DAC Quantization Noise Reduction for Servo Control Systems in Hard Disk Drives

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1 DAC Quantization Noise Reduction or Servo Control Systems in Hard Disk Drives Wei-Min u Member IEEE Roger Wood Member IEEE and Mantle Yu Abstract In hard disk drive (HDD) servo control systems quantization noises due to the inite precision o the D/A converter (DAC) driving the VCM contribute a signiicant portion o the total track mis-registration (MR) In this paper a quantization error eedback (QEF) technique to reduce MR due to DAC quantization noises is introduced) he QEF technique oers a simple method o reshaping the spectrum o this noise to minimize its contribution to MR Despite the name QEF is an eectively eed-orward technique that means that it does not aect or degrade the servo loop response In addition the limitation and optimality o dierent QEF schemes are examined in detail Simulations reveal that contribution to MR on an HDD is reduced by more than a actor o ten ests have veriied the improvement I INRODUCION n hard disk drives (HDD) the digital to analog converter (DAC) driving the voice coil motor (VCM) I has limited accuracy Quantization noises (or roundo errors) due to the inite precision o the DAC disturb the servo control loop and degrade servo control perormance he DAC quantization noises contribute a signiicant portion o the total track-misregistration (MR) in HDD products Furthermore or ixed mechanics and servo control bandwidth in a HDD the power spectrum o MR due to DAC noises is ixed and the MR does not scale with the track density hereore the DAC quantization noise will be a recurring problem as the track density goes higher; techniques to eectively reduce such MR are desirable his paper describes a numerical technique which goes by the general name o quantization error eedback or QEF to reduce the HDD MR due to quantization noises arising rom a DAC with ixed precision without degrading other desired servo properties hese QEF schemes are implemented entirely within the DSP In the digital servo control loop the internal precision o the digital signal processor (DSP) is typically higher (eg 6 bits) than that o the DAC (eg bits) he lower order bits have to be dropped when the calculated control signal is sent to DAC In the QEF schemes the truncated lower order bits which are reerred as the quantization errors is monitored and accumulated in the DSP When suicient error has accumulated the bits eeding the DAC can be modiied in such a way as to largely cancel the eects o the error In the requency domain the Manuscript received September 3 he authors are with the Advanced echnologies Hitachi GS Inc San Jose CA 9593 ( wei-minlu@hgstcom; rogerwood@hgstcom; mantleyu@hgstcom) eect o the process is to reshape the spectrum o the quantizing noise he noise seen at the DAC may increase at some requencies but its overall contribution to (the mean square value o) MR is dramatically reduced Simulations with real HDDs reveal that the contribution to MR can be reduced by more than a actor o ten with optimal QEF schemes (see Fig 5) ests in real HDD products show the MR improvement A kind o QEF technique was also used in the data transmission systems to reduce low requency content o quantizing errors [3] he purpose and implementation o the QEF technique introduced here or HDDs are dierent In this paper we will examine the QEF schemes rom control theoretical point o view in order that we ully understand the options available or reducing MR due to DAC quantization noise Despite the name QEF is an eective eed-orward technique; thereore it does not aect the servo loop perormance In addition we will also examine the limitation and optimality o the QEF technique; an algorithm that gives optimal QEF schemes is developed It should be pointed out that even though the QEF technique is proposed or dealing with quantization noise in HDD servo control system the methodologies and results can be applied to general digital control systems to deal with noises in the control input he structure o this paper is organized as ollows: In Section we will examine the basic properties o DAC quantization error and motivate the use o quantization error reduction techniques In Section 3 we will introduce the basic concepts o the QEF technique In Section 4 we will give more detailed analysis about QEF technique with general structures in particular we will discuss the limitations o using the QEF technique to reduce the impact o quantization noises and develop an algorithm to derive the optimal QEF ilter resulting in maximal MR reduction In Section 5 we will give some test results o the QEF technique on real HDDs II DAC QUANIZAION NOISE D/A Quantization Error A simpliied block diagram o the HDD servo system is given in Fig In HDD servo system the DSP has higher precision than the A/C converter (DAC) does hereore only the most signiicant bit (MSB) part o the calculated control signal is sent to the D/A converter the least signiicant bit (SB) part is truncated (rounded o) he error (SB) due to the DAC truncation is reerred as

