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IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017 3301813

Adaptive Loop Shaping for Wideband Disturbances Attenuation

in Precision Information Storage Systems

Liting Sun1, Tianyu Jiang2,andXuChen

2

1Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720 USA

2Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269 USA

Modern hard disk drive (HDD) systems are subjected to various external disturbances. One particular category, deﬁned as wide-

band disturbances, can generate vibrations with their energy highly concentrated at several frequency bands. Such vibrations are

commonly time-varying and strongly environment/product-dependent, and the wide spectral peaks can occur at frequencies above

the servo bandwidth. This paper considers the attenuation of such challenging vibrations in the track-following problem of HDDs.

Due to the fundamental limitation imposed by the Bode’s integral theorem, the attenuation of such wide-band disturbances may

cause unacceptable ampliﬁcations at other frequencies. To achieve a good performance and an optimal tradeoff, an add-on adaptive

vibration-compensation scheme is proposed in this paper. Through parameter adaptation algorithms that online identify both the

center frequencies and the widths of the spectral peaks, the proposed control scheme automatically allocates the control efforts with

respect to the real-time disturbance characteristics. The effect is that the position error signal in HDDs can be minimized with

effective vibration cancelation. Evaluation of the proposed algorithm is performed by experiments on a voice-coil-driven ﬂexible

positioner system.

Index Terms—Adaptive loop shaping, hard disk drives (HDDs), parameter adaptation algorithm (PAA), vibration compensation,

wideband disturbance.

I. INTRODUCTION

THE production of data is expanding at an astonishing

pace. As the main cost-effective data storage device,

modern hard disk drive (HDD) systems are facing ever

stringent requirements to further increase the storage capac-

ity (track density) and the data access speed. However, as

the track density increases, the regulation of the read/write

head becomes more difﬁcult, particularly in vibration-intensive

environments. For instance, in all-in-one compact comput-

ers, the HDDs are subjected to signiﬁcant audio vibrations

generated by the audio systems [1]. Similar problems occur

in smart TVs, data centers, and cloud storage. In the latter

cases, a large number of HDDs are stacked and mounted in

compact enclosures; it is thus crucial to attenuate the vibration

disturbances that get excited and transmitted among the drives

and the cooling fans.

Extensive research efforts in the ﬁelds of design, control,

and signal processing have been made to attenuate external

disturbances in HDDs in the past decades. From the hard-

ware viewpoint, Hirono et al. [2] installed a spoiler between

the disks and studied the effect of the spoiler in reducing

the ﬂow-induced vibrations. Choi and Rhim [3] applied a

herringbone groove pattern to the plane disk damper and

effectively suppressed the axial vibrations. Mou et al. [1]

analyzed the structurally transmitted vibrations in HDDs

induced by speakers, fans, and CD/DVD drives in notebook

PCs and concluded that soft chassis and mounting could

be helpful to suppress such vibrations at frequencies above

1000 Hz. Certainly, the hardware solutions are potentially

Manuscript received September 2, 2016; revised November 28, 2016;

accepted January 12, 2017. Date of publication January 17, 2017; date of

current version April 17, 2017. Corresponding author: X. Chen (e-mail:

xchen@uconn.edu).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TMAG.2017.2654200

time-consuming and cost-expensive. They are also subject to

performance loss due to limited adaptiveness in the presence

of time-varying or environment/product-dependent vibrations.

On the other hand, control designs (software solution), par-

ticularly those with adaptive vibration compensation algo-

rithms, offer the opportunity of more ﬂexibilities and good

performance with lower cost. For instance, through adap-

tive noise cancelation (ANC) technique [4], the unknown

transmission paths from external disturbances to the HDD

systems can be adaptively identiﬁed such that effective can-

celation signals can be generated based on the measured

disturbance signals [5], [6]. As an adaptive feedforward

control approach, ANC requires additional sensors. Adap-

tive feedback control schemes have also been developed

to deal with time-varying disturbances in HDDs. Examples

include: 1) adaptive repetitive control, which cancels repeti-

tive disturbances based on internal model principle [7], [8];

2) loop shaping via Youla parameterization with an adaptive

Qﬁlter [9]–[11] or special state-space formulations [12], [13];

3) adaptive peak ﬁlters [14], [15]; and 4) adaptive disturbance

observer (DOB) [16]–[21].

Although we have successfully dealt with many con-

ventional problems (such as basic noise cancelation and

rejection of internal vibrations) and achieved relatively

good servo performance of HDDs through the above-listed

approaches, one particular category, deﬁned as wide-band

disturbance hereinafter, remains difﬁcult to effectively com-

pensate. Such wide-band disturbances include the audio

and chassis vibrations described at the beginning of this

section. They can induce widely spanned spectral peaks

in contrast to the conventional single-frequency excitations

(an example is provided in Fig. 1). Moreover, the spectral

peaks are time-varying, product-dependent, and can appear

at frequencies above the bandwidth of the baseline servo

loop [22], [23].

0018-9464 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017

Fig. 1. Examples of the disturbance spectrums.

One of the greatest challenges in compensating these wide-

band disturbances stems from the fundamental “waterbed”

limitation imposed by the Bode’s integral theorem. It states

that in linear feedback control systems, it is impossible to

attenuate disturbances at all frequencies unless for some

special cases that are rarely feasible in motion control applica-

tions. The attenuation of disturbances at one frequency range

will inevitably cause ampliﬁcations at other spectral regions.

Although approaches 3) and 4) can theoretically attenuate dis-

turbances at any customer-speciﬁed frequencies (even above

the bandwidth of the servo loop), they focus on the rejection

of narrow-band disturbances [see Fig. 1(a)] or single-peak

band-limited disturbances where the “waterbed” ampliﬁcation

is relatively small. Hence, in these approaches, the adaptation

focuses mainly on ﬁnding the optimal attenuation frequen-

cies. Yet for multiple wide-band disturbance attenuation, such

undesired ampliﬁcations are much more dangerous, and should

be systematically considered in controller design to avoid

deterioration of the overall performance.

