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Pseudo Youla-Kucera Parameterization with Control of the

Waterbed Eﬀect for Local Loop Shaping

Xu Chen?a, Tianyu Jiang a, Masayoshi Tomizuka b

aDepartment of Mechanical Engineering, University of Connecticut, Storrs, CT, 06269, USA.

bDepartment of Mechanical Engineering, University of California, Berkeley, CA, 94720, USA.

Abstract

This paper discusses a discrete-time loop shaping algorithm for servo enhancement at multiple wide frequency bands. Such

design considerations are motivated by a large class of practical control problems such as vibration rejection, active noise

control, and periodical reference tracking; as well as recent novel challenges that demand new design in the servo technologies.

A pseudo Youla-Kucera parameterization scheme is proposed using the inverse system model to bring enhanced control at

selected local frequency regions. Design methodologies are created to control the waterbed ampliﬁcations that come from

the fundamental limitations of feedback control. Finally, simulation and experimental veriﬁcation are conducted in precision

control and semiconductor manufacturing.

Key words: digital control, vibration rejection, loop shaping, active noise control, Youla-Kucera parameterization

1 Introduction

Recent technology innovations are urging the penetra-

tion of customized controls in modern precision systems

and advanced manufacturing. For instance, in 2011, the

wafer scanning process in semiconductor manufacturing

requires a control precision which can be mimicked by

repetitively driving on a highway segment, under mm

scale error tolerance between consecutive iterations. In

the disk drive industry, new external disturbances—such

as vibrations from high-power audio speakers and ad-

jacent drives (in data center and cloud storage)—have

become the most important source of the position error

in data reading and writing [1]. Both examples demand

a control precision at nm scale, with extremely low sen-

sitivity against vibrations and noises.

To reach the required servo goals, enhanced control at

selected frequency (or frequency ranges) has been recog-

nized to be essential in many ﬁelds including precision

mechatronics [2], active suspensions [3], cooling systems

[4], and the aforementioned semiconductor manufactur-

ing [5] and information storage systems [6]. Based on

?Tel. +1-860-486-3688.

Email addresses: xchen@engr.uconn.edu (Xu Chen?),

tianyu.jiang@uconn.edu (Tianyu Jiang),

tomizuka@me.berkeley.edu (Masayoshi Tomizuka).

how the enhancements are placed, control algorithms

have been proposed to cover: (i) integer multiples of a

fundamental frequency, using repetitive control and its

variants [7,8,9,10]; (ii) nonrestrictive multiple frequen-

cies, via peak ﬁlters [11], adaptive feedforward cancel-

lation [12], disturbance observers [13,6], and internal-

model-principle based Youla-Kucera parameterization

[14,15,16,17,3]; (iii) broadband frequencies, based on,

e.g., adaptive variance minimization [18], and sensor-

based adaptive noise cancellation [19]. From the view-

point of shaping the response of the servo loop, such

local loop shaping (LLS) algorithms can be classiﬁed

into: (i) shaping of the open-loop frequency response

[11]; (ii) design based on internal model principle and/or

time-domain cancellations [9,12,14,15,16,17,6]; (iii) di-

rect shaping of the sensitivity function [18,13,3].

A major recognition from the LLS literature is the sig-

niﬁcant challenge raised from Bode’s Integral Theorem,

which guarantees (under mild assumptions that are

commonly satisﬁed in practice) the occurrence of error

ampliﬁcations during loop shaping. Such “waterbed”

limitation—particularly important for LLS at multiple

frequency locations—has attracted great research at-

tention, and recently been extensively investigated in a

benchmark on adaptive narrow-band vibration rejection

[3]. Evolving from earlier design of “achieving enhanced

local servo performance while maintaining the system

Preprint submitted to Automatica 1 October 2015

stability,” more and more algorithms are emerging with

the philosophy of achieving LLS with bounded waterbed

eﬀect. Examples in the area of narrow-band LLS include

[17,3,13,10,6]. Based on the results, inﬁnite-impulse-

response (IIR) ﬁlters are being more and more adopted

over the conventional choice of ﬁnite-impulse-response

(FIR) ﬁlters for waterbed mitigation.

