Loop-Shaping Controller Design with the RBode Plot for Hard Disk ...

2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010

ThA08.5

Loop-Shaping Controller Design with the RBode Plot for Hard Disk Drives Takenori Atsumi and William C. Messner Abstract— The controller for the head positioning system of a hard disk drive must have both high performance and high robustness. This paper presents a user-friendly loopshaping design method employing the Robust Bode (RBode) plot for the head positioning system. By using the RBode plot, control engineers can easily design controllers that can suppress disturbances and account for perturbations of the controlled object with frequency response data alone. A transfer function-based nominal model is not required. As a result, the engineer can avoid the excessively conservative designs while accommodating perturbations in the plant and improve the controller performance. The design results for a benchmark hard disk drive problem show the usefulness of proposed method.

I. I NTRODUCTION Hard disk drives (HDD) are inexpensive mass-production products and are used in a range of environments. As a result, the head-positioning control system in HDDs must be robust perturbations of the mechanical system, due to manufacturing variations, temperature changes, aging, etc. At the same time, the head-positioning control systems require nanometer-scale positioning performance even in the presence of numerous disturbances, such as mechanical vibrations and aerodynamic forces from airflow induced by high-speed disk rotations. Therefore, the head-positioning system must have both high performance and high robustness. The problem of designing for both high performance and high robustness is the so-called mixed sensitivity problem. H∞ control theories are widely used for analyzing or synthesizing controllers of the head-positioning control systems of HDDs [1], [2], [3], [4]. These techniques are automated methods for designing controllers to achieve robust performance in the presence of plant uncertainties. However, these automated tools often do not provide much insight into the relationship between the open-loop frequency response and the performance. Also these tools require that the plant and the uncertainties and performance specifications be represented by realizable transfer functions, which sometimes are hard to construct and may result in conservative designs. On the other hand, Quantitative Feedback Theory (QFT) can represent the plant and uncertainties as frequency response data alone, but it uses the Nichols chart, for which frequency is a parametric variable, and therefore hidden [5].

The Robust Bode plot (RBode plot) was developed to provide a user-friendly way of designing controllers for robust performance without the need for transfer function representations of the plant or the performance and robustness specifications [7], [8]. The RBode plot represents the robust performance criterions as allowable and forbidden regions on the open-loop Bode plot. The RBode plots can help the designer achieve desired closed-loop performance and robustness specifications by showing on the open-loop Bode plot how a particular frequency response feature relates to the performance and robustness characteristics. The strategy for compensator design with loop-shaping using the RBode plot is to shape the open-loop response to assure that no intersections occur between the forbidden regions and the open-loop frequency response at any frequency. By using Bode plot on which frequency is explicit, rather than the Nichols chart, the technique is more intuitive and easier to use than QFT. This paper presents a loop-shaping design method using the RBode plot for the head-positioning control system in HDDs. We compare the results for the case where the nominal model is a transfer function and the case where the nominal model is simply the average of frequency response data. An interesting aspect is that the use of the complex lead compensator greatly facilitates the loop-shaping process [10]. The benefits of proposed method, particularly for the use of frequency response data, are verified by using an open-source HDD benchmark problem that for time-domain simulations of head-positioning control in an HDD [11]. II. HDD H EAD -P OSITIONING C ONTROL S YSTEM A. Features of the head-positioning system An HDD is comprised of a voice coil motor (VCM), magnetic heads, disks, and a spindle motor, as shown in

T. Atsumi is with Research & Development Group, Hitachi, Ltd., 1 Kirihara, Fujisawa, Kanagawa 252-8588 Japan,

[email protected] W. C. Carnegie

Messner Mellon

is with the Data Storage Systems Center, University, Pittsburgh, PA 15213 USA,

[email protected]

978-1-4244-7427-1/10/$26.00 ©2010 AACC

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Fig. 1.

