Mechatronics 20 (2010) 453–463

Contents lists available at ScienceDirect

Mechatronics

journal homepage: www.elsevier .com/ locate/mechatronics

Switching control in blu-ray disk drives

Marcel Heertjes a,*, George Leenknegt b

a Eindhoven University of Technology, Department of Mechanical Engineering, PO Box 513, 5600 MD Eindhoven, The Netherlandsb TMC, Mechatronics, PO Box 700, 5600 AS Eindhoven, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 October 2009Accepted 31 March 2010

Keywords:Circle criterionInput-to-state stability(Nonlinear) observerOptical storage drivesSwitching control

0957-4158/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.mechatronics.2010.03.004

* Corresponding author.E-mail addresses: m.f.heertjes@tue.nl (M. Heertj

(G. Leenknegt).

To improve the nominal PID-controlled servo performances of a blu-ray disk drive, a switching observeris studied. Switching is done using a deadzone switching gain having the following properties: large-magnitude signals induce extra gain, small-amplitude signals do not. Additionally the observer partassures that large-magnitude signals induce extra low-pass filtering. As a result the design has low-fre-quency disturbance rejection properties resulting from high-gain feedback but with a favorable high-fre-quency noise response related to the properties of low-gain feedback. Robust stability is guaranteed usinga Lyapunov characterization of input-to-state stability, whereas improved performances under vibrationsand disk surface defects are demonstrated using two blu-ray disk drive setups.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In optical storage drives, proportional-integrator-derivative(pid) control continues demonstrating its effectiveness. Examplesinclude compact disk (cd), digital versatile disk (dvd), and nowa-days blu-ray disk (bd). Robust stability and performance, the tun-ing possibilities, and the straightforward implementation are keyfeatures which in practice appear difficult to compete with. Be-sides, control alternatives are generally subject to similar designlimitations [6,8,9]. Consider for example automotive applications,where high-gain feedback provides the means to more effectivelyhandle low-frequency shock and vibration. But in the presence ofscratches or fingerprints on the disk surface a low-gain feedbackinduces favorable noise response.

To deal with this trade-off, recently a switching control designwas reported, wherein the controller gain continuously operatesin on–off state [22]; see [1,2,4,20] for related advances in variablegain control. Subject to low-frequency shock and vibration, largeerror signals induce large controller gains. In the absence of shockand vibration, but in the presence of for example high-frequencydisk surface defects the original controller gain is restored and soare the original design properties in terms of noise amplification.Evidently this gives flexibility in dealing with the trade-off be-tween disturbance rejection and measurement noise sensitivity.But the problem of noise amplification under increased gain re-mains. This hampers the effectiveness of the switching when dis-

ll rights reserved.

es), george.leenknegt@tmc.nl

turbances like shocks, vibrations and disk surface defects occursimultaneously. As a solution to the problem, a switching observeris studied [11,15]; regarding drive disturbances other approachesare given in [21,24]. Different however from the numerical (waferstage) studies in [11] this paper considers the analysis, design,and demonstration of the switching observer applied to a blu-raydisk drive. The main contributions are: the implementationaspects and filter design of the observer, a rigorous stability analy-sis using a Lyapunov characterization of input-to-state stability(ISS), and an in-depth experimental verification on a blu-ray diskdrive.

The observer is based on a linear time-invariant part and a non-linear part consisting of a deadzone-based switching gain; see[3,14] for similar so-called circle-criterion observer designs. Thegain balances in a continuous manner the trade-off between low-frequency tracking under shock and vibration and high-frequencyroll-off in view of noises. Different from the fixed manner corre-sponding to any linear observer and depending on the servo erroramplitudes at hand, the observer gain is freely to adapt and actadequately in view of the time-varying mix of disturbances. Thisprovides more design freedom which can effectively be used to im-prove servo performances.

The paper is further organized as follows. In Section 2, an intro-duction is given toward modelling and (switching) control of stor-age drives. In Section 3, stability of the switching control design isstudied, where explicit reference is made to a Lyapunov character-ization of input-to-state stability. Section 4 contains a performanceassessment conducted on an industrial blu-ray disk drive. Thedrive is subjected to different disturbance situations combiningharmonic and broad-band noise disturbances with the occurrence

454 M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463

of disk scratches and fingerprints. In Section 5, a summary of themain conclusions is given.

Fig. 2. Block diagram of the simplified control scheme of a blu-ray disk drive inradial direction.

2. Blu-ray disk drives, control design, and motivation

Blu-ray disk drives (or more generally speaking optical storagedrives) are designed to read/write information on a disk via opticalmeans; see [17] for a general introduction. Hereto an objective lensis positioned about a disk track; see Fig. 1a for the nanometer scalein which this positioning occurs. The track contains digital infor-mation stored in lands and pits which both result from a fixedset of specified length variations. The objective lens is controlledseparately in (at least) two directions: the focus direction andthe radial direction, see Fig. 1b. The rotation frequency in the con-sidered blu-ray disk application is fixed at 40 Hz. In combinationwith disk eccentricity (less than 300 tracks) this forced excitationacts as one of the main sources of disturbance that needs to be re-jected by the control system.