Many novel applications, including the two examples

at the beginning of this section, demand enhanced

servo at not only single frequencies but also wide fre-

quency bands. Motivated by the aforementioned the-

oretical challenges in conjunction with the rise of new

applications that place more stringent performance re-

quirements, this paper provides a pseudo Youla-Kucera

(YK) parameterization scheme for narrow- and wide-

band LLS. Several theoretical limits of performance

are identiﬁed, which explains the diﬀerence and funda-

mental challenge of wide-band LLS compared with its

narrow-band analogue. Using a robust pseudo version of

YK parameterization—a formulation of all stabilizing

controllers—we construct an LLS scheme and discuss

its design ﬂexibility and intuitions. A particular contri-

bution of the paper is a systematic design methodology

to control the fundamental waterbed constraint in LLS.

We show that with an add-on pole/zero modulation in

the inverse-based YK scheme, error ampliﬁcations in

LLS can be ﬂexibly controlled over the frequency do-

main, to accommodate diﬀerent servo requirements and

disturbance spectra.

The proposed algorithm focuses on obtaining an ana-

lytic LLS solution of the central Q ﬁlter in YK param-

eterization, instead of an optimal implicit one, from

e.g. weighted sensitivity minimization via optimization

and H∞theories. Diﬀerent from conventional servo

problems, LLS highly depends on the speciﬁc distur-

bance proﬁles, which can vary among systems and even

change with respect to time. The solution approach in

the present paper is made for more convenient incorpo-

ration of features such as adaptive control (to attenuate

unknown or time-varying disturbance spectra), and

industrial tuning (where a direct controller parameteri-

zation can be easier for implementation across diﬀerent

product platforms).

Notations: We focus on controlling single-input single-

output (SISO) systems. Throughout the paper, the cal-

ligraphic Sand Rdenote, respectively, the set of stable

proper rational transfer functions; and the set of proper

rational transfer functions. When a linear time invari-

ant (LTI) plant Pis stabilized by an LTI controller C

(in a negative feedback loop), S(,1/(1 + P C)) and

T(,P C/(1 + P C )) denote, respectively, the sensitiv-

ity 1and the complementary sensitivity functions. Fi-

1the transfer function from the reference to the feedback

error and from the output disturbance to the plant output.

nally, if n= 0, then Pn

i=1 ai= 0 and Qn

i=1 ai= 1.

2 Review of YK Parameterization

Let G∈ R.(N, D)is called a coprime factorization of

Gover Sif: (i) G=ND−1, (ii) N(∈ S)and D(∈ S )are

coprime transfer functions, and (iii) D−1∈ R.

Theorem 1 (YK parameterization) [20,21] If a

SISO plant P=N/D can be stabilized by a negative-

feedback controller C=X/Y , with (N,D) and (X,Y)

being coprime factorizations over S, then any stabiliz-

ing feedback controller of Pcan be parameterized as 2

Call =X+DQ

Y−NQ :Q∈ S, Y (∞)−N(∞)Q(∞)6= 0 (1)

Remark 1 YK parameterization advantageously

changes the principle of feedback design by rendering

the new sensitivity function to:

˜

S=1

1 + P Call

=1

1 + P C 1−N

YQ(2)

Literature often additionally normalizes the coprime

factorization, such that NX +DY = 1 and thus

˜

S=D(Y−NQ). For LLS which is commonly con-

ducted upon an existing closed loop that operates under

regular servo performance, we keep the structure of

(2), and interpret Theorem 1 as an add-on scheme that

decouples Sto the product of the baseline sensitivity

1/(1 + P C)and the add-on aﬃne module 1−N Q/Y .

3 LLS with Discrete-time Pseudo YK Design

We discuss ﬁrst a special discrete-time case of Theorem

1 and its generalization for LLS, then provide the corre-

sponding design of Qand control of the waterbed.

Proposition 2 Consider a stable discrete-time nega-

tive feedback loop consisting of a controller C(z), and

a plant P(z)whose relative degree is m. If P−1(z)and

C(z)are stable, then

Call (z) = C(z) + z−mP−1(z)Q(z)

1−z−mQ(z), Q(z)∈ S (3)

parameterizes all stabilizing controllers for P(z), and

the sensitivity function is

S(z) = 1−z−mQ(z)

1 + P(z)C(z)

,So(z)1−z−mQ(z)(4)

Proof follows by letting X(z) = C(z),Y(z)=1,

N(z) = z−m, and D(z) = z−mP−1(z)in Theorem 1.

2Y(∞)denotes, respectively, Y(s)|s=∞and Y(z)|z=∞, in

the continuous- and the discrete-time cases.

