Limits of Performance in Systems withPeriodic Irregular Sampling and Actuation

Rates

Behrooz Shahsavari ∗ Richard Conway ∗ Ehsan Keikha ∗∗

Roberto Horowitz ∗

∗Mechanical Engineering Department, UC Berkeley, CA 94720 USA(e-mails: behrooz@berkeley.edu, rconway345@gmail.com,

horowitz@berkeley.edu)∗∗Department of Electrical and Computer Engineering, National

University of Singapore, Singapore 117576 (e-mail:ehsan.keikha@nus.edu.sg)

Abstract: This paper examines the limits of performance in systems with periodic irregularsampling rate when the actuation is not necessarily synchronized with the sampling. For such asystem, three sampling and actuation schemes are considered: when the sampling and controlrate are both regular, when they are both irregular, and when the sampling rate is irregular whilethe control rate is regular. To ascertain the limits of performance of this type of systems undereach sampling and actuation scheme, the system is modeled as a linear periodically time-varying(LPTV) system; optimal LQG control design with a variance constraint is applied to find thesmallest achievable mean variance of the performance signal subject to a constraint on the meancontrol effort variance. In addition, to deal with the computational delay of the controller, aninnovative discretization method is proposed which does not introduce any extra states intothe state space model. The proposed method is exploited to determine the performance of ahard disk drive (HDD) in track-following mode. A simulation study demonstrates that in thepresence of 30% irregularity in sampling time, using the irregular sampling and regular controlaction scheme for the HDD achieves an RMS 3σ value of the position error signal (PES) thatis 40% smaller than the corresponding value achieved by using a controller provided by ourindustry partner, in which both the sampling and control rates are irregular.

Keywords: Limits of performance, hard disk drives, irregular sampling, LQG control, periodicsystems.

1. INTRODUCTION

In the field of mechatronics, it is useful to have a designtool that determines the limitations of performance ofa given system configuration. The limits of performanceof a dynamic system can be determined by designing anoptimal controller and calculating the performance of thecorresponding closed-loop system. This can be used tocompare several different hardware setups by determiningthe best possible closed-loop performance that can beachieved for each one. Furthermore, the optimal controllerthat achieves the limits of performance can be used as aguide for other types of control design, e.g. the frequencyresponse of the closed-loop system for such an optimalcontroller can be used in loop-shaping methods like H∞optimal control design.

For a system with irregular sampling, there are two mean-ingful schemes for updating the control action. In the firstscheme, the control action is updated as quickly as possibleafter a measurement is obtained, which causes the controlrate to also be irregular. In the second scheme, the controlaction is updated at a regular rate. Since the outputof the controller is usually passed through one or more

notch filters before being applied to the plant, this secondscheme is attractive because it allows the notch filters to bedesigned independent of the sampling rate variation. Weconsider three sampling and actuation schemes in linearsystems and determine the limits of performance for eachof these cases. These sampling and actuation schemesare as follows: when the sampling and control rate areboth regular, when they are both irregular, and when thesampling rate is irregular but the control rate is regular.

To assess the limits of performance associated with eachsampling and actuation scheme, the optimal control designmethodology we use is LQG optimal control with outputmean variance constraints (e.g. Conway et al. (2008)). Thiscontrol design methodology allows us to determine thesmallest possible mean variance of the PES while explicitlyimposing constraints on the mean variance of the controlsignal.

To demonstrate the effectiveness of the proposed method,we then consider calculating the limits of performance forhard disk drives with different sampling and actuationconditions. The areal data density of HDDs has beendoubling roughly every 18 months for several decades,

6th IFAC Symposium on Mechatronic SystemsThe International Federation of Automatic ControlApril 10-12, 2013. Hangzhou, China

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as predicted by Kryder’s law. In order to increase highdata storage density, it is necessary to write the datatracks very close together, which requires very accurateservo-controllers to move the read/write heads on topof the disks. Currently, it is a goal of the magneticrecording industry to achieve an areal storage density of 4terabits/in2. It is expected that the track width requiredto achieve this data density is 23 nm, which means thatthe 3σ value of the closed-loop position error signal (PES)should be less than 2.3 nm to achieve this specification.