A simplified control scheme of the objective lens dynamics inradial direction is represented by the block diagram of Fig. 2. Thescheme is centered about the plant transfer PðsÞ of which a stan-dard model is given by

PðsÞ ¼ 1ms2 þ bsþ k|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

objective lens

� xlp;1

sþxlp;1|fflfflfflfflffl{zfflfflfflfflffl}actuator

; ð1Þ

with s 2 C the Laplace variable [22]. The first part of this model con-tains the objective lens unit mass m = 0.5610�4 kg together with themechanical properties of the objective lens support, i.e., viscousdamping b = 0.01 Nsm�1, and stiffness k = 49 Nm�1, respectively.The second part of the model represents a first-order low-pass filterwith cut-off frequency xlp,1 = 2p � 20,000 rad s�1 related to actua-tor inductance in the transfer from voltage to current [5]. Input toP is the sum of forces generated by the feedback controller Cfb

and disturbances f. Output to P is given by y. Subtracted from thereference signal r this gives the radial error signal e. The feedbackcontroller Cfb is given by

CfbðsÞ ¼kpðs2 þ ðxd þxiÞsþxdxiÞ

xds|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}PID filter

�x2

lp;2

s2 þ 2bxlp;2sþx2lp;2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

low-pass filter

; ð2Þ

hence a proportional-integrator-derivative (PID) based filter withkp = 6.7 � 104 Nm�1 the radial loop gain, xi = 2p � 400 rad s�1 thecut-off frequency of a lag filter, xd = 2p � 1115 rad s�1 the cut-off

Fig. 1. Blu-ray di

frequency of a lead filter, xlp,2 = 2p � 20,000 rad s�1 the cut-off fre-quency of a low-pass filter, and b = 1 the corresponding dimension-less damping coefficient. The actual controller implementation hasmore functionality. For example, a repetitive controller is used tosuppress the periodic effect of disk eccentricity in combination withdisk rotation [18].

The series connection between controller CfbðjxÞ and plantPðjxÞ represented by the open-loop frequency response functionsCfbðjxÞPðjxÞ is depicted in Bode representation in Fig. 3. From200 Hz to 20 kHz it can be seen that a good correspondence is ob-tained between model and drive. Essentially this range relates tothe mass characteristics of the actuator, the latter having its firstresonance frequency about 50 Hz and the second (parasitic) reso-nance frequency at 50 kHz. Up to 1 kHz the effects of repetitivecontrol are visible; treatment of the repetitive controller itselffalls outside the scope of this paper [18]. In the industrial imple-mentation of the PID controller a different sampling frequency isused for the PI-part (44 kHz) than for the D-part (4 MHz); the for-mer stems from the industrial signal processor, whereas the latteris obtained through direct implementation in hardware; both areimplemented in commercially available hardware logic: PhilipsPNX7866 (Centaurus 2.2). As a result the open-loop characteris-tics in Fig. 3 shows no phase lag related to time-discretization.Robust stability and performance are met given the controllerbandwidth of 2879 Hz, phase margin of 37�, and gain margin of15 dB.

A drawback of the presented nominal design is the assumedstationarity of its disturbances. In automotive applications, dis-turbances vary in time due to road surface excitation and enginevibrations [24]. Similarly disk surface contamination and defectslike fingerprints and/or scratches generally occur incidentallyand in a time-varying manner [13]. Addressing non-stationarityof disturbances in the control design is therefore expected to sig-nificantly improve upon servo performances. For this purpose,the control scheme in Fig. 4 is proposed, where the input in

sk and drive.

Fig. 3. Bode plot of the open-loop characteristics CfbðjxÞPðjxÞ of a blu-ray disk drive with a bandwidth of 2879 Hz, a phase margin of 37 �, and a gain margin of 15 dB.

Fig. 4. Block diagram of the switching observer.

M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463 455

time-domain is given by the radial error signal e = e(t) and theoutput by the filtered radial error signal u = u(t). Note that thisfeedback is used as add-on to the nominal control design inFig. 2 thereby preserving the favorable properties associatedwith the nominal design. Key to the scheme is the fact thatthe nonlinear controller output u is used to construct the ob-served radial error signal e. As a result, a switching observer isobtained that satisfies the following frequency-domain relations

LfeðtÞg ¼F2ðsÞLfeðtÞ � eðtÞg �LfuðtÞg¼F2ðsÞLfeðtÞ � eðtÞg �F1ðsÞLfe/ðtÞg

¼ 1þF2ðsÞ þF1ðsÞLf/ðeðtÞÞgLfeðtÞg

� ��1

F2ðsÞLfeðtÞg; ð3Þ

where the signal e/ defined in time-domain is given by e/ðtÞ ¼/ðeðtÞÞeðtÞ and LfeðtÞg represents the Laplace transform: the inte-gral transform of time signal eðtÞ with kernel e�st. The switchinggain /ðeðtÞÞ is defined by a deadzone function