In view of the needs and challenges, an adaptive loop

shaping scheme is proposed in this paper for multiple wide-

band disturbance attenuation. By considering adaptations with

respect to not only the attenuation frequencies but also the

attenuation widths, this scheme automatically tunes for the

optimal controller parameters that offer better overall per-

formance. Such a two-degree-of-freedom (2-DOF) adaptation

allows an optimal allocation of control efforts over all fre-

quencies. It balances the preferred disturbance attenuation and

undesired ampliﬁcation with minimum position errors. As an

extension to our previous work [18]–[20], [24]–[26], this paper

contributes in three aspects: 1) the adaptive controller design

covers both single and dual-stage HDDs; 2) wideband distur-

bances with the important extension to multiple spectral peaks

are addressed; and 3) experimental veriﬁcation on a voice-coil-

driven ﬂexible positioner (VCFP) system is performed.

The remainder of this paper is organized as follows.

Section II presents the proposed adaptive control structure for

both single and dual-stage HDDs. Section III discusses the

performance-oriented Qﬁlter design with tunable passband

widths. In Section IV, the 2-DOF parameter adaptation algo-

rithms (PAAs) are formulated. Stability analysis is given in

Section V, followed by the experimental results on a VCFP

Fig. 2. Structure of proposed control scheme for single-stage HDDs.

Fig. 3. Typical magnitude response of the baseline sensitivity function.

system in Section VI. Finally, Section VII concludes this

paper.

II. PROPOSED ADAPTATION STRUCTURE WITH MODEL

INVERSION

A. General Structure

Throughout this paper, we focus on the track-following

problem of HDD systems in the presence of the above

discussed wideband disturbances. Fig. 2 shows the proposed

adaptive control structure in single-stage HDDs. Without the

add-on compensator, it reduces to a basic feedback loop

where the system P(z−1)is stabilized by a controller C(z−1),

which achieves a baseline sensitivity function whose magni-

tude response is similar to the one shown in Fig. 3. Such a

baseline servo design is typical in practice, and can commonly

achieve a bandwidth of around 1000 Hz for single-stage

HDDs and around 2000 Hz for dual-stage HDDs. Above the

bandwidth, disturbances are ampliﬁed due to the “waterbed”

effect. Therefore, new customized compensator is desired for

enhanced disturbance attenuation at high frequencies.

Within the add-on compensator in Fig. 2, z−mseparates1the

delay out of system P(z−1), such that P(z−1)z−mPm(z−1)

and Pn(z−1)represents the nominal model of Pm(z−1),i.e.,

Pn(z−1)≈Pm(z−1)at least at frequencies where distur-

bance attenuation is desired. The objective of the add-on

compensator is to generate a cancelation signal c(k),which

1For discrete-time systems, we always have m≥1 and the separation of

z−massures a realizable inverse of Pm(z−1)and its nominal model described

afterward.

SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813

can effectively cancel out the inﬂuence of d(k). Assume that

the noise signal n(k)is relatively small compared with the

dominating vibration signal d(k). Then, one can notice that

ˆ

d(k)=P−1

n(q−1)[P(q−1)(d(k)+u(k)) +n(k)]−q−mu(k)

≈P−1

n(q−1)Pm(q−1)q−m(d(k)+u(k)) −q−mu(k)(1)

and

c(k)=Q(q−1)ˆ

d(k)

≈Q(q−1)P−1

n(q−1)Pm(q−1)q−md(k)

+Q(q−1)(P−1

n(q−1)Pm(q−1)−1)u(k−m)(2)

where q−1is the one-step delay operator in time domain and

P−1

n(z−1)represents the stable inverse2of Pn(z−1). Thus, for

perfect cancelation, i.e., c(k)approximately equals to d(k),

the ﬁlter Q(z−1)should satisfy

Q(e−jω)P−1

n(e−jω)Pm(e−jω)e−jmω≈1(3)

Q(e−jω)P−1

n(e−jω)Pm(e−jω)−1≈0(4)

at all target frequencies.

Compared with the regular DOB structure where the Qﬁlter

is typically a low-pass ﬁlter [28], the formulation in (3) and

(4) offers several ﬂexibilities and beneﬁts:

1) Enhanced Performance at High Frequencies: First, with

Qbeing a low-pass ﬁlter, a regular DOB mainly focuses

on the attenuation of disturbances at low frequencies

(within the bandwidth of the Qﬁlter, to be more

speciﬁc). While through (3) and (4), the yielded Qcan

additionally have band-pass and/or high-pass properties,

depending on the available model information and the

spectral distribution of the disturbances. Second, a reg-

ular DOB structure focuses more on the magnitude and

does not explicitly consider the phase compensation that

can signiﬁcantly deteriorate the achievable performance

at high frequencies. On the contrary, (3) and (4) sys-

tematically include the phase inﬂuence for enhanced

performance.

2) Relaxed Requirements for Plant Inversion: In a regular

DOB, it is required that P−1

n(e−jω)Pm(e−jω)≈1at

all frequencies within the bandwidth of a low-pass Q

ﬁlter. As the bandwidth of the Qﬁlter increases, this

becomes difﬁcult to satisfy. While in (3) and (4), the

Qﬁlter (possibly of band-pass property) serves as a

selective factor, such that the model only needs to be

stably invertible “locally” at frequencies where enhanced

disturbance attenuation is desired. At these frequencies,

(1) and (2) reduce to

ˆ

d(k)≈d(k−m)(5)

c(k)=Q(q−1)ˆ

d(k)≈Q(q−1)d(k−m). (6)

3) “Model-Free” Adaptability: At interested frequencies,

P−1

n(e−jω)Pm(e−jω)≈1 is satisﬁed. Then, the sensi-

2When Pn(z−1)is nonminimum phase, stable inversion strategies such

as zero-phase error tracking [27] or model-matching techniques via H∞

formulation could be used.

Fig. 4. Structure of proposed control scheme for dual-stage HDDs.

tivity function of the structure in Fig. 2 becomes

S(z−1)

=(1−z−mQ(z−1))

×1

1+P(z−1)C(z−1)+Q(z−1)P−1

n(z−1)P(z−1)−z−m

≈(1−z−mQ(z−1))S0(z−1)(7)

where S0(z−1)1/[1+P(z−1)C(z−1)]is the baseline

sensitivity function without the add-on compensator.