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Fig. 1. In HDD head-positioning control systems, the control input is an input command value to a VCM drive amplifier, and the observed output is the head position. The headpositioning has two primary modes of operation: track seek for moving the magnetic head from one track to another, and up to track follow for keeping the read-write head on a single track with a high degree of accuracy. During seek, the control objective is the control of the transient characteristics of the head position. During track follow the control objective is the control of the steady-state characteristics of the head position subject to various disturbances. The focus of this study is to develop a track following controller.

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A technical committee consisting of representatives of HDD manufacturers and major universities with HDD servo research in Japan has developed an open-source HDD benchmark problem [11]. This software has been widely used for HDD research [12], [13], [14]. This paper uses the trackfollowing control system in the benchmark problem Ver. 3.1. Fig. 2 shows the block diagram of the track-following control system where S is the sampler, C is the digital controller, H is the zero-order hold, P c is the continuous-time plant, r is the reference signal, d is the disturbance signal, and y is the head-position signal. The sampling time in this system is 37.9 µs. In this benchmark problem, the continuous-time plant P c has 18 transfer functions to represent the variable characteristics of the mechanical system. Each transfer function is fifteenth order and consists of seven mechanical resonant modes and a time-delay of 10.0 µs. The controlled object P r (the real plant) is a discretized version of P c . Fig. 3 shows the frequency responses of P r . The disturbance signal d includes the periodic disturbances caused by disk rotation known as repeatable runout (RRO), the effects of the torque noise, and mechanical vibrations excited by the airflow induced by the spinning disks. Fig. 4 shows the power spectrum density of the disturbance signal used in this benchmark problem. III. L OOP S HAPING D ESIGN WITH THE RB ODE P LOT Consider the uncertain plant described by multiplicative uncertainty model shown in Fig. 5, where P r is a real plant, and Pn is a nominal plant. The multiplicative model uncertainty ∆m can be given by ∆m (ω) =

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RBode plot design is based on the well-known necessary and sufficient condition for robust performance that if the

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nominal open-loop transfer function L n ≡ Pn C is stable, |∆m (ω)| < |Wu (ω)|, and |Wu (ω)Tn (ω)| + |Ws (ω)S| |Wu (ω)Ln (ω)| + |Ws (ω)| < 1 f or all ω. (2) = |1 + Ln (ω)| where Tn ≡ Ln /(1 + Ln ), and Sn ≡ 1/(1 + Ln ), then the closed-loop system is stable for all P r and |T (ω)| ≤ |Wu (ω)|−1 .

(3)

|S(ω)| ≤ |Ws (ω)|−1 .

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and for all ω where T ≡ Pr C/(1 + Pr C) and S ≡ 1/(1 + Pn C). RBode plots partition conventional Bode plots into allowable regions that meet the specific robust performance criteria and forbidden regions that do not meet these criteria (2)[8]. The design concept is to shape the open-loop to eliminate intersections between the frequency response with the forbidden regions on the gain plot or on the phase plot. In this paper, our design method follows steps below. Step 1. Determine the frequency response of the nominal model Pn and the weighting functions W u and Ws , all of which can be represented by frequency response data only. Step 2. Design an initial controller C 0 which stabilizes the Pn . Then plot the first RBode plot using P n , |Wu |, |Ws |, and C = C0 .

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Step 3. Design the loop-shaping filter C s to satisfy the low frequency robust performance criteria on the first RBode plot. Then plot the second RBode plot using Pn , |Wu |, |Ws |, and C = C0 Cs . Step 4. Design the loop-shaping filter C u to satisfy the high frequency robust performance criteria on the second RBode plot. IV. C ONTROL S YSTEM D ESIGN In this section we consider two design cases. In the first the nominal model is a second order transfer function. In the second the nominal model is based on the average of frequency response data from the 18 transfer function models of the benchmark problem. A. Case 1: Transfer function nominal model Here we consider the situation where the nominal model is a second order transfer function, which are widely used in conventional robust control design for HDDs [1], [2]. Step 1. The frequency response of |W u | is designed from multiplicative model uncertainties ∆ m as follows. |Wu (ω)| = max |∆m (ω)|.

