbance in NRRO can be modeled as the sum of several sinusoidal signals Ehrlich and Curran 1999); Guo and Chen 2)). It is well known that any sinusoidal signal x k) satisfies 2 More precisely, k) and the multiple narrowband disturbance in ˆd k) have the same amplitude but different phases.

3 1 2 cos ω) 2) x k), where ω 2πΩT s is the frequency of x k) in radians 3. The equality can either be verified by direct expansion or by noting that the eros of the FIR filter 1 2 cos ω) 2 lie exactly at e ±jω on the unit circle. The term 1/ 1 2 cos ω) 2) is named as the internal model of x k). Extending the idea in the last paragraph, we can now develop the internal model of multiple narrowband disturbances. Assume that the signal k) contains n narrowband components. k 1) will then satisfy n 1 2 cos ωi ) 2) k 1), 4) where ω i i 1,..., n) is the frequency of the i th narrowband component in k). The polynomial on the left hand side of Eq. 4) is A ) n 1 2 cos ωi ) 2) 1 a 1 a n n 5) a 1 2n1 2n n 1 a i i 2ni) a n n 2n. The values of ω i s are unknown, a is are thus unknown, and need to be estimated for constructing A ). Choosing to directly estimate a is makes the adaptation simple in computation, since A ) is linear in a is. Notice that the coefficients of A ) have a mirror symmetric form. Therefore only n parameters need to be identified, which is the minimal possible number for n narrowband signals. To construct an adaptive estimation scheme, we substitute and expand Eq. 5) to Eq. 4), then move the terms containing k), k 1),..., k 1 2n) from the left side to the right side, to get the adaptation model: n k 1) a i [ k 1 i) k 1 2n i)] a n k 1 n) k 1 2n). 6) Introduce the parameter vector to be estimated: θ [a 1, a 2,..., a n ] T. 7) Introduce also the regressor vector at time k: φ k) [φ 1 k), φ 2 k),..., φ n k)] T, 8) where φ j k) k 1 j) k 1 2n j) 9) j 1,..., n 1 φ n k) k 1 n) 1) Eq. 6) can then be simply represented by k 1) φ k) T θ k 1 2n). 11) We can now define the a priori prediction of k 1): ẑ o k 1) φ k) T ˆθ k) k 1 2n), 12) where ˆθ k) is the predicted parameter vector at time k. 3 Ω is the frequency in H, T s is the sampling time in seconds. The a priori prediction error is then given by e o k 1) k 1) ẑ o k 1) φ k) T θ k), 13) where θ k) ˆθ k) θ is the parameter estimation error. Correspondingly, we define the following a posteriori signals for later use in the stability analysis: the a posteriori prediction of k 1): ẑ k 1) φ k) T ˆθ k 1) k 1 2n). 14) the a posteriori prediction error: e k 1) φ k) T θ k 1). 15) With the above information, the following recursive least squares RLS) parameter adaptation algorithm PAA) can be constructed Landau et al. 1998)). ˆθ k 1) ˆθ k) F k) φ k) eo k 1) 1 φ k) T 16) F k) φ k) e o k 1) k 1) ẑ o k 1) 17) ẑ o k 1) φ k) T ˆθ k) k 1 2n) 18) [ ] F k 1) 1 F k) F k) φ k) φ k)t F k) λ k) λ k) φ k) T F k) φ k) 19) To improve the convergence rate, the forgetting factor λ k) is designed to increase from.95 to 1 Ljung 1999)), obeying the rule λ k) k. As an example of the adaptation algorithm, when n 2, A ) 1 2 cos ω 1 ) 2) 1 2 cos ω2 ) 2). 2) Expanding Eq. 2) and introducing a 1 2 cos ω 1 ) 2 cos ω 2 ) ; a cos ω 1 ) 2 cos ω 2 ), we obtain A ) 1 a 1 a 2 2 a ) The unknown parameter vector is thus θ [a 1, a 2 ] T, and where k 1) φ k) T θ k 3) 22) ẑ o k 1) φ k) T ˆθ k) k 3) 23) e o k 1) k 1) ẑ o k 1) 24) φ k) [ ] k) k 2). 25) k 1) θ can then be estimated according to Eqs. 1619). 3.2 Stability and Convergence For stability analysis, we first transform the PAA to the a posteriori form. Premultipling φ T k) to Eq. 16) yields φ T k) ˆθ k 1) φ T k) ˆθ k) φt k) F k) φ k) 1 φ T k) F k) φ k) eo k 1). 26) Subtracting φ T k) θ from each side in Eq 26), and substituting in Eqs. 15) and 17), we have e o k 1) e k 1) 1 φ T k) F k) φ k). 27)

4 Substituting Eq. 27) back to Eq. 16), we arrive at the PAA in the a posteriori form: ˆθ k 1) ˆθ k) F k) φ k) e k 1) 28) e k 1) φ k) T θ k 1) 29) Subtracting θ from each side in Eq. 28) yields θ k 1) θ k) F k) φ k) e k 1). 3) Combining Eqs. 27) and 3), we can construct the equivalent feedback loop for the adaptive system as shown in Fig. 4. w k) 1 θ k 1) Fk) e k 1) φ T k) 1/2 1/2 e k 1) L φ k) NL s k) Fig. 4. Equivalent feedback loop of the adaptive system The nonlinear block NL in Fig. 4 is shown to be passive and satisfies the Popov Inequality section of Landau et al. 1998)). The linear block L 1 1/2 is strictly positive real. Therefore, the parameter adaptation algorithm is asymptotically hyperstable. Applying further theorem from Landau et al. 1998), we have lim k Substituting Eq. 29) to the above gives φ k 1) T θ k) n e k). 