Hence, such formulation offers an approximately model-

free shaping factor 1 −z−mQ(z−1)that allows us to

shape the sensitivity function without inﬂuencing the

nice properties achieved by S0(z−1).3Moreover, in the

presence of unknown time-varying disturbances, such a

“model-free” Qﬁlter in Fig. 2 is easily adaptable, not

only to automatically identify the center frequencies of

the spectral peaks, but also to optimally allocate the

attenuation strength over frequencies to minimize the

position errors.

Remark: Note that the control structure in Fig. 2 treats

d(k)as a lumped equivalent input disturbance. The effects

of different error sources on the system output, for instance,

input disturbance, output disturbance, and model nonlineari-

ties/uncertainties [29], are lumped into d(k), such that they can

be compensated by c(k)from the input side of the plant. For

the general case with y(k)=P(z−1)u(k)+Md(z−1)d(k),the

output disturbance becomes Md(z−1)d(k), and an equivalent

input disturbance compensation signal c(k)is generated to

provide the compensation.

B. Extension to Dual-Stage HDDs

The control scheme in Fig. 2 can be easily extended to

dual-stage HDD systems using the decoupled DOB technique

as in [19]. Fig. 4 shows one example design for the dual-stage

case, where Pi(z−1)and Ci(z−1)(i=1,2) represent the two

actuators (voice coil motor for the ﬁrst stage and piezoelec-

tric actuator for the second) and the corresponding baseline

feedback controllers, respectively. Similar to Fig. 2, Pi,n(z−1)

3More detailed analysis about this can be found in our previous work [18].

3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017

denotes the delay-free nominal model of Pi(z−1),andm2is

the delay steps in P2(z−1), i.e., P2(z−1)=z−m2P2m(z−1).

Then, the corresponding requirements for Qﬁlter in (3)

and (4) become

Q(e−jω)P−1

2n(e−jω)P2m(e−jω)e−jm2ω≈1(8)

Q(e−jω)P−1

2n(e−jω)P2m(e−jω)−1≈0(9)

and the sensitivity function satisﬁes [19]

S(z−1)≈(1−z−m2Q(z−1))S

0(z−1)

1−z−m2Q(z−1)

1+P1(z−1)C1(z−1)+P2(z−1)C2(z−1).(10)

One can notice that the Qﬁlter design in both single and dual-

stage HDDs are quite similar. Therefore, the structure in Fig. 2

will be used as an example for analysis and implementation

hereinafter.

III. QFILTER DESIGN

A. Performance-Oriented Formulation

This section discusses the Qﬁlter design for loop shaping.

Recalling Fig. 2 and (7), the relationship between the steady-

state position error e(k)and the disturbance signal d(k)can

be expressed as

e(k)=−(1−q−mQ(q−1))S0(q−1)P(q−1)d(k). (11)

Hence, to attenuate wideband disturbances with, for example,

nspectral peaks centered at fi(i=1,2,...,n,inHz),itis

necessary to generate nnotches/stopbands centered at these

frequencies with proper widths. Namely, the add-on shaping

factor 1−z−mQ(z−1)should generate a structure N(z−1)with

nwide notches/stopbands such that N(q−1)d(k)≈0.

Deﬁne such a stable structure as N(z−1)=

BN(z−1)/AN(z−1)and let Q(z−1)=BQ(z−1)/ AQ(z−1).

Then, for effective cancelations, we should have

1−z−mQ(z−1)=N(z−1)J(z−1)(12)

where J(z−1)is additional polynomial for the existence

of solutions. To further simplify the solution, we let

AQ(z−1)=AN(z−1), and then, (12) reduces to

z−mBQ(q−1)+BN(z−1)J(z−1)=AN(z−1)(13)

which is a Diophantine equation with unknowns BQ(z−1)and

J(z−1). Compared with directly assigning the parameters of

the Qﬁlter (e.g., by assuming some bandpass structures based

on the theory and practice of digital ﬁlters), the Diophantine-

based indirect approach beneﬁts from: 1) more intuitive

performance-oriented designs, i.e., it converts the problem into

ﬁnding an optimal N(z−1)that cancels d(k)based on its

spectral characteristics; 2) the bandpass property and phase

compensation in (3) are systematically included in the solution

of (13); and 3) stability of the Qﬁlter is guaranteed with

astableN(z−1)and hence the closed-loop stability to be

discussed later in Section V.

Fig. 5. Example magnitude responses of direct-form notch ﬁlters.

B. N(z−1)With Tunable Notch/Stopband Widths

To construct N(z−1), recall from digital ﬁlter design, that

there are two main types of notch ﬁlters (a.k.a. band-stop

ﬁlters): 1) direct-form notch ﬁlter (denoted with superscript d)

and 2) lattice-form notch ﬁlter (denoted with superscript l).

The structures of the two are given by (with single notch)

Nd(z−1)=1−2cosω0z−1+z−2

1−2αdcos ω0z−1+(αd)2z−2(14)

Nl(z−1)=1−2cosω0z−1+z−2

1−(1+αl)cos ω0z−1+αlz−2(15)

where ω0=2πf0Ts(Tsis the sampling time) controls the

location of the notch and αd,l∈(0,1)controls the notch width

(denoted as NW, which is the difference between the upper

and lower frequencies where the magnitudes are −3dB)by

the following relationships:

NWd≈π(1−αd)(16)

NWl≈2arctan1−αl

1+αl.(17)

Figs. 5 and 6 give some examples of the two types of notch

ﬁlters with different center frequencies and notch widths. It can

be seen that consistent with (16) and (17), a smaller αd,lgives

a wider notch width. Meanwhile, as the notch width grows, the

distortions (i.e., nonunity gains) at other frequencies become

more signiﬁcant.

Most of the previous studies on adaptive notch ﬁlter or

adaptive DOB have adopted the direct form for online identi-

ﬁcation of the unknown center frequency (ω0)ofthespectral

peak in disturbances, treating αdas a ﬁxed predesigned

parameter. For wideband disturbance attenuation, we need to

perform loop shaping based on not only ω0but also αd,l.

The latter controls the strength of desired attenuation and

undesired “waterbed” ampliﬁcations. Therefore, control para-

meters should be adaptively updated with respect to both ω0

and αd,l. The nonlinear term (αd)2in (14), however, makes

the direct form difﬁcult to be adaptable in such scenarios.

SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813

Fig. 6. Example magnitude responses of lattice-form notch ﬁlters.

Moreover, as shown in Fig. 5, the magnitude response of

the direct-form notch ﬁlter is not symmetric at low and high

frequencies, i.e., it intrinsically brings more ampliﬁcations

at high frequencies. On the contrary, the lattice-form notch

ﬁlter only contains linear terms with respect to parameter αl

[see (15)] and its magnitude response (as shown in Fig. 6) is

symmetric at low and high frequencies. Mathematically, we

have |Nl(z−1)|z=1,−1=2/(1+αl), and namely, the ﬁlter has

the same gain at dc and Nyquist frequencies.

Therefore, for the adaptation of notch width, the lattice-form

notch ﬁlter in (15) is adopted. To generate multiple notches at

nfrequencies fi,i=1,...,n,N(z−1)in (12) is given by

N(z−1)=

n

i=1

Nlωi,α

l

i,z−1

=

n

i=1

1−2cosωiz−1+z−2

1−1+αl

icos ωiz−1+αl

iz−2(18)

with ωi=2πfiTs.

Recalling (12) and (13), one can notice that the Qﬁl-

ter is an implicit function of the parameter vector θ

[cos ω1,...,cos ωn,α

l

1,...,α

l

n]T,thatis

Q(z−1)=(θ) =cos ω1,...,cos ωn,α

l

1,...,α

l

n

(19)

where cos ωiand αl

i(i=1,...,n), respectively, control the

center frequencies and widths of the passbands. For example,

if m=1 and only a single notch is needed, namely, N(z−1)=

Nl(z−1)as in (15), then solving (13) yields

Q(z−1)=(1−αl)cos ω0+(αl−1)z−1

1−(1+αl)cos ω0z−1+αlz−2(20)

which is a bandpass ﬁlter whose center frequency and width

of passband depend on cos ω0and αl, as shown in Fig. 7.

Fig. 7. Frequency responses of the bandpass Q ﬁlters.

IV. ADAPTIVE ATTENUATION ALGORITHMS

In this section, the adaptation algorithm for Qﬁlter is

presented. Recalling the structure in Fig. 2 and (11), and

by letting F(z−1)=Pn(z−1)/(1+z−mPn(z−1)C(z−1)) and

dF(k)=F(q−1)ˆ

d(k)≈F(q−1)d(k−m), one can approxi-

mately represent the steady-state position error signal (PES)

as

e(k)=−(1−q−mQ(q−1))S0(q−1)P(q−1)d(k)

≈−(1−q−mQ(q−1))S0(q−1)Pn(q−1)d(k−m)

≈−(1−q−mQ(q−1))F(q−1)d(k−m)

≈−(1−q−mQ(q−1))dF(k). (21)

Suppose that d(k)generates vibrations with nunknown

wide spectral peaks, then based on (19), (21) suggests

that e(k)is also a function of parameters in θ=

[cos ω1,...,cos ωn,α

l

1,...,α

l

n]T. Our control goal is to adap-

tively tune these parameters, such that |e(k)|, i.e., |(1−

q−1Q(q−1))dF(k)|is minimized.

The cost function, however, may not be convex with respect

to the parameter vector θ. As shown in (18)–(20), if we

separate θinto θf[cos ω1,...,cos ωn]Tand θW

[αl

1,...,α

l

n]Tthat, respectively, controls the center frequen-

cies and passband widths of the Qﬁlter, then the cross-product

term of θfand θWmakes it difﬁcult to simultaneously update

both parameter vectors through linear adaptation algorithms.

To cope with the challenge, we separate the PAA into two

stages.

1) Stage I: Adaptation with respect to θfwith preas-

signed θW.

2) Stage II: Adaptation with respect to θWwith converged

θfin Stage I.

A. Adaptation of the Center Frequencies

In this stage, the width parameter vector θWis ﬁxed.

Moreover, to help the center frequency parameters in θf

3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017

converge to the dominant spectral peaks, narrow notches are

desired, which makes it necessary and practically efﬁcient

to set all αl

i(i=1,...,n)close to 1 and identical, e.g.,

αl

i=αl∈[0.9,0.995], given the relationship in (17). Then,

the adaptation problem with respect to θfreduces to one that

is similar to adaptive narrow-band disturbances attenuation

in [18].

Combining (12) and (18), one can obtain (1−

q−mQ(q−1)) =J(q−1)(BN(q−1)/ AN(q−1)) where

BN(q−1)=

n

i=1

(1−2cosωiq−1+q−2)

=1+

n−1

i=0

ai(q−i+q−2n+i)+anq−n+q−2n

(22)

AN(αl,q−1)=

n

i=1

(1−(1+αl)cos ωiq−1+αlq−2)

=1+

n−1

i=1

bi(q−i+(αl)n−iq−2n+i)+bnq−n

+(αl)nq−2n.(23)

In (22) and (23), we have used the fact that coefﬁcients of

1andq−2are the same in 1 −2cosωiq−1+q−2and differ

by a scalar gain of αlin 1 −(1+αl)cos ωiq−1+αlq−2.

Moreover, due to the afﬁne property, we can establish afﬁne

relationships between the coefﬁcients biand ai,i=1,...,n.

For example, after some algebra, it has the following.

1) n=1

a1=−2cosω1

b1=(1+αl)a1/2.

2) n=2

a1=−2(cos ω1+cos ω2)

a2=(2+4cosω1cos ω2)

b1=(1+αl)a1/2

b2=(a2−2)(1+αl)2/4+2αl.

3) n=3

a1=−2(cos ω1+cos ω2+cos ω3)

a2=3+4(cos ω1cos ω2+cos ω1cos ω3

+cos ω3cos ω2)

a3=−4(cos ω1+cos ω2+cos ω3)

−8cosω1cos ω2cos ω3

b1=(1+αl)a1/2

b2=(1+αl)2

4a2+3αl−3(1+αl)2

4

b3=(1+αl)3

8a3+αl(1+αl)−(1+αl)3

4a1.