2 DAC quantization error denoted as QE in the ollowing discussion Fig Digital Servo Systems or HDD According to Widrow the quantization noise QE( can be modeled as a white random process having a uniorm probability density in [-q/ q/] where q l c with c being the quantization resolution in DSP and l being the number o SB bits [4] In particular its mean value is: E(QE() and its variance satisies q δ QE E{( QE E( QE)) } he autocorrelation unction is given as ollows R ( ) E{ QE( QE( t )} q QE τ + τ i τ ; otherwise R QE ( τ ) Power density spectrum is given as ollows: iτω q SQE RQE( τ ) e RQE() τ Impact o the Quantization Noise on MR he block diagram in Fig can be redrawn as the block diagram in Fig Fig Equivalent Block Diagram or HDD Servo Systems hereore the MR due to quantization error o D/A can be represented by PES(H( QE( where H( is the transer unction rom QE to PES: P( H ( P( C( P( is the transer unction o the plant which is dominated by a double-integrator; in this section we will assume it is a double integrator he magnitude o MR or position error signal (PES) is measured by its mean square value (or variance) which can be represented by ωn E( S dω ω n PES where ω n is the Nyquist requency which is normalized as in this paper hus the mean square value o MR is the average power o the signal which is determined by its power spectrum density unction: iω S PES H ( e ) SQE As the quantization noise is represented as white noise as discussed in the previous subsection the shape o power spectrum o MR depend solely on the requency response o the system: q iω SPES ( ω ) H ( e ) he typical requency response o the closed loop system rom QE to PES has a shape o a low pass ilter Fig 3 shows the typical requency response rom quantization error to MR or a HDD Magnitude 4 x Frequency [rad/sec] x 4 Fig3 Frequency Response rom Quantization Noise to MR he power o MR due to DAC quantization noise is concentrated at lower requencies hereore the reduction o the average power o PES can be achieved by reshaping the power spectrum density unction o MR Since any alteration o the requency response o the closed loop transer unction may change other desired properties o the servo systems desirable techniques or MR reduction should be those that are able to reshape the power spectrum o the input noise QE and reduce the average power o PES in the requency range o interest In the ollowing we will ocus on such a technique: QEF III QUANIZAION NOISE FEEDBACK 3 What is QEF? he idea o QEF is illustrated with the block diagram in Fig 4 With this technique the quantization error is monitored and accumulated (integrated) in the DSP When suicient error has accumulated ie an MSB is generated it is added to the original MSB eeding the DAC Note that in this QEF scheme the QE signal is integrated once we will call this scheme single-integrator QEF It is also noted that a double integrator scheme also be used or reducing MR [6]

3 Fig 4 QEF echnique or MR Reduction: Single Integrator Fig 5 shows the ideal power spectrums o MR due to quantization error with single-integrator QEF (lower curve) and without any QEF (upper curve) respectively or an HDD he MR reduction is signiicant Magnitude x -8 stochastic properties as QE under Widrow s assumption in particular q SQN ( ω ) S ( ) QE ω he control scheme shown in the block diagram is actually eedorward control which does not alter closed loop transer unction As the preilter K( in this case is a single discrete time integrator In the ollowing we reer the QEF technique in Fig3 as single integrator QEF In act the single integrator QEF is an optimal eedorword control scheme in the sense that the preilter K(I( in Fig 4 is optimally chosen such that the resulting PES has minimal value [6] IV IMIAION AND OPIMAIY OF QEF 4 A General Structure o QEF In the previous section we introduced a single integrator QEF scheme A general structure or QEF schemes is shown in Fig4 Note that the eedback loop in the error accumulation operation has at least one pure delay so that the operation is implementable Frequency [rad/sec] x 4 Fig5 MR due to Quantization Error with/without QEF 3 QEF is a Feedorward Scheme o help explain the QEF technique the QEF scheme discussed earlier (Fig4) is equivalently represented as the block diagram in Fig6 Fig 4 QEF echnique or MR Reduction In this block diagram QN is truncation error (SB) o QE ater iltering Note that QN is white noise with similar stochastic properties to QE under Widrow s assumption in particular S ( ω ) S q / QN QE Fig 6 QEF: Equivalent Block Diagram In this block diagram the double-integrator plant is decomposed into two integrators: P(k I ( and K(I( where I( is a discrete-time integrator: I ( QN can also be interpreted as the z truncation error (SB) o QE ater integration in the above block diagrams; as the internal signal driven the block ki( should be MSB Note thatqn is exactly the same signal as QN in the diagram in Fig 3 In the ollowing analysis QN is white noise with the similar 4 Optimal QEF Filter and Approximations Consider the general QEF scheme in Fig 4 In this structure the eedback loop gain is z F( where F( is a ilter whose transer unction is assumed to be a real rational unction which is analytic and bounded in { z : z } o explore the limitation o the general QEF schemes let s look at the possible optimal choice o the ilter F( With some block diagram manipulation one can equivalently transorm the block diagram in Fig 4 to the eedorward block diagram in Fig5 It is noted that in the above igure the quantization noise QN is reshaped by the general ilter z F( resulting in disturbance GQE entering the system as GQE( ( z F( ) QN (