/ðeðtÞÞ ¼0; if jeðtÞj 6 d

a� adjeðtÞj ; if jeðtÞj > d;

8<: ð4Þ

with d a switching length and a a gain. It satisfies 0 6 /ðeðtÞÞ 6 a.The nonlinear observer characteristics in Eq. (3) can be partly as-sessed through the following low- and high-gain transfers,respectively,

eðsÞeðsÞ

� �/ðeðtÞÞ¼0

¼ F2ðsÞ1þF2ðsÞ

and

eðsÞeðsÞ

� �/ðeðtÞÞ¼a

¼ F2ðsÞ1þF2ðsÞ þ aF1ðsÞ

: ð5Þ

For small-amplitudes the observed error signal e falls inside theswitching length d, i.e., limt!1 sup jeðtÞj 6 d. If F2 has low-passcharacteristics and F1 does not, the observer tends to the low-gaincharacteristics determined by F2, see the first part of Eq. (5). Con-trarily for large-amplitudes, i.e., limt!1 sup jeðtÞj � d, the observertends to the high-gain characteristics given by F2ð1þ aF1Þ�1, seethe second part of Eq. (5). Switching the gain /ðeðtÞÞ ideally givesthe possibility to obtain improved low-frequency disturbancerejection without the high-frequency noise amplification whichwould otherwise result from using the more straightforwardswitching controller [22]:

LfuðtÞg ¼F1ðsÞLf/ðeðtÞÞeðtÞg: ð6Þ

Given /(e(t)) = a > 0 the transfer between e and r in Fig. 2 underswitching in Eq.(6) is then given by

eðsÞrðsÞ

� �/ðeðtÞÞ¼a

¼ 11þ ð1þ aF1ðsÞÞCfbðsÞPðsÞ

: ð7Þ

Since this sensitivity function expresses the ability to keep thelow-frequency servo errors small in view of input disturbances, itfollows that a factor of 1þ aF1 extra low-frequency disturbancerejection is induced when compared to the case of a = 0. However,this improvement comes at the price of an increased sensitivity tohigh-frequency noises. This follows from the transfer between yand r which reads

yðsÞrðsÞ

� �/ðeðtÞÞ¼a

¼ ð1þ aF1ðsÞÞCfbðsÞPðsÞ1þ ð1þ aF1ðsÞÞCfbðsÞPðsÞ

: ð8Þ

So increased low-frequency disturbance rejection under high-gain feedback by a factor of 1þ aF1 corresponds to increasedhigh-frequency noise sensitivity by the same amount. This is the

456 M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463

waterbed effect. It is the aim of the switching observer to avoid thiseffect. Herein the choice of F1; F2, and /ðeðtÞÞ and its parametersis key and as such will be addressed in the context of stability inSection 3 and in the context of performance in Section 4.

3. Design and stability of the switching observer

Stability of the nominal feedback controlled system in Fig. 2having the switching observer in Fig. 4 strongly depends on theproperties of its filters: F1 and F2. The design of F1 aims at alow-frequency gain increase to improve the disturbance rejectionproperties of the controller. Contrarily, the design of F2 aims ata high-frequency gain reduction as to improve the controller’s sen-sitivity to noise. Given the circle criterion to assess stability, it willbe shown that both aims not necessarily conflict. Additionally thecircle criterion is used as a tool for filter design [11,16]. For thispurpose, the absolute stability representation of Fig. 5 is adopted,where GðsÞ represents the transfer of the nominal complementarysensitivity function, or

GðsÞ ¼ CfbðsÞPðsÞ1þ CfbðsÞPðsÞ

: ð9Þ

Closed-loop stability is guaranteed on the basis of the nextresult.

Theorem 1. Assume PðsÞ in Fig. 2 is globally asymptotically stabilizedby CfbðsÞ for uniformly bounded disturbances r(t) and f(t). Moreoverassume F1ðsÞ; F2ðsÞ, and ð1þF2ðsÞÞ�1 Hurwitz. Then any switch-ing controller of the form as considered in Fig. 4 with 0 6 /(�) 6 astabilizes PðsÞ if

RF1ðsÞF2ðsÞGðsÞ �F1ðsÞ

1þF2ðsÞ

� �P �1

a: ð10Þ

The proof is constructed as follows. If CfbðsÞ globally asymptot-ically stabilizes PðsÞ then the closed-loop transfer GðsÞ is Hurwitz.Furthermore from the feedback connection of Fig. 5 a stable trans-fer G0ðsÞ follows that reads

G0ðsÞ ¼ F1ðsÞ1þF2ðsÞ

F2ðsÞGðsÞ � 1f g: ð11Þ

The first part in this series connection is Hurwitz. The secondpart is also Hurwitz because the poles of F2ðsÞGðsÞ � 1 are deter-mined by the poles of F2ðsÞGðsÞ. Hence G0ðsÞ is Hurwitz whichcombined with the fact that 0 6 /(�) 6 a allows for proving abso-lute stability through the circle criterion [23]. It is important tonote that stability through (10) is satisfied regardless the value ofthe switching length d in (4). Stability-invariance thus rendersthe switching length tuning performance-based only. This tuningcan be done empirically, see for example [22], where d ¼limt2½t1 ;t2 � sup jeðtÞj, i.e., the smallest absolute value that encapsu-lates the noise in the interval [t1, t2], or more rigorously as donein [12] using an extremum seeking algorithm.