Substituting such afﬁne relationships into (23), it then

becomes

AN(αl,q−1)=1+

n

i=1

aigi(αl,q−1,...,q−2n+1)

+(αl)nq−2n(24)

where gi(αl,q−1,...,q−2n+1), i=1,...,nare polynomials

whose coefﬁcients are nonlinear functions of predesigned

parameter αl. Additionally, the speciﬁed conditioning on αl

i

allows us to drop J(q−1)in the adaptation, since its inﬂuence

to narrow notches is small. Therefore, our adaptation model

becomes y(k)=−e(k)=(BN(q−1)/ AN(q−1))dF(k),which

is equivalent to

AN(q−1)y(k)=BN(q−1)dF(k). (25)

With (22)–(24), (25) yields the following difference

equation:

y(k)=ψ(k−1)Tθa+dF(k)+dF(k−2n)

−(αl)ny(k−2n)(26)

where θa[a1,...,an]Tis the unknown parameter vector

and ψ(k−1)[ψ1(k−1),...,ψ

n(k−1)]Tis the regressor

vector satisfying

ψi(k−1)=dF(k−i)+dF(k−2n+i)−gi(αl,y(k−1),

...,y(k−2n+1)), i=1,...,n−1 (27)

ψn(k−1)=dF(k−n)−gn(αl,y(k−1),

...,y(k−2n+1)). (28)

Here, gi(αl,y(k−1),...,y(k−2n+1)),i=1,...,n,

is obtained by replacing q−1,...,q−2n+1with

y(k−1),...,y(k−2n+1)in gi(αl,q−1,...,q−2n+1), i=

1,...,n.

Notice that in regulation problems in HDDs, the desired

output is zero, which means that the output y(k)also serves

as the estimation error (k). Replacing θain (26) with its

adaptive version, the difference equation for the adaptive

system becomes

(k)=ψe(k−1)Tˆ

θa(k)+dF(k)+dF(k−2n)

−(αl)n(k−2n)(29)

where ψe(k−1)is deﬁned similar to ψ(k−1), with y(k)

replaced by (k).

Hence, we can see that (k)and ˆ

θa(k)in (29) are actually

the a posteriori estimation error and the a posteriori parameter

estimate. The aprioriestimation error is similarly deﬁned as

0(k)=ψe(k−1)Tˆ

θa(k−1)+dF(k)+dF(k−2n)

−(αl)n(k−2n). (30)

Based on (26)–(30), an output-error-based PAA with a ﬁxed

compensator can be adopted for its simplicity and good

performance in noisy environments [18]. The implementation

of the algorithm is summarized as follows.

SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813

1) Select a ﬁxed compensator4C(q−1)1+(αl)nq−2n+

n

i=1cigi(αl,q−1,...,q−2n+1)that guarantees the

strict positive realness of (C(q−1)/AN(q−1)) −(1/2).

2) Deﬁne the augmented a posteriori and aprioriestima-

tion errors

v(k)=(k)+(αl)n(k−2n)

+

n

i=1

cigi(αl,(k−1),...,(k−2n+1))

(31)

v0(k)=0(k)+(αl)n(k−2n)

+

n

i=1

cigi(αl,(k−1),...,(k−2n+1)).

(32)

3) Update the parameter estimation by

ˆ

θa(k)

=ˆ

θa(k−1)−F(k−1)ψe(k−1)v0(k)

1+ψe(k−1)TF(k−1)ψe(k−1)

(33)

F(k)

=1

λ(k)F(k−1)

−F(k−1)ψe(k−1)ψT

e(k−1)F(k−1)

λ(k)+ψT

e(k−1)F(k−1)ψe(k−1)

(34)

where λ(k)=1−0.99(1−λ(k−1)) with λ(0)=0.97

is a slowly growing forgetting factor in the region of

(0,1), which helps to increase the convergence speed.

Once the parameter vector θa=[a1,...,an]Tconverges,

we can then get bi,i=1,...,nin (23), and the actual

center frequency parameter vector θfand real frequencies

ωi(or fi), i=1,...,ncan be calculated by solving nfunc-

tions. For instance, when n=2, from (22), ω1and ω2are

found by solving

a1=−2(cos ω1+cos ω2)

a2=(2+4cosω1cos ω2).

B. Adaptation of the Passband Widths

Once ωi,i=1,...,nare identiﬁed in Stage I, the

next step is to adaptively tune the passband width para-

meter vector θW=[αl

1,...,α

l

n]Tthat minimizes the PES.

Recalling (18), (21), and the fact that J(q−1)depends5on

N(q−1), one can obtain

y(k)=(1−q−mQ(q−1))dF(k)

=

n

i=1

Nl(ωi,α

l

i,q−1)J(θW,q−1)dF(k). (35)

4The selection of the compensator can be done manually or by adopting an

initial adaptation process [18].

5When m=1, J(q−1)=1. However, for more gen-

eral case, J(q−1), as a solution of the Diophantine equation

in (13), depends on the parameter

vector θW.

Similarly, if we replace αl

iand θWwith their adaptive versions

ˆαl

i(k−1)and ˆ

θW(k−1)in (35), then y(k)will also serve as

an aprioriestimation error (deﬁned as 0(k)). Therefore, the

adaptive system is given by

0(k)=

n

i=1

Nlωi,ˆαl

i(k−1), q−1Jˆ

θW(k−1), q−1dF(k)

(36)

where ˆαl

i(k−1), i=1,...,nare the aprioriestimates of

the unknown passband width parameters and ˆ

θW(k−1)=

[αl

1(k−1),...,α

l

n(k−1)]T.

In regard of the nonlinear relationships with respect to

the unknown parameter vector θW, recursive prediction-error

method (RPEM; [30, Ch. 11]) approach is adopted for unbi-

ased local convergence. The recursive PAA is given as

ˆ

θW(k)=ˆ

θW(k−1)+F(k−1)φ(k−1)e0(k)(37)

F(k)=1

λ(k)

×F(k−1)−F(k−1)φ(k−1)φT(k−1)F(k−1)

λ(k)+φT(k−1)F(k−1)φ(k−1)

(38)

where φ(k−1)is the negative gradient of 0(k)with

respect to the latest parameter estimate ˆ

θW(k−1),

i.e., φ(k−1)=[φ1(k−1),...,φ

n(k−1)]T

−[((∂0(k))/(∂ ˆαl

1(k−1))), . . . , ((∂0(k))/(∂ ˆαl

n(k−1)))]T.