If the sampling rate in a HDD is regular, the servosystem can be modeled as a LTI system and LTI controldesign tools can be exploited for this system (e.g. Chenet al. (2006)). However, sampling intervals for hard diskdrive servo systems are not always equidistant over arevolution of the disk. For instance, when the center ofthe servo tracks does not exactly coincide with the centerof rotation of the disk, there can be large variationsin the sampling rate of the position error signal (PES)(Shahsavari et al. (2012)). The other factor that causesirregularity in sampling rate in a HDD is the existence ofmissing sectors (Nie et al. (2011)). For example, false PESdemodulation, due to incorrect servo address mark (SAM)detection (Ehrlich (2005)) or damaged servo patterns inseveral servo sectors, makes the feedback PES unavailablein those servo sectors, resulting in an irregular samplingrate.

Since HDDs are naturally periodic, with period equal tothe time for one rotation of the disk, the servo systemfrom the control input to the PES can be represented bya linear periodically time-varying (LPTV) system whoseperiod is the number of servo sectors, N (Nie et al. (2011)).When the HDD has regular sampling and actuation, themeasures of performance we use are the variance of thePES and the variance of the control effort. However, inthe other two sampling and actuation schemes, the closed-loop system is LPTV. As a result, the closed-loop signalsare not stationary; they instead have periodic second-orderstatistics. Thus, to measure the performance of the closed-loop system in these cases, we use the mean variances (overtime) of the PES and the control effort.

The paper is organized as follows: section 2 discussesthe discretization methods for the three aforementionedsampling and actuation schemes and describes how toform LPTV discrete-time models of systems with thesesampling and actuation characteristics from a continuous-time model. Section 3 presents the details of the optimalcontrol design for the discrete-time systems discussed insection 2. Section 4 examines the limits of performanceof a HDD in track-following mode for each of the threesampling and actuation schemes; and finally, conclusionsare given in section 5.

2. MODEL DISCRETIZATION

In this section, we will describe each of the three samplingand actuation schemes and describe how to form LPTVdiscrete-time models of systems with these sampling andactuation characteristics from a continuous-time model.Throughout this section, the continuous-time state spacemodel of the system is given by

xc(t) = Acxc(t) +Bcuc(t) (1)

yc(t) = Ccxc(t) +Dcuc(t) . (2)

The signals xc(t), uc(t), and yc(t) respectively have dimen-sion nx, nu, and ny. The computational delay associatedwith the controller will be denoted as δ. We also define forT > 0

Ad(T ) := eAcT , Bd(T ) :=

(∫ T

0

eAcτdτ

)Bc .

Note in particular that these matrices are the discrete-time state space matrices that result when discretizingthe continuous-time state dynamics using a zero-order holdwith period T . Throughout this section, we will use a zero-order hold approach to discretizing the model.

2.1 Regular Sampling and Actuation

The first case we consider is when the system generatesmeasurements at a regular rate and the controller updatesthe control signal at the same rate. Due to computationaldelay, the control signal is not updated at the same time asmeasurements are obtained. We will assume for simplicitythat δ < T , where T is the sample period.

The simplest approach to discretizing this system is to usethe c2d command in MATLAB. To incorporate the com-putational delay, either the InputDelay or OutputDelayproperty of the continuous-time model should be set to δbefore using the c2d command. Since δ < T , this methodintroduces either nu or ny extra states to the model aspart of the discretization process, depending on whetherthe InputDelay or OutputDelay property was used.

We now describe an alternate discretization method thatdoes not introduce any extra states. We first choose t = 0so that the control is updated at time instances

tu,k = kT, k ∈ Zwhich means that the measurements are obtained at timeinstances

ty,k = (k + 1)T − δ, k ∈ Z .

Note that we have shifted the time index associated withthe measurements by one step.

We now define

xk := xc(tu,k) (3)

uk := uc(tu,k) (4)

yk := yc(ty,k) . (5)

The zero-order hold assumption yields that

u(t) = uk, t ∈ [tu,k, tu,k+1)

which means that (1) discretizes as

xk+1 = Ad(T )xk +Bd(T )uk . (6)

To find the discrete-time representation of the measure-ments, we first note that

ty,k = tu,k + (T − δ)which means that

yk = Ccxc(tu,k + (T − δ)) +Dcuc(tu,k + (T − δ)) .By the zero-order hold assumption

uc(tu,k + (T − δ)) = ukxc(tu,k + (T − δ)) = Ad(T − δ)xk +Bd(T − δ)uk

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Fig. 1. Schematic timeline for irregular sampling (circles)and actuation (crosses) rate

which implies that

yk = [CcAd(T − δ)]xk + [CcBd(T − δ) +Dc]uk . (7)

Note that (6) and (7) constitute an LTI discrete-timemodel. This model describes the continuous-time model(1)–(2) under a zero-order hold on the input withoutintroducing any additional states during the discretizationprocess.