2

Proposition 2 reduces the add-on module in the sensitiv-

ity function to 1−z−mQ(z)in (4). The inverse-based pa-

rameterization has made the added module simple and

depend little on C(z)and P(z)(only the delay z−map-

pears here). This was achieved by conﬁning to plants

with stable inverses, and closed loops with stable base-

line controllers. These two conditions are relaxed in a

pseudo YK scheme in the next result.

Proposition 3 Consider a stable discrete-time nega-

tive feedback loop consisting of a controller C(z), and a

plant P(z)whose relative degree is m. Let

˜

C(z) = C(z) + z−mˆ

P−1(z)Q(z)

1−z−mQ(z), Q(z)∈ S (5)

where ˆ

P−1(z)is chosen stable. If P(z) = ˆ

P(z), then

the new feedback loop consisting of P(z)and ˜

C(z)has

guaranteed stability. Otherwise, the new feedback loop is

stable if the roots of the following characteristic equation

are all inside the unit circle:

zmAQ(z)Bˆ

P(z)[AC(z)AP(z) + BC(z)BP(z)]

+AC(z)BQ(z)[Aˆ

P(z)BP(z)−AP(z)Bˆ

P(z)] = 0 (6)

where BG(z)and AG(z)denote the coprime numerator

and denominator polynomials of a transfer function G.

Proof With P(z) = BP(z)/AP(z),C(z) = BC(z)/AC(z),

ˆ

P(z) = Bˆ

P(z)/A ˆ

P(z), and Q(z) = BQ(z)/AQ(z), (5)

transforms to

˜

C(z) = AQ(z)Bˆ

P(z)BC(z) + z−mAC(z)Aˆ

P(z)BQ(z)

Bˆ

P(z)AC(z) [AQ(z)−z−mBQ(z)] (7)

With (7), the closed-loop characteristic equation is (6).

The root condition in the second half of the proposition

then readily follows. If P(z) = ˆ

P(z), (6) reduces to

AQ(z)Bˆ

P(z)[AC(z)AP(z) + BC(z)BP(z)] = 0 (8)

Hence the closed-loop poles are composed of the baseline

closed-loop poles and the poles of Q(z)and ˆ

P−1(z).

As the baseline feedback loop, Q(z), and ˆ

P−1(z)are all

stable, the new closed loop is thus stable. 2

Proposition 3 relaxes the requirements on stable Cand

P−1by focusing on LLS and dropping the attempt to

parameterize all the stabilizing controllers. More specif-

ically, the subclass of all stabilizing controllers (5) can

be seen to always retain the unstable poles (which can

occur for stabilizing certain unstable plants) of C, if any.

On the other hand, from the viewpoint of implemen-

tation, a perfect plant model is unrealistic in practice

(due to high complexities or system uncertainties). In

this sense, a practical YK parameterization has to be an

approximation, or a robust version, of the ideal cases in

Section 2 and Proposition 2. As a perfect plant model is

not available anyway, a stable nominal inversion ˆ

P−1(z)

is adopted in Proposition 3. This is one constraint that is

achievable in a large class of practical systems. 3More-

over, the next two paragraphs will show that the mis-

match between ˆ

P−1and P−1, if any, can actually be al-

lowed in the frequency regions that do not require servo

enhancement in LLS.

With (5), the sensitivity function 1/(1 + P(z)˜

C(z)) is

S(z) = 1−z−mQ(z)

1 + P(z)C(z) + z−mQ(z) ( ˆ

P−1(z)P(z)−1) (9)

If ˆ

P−1(ejω )P(ejω )=1, namely, at frequencies where the

inverse model is accurate, (9) gives

S(ejω ) = 1−e−mjω Q(ejω)

1 + P(ejω )C(ejω )(10)

i.e. the decoupling of sensitivity in (4) remains valid

in the frequency domain. Enhancing the closed-

loop performance at ωithus translates to design-

ing e−mjωiQ(ej ωi)=1, which gives perfect distur-

bance rejection (S(ejωi)=0) and reference tracking

(T(ejωi) = 1) at ωi. Meanwhile, at ωkwhere there

are large model uncertainty and mismatches, high-

performance control intrinsically has to be sacriﬁced

for robustness based on robust control theory. We

will thus make Q(ejωk)≈0, to keep the inﬂuence

of the mismatch element z−mQ(z)( ˆ

P−1(z)P(z)−1)

small in (9). More formally, as Q(z)and the base-

line sensitivity 1/(1 + P(z)C(z)) are both stable, if

|Q(ejω )(P(ejω )ˆ

P−1(ejω )−1)|<

1 + P(ejω )C(ejω )

in

(9) ∀ω,S(z)will have guaranteed stability.