To get φi(k−1), i=1,...,n, notice that

φi(k−1)

=− ∂0(k)

∂ˆαl

i(k−1)

=−

∂n

i=1Nlωi,ˆαl

i(k−1)J(ˆ

θW(k−1))

∂ˆαl

i(k−1)dF(k)

=−

∂n

i=1Nlωi,ˆαl

i(k−1)

∂ˆαl

i(k−1)J(ˆ

θW(k−1))dF(k)

−

n

i=1

Nlωi,ˆαl

i(k−1)∂J(ˆ

θW(k−1))

∂ˆαl

i(k−1)dF(k)

φi,1(k−1)+φi,2(k−1). (39)

Note that rewriting (36) yields

dF(k)=n

i=1

Nlωi,ˆαl

i(k−1), q−1J(ˆ

θW(k−1), q−1)

−1

×0(k). (40)

Substituting (40) into (39), one can obtain

φi,1(k−1)

=−

∂n

i=1Nlωi,ˆαl

i(k−1)

∂ˆαl

i(k−1)J(ˆ

θW(k−1))dF(k)

=−

∂Nlωi,ˆαl

i(k−1)

∂ˆαl

i(k−1)

n

j=i

Nlωj,ˆαl

j(k−1)J(ˆ

θW(k−1))

3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017

×n

i=1

Nlωi,ˆαl

i(k−1)J(ˆ

θW(k−1))−1

0(k)

=−

∂Nlωi,ˆαl

i(k−1)

∂ˆαl

i(k−1)Nl−1ωi,ˆαl

i(k−1)0(k)

=q−1(q−1−cos ωi)

1−1+ˆαl

i(k−1)cos ωiq−1+ˆαl

i(k−1)q−20(k).

(41)

Proof of the last equality in (41) is provided in

Appendix A.

For the second term φi,2(k−1)in (39), note that with

given nand m, the analytical expression of J(ˆ

θW(k−1), q−1)

can be found by solving the Diophantine equation in (13). It

yields a polynomial of order m−1 whose coefﬁcients are func-

tions of ˆ

θW(k−1). Hence, ∂J(ˆ

θW(k−1), q−1)/∂ ˆαl

i(k−1),

i=1,2,...,ncan be explicitly calculated, all of which are

polynomials of order m−1.

To further improve the estimation performance and increase

the convergence rate, the a posteriori estimation error

(k)n

i=1Nl(ωi,ˆαl

i(k), q−1)J(ˆ

θW(k), q−1)dF(k)is used

to update φi(k−1)in (39) and the PAA can be summarized

as follows.

1) Initialization:

ˆαi(0)=0.9,i=1,...,n

F(0)=30In/2(0)

φ(0)=φ(−1)=[0,...,0]T∈Rn

(−1)=(−2)=0.

2) Main Loop: For k=1,2,...

0(k)=

n

i=1

Nlωi,ˆαl

i(k−1), q−1

×J(ˆ

θW(k−1), q−1)dF(k)(42)

ˆ

θW(k)=ˆ

θW(k−1)+F(k−1)φ(k−1)0(k)(43)

F(k)=1

λ(k)F(k−1)

−F(k−1)φ(k−1)φT(k−1)F(k−1)

λ(k)+φT(k−1)F(k−1)φ(k−1)

(44)

φ(k−1)=[φ1(k−1),...,φ

n(k−1)]Tgiven by (39)(45)

(k)=

n

i=1

Nlωi,ˆαl

i(k), q−1J(ˆ

θW(k), q−1)dF(k)(46)

Update φ(k−1)by ˆαl(k)and (k). (47)

In the above 2-DOF adaptation algorithm, suppose that

there are nunknown frequency peaks in the disturbances,

then by explicitly utilizing the afﬁne relationships between

aiand bivalues in (22) and (23), one can see that only

nparameters (a1,a2,...) need to be identiﬁed in the center

frequency adaptation stage. Meanwhile, in the passband width

adaptation process, the number of adaptive parameters is also

n,i.e.,αl

1,α

l

2,.... Moreover, the analytical expression of φi(k)

can be found and calculated via (39)–(41) in a recursive way,

where in each time step, a system with a maximum order of

2n+m−1 is handled.

Remark:

1) For implementation, both terms in (39) can be trans-

formed into state-space forms such that in (47), all the

states are updated with the a posterior error and the

a posterior parameter estimate.

2) RPEM algorithm is advantageous in ﬁnding the local

minima under noisy environments. Initial values of αl

i

will thus inﬂuence the adaptation results. It is common

practice to set reasonable lower and upper bounds of

the parameters, based on the engineering problem in

practice.

V. STABILITY OF THE CONTROLLER

This section discusses the stability of the proposed con-

trol scheme. Based on the previous discussion, we have the

following.

1) The baseline controller C(z−1)stabilizes the closed-loop

system, i.e., all baseline closed loop poles are within the

unit circle.

2) At frequencies where disturbance attenuation is desired,

P(e−jω)≈e−jmωPn(e−jω)holds, i.e., the mismatch

between the plant and its nominal model is negligible in

the interested spectral regions.

Based on these, when P(z−1)≈z−mPn(z−1)holds, the

structure in Fig. 2 is essentially a pseudo Youla parame-

terization [31] and the closed-loop stability depends on the

stability of the Qﬁlters. Recall the fact that Q(z−1)=

BQ(z−1)/AQ(z−1)where AQ(z−1)=AN(z−1)=n

i=1(1−

(1+αl

i)cos ωiz−1+αl

iz−2). Hence, the poles of the Qﬁlter

are located at

zi

1,2=1+αl

icos ωi

2±j1+αl

i2cos2ωi−4αl

i

2i=1,...,n

with |zi

1,2|<1 if (proof provided in Appendix B)

1+αl

i2cos2ωi<4αl

i,i=1,...,n.(48)

Therefore, (48) provides a sufﬁcient condition for the sta-

bility of the Qﬁlters as well as the closed-loop stability. For

implementation, it is easy to satisfy (48) in practice. In Stage I,

for the adaptation of ωivalues, αl

i≈1 is set, which makes

the inequality automatically satisﬁed, since we focus on the

disturbance whose spectral peaks are well below the Nyquist

frequency, i.e., cos2ωi<1. In Stage II, with the converged

ωivalues, (48) is utilized to calculate the lower and upper

bounds on αl

iduring adaptation.