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disk flutter (from 800 Hz to 1000 Hz) and other mechanical vibrations. Step 2. The form of the controller that stabilizes the control system with the nominal model is a “PI-lead-lag filter”, and the design of the continuous-time controller C pill is based on the break frequency design of [15]. The C pill below achieves an open-loop 0 dB crossover frequency of 600 Hz, a phase margin greater than 30 ◦ , and a gain margin of more than 4.5. Cpill (s) = 0.0107

(s + 314)(s + 925)(s + 34500) . s(s + 20100)2

(6)

Fig. 8 shows the RBode plot where C 0 is Cpill discretized by the Tustin transformation. Step 3. We focus on the loop shaping below 1.5 kHz. Fig. 8 shows the open-loop gain at the RRO frequencies (120 Hz, 240 Hz, and 360 Hz) must be increased, because there is no allowable region on the phase plot at those frequencies. We choose the peak filter C s1 to be

(5)

where the maximum is taken over the 18 variations provided by the benchmark problem. Fig. 7(a) shows the frequency response of |Wu (ω)| (dashed line ). The frequency response of |W s | is designed by taking a 50 point moving average of the power spectrum density of the disturbance signal shown in Fig. 4 and multiplying by a positive constant (in this case 0.1). Fig. 7(b) shows |W s (ω)|. The weighting function includes the effects of RRO (at 120 Hz, 240 Hz, and 360 Hz), torque noise (below 300 Hz),

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Cs1 (s) =

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where, ωp1 = 2π · 120, ωp2 = 2π · 240, ωp3 = 2π · 360, ζpd = 0.01, and ζpd = 0.0001. Fig. 8 also shows that openloop gain and/or phase must be increased around 900 Hz. A complex lead filter is well suited to this purpose because it realizes a narrow band phase-lead effect by using complex poles and zeros [10]. The continuous-time complex lead filter Cs2 provides 60 ◦ of phase at 891 Hz with a damping ratio

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of co-sensitivity function using nominal model (solid line) and |Wu |−1 (dashed line). These figures indicate that this control system satisfies (3) and (4).

of 0.3 to increase the phase to −60 ◦ . s2 + 2830s + 2.22 × 107 . (8) s2 + 3990s + 4.43 × 107 Fig. 9 shows the RBode plot where C s is Cs1 Cs2 discretized with the Tustin transformation. Step 4. We focus on the loop-shaping above 1.5 kHz, choosing our compensators to have essentially no effect on the gain and phase at low frequencies. Fig. 9 shows that the open-loop gain at the frequencies of some mechanical resonance frequencies (4 kHz, 5 kHz, 7 kHz, and 12.3 kHz) must be decreased, because there is no allowable region on the phase plot at these frequencies. We employ a cascade of notch filters Cn . Cs2 (s) = 1.99

Cn (s) =

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where, ωn1 = 2π · 4000, ζnn1 = 0.06, ζnd1 = 0.25, ωn2 = 2π · 5000, ζnn2 = 0.005, ζnd2 = 0.05, ωn3 = 2π · 7000, ζnn3 = 0.01, ζnd3 = 0.1, ωn4 = 2π · 12300, ζnn4 = 0.01, ζnd4 = 0.05, ωn5 = 2π · 1800, ζnn5 = 0.015, and ζnd5 = 0.03. Fig. 15 shows the frequency response of the complete C as a dashed line, and Fig. 10 show the RBode plot after the fourth step. Fig. 11(a) shows the magnitude responses of the sensitivity function using nominal model (solid line) and |W s |−1 (dashed line). Fig. 11(b) shows the gain frequency responses