However, there is one subtle detail that remains: with theconvention we have chosen for defining yk, we cannot useany arbitrary control scheme to control this model. Inparticular, for continuous-time causality to hold, uk canonly depend on yk−1, yk−2, yk−3, . . ., i.e. a discrete-timecontroller satisfies continuous-time causality if and only ifit is strictly causal in the discrete-time domain.

2.2 Irregular Sampling and Actuation

We now consider the case when the measurements aresampled at an irregular (but periodic) rate and the controlaction is updated as quickly as possible after a measure-ment is obtained. Figure 1 shows a schematic timelinefor this sampling and actuation scheme. The circles andcrosses in this figure respectively represent the momentsthe measurements are obtained and the control signal isupdated. In particular, we consider the case when themeasurements are obtained at time instances ty,k, k ∈ Zwhere the sequence

Tk := ty,k − ty,k−1is periodic with period N . For simplicity, we assume thatTk > δ, ∀k ∈ Z. The time instances when the controlis updated will be denoted by tu,k, k ∈ Z. Due tocomputational delay, we have the relationship (see Fig. 1)

ty,k−1 = tu,k − δ .

Notice that, as in subsection 2.1, we have shifted the timeindex associated with the measurements by one step. Alsonote that we have the relationship

tu,k+1 − tu,k = Tk .

We now define xk, uk, and yk as in (3)–(5) and notice that,by the zero-order hold assumption

xk+1 = Ad(Tk)xk +Bd(Tk)uk . (8)

Since Tk varies with time, these dynamics are time-varying. However, because Tk is periodic with period N ,we see that Ad(Tk) and Bd(Tk) are periodic with periodN , i.e. these dynamics are LPTV.

To find the discrete-time representation of the measure-ments, we first note that

ty,k = tu,k + (Tk − δ)which means that

yk = Ccxc(tu,k + (Tk − δ)) +Dcuc(tu,k + (Tk − δ)) .

Fig. 2. Schematic timeline for irregular sampling (circles)and regular actuation (crosses) rates

Using the zero-order hold assumption along with themethod in section 2.1, we obtain

yk = [CcAd(Tk − δ)]xk + [CcBd(Tk − δ) +Dc]uk . (9)

Note that (8) and (9) constitute a discrete-time state spacemodel. Also note that all of the model parameters areperiodic with period N , which implies that this state spacemodel is LPTV. This model describes the continuous-time model (1)–(2) under a zero-order hold on the inputwith sampling and actuation conditions described at thebeginning of this section. As in subsection 2.1, a discrete-time controller satisfies continuous-time causality if andonly if it is strictly causal in the discrete-time domain.

2.3 Irregular Sampling with Regular Control Updates

We now consider the case when the measurements aresampled at an irregular (but periodic) rate. However,unlike subsection 2.2, the control action is updated at aregular rate. Figure 2 shows a schematic timeline thatrepresents the instances measurements (shown by circles)arrive and control signal (shown by crosses) is updated.We choose t = 0 so that the control is updated at timeinstances

tu,k = kT, k ∈ Z .

In this system architecture, the role of computational delayis fundamentally different than in the previous sections.In particular, since the control updates are scheduled intime rather than the control being updated as quicklyas possible after receiving a measurement, we treat thecomputational delay as the constraint that uk can onlydepend on any measurements that arrive in the timeinterval (−∞, tu,k − δ]. Equivalently, the value of uk willbe updated at time step k+1 by measurements that arrivein the time interval (see Fig. 2)

Sk := (tu,k − δ, tu,k+1 − δ] .

We now define xk and uk as given in (3)–(4). Since thecontrol is updated regularly in time, we discretize (1) as

xk+1 = Ad(T )xk +Bd(T )uk .

These dynamics are LTI and do not depend on the mea-surement characteristics of the system.