31) k i) k 2n i)) ã i k) k n) ã n k) n ) i 2ni) ã i k) n ã n k) k) as k. 32) Based on the assumption that k) has n independent frequency components, the Frequency Richness Condition for Parameter Convergence holds. Therefore, the only solution to the above equation is lim k ã i k), i.e., the parameters converge to their true values. 4. MULTIPLE BANDPASS QFILTER DESIGN With the estimated parameters a is, we are ready to design the Qfilter and turn on the adaptive DOB for the disturbance compensation. The Qfilter used in single narrowband disturbance rejection Zheng and Tomiuka 28)) is given by Q ) 1 α) 1 α 2) 1 α 2 cos ω) α 2, 33) 2 where the shaping coefficient α is a real number close to but smaller than 1. The above Qfilter has two poles close to e ±jω but slightly shifted towards the origin. The magnitude response of Q ) has a narrow passband centered at ω. The closer α is to 1, the narrower the passband of Q ). For multiple narrowband disturbance rejection, we extend Eq. 33) to Q ) n 1 α i ) 1 α i 2) 1 α i 2 cos ω i ) αi 2. 34) 2 For simplicity, we let α i α.998. Recall the definition of A ): A ) n 1 2 cos ωi ) 2) 1 a 1 a n n a 1 2n1 2n. 35) Eq. 34) can then be expressed as Q ) 1 α) 1 α 2) B ) Q n 1 α 2 cos ω i) α 2 2 ) 1 α) 1 α 2) B ) Q A α. 36) ) where A α ) is obtained by replacing every by α in Eq. 35), and B ) Q is a polynomial of. 4.1 The Case of two narrowband disturbances When n 2, direct expansion in Eq 34) gives Q ) 1 α) 1 α 2) 2 αa 1 2α 2 2) 1 αa 1 α 2 a 2 2 α 3 a 1 3 α 4 4, 37) where a 1 2 cos ω 1 ) 2 cos ω 2 ) and a 2 22 cos ω 1 ) 2 cos ω 2 ). Notice that α, a 1 and a 2 completely determine Q ). With the estimated â 1 and â 2 in section 3, the Qfilter can then be constructed according to Eq. 37), which has a frequency response as shown in Fig. 5. Notice that at the central frequencies, the magnitude and the phase of Q ) are 1 db) and deg, respectively. Therefore, passing a broad band disturbance ˆd k) through Q ), one gets the exact multiple narrowband signals at 5 H and 12 H. Magnitude db) Phase deg) Frequence Response of Qfilter Frequency H) Fig. 5. Frequency response of the proposed Qfilter central frequencies: 5 H and 12 H)

5 r P ES CF B 1 ) uk) dk) G 1 ) P nk) 8 6 Bode Diagram DOB ck) 1 Q ) ^dk) m 1 1 G ) n Fig. 6. Block diagram of the closed loop system with the proposed multiple narrowband DOB The error rejection function S ) a.k.a. the sensitivity function), is the transfer function from the output disturbance n k) to the position error signal P ES in Fig. 6. When the DOB is turned on, S ) can be derived as S ) 1 1 C eq ) G p ), 38) where C eq ) G ) F B Q ) G ) n 1 m Q 39) ) is the equivalent feedback controller. Figure 7 shows the frequency response of the sensitivity function for the closed loop system with the proposed DOB. With the addon compensation scheme, PES at 5 H and 12 H gets greatly attenuated due to the deep notches in the magnitude response at the corresponding frequencies, while the influence on the sensitivity at other frequencies is neglectable. Magnitude db) Phase deg) Frequency response: error rejection function/sensitivity function Frequency H) w ith DOB w ithout DOB Fig. 7. Frequency response of the sensitivity function Stability of DOB see Kempf and Kobayashi 1999)) requires the nominal model m G ) n to have no eros outside the unit circle and that Q e jω ) 1 < e jω ω, 4) ) where ) [ G p ) m G n ) ] / m G n ) represents the multiplicative model mismatch. Plotting the magnitude responses of 1/ ) and Q ) in Fig. 8, we see that the multiple narrowband DOB is stable as long as the narrowband disturbance arises below 3 H. 4.2 The Case of n narrowband disturbances For the general case of n narrowband disturbances Q ) 1 α) 1 α 2) B ) Q A α ) 41) Magnitude db) / ) Q) Frequency H) Fig. 8. Magnitude responses of 1/ ) and Q ) where A α ) 1 a 1 α a n α n n a 1 α 2n 2n1 α 2n 2n. Derivation of the B Q ) is best done by using a Computer Algebra System such as Maple or Mathematica. We have, for n 3, B Q ) 3 2αa 1 α 2 a 2 3) 2 For n 4, 2α 3 a 1 3 3α ) B Q ) 4 3αa 1 α 2 2a 2 4) 2 α 3 a 3 3a 1 ) 3 α 4 2a 2 4) 4 3α 5 a 1 5 4α ) By induction, we can get the general form of B Q ) : B Q ) n 2 b i α i i α 2n) i 2n)i) i b n α n n1, 44) where b n; b 1 n 1) a 1 ; b i n i) a i b i 2 ; i 2,... n SIMULATION RESULT The proposed adaptive compensator for multiple narrowband disturbance rejection is implemented in the