When P(z−1)=z−mPn(z−1), we apply robust stability

analysis to the problem. Let the plant model be subject

to a bounded uncertainty (z−1), such that P(z−1)=

z−mPn(z−1)(1+(z−1)). Substituting the actual dynamics

into (7) yields the closed-loop characteristic equation

1+P(z−1)C(z−1)+Q(z−1)z−m(z−1)=0.(49)

SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813

Fig. 8. Experimental setup of the VCFP system.

TAB L E I

RESONANCE PARAMETERS OF THE VCFP SYSTEM

Therefore, a sufﬁcient condition for the robust stability of

the closed-loop system is

|Q(e−jω)|<

1+P(e−jω)C(e−jω)

(e−jω)

=

1

S0(e−jω)(e−jω)

∀ω. (50)

VI. EXPERIMENTS

A. Experimental Setup and Baseline Performance

In this section, the proposed adaptive loop shaping approach

for wideband disturbance attenuation is conducted on a VCFP

system for veriﬁcation. As shown in Fig. 8, the experimental

setup includes a VCFP simulator board, a dSPACE real-

time DS1104 controller, and a user interface developed via

ControlDesk.6

The main component of the experimental setup is the VCFP

simulator board, which contains a complete analog circuit that

is designed to mimic the mechanical dynamics of HDDs based

on a lumped parameter model with three masses of inertia and

two springs with dampings [32]. The model and parameters

of the VCFP simulator are given as follows (see Appendix C

for detailed parameters):

P(s)

=2.03×1016 (s+1.25×105)2

s(s+620)(s+9407)(s2+1478s+9.33×107)(s2+1873s+1.70×108)

and the resonance parameters are listed in Table I.

With a sampling frequency of Fs=8000 Hz, Fig. 9 presents

the frequency responses of the plant and the identiﬁed discrete-

time nominal model z−3Pn(z−1). It shows that via the lumped

6ControlDesk is a software package developed by dSPACE that allows the

users to develop controllers in MATLAB Simulink and deploys them to the

dSPACE real-time controller.

Fig. 9. Frequency responses of the plant and its nominal model.

Fig. 10. Magnitude response of the sensitivity function S0(z−1).

parameter model, the input–output frequency response of the

VCFP system shares similar characteristics with HDDs, with

a damped inertia type of main dynamics and a collection of

high-frequency resonance modes.

Fig. 10 shows the magnitude response of the baseline

sensitivity function S0(z−1)achieved with a PID controller.

One can see that the bandwidth of the closed-loop system is

about 1/20 of the Nyquist frequency, which is typical in digital

control. Above that, disturbance attenuation is quite limited.

As a matter of fact, most high-frequency disturbances will be

ampliﬁed with the baseline controller.

Fig. 11 gives an example baseline closed-loop performance

in the presence of wideband disturbances.7Clearly, strong

vibrations remain in the PES. Moreover, such vibrations gen-

erate two wide spectral peaks at frequencies around 1000 and

1500 Hz, which are located above the closed-loop bandwidth

and largely ampliﬁed.

Hence, we focus on the attenuation of the wideband distur-

bances that generate the two spectral peaks (i.e., n=2) above

the bandwidth. From Fig. 9, it can be seen that, within our

interested frequency range, the nominal model approximately

equals to the actual system, i.e., P(z−1)≈z−3Pn(z−1)holds.

7The wideband disturbance data used in experiments is a scaled and resam-

pled version from actual test data of audio vibrations in HDDs. Typically,

the frequency peaks of such audio vibrations are within 1000–4000 Hz.

Considering the relatively low sampling rate, a two-peak audio vibration

proﬁle is chosen.

3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017

Fig. 11. Baseline control performance with wideband disturbances.

Fig. 12. Adaptation to the center frequencies of the spectral peaks.

B. Performance With Adaptive Compensation

1) Adaptation of the Center Frequencies: To attenuate the

remaining vibrations in Fig. 11, as discussed in Section IV, the

ﬁrst step is to correctly identify the center frequency parame-

ters f1and f2. With a ﬁxed notch width θW=[0.92,0.92]T,

the estimates of f1,2are shown in Fig. 12, where we can see

that both center frequencies have been correctly identiﬁed.

Fig. 13 shows the frequency response of the converged Q

ﬁlter. As discussed in Section III, it is of bandpass prop-

erty with two passbands, the center frequencies of which

are located at the identiﬁed frequencies. The corresponding

compensation performance is shown in Fig. 14. We can see

that, compared to the baseline performance, the large spectral

peaks in the PES have been largely attenuated, although the

cancelation is not strong enough due to the narrow notch

width.

2) Adaptation of the Notch/Passband Widths: To s h ow

the inﬂuence of notch/passband widths to the compensation

performance, a series of experiments with different notch

widths is conducted with the identiﬁed frequencies f1and f2.

As shown in Fig. 15, even if we set an identical width

parameter for both peaks, there is an “optimal” width (αl),

such that the steady-state PES can be minimized. If αlgrows

too large, the generated notch will be too narrow to be

Fig. 13. Frequency response of the Qﬁlter from center frequency adaptation.

Fig. 14. Performance comparison with and without the add-on adaptive

compensator with ﬁxed θW.

Fig. 15. Relationship between the compensation performance and the notch

width parameters.

effective. On the other hand, if αldecreases too much, the

notch becomes too wide; due to the “waterbed” effect, the

inﬂuence at other frequencies can be signiﬁcant and eventually

degrade the overall servo performance. In fact, if the notch

width becomes too wide, the robust stability condition in (50)

can be violated, leading to system instability.

To adaptively ﬁnd such “optimal” notch widths, the PAA

discussed in Stage II (Section IV) is implemented and the

SUN et al.: ADAPTIVE LOOP SHAPING FOR WIDEBAND DISTURBANCES ATTENUATION 3301813

Fig. 16. Evolution of the width parameters.