We now find the representation of a measurement at a timeinstant t ∈ Sk. There are two cases to consider. The firstcase corresponds to t ≥ tu,k. In this case, we obtain

y(t) = Ccxc(t) +Dcuc(t)

= [CcAd(t− tu,k)]xk + [CcBd(t− tu,k) +Dc]uk

The second case corresponds to t < tu,k. In this case, wenote that

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xc(t) = Ad(t− tu,k−1)xk−1 +Bd(t− tu,k−1)uk−1

= Ad(t− tu,k−1)A−1d (T )[xk −Bd(T )uk−1]

+Bd(t− tu,k−1)uk−1 .

For notational convenience, we define

A := Ad(t− tu,k−1)A−1d (T )

B := Bd(t− tu,k−1)−Ad(t− tu,k−1)A−1d (T )Bd(T )

so that the previous expression can be written as

xc(t) = Axk + Buk−1 .

This yields

y(t) = Ccxc(t) +Dcuc(t)

= CcAxk + CcBuk−1 +Dcuk .

Since this expression depends on uk−1, we need to augmentthe state vector, i.e. we write the state dynamics of thediscrete-time system as[

xk+1

uk

]=

[Ad(T ) 0

0 0

] [xkuk−1

]+

[Bd(T )I

]uk . (10)

Again, these dynamics are LTI and do not depend on themeasurement characteristics of the system. The output attime instant t ∈ Sk corresponds to

y(t) = C

[xkuk−1

]+ Duk (11)

where

C =

{[CcA CcB

], t ∈ (tu,k − δ, tu,k)[

CcAd(t− tu,k) 0], t ∈ [tu,k, tu,k+1 − δ]

(12)

D =

{Dc, t ∈ (tu,k − δ, tu,k)

CcBd(t− tu,k) +Dc, t ∈ [tu,k, tu,k+1 − δ](13)

Although (11)–(13) describe the output at an arbitrarytime instance, these relationships do not fully describe thesystem output corresponding to a given time index. Inparticular, since there is no fixed relationship between thetimes at which measurements are obtained and the timesat which the control is updated, the number of samplesin the time interval Sk is not necessarily constant over k.For simplicity, we will consider a situation in which 0, 1,or 2 measurements may be made in any time interval Sk.We thus have three cases to consider. In all three cases,we will force the discrete-time model to have two outputs.

We begin by considering the case when, for a particularvalue of k, there are two measurements made in the timeinterval Sk (e.g. Si in Fig. 2 includes two measurementsyi,1 and yi,2); we denote the time instances correspondingto these measurements as t1 and t2. In this case, we choose

yk =

[y(t1)y(t2)

]. (14)

Note that yk captures all of the information that thecontroller can use to update the value of uk to uk+1.

We now consider the case when, for a particular value ofk, there are no measurements made in the time intervalSk (e.g. Si+2 in Fig. 2). In this case, the controller shouldaccept no inputs at time step k. Equivalently, the inputinto the controller should be zero. This motivates choosing

yk =

[00

]. (15)

Choosing this form for yk allows the system to have a time-invariant structure; yk is acting here as a placeholder sothat the discrete-time model has two outputs, even at timesteps when the controller has no inputs.

Finally, we consider the case when, for a particular valueof k, there is one measurements made in the time intervalSk (e.g. Si+1 in Fig. 2 includes only one measurement,yi+1,1); we denote the time instance corresponding to thismeasurement as t. In this case, we choose

yk =

[y(t)

0

]. (16)

As in the previous case, we are using zeros as placeholdersso that the discrete-time model has two outputs, even attime steps when the controller only has one input.

Under the assumption that the sampling of the systemis periodic, (10)–(16) define an LPTV discrete-time statespace model. This model describes the continuous-timemodel (1)–(2) under a zero-order hold on the input withsampling and actuation conditions described at the be-ginning of this section. As in section 2.1, a discrete-timecontroller satisfies continuous-time causality if and only ifit is strictly causal in the discrete-time domain.

3. CONTROL DESIGN

This section presents optimal LQG control design withvariance constraints for the discrete-time systems dis-cussed in section 2. Beside the performance of a sys-tem, there are other required specifications that shouldbe achieved by a designed controller to make that designimplementable. Robustness, bandwidth frequency and re-quired memory amount for storing the controller parame-ters are some of these specifications. Since we do not con-sider robustness and bandwidth frequency in our design,we cannot guarantee a lower bound for these system spec-ifications. However, the optimal LQG controller proposedin this section can be used as a reference in other typesof design that consider these specs, e.g. the closed-loopfrequency response of this system can be used as a guidein optimal H∞ control design. When the plant is modeledas an LPTV state space model whose entries are periodicwith period N , the proposed controller will also be LPTV.Hence,when the period is large, it may not be feasible tostore all the controller parameters in the available memory.However, as mentioned in the introduction, the goal ofthe optimal LQG control design in this paper is not tofind a controller to be implemented on a real system, butrather to assess the limitations of performance of varioushardware setups.