Fig. 5. Block diagram of the nonlinear control scheme in Luré-form.

Based on Theorem 1 a more explicit stability argument can bederived using a Lyapunov characterization of input-to-state stabil-ity [19]. A clear distinction between stationary and non-stationarydisturbances, which in the context of the paper mainly refers to thedistinction between steady-state and transient (incidental) distur-bances, respectively, is then obtained from the next result.

Theorem 2. Given the uniformly bounded disturbances r(t) and f(t),the nominal controlled system in Fig. 2 using the switching observer inFig. 4 with switching function (4) is input-to-state stable if

RF1ðsÞF2ðsÞGðsÞ �F1ðsÞ

1þF2ðsÞ

� �P �1

a: ð12Þ

The proof is given as follows. Through the circle criterion,Eq.(12) implies the existence of a positive definite P 2 Rn�n thatsatisfies the positive real condition

ATPþ PA ¼ �qqT � �P;

Pb1 ¼ c�ffiffiffi2a

rq;

ð13Þ

with A 2 Rn�n; q; b1; c 2 Rn, and � > 0. By choosing the Lyapunovfunction candidate V(t),

c1kxðtÞk26 VðtÞ ¼ xTðtÞPxðtÞ 6 c2kxðtÞk2

; ð14Þ

with xðtÞ 2 Rn and c2 P c1 > 0 and which stems from the state-spacerepresentation:

_xðtÞ ¼ AxðtÞ þ b1vðtÞ þ b2rðtÞ þ b3f ðtÞ;eðtÞ ¼ cTxðtÞ;vðtÞ ¼ �/ðeðtÞÞeðtÞ;

ð15Þ

where b2;b3 2 Rn, it follows that

_VðtÞ 6 ��xTðtÞPxðtÞ � ðxTðtÞqþffiffiffi2a

rvðtÞÞ2 þ 2xTðtÞPðb2rðtÞ

þ b3f ðtÞÞ6 ��VðtÞ þ 2xTðtÞPðb2rðtÞ þ b3f ðtÞÞ; ð16Þ

where explicit use is made of the sector condition:aeðtÞvðtÞ 6 �v2ðtÞ. Since 2kb2r(t) + b3f(t)kkx(t)k 6 kkb2r(t) +b3f(t)k2 + (1/k)kx(t)k2 for any k > 0, it follows that

_VðtÞ 6 ��rVðtÞ � �ð1� rÞVðtÞ þ c2kkb2rðtÞ þ b3f ðtÞk2

þ c2

kkxðtÞk2; ð17Þ

with 0 < r < 1. Now suppose

kxðtÞkP

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2k

2

ð1� rÞ�c1k� c2

smaxfkb2k; kb3kgkrðtÞ þ f ðtÞk1; ð18Þ

with kr(t) + f(t)k1 = suptP0jr(t) + f(t)j. It then follows that

�ð1� rÞVðtÞP c2kkb2rðtÞ þ b3f ðtÞk2 þ c2

kkxðtÞk2

; ð19Þ

such that _VðtÞ 6 ��rVðtÞ for all kx(t)k satisfying (18). Hence system(15) is input-to-state stable with

kxðtÞk 6 qðkxð0Þk; tÞ þ cðkrðtÞ þ f ðtÞk1Þ; ð20Þ

where

qðkxð0Þk; tÞ ¼ c2

c1kxð0Þkexp

��c2

tð21aÞ

cðkrðtÞþ f ðtÞk1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2

2k2

ð1� cÞ�c21k� c1c2

smaxfkb2k;kb3kgkrðtÞþ f ðtÞk1:

ð21bÞ

M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463 457

In view of the distinction between stationary vibrations andnon-stationary disk surface defects, q in Eq. (21a) expresses the de-cay of the nonlinear system response and therefore associates withtransient (non-stationary) effects that are induced only inciden-tally through r(t) or f(t). Contrarily c in Eq. (21b) expresses a stea-dy-state level for the error response and as such relates to theconstant level of stationary disturbance in either r(t) or f(t).

Apart from assessing stability, the frequency-domain inequalityof Eq. (10) also gives direction toward the design and tuning of thefilters F1 and F2. This is illustrated in Fig. 6 using the Nyquist plotof the frequency response functions G0ðjxÞ resulting from eitherthe previously derived model or from blu-ray disk drive measure-ment. It can be seen that stability is guaranteed by satisfying Eq.(10) for a 6 1/0.26 � 3.8. This extra gain, inducing a potential dis-turbance suppression of 13.6 dB, is obtained with a loop-shapingfilter F1:

Fig. 6. Nyquist plot of G0 giving a graphical interpretation of Eq. (10).

