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Digital Control of Repetitive Errors in Disk Drive Systems Kok Kia Chew and Masayoshi Tomizuka Presented by Corey Drechsler as a part of the DSSC Servos and Channels Seminar Series 10 August 2011 The System The plant is an 8 in. IBM hard drive, but is still mechanically similar to modern hard drives. Spindle Speed: 3125 RPM Track Density: 1200 TPI Spindle Motor VCM Disk Head ~450mm 8-in. IBM 62PC Hard Drive The Problem : Periodic Disturbances Simple frequency response techniques can be used for track following, given constant or slowly varying disturbances. Periodic disturbances can be large and more difficult to eliminate. Additionally, there are often many harmonics of the signal present Changing existing (analog) controllers is undesirable. Controller Layout An add-on repetitive controller can reduce periodic disturbances. Computational Limits A 16 MHz Intel 80386 CPU is used for all digital processing (approximately 250 kFLOPS) Spindle speed of 3125 RPM (52 Hz) Up to 54 samples per track (356 µsec sample time) cpu _ speed _ in _ flops operations N spindle _ speed 250000 operations sec ond 89 operations 54 revolution 52 revolutions samples sample sec ond Limited to 89 operations at the highest sampling rate! Higher sample rates allow more harmonics to be removed, but allows less time for computation. Repetitive Controller Design: Overview The block diagram is redrawn from the perspective of the repetitive controller. The “system” is not just the analog controller and the plant in series, but is the closed-loop response from ur(k) to er(k). Reducing the magnitude of the output/error signal er(k) is the goal of the repetitive controller design. Repetitive Controller Design: System Identification The system transfer function can be represented in the following form: GS z 1 z d B z 1 A z 1 where Az 1 1 a1 z 1 a n z n Bz 1 b0 b1 z 1 bm z m The numerator B(z-1) is then factored into two parts, B+(z-1) and B-(z-1), which are the cancelable (stable) and uncancelable parts of B Repetitive Controller Design: System Identification Example For the test system, a least-squares identification algorithm was used. At N=54 (sample time of 356 µsec), the function was estimated as: z 3 0.0972 0.0848 z 1 G S z 1 1 1.6054 z 1 0.7791 z 2 Pole-Zero Map 1 In this case, there are no 0.8 0.6 unstable zeros, so B+ = B 0.4 and B- = 1 Imaginary Axis 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.5 0 0.5 1 Real Axis Repetitive Controller Design: Ideal Controller The ideal repetitive controller is then: GC z 1 k r z N d A z 1 B z 0 kr 2 1 z N bB z 1 b max B e j 2 ,0 A and B+ cancel the stable portion of the system. B- cancels only the phase of the unstable zeros, while b cancels their magnitude. kr sets the controller gain, but is limited to ensure robust stability. The z-N/(1-z-N) is the internal model of the disturbance and cancels the fundamental frequency and all of the un-aliased harmonics. Repetitive Controller Design: Ideal Controller Example The controller for the test system becomes: 1 GC z z 54 1 - 1.605 z -1 + 0.7791z -2 1 z 54 - 0.0972 z -3 - 0.0848 z -4 where kr=b=1 Bode Diagram Pole-Zero Map 100 1 80 0.8 Magnitude (dB) 60 0.6 40 0.4 20 Imaginary Axis 0.2 0 -20 0 720 -0.2 540 Phase (deg) -0.4 360 180 -0.6 0 -0.8 -180 -1 0 1 2 3 10 10 10 10 -1.5 -1 -0.5 0 0.5 1 Frequency (Hz) Real Axis Repetitive Controller Design: Robust Controller However, this system is only stable if the system model is sufficiently accurate. To ensure robust stability, a (typically low pass) filter q(z-1) is added to the controller, sacrificing high- frequency regulation for stability: 1 k r z N d q z 1 A z 1 B z GC z 1 q z 1 z N bB z 1 This typically does not hurt the tracking performance, as higher harmonics of the disturbance tend to be lower in magnitude. Repetitive Controller Design: Stability The system is provably stable if: Ae j B e j Ba e j 1 1 kr ,0 bB e Aa e j j qe j where Aa and Ba represent the actual system transfer function Note that if A = Aa, B = Ba (i.e. the model is perfect), and 0 < kr < 2, the left-hand side is always less than unity, and is therefore stable for any of the choices of q investigated in the paper. Repetitive Controller Design: Filter Selection The figures below show the frequency response and stability bounds for three different filters. For the test system, q2 is used, as it provides the largest range of robustness, while not significantly affecting the results. 0 Bode Diagram -2 Magnitude (dB) -4 -6 q1 = 1 q2 = (z + 2 + z-1)/4 -8 q3 = (z + 6 + z-1)/8 -10 0 1 2 3 10 10 10 10 Frequency (Hz) Repetitive Controller Design: Robust Controller Example The robust controller becomes: z 2 z 1 54 z GC z 1 4 1 - 1.605 z -1 + 0.7791z -2 z z 2 z 1 54 - 0.0972 z -3 - 0.0848 z -4 1 4 Pole-Zero Map 1 0.8 The high frequency poles 0.6 0.4 move away from the unit Imaginary Axis 0.2 0 circle, decreasing the gain -0.2 (and hence rejection) at -0.4 -0.6 higher harmonics -0.8 -1 -1.5 -1 -0.5 0 0.5 1 1.5 Real Axis Experimental Results: Original vs. Repetitive Control Experimental Results: Varying Sample Rates Conclusions Discrete time repetitive control can significantly reduce the effect of a periodic disturbance and its harmonics. This controller can be added to an existing control system without the need to modify the original controller. Computational limits restrict how many harmonics can be removed by the controller. Modeling errors also restrict the amount of reduction for high frequency harmonics. References T. Yamaguchi, Modeling and control of a disk file head- position system. Proc. Instn. Mech. Engrs., Vol 215, Part I, 2001, pp. 549-568 K. Chew and M. Tomizuka, Digital control of repetitive errors in disk drive systems. IEEE Control Systems Mag., Vol 10, Issue 1, 1990, pp. 16-19 K. Chew and M. Tomizuka, Digital control of repetitive errors in disk drive systems. Proc. Amer. Cont. Conf., Vol 1, 1989, pp. 540-548

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posted: | 8/11/2011 |

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