Fig. 1 presents a realization of the pseudo YK scheme. By

block diagram analysis, one can show that when r(k)=0

and z−mQ(z)|z=ejωi≈1,c(k)approximates −d(k)at

ωi(hence canceling the disturbance). Such time-domain

intuition can be used for tuning during implementations.

//z−mˆ

P−1(z)+

//◦

z−m

+

oooo

Q(z)

+

c(k)

? d(k)

+

r(k)+

//◦//C(z)+

//◦u(k)+

//◦//P(z)y(k)

//

−

OO

Fig. 1. Block diagram of pseudo YK parameterization

3Indeed, inverse-based design has long been used in motion

control, e.g., in feedforward designs.

3

Next we provide solutions of Qfor LLS, and then several

design tools to control the waterbed.

Prototype LLS Solution Let {ωi}n

i=1 be a set of

distinct frequencies (in rad). Let Aζ(z) = Qn

i=1(1 −

2ζcos ωiz−1+ζ2z−2)—or after multiplication—Aζ(z) =

1 + a1ζz−1+· · · +anζnz−n+· · · +a1ζ2n−1z−2n+1 +

ζ2nz−2n, where ζ=αor β,0< α < β ≤1, and βis

very close or equal to 1. Let P(z)be stabilized by (5),

with P(ejωi) = ˆ

P(ejωi), and

Q(z) = BQ(z)

Aα(z):K(z)Aβ(z)+z−mBQ(z) = Aα(z)(11)

where K(z) = 1 if m= 1;K(z) = k∈(0,1] if m=

0; and K(z)is an FIR ﬁlter when m > 1. Then at

each ωi, a notch is created in the magnitude response

of the sensitivity function. Furthermore, if β= 1, the

design achieves perfect disturbance rejection at {ωi}n

i=1

and Q(ejωi) = ej mωi.

We verify the solution along with a discussion on several

central concepts in LLS. (11) yields

1−z−mQ(z) = Aβ(z)

Aα(z)K(z)(12)

For each ωi,Aβ(z)/Aα(z)has a pair of damped pole

and zero αe±jωi, β e±jωi.αcontrols the width of the

attenuation range. The zero βejωi—on or close to the

unit circle—provides small gains to

1−e−jωm Qejω

when ωis close to ωi. If β= 1, applying cos(ωi) =

(e−jωi+ej ωi)/2gives Aβ=1(ejωi) = (1 −2 cos ωie−jωi+

e−2jωi)Qn

j=1,j6=i(1 −2 cos ωje−j ωi+e−2jωi)=0, and

hence Q(ejωi) = ej mωiin (12), which renders S(ejωi) =

0in (10) and therefore full disturbance rejection at ωi.

K(z)Aβ(z) + z−mBQ(z) = Aα(z)in (11) is a Diophan-

tine equation (see, e.g. [22]), and is always solvable us-

ing a Sylvester matrix, as Aβ(z)and z−mare coprime.

The minimum-order solution satisﬁes BQ(z) = bQ,0+

bQ,1z−1+· · · +bQ,2n−1z−2n+1 and

K(z) = k0+k1z−1+· · · +km−1z−m+1 (13)

(if m= 0,K(z) = k). For low-order problems, bQ,i ’s and

ki’s can be directly obtained by the method of undeter-

mined coeﬃcients.

Example 1 Let m= 2,β= 1, and n= 1. From (13),

K(z) = k0+k1z−1. Matching coeﬃcients of z−i’s in

the Diophantine equation (11) gives

Q(z)=(α−1)(α+ 1 −a2)−az−1

1 + aαz−1+α2z−2(14)

with a=−2 cos(ω1)and K(z) = 1 + (α−1)az−1.

Corresponding to the notch shape of Aβ(z)/Aα(z),Q(z)

from (12) is a special bandpass ﬁlter (see Example 1

and Fig. 6), which reduces the inﬂuence of model mis-

match outside the Q-ﬁlter passband for enhanced ro-

bust stability [recall (9)]. Note, however, that no practi-

cal bandpass ﬁlters are ideal, especially when the pass-

band gets wider. Along with the desired notch shape,

maxω|1−z−jmω Q(ejω )|(= ||1−z−mQ(z)||∞)will ex-

hibit the waterbed eﬀect of exceeding 1. The root cause

comes from the following fact from fundamental limita-

tions of feedback control.