Fig. 7. Bode magnitude plots of the measured (solid) and simulated (dashed) frequencfrequency response function limits (lower part) of Eq.(5); the low-gain limit (dashed) an

F1ðsÞ ¼s2 þ 2b1xnsþx2

n

s2 þ 2b1xnsþx2n; ð22Þ

with xn = 2p � 3250 rad s�1 the notch frequency and b1 = 0.216 andb2 = 5.16 its dimensionless damping coefficients. Its physical inter-pretation stems from broad-band amplification (in combinationwith a) except for the notch frequency region near 3250 Hz, whereamplification easily renders the nominal closed-loop system unsta-ble; see [10] for closed-loop circle criterion tuning and other loop-shaping strategies. In the switching observer scheme of Fig. 4, F2 isrepresented by a first-order low-pass filter:

F2ðsÞ ¼cxlp

sþxlp; ð23Þ

with xlp = 2p � 250 rad s�1 the cut-off frequency of the filter andc = 3.1 a gain. The low-pass filter is designed to give the nonlinearobserver sufficient low-pass characteristics beyond the controllerbandwidth (2879 Hz). Thus avoiding amplification of noises. It of-fers an implementation-friendly alternative toward higher-orderfilters that are often used in observer design.

Both F1 and F2 are given in Bode magnitude representationin the upper part of Fig. 7. It can be seen that given a samplingfrequency of 44 kHz for the PI-part of the feedback controllerthe continuous-time models in Eqs. (22) and (23) deviate fromthe discrete-time implementation in terms of zero-order-holdeffects. This fact is used in the design of F1. Recall from Eqs.(10), (22) and (23) that lims!1G

0ðsÞ !F1ðsÞ ! 1, hence the gainis limited at a = 1. The zero-order-hold effects introduced in theimplementation of F1 (and adding low-pass filter properties)are used in surpassing this limitation and obtain the current valueof a � 3.8. The cut-off frequencies in F1 and F2 are tunedsufficiently discriminative as to simultaneously access large extragain values and still counteract significant parts of the increasednoise response. This is shown in the lower part of Fig. 7 by the

y response functions F1ðjxÞ and F2ðjxÞ (upper part) and the measured observerd the high-gain limit (solid).

458 M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463

measured frequency response functions from e to e in terms ofthe linear observer limits; see Eq. (5). Apart from the first-orderlow-pass characteristics, it can be seen that the high-gain obser-ver gives access to extra high-frequency reduction by a factor ofð1þ aF1Þ�1. This reduction compensates for the increased noiseamplification induced under high-gain feedback. Apart from extranoise suppression, the high-gain observer limit shows reducedlow-frequency disturbance rejection. This being part of a lineardesign trade-off.

4. Experiments with the switching observer

To assess the performances of the switching observer applied toa blu-ray disk drive three sets of switching observer parameters,see Table 1 for the parameter values, are compared with the nom-inal linear controller such as given in Fig. 2; set1 has nominal low-pass characteristics; set2 has an increased cut-off frequency givingfocus to low-frequency shock suppression; set3 has a decreasedcut-off frequency giving focus to high-frequency noise response.Robust stability through Eq. (10) is kept equal for all sets. Giventhese sets the effect in terms of nonlinear controller limits is shown

Table 1Three sets of parameter values for F1 and F2.

Filter Parameter Set1 Set2 Set3

F1 xn in rad s�1 23625 26232 20420b1 0.18 0.18 0.216b2 4.3 4.3 5.16

F2 xlp in rad s�1 1571 1571 1571c 12.4 49.6 3.1

Fig. 8. Bode magnitude plots of the sensitivity (left) and complementary sensitivity (rig(with d = 0).

in Fig. 8 in Bode magnitude representation. The linear limitsobtained with d = 0 are assessed through the sensitivity S andcomplementary sensitivity Sc function, respectively, which are de-fined by

SðsÞ ¼ eðsÞrðsÞ ¼

C1ðsÞC1ðsÞ þ C2ðsÞ

;

ScðsÞ ¼yðsÞrðsÞ ¼

C2ðsÞC1ðsÞ þ C2ðsÞ

;

ð24Þ

where

C1ðsÞ ¼ 1þ aF1ðsÞ þF2ðsÞ;C2ðsÞ ¼ CfbðsÞPðsÞðC1ðsÞ þ aF1ðsÞF2ðsÞÞ:

ð25Þ

The figure clearly shows the aim in low-frequency improve-ment through the sensitivity functions under high-gain feedback(left part) without any high-frequency deterioration in the comple-mentary sensitivity functions (right part). For the considered de-sign studies, the trade-off between attenuation in the centralfrequency interval and suppression in the low-frequency interval,due to the waterbed effect, is met for each of the sets differently.Note that these limits only partly define the assessable perfor-mances associated with the observer design. The remaining partis determined by the switch itself.