Although the output signals are not stationary when theplant and controller are both LPTV, their second-orderstatistics will be periodic, with periodN . Thus, a good wayto capture the performance of a controller is to computethe RMS and maximum 3σ values of the relevant signalsover a revolution of the system. As an example, it is shownin subsection 3.1 that the control objective in a HDD is tominimize the RMS of 3σ value of PES while keeping thevariance of control less than or equal to a value specifiedby the control designer.

As mentioned in section 1, we aim to find the limits ofperformance for three different sampling and actuation

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G

K

d

y

p

u

u

Fig. 3. System structure for control design

schemes. However, regardless of the sampling and actu-ation conditions, we can model the closed-loop system,which we denote as Gcl(K), as the block diagram shownin Fig. 3. In this block diagram, K and G are respec-tively the controller and the open-loop model containingboth the discretized plant and the discretized disturbancemodels. The signals d, p, y and u in Fig. 3 are respectivelythe disturbance, performance, measurement, and controlsignals.

Since the control problem we are considering is the min-imization of RMS of the 3σ value of performance signal(e.g. PES in a HDD) with a constraint on the mean controleffort variance, we can formulate the problem as (Conwayet al. (2008))

minK‖L1Gcl(K)‖2`2sn

subject to: ‖L2Gcl(K)‖2`2sn ≤ γ(17)

where γ is a prescribed upper bound on the mean con-trol variance, Gcl(K) is the closed-loop system shown inFig. 3, and the matrices L1 and L2 are chosen such thatL1Gcl(k) and L2Gcl(K) respectively represent the closed-loop system from d to p and from d to u. The norm beingconsidered in this problem is l2 semi-norm given by∥∥L1G

LPTVcl

∥∥2`2sn

:= lim supl→∞

1

2l + 1

l∑k=−l

tr E[pdk(pdk)T ]

.

which, for single output systems, corresponds to the RMSstandard deviation of that output signal when the inputsignal is white noise with covariance equal to the identitymatrix.

To solve this optimal control problem, we use the algo-rithm given by Conway et al. (2008), which uses derivativeinformation about the l2 semi-norm costs to achieve quasi-Newton convergence. In the following subsection we showan example of this optimal control design for a HDD withdifferent sampling and actuation schemes when it is intrack-following mode.

3.1 Control Design for Hard Disk Drives

The architecture we use for track-following control designwhen sampling and control update rates in a HDD areeither both regular or both irregular in time is shownin Fig. 4. The signals r, w and n are independent whitenoises with unit variance that respectively model theeffect of the non-repeatable runout (NRRO), windage,and measurement noise. NRRO is the random lateralmovement of the disk caused by the mechanical contacts inthe bearing motor, and windage is the off track motion at

Plant

K

NRROw

w r n

n

u

u

p = y

y

Fig. 4. Control architecture for either regular sampling andactuation or irregular sampling and actuation

Plant

K

NRRO nw

w r n1 n2

n

p

u

uy1

y2

n

n3

Fig. 5. Control architecture for regular actuation, butirregular sampling

the head caused by the turbulent nature of the air betweenthe disk and the actuator(Mamun et al. (2006)).

The signals p, y and u are respectively the performance,measurement and control signals. Note that the perfor-mance and measurement signals are both equivalent to thePES. The blocks K, NRRO, σn and σw are respectivelythe controller, the NRRO model, the standard deviationof measurement noise, and the standard deviation of thewindage. It is trivial to transform the block diagram inFig. 4 into the form in Fig. 3 by choosing

d =

[wnr

].