4 and double integrator QEF We also see that the larger a is the bigger the convergent set; in particular as a the convergent requency set approaches [ /) Fig 5 General QEF: Equivalent Block Diagram hereore the MR due to quantization error becomes P( PES( GQE( H ( ( z F( ) QN( P( C( where H( is the transer unction rom QE to PES in the original block diagram QN is white noise with similar stochastic properties to QE and SPES ( e F( e )) H ( e ) SQN ; the variance o PES is as ollows: q E( SPES dω ( e F( e )) H ( One can see that q min ( ( e F( e )) H ( ) F F where the minimal is achieved (mathematically) with F ( e ) e or F(z hereore the (mathematically) optimal ilter F( is a orward shit operator hus the optimal eedback loop gain is identity without any delay However it is not implementable because the QEF eedback loop should contain at least a delay operator In the ollowing we will examine how the optimal QEF ilter can be approximated by an implementable ilter Given a > or all z satisying z < we have the a ollowing expansion : / a z z z 3 z ( + ( ) + ( ) + ( ) + ) z ( z / a) a a a a In particular the n-th order approximation F( to z yields: z z z n z n z F( z ( + ( ) + ( ) + + ( ) ) ( ) a a a a a I a then the single and double integrator QEF schemes as discussed earlier are recovered with n respectively As n the eedback loop gain z F( is convergent to the optimal ilter at the ollowing requencies: ω e j { ω : < ω } a In particular i a the convergent requencies are [ /3) his interprets why the power at the lower requencies (ω [ /3)) is reduced and enlarged at the higher requencies (ω [/3 ]) or both single integrator 43 imitations o QEF echnique In the ollowing we will assume the ilter F( used in the QEF ilter has the ollowing expansion: F ( + z + z + he reduction o the power o quantization error at some requencies is at the cost o enlargement o the power at other requencies In act this is a undamental limitation or implementable QEFs his observation is stated in the ollowing statement [6] QEF imitation (Spectrum Shaping) For all implementable QEF schemes ie the ilter F( in Fig is a real rational unction analytical and bounded in {z : z } the power spectrum S GQE (ω ) o shaped noise GQE( ( z F( ) QN( with QN being the white quantization noise always satisies: ω ω ln( S GQE ( )) d constant where the constant is independent o the choice o the ilter F( he proo o the above statement is given in the [6] In the ollowing we will examine the limitation o the MR reduction he PES with the general structure o the QEF ilter is represented as ollows: PES ( H ( ( z 3 F ( ) QN ( : Θ( QN ( where H ( is the transer unction rom QE to PES in the original block diagram he power spectrum o PES is S ( e F( e )) H ( e ) S PES the variance o PES can be obtained by the ollowing ormulation: q ω q E( ( e F( e )) H( Θ( e QN j ) dω I the transer unction Θ( rom QE to PES is represented as: Θ( z ) p + pz + pz + ie its impulse response is { p i} i then rom Parseval s identity one has Θ( p i et the closed loop transer unction H( satisies k bk z + + b z + b H( z ( h + ) : z H ( k k z + ak z + + az + a or some nonnegative integer m; ie there are m pure delays in the closed loop transer unction hus k bk z + + b z + b H ( h + k k z + ak z + + az + a ( k ) k bk z + + b z + b z h + ( k ) k + ( ak z + + az + az ) h + h z +