For the experiments, two setups are used: a test setup (seeFig. 9a) used for disturbance injection directly at the objectivelens, and a shaker setup (see Fig. 9b) used for vibration excitationof the entire blu-ray disk drive module. Both setups exploit aprogramable digital signal processor (dsp) in combination witha micro-processor (as implemented in the Philips PNX7866) anda computer to apply the desired control designs. A measurementboard is used to apply the controller commands and monitor the

ht) frequency response functions for the nonlinear controller limits: a = 0 and a = 3

Fig. 9. Measurements with an industrial blu-ray disk drive.

M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463 459

desired (analog) signals. A dsp debug board is used to monitor thedesired (digital) signals obtained from the digital signal proces-sor; boards and drive use external power supply. The voice-coilshaker having a low-friction hydraulically supported vibrationtable is capable of generating accelerations over 100 ms�2 witha bandwidth beyond 1 kHz.

4.1. Harmonic excitation in the presence of disk scratches

To study the potential of the switching controller in suppress-ing vibrations an harmonic signal is injected to the radial controlloop of the blu-ray disk drive while operating in the presence ofscratches on the disk surface; this is done at the test setup ofFig. 9a. The results are shown in the left part of Fig. 10 bytime-series measurement of the considered signals in Fig. 4given the parameter sets in Table 1. The drive is controlled onthe basis of the (nominal) linear controller Cfb in Eq. (2). Theinput of the switching observer in Fig. 4 is connected to the out-

Fig. 10. Time-series measurement of the observer error signals: e(t), eðtÞ; e/ðtÞ, and u(t)when subjected to two disturbances: a 100 Hz harmonic disturbance and disk scratches

put of the nominal controlled drive in Fig. 2 but the output ofthe switching observer is not yet applied to the nominal feed-back loop. As a result, the error signals e measured with differ-ent switching observer parameter sets are identical apart fromnoise contributions; this is the upper left part of Fig. 10. Theswitching length is fixed at d = 18 nm. It comprises a trade-offin which noises sufficiently fall inside the deadzone band,whereas the responses that need to be suppressed clearly exceedit. The responses result from a 100 Hz harmonic excitation andthe repetitive occurrence of several scratches on the disk. For again of a = 2, the second part in the (left) figure shows the effectof the switching low-pass filter in terms of the filtered errorsignals e. Clearly set3 induces less noise as compared to set2.Also indicated are the switching incidents. The third part ofthe (left) figure shows the effect of the switching gain in termsof the signals e/. Outside the interval indicated by the switchinglength extra gain is induced. Inside the interval, e/ becomes zero.The output of the switching controller is shown in the lower (left)

, see also Fig. 4, under nominal control (left part) and switching control (right part).

460 M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463

part of Fig. 10. The largest suppression of harmonic vibrations isexpected from set2 which generates the largest output. The ef-fect of the considered switching observers is shown in the rightpart of Fig. 10. The switching observer output u is connected tothe nominal feedback loop, see Fig. 2 and Fig. 4. A significantreduction is demonstrated in the amplitude of the harmonic er-ror responses e; the amplitudes drop from �97.5 nm to �44.5nm, hence a factor of 2.3 improvement. All sets induce compara-ble vibration suppression without significant differences in noiseresponse. This follows from e and u but also from the internalsignals e and e/. As such, vibration suppression is hardly com-promised by the switching observer: small c that via (23) implyreduced loop gains (lower left part of Fig. 10) seemingly yieldequally improved error levels e(t); compare the switching con-trolled upper right part of Fig. 10 with the nominal controlledupper left part.

Fig. 11. Cumulative power spectral density analysis of the measured signals: eðtÞ; eðtÞ; e/

to broad-band noise disturbances; the switching control parameter sets stem from Tabl

Fig. 12. Wavelet-based (multi-resolution) frequency-time analysis of the radial error sign(right part) subjected to two disturbances: a 100 Hz harmonic disturbance and a heavy

4.2. Broad-band noise excitation

To study the behavior of the switching observer in terms of noisesensitivity, a broad-band noise signal is injected to the radial errorcontrol loop of the blu-ray disk drive; this is done using the test set-up in Fig. 9a. For the considered parameter sets the effect is not atall clear in time-domain. The effect, however, becomes apparentin frequency-domain. This is shown in Fig. 11, where a cumulativepower spectral density (cpsd) representation is adopted. In terms ofe; e; e/, and u the effect is clear. Lowering the cut-off frequencies inthe switching controller induces less noise in the output u. Depend-ing on the closed-loop characteristics and the disturbances at handthe effect also applies to the closed-loop errors e, where it can beseen that set3 approximates the noise response corresponding tothe nominal control system. As a result, the increase in noise re-sponse under switching can remain fairly limited.

ðtÞ, and u(t), see also Fig. 4, under both nominal and switching control and subjectede 1.

al e(t) under nominal control (left part) and switching control using parameter set2fingerprint.