The control structure considered for a HDD with irregularsampling time and regular control update is shown inFig. 5. As in section 2.3, we assume that the numberof measurements arriving between two consecutive controlupdates is less than or equal to two; yi and σi are respec-tively the i-th component of yk discussed in section 2.3and its corresponding measurement noise, for i ∈ {1, 2}.The signals w, r and u have the same definition as thesignals in Fig. 4. The signals p and n3 are respectively theperformance signal and the noise contaminating p. Here,p represents the position error of the head at the timeinstances when the control signal is updated. Since thesetime instances do not coincide with the time instances atwhich the PES is measured, it is not, strictly speaking, thePES. However, it is a better measure of performance thanthe PES because it is sampled regularly in time. In par-ticular, if we were to use the PES itself as the measure ofperformance, taking the mean of the PES variance would,from a spatial perspective, weight the mean towards theparts of the disk for which measurements are more closelyspaced. For example, if 3/4 of the PES measurements

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occur over half of a disk revolution, the mean PES variancewould be strongly weighted toward the measurements inthat half of the disk in which the measurements are closelyspaced. Since making measurements more frequently tendsto yield better optimal performance, the mean PES wouldbe weighted towards the half of the disk in which theperformance is best. As an added benefit of choosing ourperformance signal as p rather than the PES, the statespace entries of the plant that correspond to the signal pare time-invariant. We can transform the block diagram inFig. 5 into the form in Fig. 3 by choosing

d =

wn1n2n3r

, y =

[y1y2

].

4. LIMITS OF PERFORMANCE IN HARD DISKDRIVES WITH IRREGULAR SAMPLING

This section examines the limits of the performance of anHDD in track-following mode for the three sampling andactuation schemes discussed in section 2. To this end, weare using LQG control design with a variance constraintto find the smallest achievable mean variance of p subjectto a constraint on the mean control effort variance. Per-formance results for six different system configurations arestudied in this section. These configurations are differentin terms of the controller used as well as the irregularityin the sampling and control rates. We will denote eachconfiguration in the form SR-CR-K. The first entry, SR,can either be R or I; these respectively denote a regularand an irregular sampling rate. Similarly, CR can either beR or I; these respectively denote a regular and an irregularcontrol update rate. Finally, the third entry K, representsthe controller used in that system. For example, I-R-K1

means that the sampling time is irregular, the actuationrate is regular, and the controller being used is K1.

In section 4.1 we compare the performance of the controllerprovided by an industry partner with the performance of adesigned controller when the sampling and actuation ratesare both regular. We then compare these two controllerswhen they are applied to a system with irregular samplingand actuation in section 4.2. Finally, section 4.3 is dedi-cated to control design and performance analysis when thesampling time is irregular, while the actuation rate is keptregular.

For LPTV systems, frequency response methods do notapply. Despite this, we quantified the performance of theclosed-loop system in frequency domain by using what wecall an “empirical Bode magnitude plot,” which uses aswept-sine approach to find an approximate frequency re-sponse. The assumption here is that inputting a sine waveinto the system produces a negligible response at otherfrequencies. Note that for a LTI system, the empiricalBode magnitude plot and Bode magnitude plot are equal.Since the systems in section 4.2, and section 4.3 are allLPTV, the empirical Bode magnitude plot is used to showthe behavior of these systems in frequency domain.

102

103

104

−40

−30

−20

−10

0

10

Hz

dB

R−R−KI

R−R−KLtiLqg

Fig. 6. Bode magnitude plot of the sensitivity function forR-R-KI and R-R-KLtiLqg, measured at the output ofthe controller

4.1 Regular Sampling and Actuation

In order to validate the accuracy of the continuous-timeplant model used for the chosen HDD, we compare the 3σvalue of PES computed based on real PES measured inthe HDD with the corresponding value simulated basedon our plant model. Since we have access to the PESduring track following in this particular setup, we are ableto find the 3σ value of PES in this HDD. Due to thepresence of disturbances during servo track writing, thecreated reference for the track-center deviates from beingperfectly circular in shape. These deviations in the trackreference constitute the repeatable runout (RRO) for thehead positioning servomechanism(Mamun et al. (2006)).We measure the PES at 32000 consecutive sectors duringtrack-following and compute the RRO at each sector byaveraging all of the PES values measured at that particularsector. Once the RRO is computed, we subtract it from thePES data to form the NRRO, and then find the 3σ valueof the NRRO. For this particular setup, this value is 4.95%of the track width.

Since we are interested in comparing the actual 3σ value ofPES with the corresponding value computed in simulation,we discretize the plant model by using a zero-order holdmethod, and use the track-following controller provided byour industry partner to calculate the 3σ value of the closed-loop PES. Here, the sampling time used for discretizationis constant and equal to the HDD sampling time.