Corollary 4 Let Q(z)∈ S. Then

Zπ

0

ln |1−e−mjω Q(ejω )|dω=π l

X

i=1

ln |γi| − ln |σ+ 1|!

where γi(l≥0) are the unstable (outside the unit circle)

zeros of 1−z−mQ(z), and σ= limz→∞{z−mQ(z)/(1 −

z−mQ(z))}.

The proof follows by invoking Bode’s Integral Theorem

[23] and treating LQ=z−mQ(z)/(1 −z−mQ(z)) as the

open-loop transfer function in a negative feedback loop.

As a particular case, if m > 0and 1−z−mQ(z)has no

unstable zeros, then Rπ

0ln |1−e−mjω Qejω |dω= 0.

In Example 1, reducing αincreases the LLS bandwidth;

however, the zero of K(z)also becomes unstable if |(α−

1)2 cos ω1|>1, which increases the sensitivity integral

in Corollary 4. Such theoretical challenge of wide-band

LLS is seen to match the intuition from the perspective

of ﬁlter shapes in the paragraph after Example 1.

Remark 2 (Overcoming the waterbed eﬀect) If

m= 0, (12) simpliﬁes to 1−Q(z) = kAβ(z)/Aα(z).

Letting k∈(0,1) gives Q(∞) = 1 −kand then

ln |σ+ 1|= ln |1/k|>0in Corollary 4. Additionally,

1−Qhas no unstable zeros. Hence the Bode’s integral

is negative. Independent from the baseline design of C,

the new sensitivity magnitude can thus be reduced at

all frequencies. In common plants, usually m≥1. For

certain non-conventional systems without delays, one

observes that LLS design is signiﬁcantly simpliﬁed. In

the remainder texts, we focus on the cases with m≥1.

Although the overall area integral is constrained by

Corollary 4, depending on the disturbance spectrum,

performance goals, and robustness of the system in

diﬀerent regions, the waterbed can be controlled via

structural designs in Q(z), as shall be discussed next.

3.1 Detuning

One direct approach to reduce the overall waterbed am-

pliﬁcation is to design ﬁrst a regular Q ﬁlter in (11) and

then detune via ˜

Q(z) = gQ (z), g ∈(0,1). By reducing

|˜

Qejω |,|1−e−jmω ˜

Qejω |is closer to unity outside

4

the pass bands of Q(z). Mathematically, the next result

provides the amount of LLS enhancement under detun-

ing and the reduction of waterbed ampliﬁcation.

Proposition 5 Let the plant be stabilized by (5).

Let β= 1;Q(z)be the prototype solution (11);

˜

Q(z) = gQ (z),g∈(0,1); and P(ejωi) = ˆ

P(ejωi). Then

|S(ejωi)|= (1 −g)|So(ej ωi)|and ||1−z−m˜

Q(z)||∞≤

1 + g(||1−z−m˜

Q(z)||∞−1) <||1−z−mQ(z)||∞.

Proof The prototype LLS solution satisﬁes Q(ejωi) =

ejmωi. Hence 1−e−j mωi˜

Q(ejωi)=1−g, and

|S(ejωi)|= (1 −g)|So(ejωi)|from (10). The sec-

ond assertion follows from basic properties of H∞

norm: let Qu denote the output of Qw.r.t. the in-

put signal u, then ||1−z−m˜

Q||∞= supu6=0{||[1 −

z−m˜

Q]u||2/||u||2}= supu6=0{||[1−z−mgQ]u||2/||u||2} ≤

supu6=0{[||(1 −g)u||2+||g(1 −z−mQ)u||2]/||u||2}=

1−g+g||1−z−mQ||∞<||1−z−mQ||∞, where the last

inequality is due to ||1−z−mQ||∞>1from Corollary

4 and g∈(0,1).2

The root reason that detuning relaxes the waterbed ef-

fect comes from the fact that unstable zeros of 1−

z−mQ(z), if exist, can be pulled into the unit disk by

cascading the gain g. From root locus analysis, regard

−z−mgQ(z)as the open-loop transfer function, then as g

moves from 1 to 0, poles of 1/(1−gz−mQ(z))—i.e. zeros

of 1−gz−mQ(z)—move from poles of 1/(1 −z−mQ(z))

to poles of z−mQ(z), which are all stable.