M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463 461

4.3. Harmonic excitation in the presence of fingerprints

To study the combined effect of the switching controller inimproving low-frequency disturbance suppression without deteri-orated high-frequency noise response, an harmonic excitation isapplied to the radial error loop of a blu-ray disk drive in the pres-ence of an artificial fingerprint on the disk surface; the experiment

Fig. 13. Wavelet-based (multi-resolution) frequency–time analysis of the radial error sig(right part) subjected to two disturbances: a 100 Hz harmonic disturbance and a heavy

Fig. 14. Time-series measurement (upper part) and cumulative power spectral density acontrol and subjected to 100 Hz shaker vibrations combined with a heavy fingerprint; t

is conducted at the test setup of Fig. 9a. Via time-frequency analy-sis, the switching observer (using parameter set1 and set2 from Ta-ble 1) is compared with the nominal control design from Fig. 2. Theeffect under nominal control is shown in Fig. 12a. In the lower partof the figure, the considered (time-domain) error signal e is de-picted. In the upper part, the time-domain (and so-called) MIRN-signal is depicted. This signal is a measure for the total amount

nal e(t) under switching control using parameter set2 (left part) and parameter set1fingerprint.

nalysis (lower part) of the radial error signal e(t) under both nominal and switchinghe switching control parameter sets stem from Table 1.

462 M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463

of reflected light coming from the disk and as such gives a level ofintensity that drops as soon as the light reflective properties of thedisk deteriorate. In the upper part of Fig. 12a this drop is used todetect the occurrence of a heavy fingerprint; the heavy fingerprintis such that no defect detection or other nonlinear mechanism inthe PID controller is activated. The central part of Fig. 12a showsan amplitude-based time-frequency representation of the errorsignal e using a multi-resolution time-frequency decompositionon the basis of wavelet analysis [7]. Clearly visible is the 100 Hzharmonic excitation. Less visible is the effect of the fingerprinton the radial error response under nominal control. For the switch-ing controller and using parameter set2 from Table 1 the effect isshown in Fig. 12b. The rms-valued amplitude of the harmonic re-sponse is significantly reduced (from �80 nm in Fig. 12a to�35 nm in Fig. 12b; a reduction of 230%) but the system acts hea-vier on the occurrence of the fingerprint (compare the central partof Fig. 12a with the central part of Fig. 12b). Since set2 essentiallyaims at high-gain feedback, i.e., putting less weight on keeping asmall noise response, a more optimal parameter set for the consid-ered disturbances is given by set1. This is shown in Fig. 13 by com-paring the results corresponding to set1 (right part) with theresults from set2 (left part). Favorable disturbance rejection prop-erties (compare the amplitudes in the lower part of Fig. 13b forset1 with the amplitudes in the lower part of Fig. 12a under nom-inal control) are obtained without deteriorating the noise responseunder fingerprint disturbances. Design freedom in the switchingobserver is therefore used to deal with the mixed set of distur-bances. For a similar experiment but performed at the shaker setupof Fig. 9b this also follows from Fig. 14. In the experiments the blu-ray disk drive module is exposed to a 100 Hz harmonic excitationwhile operating under a heavy fingerprint. Time-series synchroni-zation of the fingerprint is done with the MIRN-signal; see theupper part of Fig. 14. For both nominal and switching control(the latter using parameter values from Table 1) the effect on theradial error signals e (this is the central part of Fig. 14) is best seenthrough cumulative power spectral density analysis; this is thelower part of Fig. 14. Here the fact is used that the (cumulative)power spectral density evaluation obtained from the error signalse and denoted at each frequency sample i 2 {1, . . .,n} by (c)psd[i]can be written as

cpsd½i� ¼Xn

i¼1

psd½i� ¼Xn

i¼1

w½i�psd½i� þXn

i¼1

ð1� w½i�Þpsd½i�; ð26Þ

with n = 128 the number of equally-spaced frequency points withfrequency resolution of 97.84 Hz, and w a frequency separationfunction defined by

w½i� ¼1; if i 6 k

0; if i > k;

�ð27Þ

with k = 11 an integer value used to separate the frequency contri-butions on either side of 11 � 97.84 � 1 kHz. For frequency contri-butions below 1 kHz (left part) or beyond 1 kHz (right part) theeffect is clear: the switching observer can maintain a nominal(low-gain) noise response with superior low-frequency (and high-gain) disturbance rejection properties.

5. Conclusions

Switching control provides the possibility to enhance low-fre-quency disturbance rejection while largely keep a favorable high-frequency noise response. Key to the design is the switchingobserver. Small-magnitude error signals which are considerednoises do not induce extra gain, whereas large-magnitude errorsignals do because of a deadzone switching. At the same time,large-amplitude signals induce more low-pass filtering. As such

low-frequency servo signals that benefit from high-gain feedbackare suppressed effectively under increased gain, whereas high-frequency servo signals that do no benefit from high-gain feedbackhardly induce any extra gain, thus keeping a favorable noise response.Stability of the switched system is guaranteed on the basis ofLyapunov arguments. Using the circle criterion, the tuning anddesign of the switching filters F1 and F2 is shown to add sufficientrobustness to model uncertainty. Roll-off in both filters due totime-discretization provides the means to force the circle criterionevaluation to zero for x ?1 and at the same time balance perfor-mances. The former to give access to extra gain, the latter to allowfor extra (incidental) low-pass filtering. A Lyapunov characteriza-tion of input-to-state stability expands the notion of stability tononlinear performance because of the distinction between tran-sient and steady-state responses and therefore explicitly addressesthe non-stationarity of disturbances. In view of this distinctionthe effectiveness of the switching controller is demonstrated ontwo blu-ray disk drive setups. Subject to steady-state harmonicexcitation with incidental disturbances caused by either diskscratches or fingerprints, the ability is shown to obtain improvedlow-frequency disturbance rejection properties with a favorablehigh-frequency noise response.