The system just described is the first considered case andwe denote it as R-R-KI whereKI is the controller providedby our industry partner. The calculated 3σ value of PESin R-R-KI is listed in Tab. 1 and the Bode magnitudeplot is shown in Fig. 6. The 3σ value of PES in R-R-KI is8% less than the value calculated based on measurement.This can be due to two main reasons. First, unmodeledhigh frequency resonance modes decrease the performanceof the closed-loop system and second, the sampling timein the hard disk drive is not perfectly constant, which candecrease the performance. However, 92% accuracy in 3σvalue of PES validates the accuracy of our plant model.

Based on the method explained in §3 an optimal controlleris designed for the same open-loop plant as the one usedin R-R-KI . We denote this controller as KLtiLqg and theclosed-loop system as R-R-KLtiLqg. Since we are interestedin comparing the performance of KLtiLqg with KI underthe same conditions, we designed KLtiLqg such that thecontrol effort variance of KLtiLqg is less than or equal to

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the control effort variance used by KI . Thus, we computedthe control variance associated with the R-R-KI setupand used it as a variance constraint in the optimal LQGcontrol design. The performance of R-R-KLtiLqg, whichis listed in Tab. 1, shows that the calculated 3σ value ofPES is 20% less than the corresponding value in R-R-KI

while the control variances are equal. Since the PESperformance of R-R-KLtiLqg can be interpreted as thelimit of performance when the control effort variance isless than or equal to the control effort variance used byKI , the PES performance of R-R-KLtiLqg can be used asa good metric to evaluate the performance associated withother sampling and actuation schemes.

Table 1. Closed-loop performance for each sys-tem configuration

Control PES (nm)

Mean 3σ Max 3σ Mean 3σ Max 3σ

R-R-KI 1.00 1.00 3.21 3.21

R-R-KLtiLqg 0.96 0.96 2.56 2.56

I-I-KI 1.67 4.86 4.33 6.09

I-I-KLtiLqg ∞ ∞ ∞ ∞I-I-KLptvLqg 1.00 1.99 2.61 3.25

I-R-KLptvLqg2 0.95 1.62 2.57 2.85

4.2 Irregular Sampling and Actuation

In order to investigate the influence of irregularity onperformances of the two aforementioned controllers, theplant and disturbance models were discretized with themethod explained in §2.2 so that the sampling and controlupdate rate are both time-varying. Here we assumed thatthe elapsed time between successive measurements variedas T (1 + 0.3 sin(2πk/N), where T is the sampling timeof the chosen HDD and k is the discrete-time index.The closed-loop system containing KLtiLqg and mentionedopen-loop model is denoted as I-I-KLtiLqg and the onecontaining KI and the same open-loop model as theprevious case is called I-I-KI .

The results for these two systems are listed in Tab. 1,which show that KLtiLqg is not stabilizing when we have30% sampling time variation, and the performance ofKI is 35% degraded. We now examine the limits ofperformance of this system architecture by designing anoptimal LQG controller with the same variance constraintas KLtiLqg for the LPTV open-loop model. We denotethe closed-loop system corresponding to this controlleras I-I-KLptvLqg. The performance of this system listedin Tab. 1 shows that the LPTV LQG controller givesclosed-loop PES performance almost equal to the limitof performance for regular sampling and actuation, whichis 20% better than achieved by our industry partner’scontroller. Figure 7 shows the empirical Bode magnitudeplots for the sensitivity function of I-I-KI and I-I-KLptvLqg

measured at the output of the controller. As mentionedbefore, the 3σ value of PES is periodic when the systemis LPTV. For I-I-KI and I-I-KLptvLqg the stationary 3σvalue of PES measured at each sector in one revolution isshown in Fig. 8.

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Fig. 7. Empirical Bode magnitude plot of the sensitivityfunction for I-I-KI and I-I-KLptvLqg

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Fig. 8. 3σ PES value at each servo sector for each systemconfiguration

4.3 Irregular Sampling and Regular Actuation

All the five aforementioned systems are dealing withsampling time and control update rate that are eitherboth time-varying or both time-invariant. As explainedin §2.3, we are interested in designing a controller witha constant update rate for a system with an irregularsampling time. To evaluate the performance of a LPTVLQG controller that keeps the control effort variance lessthan or equal to the corresponding value in the previouslydiscussed systems, the continuous-time plant model usedfor previous cases is discretized by the method discussedin §2.3, i.e. the state and control updates in this systemare occurring regularly while the measurement arrivesirregularly. The performance of the closed-loop system,which is denoted as I-R-KLptvLqg2, is listed in Tab. 1.These results show that the optimal 3σ value of PESfor this architecture is 17% less than R-R-KI . Figures 8and 9 respectively show the stationary 3σ value of PESat each sector and the empirical Bode magnitude plotof I-R-KLptvLqg2. It should be mentioned that since theopen-loop plant with regular sampling time and irregularcontrol has more than one component in the signal y, wecannot drive it by the single-input single-output (SISO)KI . Hence, it is not possible to construct an I-R-KI systemto compare with I-R-KLptvLqg2.