5 or some { h i} i One immediately conclude QEF imitation (MR Reduction) For the QEF schemes considered in this section the mean square value o PES always satisies q E( h he proo o the statement is given in [6] It gives a MR lower bound using the QEF techniques 44 Optimal MR Reduction via QEF In this section we will examine with given QEF structure what is the best QEF ilter in the sense the MR due to quantization noise is reduced In the ollowing we will consider the case where the QEF ilter F( is an FIR ilter: n F( + z + + n z Denote [ n n ] then we need to ind an optimal vector such that * q arg arg min ( Θ( ) o ind out the optimal solutions we irst show how to compute the mean square value o PES in terms o the Parseval s identity rom transer unction We irst need to derive the time domain equation or the transer unction: Θ( H ( ( z F( ) ( z F( ) H ( where the transer unction o the closed loop system H( has m pure delays as given in the previous subsection: k bk z + + b z + b H( z ( h + ) : z H ( k k z + ak z + + az + a et Θ ( z F( ) H ( ) then ( z z m Θ ( ) Deine Θ( z and A M C A M a M n n B M n [ ] n n D a a M a k B M [ b b b ] D k C k k h hereore the (n+k)-th order state space realization o the transer unction Θ ( z F( ) H ( ) is: ( z x( t + ) Ax( + Bu( y( Cx( + Du( where state x x and coeicient matrix: x A BC B D A B C A B [ C D C ] D D D Now with the state space equation one can calculate the impulse response o the system; ie or the input {u(} with u() and u( or t > the output response {y(} can be calculated interms o A B and C hereore jmω Θ( ( ) e Θ e dω Θ( pi y ( i) D + : D i D + C( A i + CC i i CA BB ( A ) C i BB ( A ) ) C where i i A BB ( A ) satisies yapunov equation: AA + BB Notice that in the state space equation the matrices A B and D are known so can be solved in the yapunov equation; part o the matrix C depends on the unknown in act C [ C DC ] [ C] Suppose the matrix > is partitioned as ; also notice that D DD h one has q j q E( ( ) ( ω Θ e dω D q ( C + CC + h ) * hen the optimal solution satisies: or * C and E ( q * ( C ) + CC ) q ( C( ) C + h ) is positive deinite then so > q ) C + h ) h q ( C( which again conirms the QEF MR imitation statement Optimal QEF Algorithm: Construct state space matrices B D A and B Solve the ollowing yapunov equation: AA + BB to get the solution

6 3 Find the optimal ilter coeicients: * C 4 he optimal solution is given by: q ( C( ) C + h ) he above algorithm is used to calculate the optimal QEF schemes or Sailin HDD products he reduction o MR vs the order o optimal ilter is illustrated in igure NRRO (RMS) Reduction vs Parameter (/a) /a RMS MR Reduction [db] Order o Optimal Filter Fig 5 MR Reduction vs Order o QEF Filter V ES WIH REA HDDS In this section we will present some test results o QEF techniques on Mako HDD Several irst order QEF schemes are implemented All o the data are taken at OD o the Mako ile he PES (MR) signals are total NRROs including MR due to quantization noise QEF schemes only reduce MR due to quantization noises hereore the reduction o the total NRROs is solely due to the reduction o the MR due to quantization errors Fig 7 shows the power spectra o NRROs with QEF (a) and without any QEF respectively Frequency (H Fig 7 Power Spectra o NRROs: Red Curve: a Blue Curve: No QEF For a changing rom 4 to the reduction on root o mean square value (RMS) o the NRRO can be shown in the ollowing diagram in Fig 9 One can see that the total NRRO RMS is reduced about 38% using the single integrator QEF scheme (a) In act the optimal parameter a in the irst order QEF is a * 358 Fig 9 he NRRO (RMS) Reduction vs /a VI CONCUSIONS In this report we have introduced a QEF technique to reduce the MR o HDD due to DAC quantization noise without altering or degrading the other desired servo perormances We have examined the limitations o QEF techniques and give a lower bound or the mean square value o MR using QEF schemes We have also derived the optimal QEF ilter which minimizes the mean square value o MR and an algorithm is provided From simulation on Sailin HDD simulation model the MR has a RMS reduction by a actor o more than with the optimal QEF schemes ests on Mako HDD have veriied the improvement (more than 3% reduction on the total NRRO with irst order QEF scheme) REFERENCES [] Abramovitch D Hurst and D Henze Decomposition o Baseline Noise Sources in Hard Disk Position Error Signals Using PES Pareto Method Proc o 997 American Control Conerence Albuquerque MN June 997 [] Cutler C C ransmission Systems Employing Quantization US Patent No9796 March 96 [3] Eddy K J Steele and W Messner Bias in Disk Drive Rotary Actuators May 997 [4] Franklin GF J D Powell and M Workman Digital Control o Dynamic Systems Addison-Wesley 99 [5] ang S Complex Analysis Springer-Verlag 985 [6] u W-M R Wood and M Yu Method and apparatus or reducing track misregistration due to digital-to-analog converter quantization noise US Patent # September [7] Spang III HA PM Schultheiss Reduction o Quantizing Noise by Use o Feedback IRE rans Comm Syst pp December 963 [8] Yu M G Herbst and S-M Shih NODAC Servo System Architecture IBM Internal Report San Jose 994 [9] Zhou K JC Doyle and K Glover Robust and Optimal Control Upper Saddle River NJ: Prentice Hall 996

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