References

[1] Al-Sweiti Y, Söffker D. Modeling and control of an elastic ship-mounted craneusing variable gain model-based controller. J Vib Contr 2007;13(5):657–85.

[2] Armstrong BSR, Gutierrez JA, Wade BA, Joseph R. Stability of phase-based gainmodulation with designer-chosen switch functions. Int J Robot Res 2006;25(8):781–96.

[3] Arcak M. Certainty-equivalence output-feedback design with circle-criterionobservers. Trans Autom Contr 2005;50(6):905–9.

[4] Baek JS, Chung CC, Tomizuka M. Anti-shock controller design for optical diskdrive systems with a nonlinear controller. In: Proceedings of American controlconference, Minneapolis, Minnesota, USA; 2006. p. 1982–9.

[5] Bittanti S, Dell’Orto F, Di Carlo A, Savaresi SM. Notch filtering and multiratecontrol for radial tracking in high-speed DVD-players. Trans Consum Electr2002;48(1):56–62.

[6] Braslavsky JH, Seron MM, Mayne DQ, Kokotovic PV. Limiting performance ofoptimal linear filters. Automatica 1999;35(2):189–99.

[7] Büssow R. An algorithm for the continuous Morlet wavelet transform. MechSyst Signal Process 2007;21(8):2970–9.

[8] Freudenberg JS, Hollot CV, Middleton RH, Toochinda V. Fundamental designlimitations of the general control configuration. Trans Autom Contr2003;48(8):1355–70.

[9] Goodwin GC, Salgado ME, Yuz JI. Performance limitations for linear feedbacksystems in the presence of plant uncertainty. Trans Autom Contr2003;48(8):1312–9.

[10] Heertjes MF, Steinbuch M. Circle criterion in linear control design. In:Proceedings of American control conference, Seattle, Washington, USA;2008. p. 3176–81.

[11] Heertjes MF, Leenknegt G, Van Goch B, Nijmeijer H. Improved noise sensitivityunder high-gain feedback in nano-positioning motion systems. In: Proceedingsof American control conference, St. Louis, Missouri, USA; 2009. p. 283–8.

[12] Heertjes MF, Van Goch B, Nijmeijer H. Optimal switching in the control ofwafer stages. In: Proceedings of symposium on mechatronic systems,Cambridge, Massachusetts, USA; in press.

[13] Helvoirt van J, Steinbuch M, Leenknegt GAL, Goossens HJ. Disk defectclassification for optical disc drives. IEEE Trans Consum Electr 2005;51(3):856–63.

[14] Ibrir S. Nonlinear circle-criterion observers in discrete-time. In: Proceedings ofAmerican control conference, New York City, USA; 2007. p. 254–9.

[15] Johansson R, Robertsson A, Shitiaev A. Observer-based strict positive real (SPR)switching output feedback control. In: Proceedings od conference on decisionand control, Atlantis, Paradise Island, Bahamas; 2004. p. 2811–6.

[16] Rahiman W, Li B, Wu B, Ding Z. Circle criterion based nonlinear observerdesign for leak detection in pipelines. In: Proceedings of conference on controland automation, Guangzhou, China; 2007. p. 2993–8.

[17] Stan SG. The CD-ROM drive; a brief system description. Dordrecht: KluwerAcademic Publishers; 1998.

[18] Steinbuch M. Repetitive control for systems with uncertain period-time.Automatica 2002;38(12):2103–9.

[19] Sontag ED. Mathematical control theory – deterministic finite dimensionalsystems. New York: Springer; 1998.

[20] Tahboub KA. Active nonlinear vehicle-suspension variable-gain control. In:Proceedings of conference on control and automation, Limassol, Cyprus; 2005.p. 569–74.

M. Heertjes, G. Leenknegt / Mechatronics 20 (2010) 453–463 463

[21] White MT, Tomizuka M, Smith C. Improved track following in magnetic diskdrives using a disturbance observer. Trans Mech 2000;5(1):3–11.

[22] Van de Wouw N, Pastink E, Heertjes MF, Pavlov AV, Nijmeijer H. Performanceof convergence-based variable-gain control of optical storage drives.Automatica 2008;44(1):15–27.

[23] Yakubovich VA, Leonov GA, Gelig AKh. Stability of stationary sets in controlsystems with discontinuous nonlinearities. Singapore: World Scientific; 2004.

[24] Yokoyama E, Nagasawa M, Katayama T. A disturbance suppression controlsystem for car-mounted and portable optical disk drives. Trans Consum Electr1994;40(2):92–9.