Fig. 17. Frequency response of the converged Qﬁlter with widths adaptation.

Fig. 18. Baseline error and compensated error with adaptively tuning widths.

results are shown in Figs. 16–18. The width parameters

for different spectral peaks converge to optimal values in

Fig. 16, such that the resulted Qﬁlter in Fig. 17 generates

two passbands that are wide enough to cancel the major

vibration components, but still not too aggressive to amplify

the disturbances at other frequencies, i.e., the gains of the Q

ﬁlter at other frequencies are reasonably small.

With the converged width parameters, the compensation

performance of the proposed structure is given in Fig. 18.

It shows that, compared to the baseline error and the one with

TAB L E I I

MODEL PARAMETERS OF THE VCFP SYSTEM

preselected width parameters (Fig. 14), the proposed algorithm

can automatically tune for better passband width parameters to

minimize the error. The large spectral peaks around 1000 and

1500 Hz have been effectively attenuated in the compensated

error spectrum, as shown in Fig. 18.

Remark:

1) Note that although the lumped-parameter-model-based

VCFP system has difference with the structured dynam-

ics of HDDs, yet it captures the key input–output

frequency features of that of HDDs (a damped iner-

tia type of main dynamics and a collection of high-

frequency resonances). Moreover, from an algorithmic

viewpoint, the adaptation algorithm is greatly decoupled

from the model information, due to the nearly “model-

free” formulation in (7). As a result, the key results

on Qﬁlter design and the 2-DOF adaptation algorithm

addressed above will remain representative in real HDD

applications.

2) In determining the number of frequency bands, i.e.,

hyperparameter n, two ways can be adopted: 1) empir-

ical estimate based on system model and typical fre-

quency peaks of the audio vibrations (our experience

from a large amount of actual test data is that two

bands are commonly sufﬁcient) and 2) simultaneously

run the proposed algorithm with different n(the number

of center frequencies) and choose the minimum one that

can achieve acceptable PES. More detailed discussion

about this point can be found in [18].

VII. CONCLUSION

In this paper, the problem of wideband disturbance atten-

uation at high frequencies in precision information storage

systems is addressed. An adaptive loop shaping approach is

proposed. Control parameters with respect to both the center

frequencies and the widths of the spectral peaks are devised in

an “optimal” way, such that the PES can be minimized. The

PAAs for both are discussed. Experiments are conducted on

3301813 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 5, MAY 2017

d¯x(t)

dt =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

−d1−d12

J1

−c12

J1

d12

J1

c12

J100

kM

J1

10 0 0 000

d12

J2

c12

J2

−d12 −d23 −d2

J2

−c12 −c23

J2

d23

J2

c23

J20

00 1 0 000

00 d23

J3

c23

J3

−d23 −d3

J3

−c23

J30

00 0 0 100

−kE

L00 000

−R

L

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

¯x(t)+

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

0

0

0

0

0

0

1/L

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

u(t)

y(t)=[

00000krkx0]¯x(t)(54)

a VCFP system. The results show that the proposed adaptive

approach can effectively attenuate wideband disturbances with

correctly identiﬁed center frequencies and automatically tuned

optimal notch widths.

APPENDIX A

PROOF OF THE LAST EQUALITY IN (41)

Given

Nlωi,α

l

i,q−1=1−2cosωiq−1+q−2

1−1+αl

icos ωiq−1+αl

iq−2

B(ωi,q−1)

Aωi,α

l

i,q−1

we have

∂Nlωi,ˆαl

i(k−1)

∂ˆαl

i(k−1)=

∂B(ωi,q−1)

∂ˆαl

i(k−1)Aωi,ˆαl

i(k−1), q−1

A2ωi,ˆαl

i(k−1), q−1

−

B(ωi,q−1)∂Aωi,ˆαl

i(k−1),q−1

∂ˆαl

i(k−1)

A2ωi,ˆαl

i(k−1), q−1(51)

where ∂B(ωi,q−1)/∂ ˆαl

i(k−1)=0and

∂A(ωi,ˆαl

i(k−1), q−1)/∂ ˆαl

i(k−1)=−cosωiq−1+q−2.

Substituting them into (51), one can obtain

∂Nlωi,ˆαl

i(k−1)

∂ˆαl

i(k−1)=B(ωi,q−1)(cos ωiq−1−q−2)

A2ωi,ˆαl

i(k−1), q−1.(52)

Note that Nl−1(ωi,ˆαl

i(k−1), q−1)=

((A(ωi,ˆαl

i(k−1), q−1))/(B(ωi,q−1))), hence, we have

−∂Nl(ωi,ˆαl

i(k−1))

∂ˆαl

i(k−1)Nl−1(ωi,ˆαl

i(k−1), q−1)

=(q−2−cos ωiq−1)

A(ωi,ˆαl

i(k−1), q−1)(53)

which yields the ﬁnal result in (41).

APPENDIX B

PROOF OF CONDITION IN (48)

Given AQ(z−1)=n

i=1(1−(1+αl

i)cos ωiz−1+αl

iz−2),

if (1+αl

i)2cos2ωi<4αl

i,i=1,...,n[i.e., (48) holds], then

the poles of the Qﬁlter will be located at

zi

1,2=1+αl

icos ωi

2±j1+αl

i2cos2ωi−4αl

i

2i=1,...,n

whose magnitudes are given by

1+αl

i2cos2ωi

4+1+αl

i2cos2ωi−4αl

i

4,i=1,...,n

=21+αl

i2cos2ωi−4αl

i

4=1+αl

i2cos2ωi−2αl

i

2.

Therefore, |zi

1,2|<1,i=1,...,nis equivalent to

1+αl

i2cos2ωi<2+2αl

i,i=1,...,n.

⇔1+αl

icos2ωi<2,i=1,...,n

which obviously holds with αl

i∈(0,1)and cos2ωi<1for

all i=1,...,n.

APPENDIX C

PLANT PARAMETERS OF THE VCFP SYSTEM

The VCFP system is built based on a lumped parameter

model with two masses of inertia connected by three springs

with dampings. Its state-space dynamics representation is

given in (54) as shown at the top of this page, with parameters

listed in Table II.

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