HDD benchmark simulation package Hirata 27)). The baseline control system is as shown in Section 2. The disturbances include the torque disturbance, the disk flutter disturbance, the RRO, and the measurement noise. The system has a sampling time of sec. Two narrowband disturbances at 5 H and 12 H were injected at the input to the plant. In the simulated track following, the first five revolutions were run without compensation. It is seen in Fig. 9 that the peak values of PES exceeded the standard PES upperbound of 15% Track Pitch TP). The dotted line in Fig. 1 presents the spectrum of the PES without compensation. We can see that the PES had strong energy components at 5 H and 12 H. Without compensation, the Track MisRegistration TMR), defined as 3 times the standard deviation of the PES, was 21.87% TP. The proposed algorithm was applied to improve the HDD track following performance. The multiple bandpass filter

6 PES %TP) Compensation starts Revolution Estimated parameters Revolution Fig. 9. PES time trace Magnitude PES Amplitude Spectrum w/ compensation 3σ %TP w/o compensation 3σ %TP Frequency H) Fig. 1. PES spectrum with and without the proposed compensator BP ) was designed using the Signal Processing Toolbox in MATLAB. BP ) has a magnitude response as shown in Fig. 11. The estimation of the parameters was turned on at the beginning of the simulation. The initial guess of the parameter vector was set to half of its true value. Figure 12 shows the estimated parameters â 1 and â 2 converged to their true values within half a revolution, i.e.,.415 sec. With the estimated parameters â 1 and â 2, the Qfilter was constructed and turned on at the fifth revolution. Figure 9 shows the resulting PES time trace. It is seen that the PES was reduced now to less than 1% TP. In Fig. 1, we observe that the strong energy concentrations at 5 H and 12 H were greatly attenuated, while the spectrum of the PES at other frequencies was almost identical to that without compensation. The TMR was reduced to 11.86% TP, implying a 45.8% improvement. Magnitude db) Frequency kh) Fig. 11. Magnitude response of the multiple bandpass filter 6. CONCLUSION In this paper, an indirect adaptive control scheme was proposed to reject multiple narrowband disturbances in HDD track following. Simulation on a realistic opensource HDD Fig. 12. Online parameter estimation of the internal model for two narrowband signals benchmark problem showed that the proposed algorithm significantly reduced PES and TMR. The proposd method is suitable for compensating disturbances within narrow frequency regions. REFERENCES Ehrlich, R. and Curran, D. 1999). Major HDD TMR sources and projected scaling with tpi. IEEE Transactions on Magnetics, 352), Guo, L. and Chen, Y. 2). Disk flutter and its impact on hdd servo performance. In Proceedings of 2 Asia Pacific Magnetic Recording Conference, TA2/1 TA2/2. Hirata, M. 27). NSS benchmark problem of hard disk drive system. Kempf, C. and Kobayashi, S. 1999). Disturbance observer and feedforward design for a highspeed directdrive positioning table. IEEE Transactions on Control Systems Technology, 75), Kim, Y., Kang, C., and Tomiuka, M. 25). Adaptive and optimal rejection of nonrepeatable disturbance in hard disk drives. In Proceedings of 25 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, volume 1, 1 6. Landau, I.D., Loano, R., and M Saad, M. 1998). Adaptive Control. SpringerVerlag New York, Inc. Landau, I.D., Constantinescu, A., and Rey, D. 25). Adaptive narrow band disturbance rejection applied to an active suspension an internal model principle approach. Automatica, 414), Ljung, L. 1999). System Identification: Theory for the User. Prentice Hall PTR, 2 edition. McAllister, J. 1996). The effect of disk platter resonances on track misregistration in 3.5 inch disk drives. IEEE Transactions on Magnetics, 323), Sacks, A., Bodson, M., and Messner, W. 1995). Advanced methods for repeatable runout compensation disc drives. IEEE Transactions on Magnetics, 312), Tomiuka, M. 1987). Zero phase error tracking algorithm for digital control. Journal of Dynamic Systems, Measurement, and Control, 191), Zheng, Q. and Tomiuka, M. 27). Compensation of dominant frequency components of nonrepeatable disturbance in hard disk drives. IEEE Transactions on Magnetics, 439), Zheng, Q. and Tomiuka, M. 28). A disturbance observer approach to detecting and rejecting narrowband disturbances in hard disk drives. In Proceedings of 28 IEEE International Workshop on Advanced Motion Control,