5. CONCLUSIONS AND FUTURE WORK

In this paper, we have investigated the limits of perfor-mance for HDDs with various sampling and actuationschemes. For the HDD considered in this paper, there

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Fig. 9. Bode magnitude plot of the sensitivity functionfor I-R-KLptvLqg2, measured at the output of thecontroller

was significant performance degradation from the R-R-KI

case to the I-I-KI case and from the R-R-KLtiLqg case tothe I-I-KLtiLqg case. In other words, if there is significantvariation in the sample period over a revolution of the disk,simply ignoring that variation and using an LTI controllercan lead to significant degradation in performance.

Also, it turned out that the I-R-KLptvLqg2 configura-tion achieved a smaller RMS 3σ PES value than theI-I-KLptvLqg configuration. This means that, although wewere primarily interested in the I-R-KLptvLqg2 configura-tion because it did not complicate the design of notchfilters for the plant, this configuration turned out to yieldbetter performance than the I-I-KLptvLqg configuration forthis particular HDD. We suspect that the I-I-KLptvlqg casedid not perform as well because the control update rateslows down over the portions of the disk for which themeasurements are made as a slower rate. In other words,the increase in control update rate over some portionsof the disk (which would tend to improve performancein those portions of the disk) did not make up for thedecrease in the control update rate over the remainderof the disk (which would tend to degrade performance inthose portions of the disk).

Also in this paper, we discussed a novel method of dis-cretizing the continuous-time plant dynamics. For theregular measurement and control update scheme and theirregular measurement and control update scheme, thediscretization process did not introduce any addition statesinto the state space model. For the scheme with irregularmeasurements and regular control updates, the discretiza-tion process did introduce nu additional states into thestate space model. These additional states were unavoid-able essentially because in the worst-case scenario, thecomputational delay causes there to be more than a fullstep delay between when a measurement arrives and whenthe controller can use that measurements to update thecontrol signal.

In an physical HDD, there is a limited amount of memoryavailable to store the controller parameters. As a result,the controllers designed in this paper are not directlysuitable for implementation. However, recall that, in theI-R-KLptvLqg2 configuration, both the state update equa-tion and the equation for p are time-invariant in the plant.As a result of this, some of the controller parametersare time-invariant, which significantly reduces the amountof storage required for the controller parameters. If afurther decrease in the number of controller parametersis required, a technique such as the one given by Nie

et al. (2011) can be used to approximate the time-varyingcontroller parameters by a reduced set of controller pa-rameters. This will be discussed in more detail in a futurepaper.

6. ACKNOWLEDGMENT

This work was supported in part by the UC Berke-ley Computer Mechanics Laboratory, and the author,Ehsan Keikha, was supported by research grant from Na-tional Research Foundation (NRF), Singapore, under CRPAward No. NRF-CRP-4-2008-06.

REFERENCES

B. Chen, T. Lee, and V. Venkataramanan. Hard disk driveservo systems. Industrial Control Series, Springer, NewYourk, 2006.

B. Shahsavari, R. Conway, E. Keikha, R. Horowitz. Ro-bust control design for hard disk drives with irregularsampling. APMRC, Singapore, Oct, 2012.

R. Ehrlich. Methods for improving servo-demodulationrobustness. U.S. Patent 694398IB2.

J. Nie, E. Sheh, R. Horowitz. H∞ control for hard diskdrives with an irregular sampling. American ControlConference, San Francisco, June, 2011

R. Conway, and R. Horowitz. A quasi-Newton algorithmfor LQG controller design with variance constraints.ASME Dynamic Systems and Control Conference, SanFrancisco, Oct, 2008.

A.A. Mamun, G. Guo, Ch. Bi Hard disk drive: mechatron-ics and control. CRC Press 2006, pages 51–99, c2007.

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