B. Case 2: Frequency response data nominal model In this section we show the advantage of using the frequency response based nominal model compared with the transfer function based nominal model. We consider the frequency response that is the average of the frequency response of the 18 transfer functions provided by HDD benchmark problem. Capturing all of the features of this averaged frequency response would result in a single transfer function would result in a very high order model. The computational burden of such a high order model could overwhelm transfer function based design methods. Step 1. The solid line in Fig. 6 shows the frequency response of the nominal model. The solid line in Fig. 7(a) shows the weighting function |W u | calculated by (5). Fig. 7(b) shows the weighting function |W s |, which is the same as the one used for the transfer function model. Step 2. The frequency response of the transfer function based nominal model and the frequency response data model are essentially identical below 1 kHz. Therefore, we apply the same initial controller C 0 as was used for the transfer function model. Fig. 12 shows the RBode plot after the second step for the frequency response data model. Step 3. We focus on shaping the loop below 1.5 kHz. Fig. 12 shows the open-loop gain at the RRO frequencies (120 Hz, 240 Hz, and 360 Hz) must be increased. Since

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the frequency responses of the open-loop for the transfer function model and the frequency response data model are essentially the same up to 1 kHz, we use the same peak filter Cs1 as used in case 1. Fig. 12 also shows that the open-loop gain or phase must be increased around 900 Hz. Therefore, we use the complex lead filter C s2 which provides 60 ◦ of phase at 883 Hz with a damping ratio of 0.2 to raise the open-loop phase to −70 ◦ at 900 Hz. Cs2 is given by Cs2 (s) = 1.59

s2 + 1980s + 2.45 × 107 . s2 + 2491s + 3.88 × 107

(10)

Fig. 13 shows the RBode plot after at the completion of Step where Cs is Cs1 Cs2 discretized by the Tustin transformation. Step 4. We focus on the shaping the loop above 1.5 kHz. Fig. 13 shows the open-loop gain at the resonance frequencies 5 kHz, 7 kHz, and 12.3 kHz must be decreased the gain because there is no allowable region on the phase plot at these frequencies. Fig. 13 also shows that the primary resonance around 4 kHz can be stabilized by a small phaselag effect, and the gain at 1800 Hz should be a little lower. Therefore, we set the notch filter C n as follows. Cn (s) =

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(11)

where, ωn1 = 2π · 5000 ζnn1 = 0.005, ζnd1 = 0.05, ωn2 = 2π·7000, ζnn2 = 0.02, ζnd2 = 0.1, ωn3 = 2π·12300, ζnn3 =

0.01, ζnd3 = 0.05, ωn4 = 2π · 1800, ζnn4 = 0.015, and ζnd4 = 0.02. Note that one fewer notch is needed, because the phase lag of the notch at 5 kHz stabilizes the resonance at 4 kHz. The final controller C is given by C[z] = C0 [z]Cs [z]Cu [z],

(12)

where Cu is Cn discretized by a matched pole-zero method. Fig. 14 shows the RBode plot after Step 4. The solid line in Fig. 15 shows frequency response of C. Fig. 16(a) shows the gain frequency responses of sensitivity function using nominal model (solid line) and |W s |−1 (dashed line). Fig. 16(b) shows the gain frequency responses of co-sensitivity function using nominal model (solid line) and |Wu |−1 (dashed line). These figures indicate that this controller satisfies (3) and (4). V. P ERFORMANCE E VALUATION To verify the designed two control systems, we simulated the control system using the 18 different transfer functions of the benchmark problem. Figs. 17(a) and (b) show the Nyquist diagrams for the 18 variations of the real plant for the two cases. These figures show that the control system for the frequency response data model realizes a phase-stabilized design. Thus the control system for the frequency response data model can decrease vibrations of the primary resonance by feedback control and increase the gain margin as indicated in Table I because of the smaller phase delay of the notch filters[15].

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TABLE I 15

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VI. C ONCLUSION This paper presents an easy-to-use loop-shaping design method employing the RBode plot for head-positioning control system in HDDs that can suppress disturbances and accommodate perturbations of the controlled object. Significantly, the nominal model need only be represented by frequency response data–no transfer function model is required. As a result, the designed controller can avoid the excessive conservativeness and improve the control performances. R EFERENCES [1] M. Hirata, K. Z. Liu, T. Mita, and T. Yamaguchi, “Head positioning control of a hard disk drive using H∞ theory,” in Proceedings of the 31st IEEE Conference on Decision and Control, 1992, pp. 2460–2461. [2] M. Hirata, T. Atsumi, A. Murase, and K. Nonami, “Following Control of a Hard Disk Drive by using Sampled-Data H∞ Control,” in Proceedings of The 1999 IEEE International Conference on Control Applications, 1999, pp. 182–186. [3] T. Semba, “An H∞ Design Method for a Multi-Rate Servo Controller and Applications to a High Density Hard Disk Drive,” in Proceedings of The 41th IEEE Conference on Decision and Control, 2001, pp. 4693–4698. [4] M. Hirata and Y. Hasegawa, “High Bandwidth Design of TrackFollowing Control System of Hard Disk Drive Using H∞ Control Theory,” in Proceedings of the 16th IEEE International Conference on Control Applications, 2007, pp. 114–117. [5] I. Horowitz, “Quantitative feedback theory,” in Proceedings of the IEE, vol. 129, pt. D, no. 6, pp. 215–226, 1982.

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Fig. 18(a) and (b) shows the maximum gains of sensitivity function and co-sensitivity function respectively with the 18 variations of the real plant. The maximum H ∞ norm of sensitivity and co-sensitivity function are listed in Table I. The maximum sensitivity and co-sensitivity function magnitudes are lower for the frequency response based nominal model. Fig. 19 shows position error signal (PES) from timedomain simulations using the benchmark software. The 3σ values of PES listed in Table I show that using frequencyresponse based nominal model can improve the positioning accuracy by about 5% in the time domain.

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[6] W. Messner, “Some Advances in Loop Shaping Controller Design with Applications to Disk Drives,” The IEEE Transactions on Magnetics, vol. 37, no. 2, pp. 651–656, 2001. [7] L. Xia and W. Messner, “Loop shaping for robust performance using the RBode plot,” in Proceedings of the 2005 American Controls Conference, 2005, pp. 2869–2874. [8] ——, “An Improved Version of the RBode Plot,” in Proceedings of the 2008 American Controls Conference, 2008, pp. 4940–4945. [9] W. Messner, “Formulas for Asymmetric Lead and Lag Compensators,” in Proceedings of The 2009 American Control Conference, 2009, pp. 3769–3774. [10] W. C. Messner, M. D. Bedillion, L. Xia, and D. C. Karns, “Lead and Lag Compensators with Complex Poles and Zeros,” IEEE Control Systems Magazine, vol. 27, no. 1, pp. 44–54, 2007. [11] Benchmark Problem, Ver. 3.1. The Institute of Electrics Engineers of Japan, Technical Committee for Novel Nanoscale Servo Control. [Online]. Available: http://mizugaki.iis.utokyo.ac.jp:80/nss/MSS bench e.htm [12] Q. Zheng and M. Tomizuka, “A Disturbance Observer Approach to Detecting and Rejecting Narrow-Band Disturbances in Hard Disk Drives,” in Proceedings of The American Control Conference, 2008, pp. 254–259. [13] S. Oh and Y. Hori, “Position Error Signal based Control Designs for Control of Self-servo Track Writer,” in Proceedings of the 17th World Congress The International Federation of Automatic Control, 2008, pp. 845–850. [14] Y. Kinoshita, Y. Chida, and Y. Ishihara, “Feedforward Control Design for Seek Control Using NME Profiler and Input Shaping,” in Proceedings of The 2009 JSME-IIP/ASME-ISPS Joint Conference on Micromechatronics for Information and Precision Equipment, no. SVC-02, 2009. [15] T. Atsumi, T. Arisaka, T. Shimizu, and T. Yamaguchi, “Vibration Servo Control Design for Mechanical Resonant Modes of a Hard-Disk-Drive Actuator,” JSME International Journal Series C, vol. 46, no. 3, pp. 819–827, 2003.

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