Besides the overall detuning to reduce ||1−z−mQ(z)||∞,

Q(z)can be locally manipulated to reduce |1−

e−jmω Q(ejω )|at designer-assigned frequencies. Such

localization can be used to build robustness against

model mismatch/uncertainties and tackle some partic-

ular disturbance spectra. Without loss of generality, we

will assume β= 1 and then apply detuning to control

the overall Q-ﬁlter gain, if needed.

3.2 Modulation of Zeros

(11) directly embedded the denominator Aα(z)in

Q(z). but did not specify the structure of the numera-

tor BQ(z). One can enforce constrained magnitude by

adding ﬁxed zeros such that

BQ(z) = B0(z)B0

Q(z).(15)

Designing B0(z)=1−z−1for example, will embed a

scaled diﬀerentiator in Q(z), yielding Q(ej ω)

ω=0 = 0

(zero DC gain). Fig. 2 presents the eﬀect of such a de-

sign, with β= 1 and g= 0.8. The enhanced small

gain at low frequency is seen to successfully reduce |1−

e−jmω Qejω |in the highlighted region.

100101102103104

−20

−10

0

Magnitude (dB)

1−z−mQ(z)

100101102103104

−60

−40

−20

0

Magnitude (dB)

Frequency (Hz)

Q(z)

B0(z) = 1 − z−1

B0 = 1

Fig. 2. Eﬀect of a ﬁxed zero at low frequency

Similarly, introducing a ﬁxed zero near z=−1provides

enhanced small gains for Q(z)in the high-frequency re-

gion. Extending this idea, one can place magnitude con-

straints at arbitrary desired frequencies, by designing

B0(z)=1−2ρcos ωpz−1+ρ2z−2, which places ﬁxed

zeros ρe±jωpin (15) to penalize |Q(ej ω )|near ωp. Com-

binations can be made, for instance, to form the en-

hancement in Fig. 3, using B0(z) = (1 + 0.7z−1)(1 −

0.86z−1)(1 −1.6 cos(2π×6000Ts)z−1+ 0.82z−2)(1 −

1.4 cos(2π×10000Ts)z−1+0.72z−2). There, reduced gain

is achieved at almost all frequencies outside the pass

bands of Q(z).

100101102103104

−20

−10

0

Magnitude (dB)

1−z−mQ(z)

100101102103104

−30

−20

−10

0

Magnitude (dB)

Frequency (Hz)

Q(z)

enhanced

baseline

Fig. 3. Eﬀect of combined zeros at diﬀerent frequencies

With (15), Q(z)is obtained by solving the Diophan-

tine equation K(z)Aβ(z) + z−mB0(z)B0

Q(z) = Aα(z).

Let the order of B0(z)be nB0. If Aβ(z)and B0(z)are

coprime, the minimum-order solution satisﬁes B0

Q(z) =

bQ,0+bQ,1z−1+· · · +bQ,2n−1z−2n+1 ;K(z) = k0+

k1z−1+· · · +k(m+nB0−1) z−(m+nB0−1).

5

Note that the proposed zeros of B0(z)are all inside

the closed unit ball. Unstable zeros in Q(z)raise funda-

mental limitations on controlling |1−e−jmωQ(ej ω)|, as

shown next.

Corollary 6 If Q(z)(∈ S)has an unstable zero zuthen

k1−z−mQ(z)k∞≥

Np

Y

i=1

|zu¯pi−1|

|zu−pi|(16)

where pi’s are Np(≥0) unstable zeros of 1−z−mQ(z)

and ¯piis the complex conjugate of pi. If Np>0, the

right hand side of (16) is strictly larger than 1; if Np=

0, the result simpliﬁes to ||1−z−mQ(z)||∞≥1.

Proof Applying all-pass factorization such that

1−z−mQ(z) = Mm(z)QNp

i=1(z−pi)/(z¯pi−1), where

Mm(z)is minimum-phase. Then ||1−z−mQ(z)||∞=

||Mm(z)||∞and Mm(zu)QNp

i=1(zu−pi)/(zu¯pi−1) = 1.

For Mm(z),||Mm(z)||∞≥ |Mm(zu)|by maximum mod-

ulus principle. Combining the last three results proves

(16). If Np= 0 then QNp

i=1(z−pi)/(z¯pi−1) = 1.

Otherwise straightforward complex analysis gives that

|zu¯pi−1|/|zu−pi|>1, due to |zu|>1and |pi|>1.

3.3 Enhancement by Cascaded IIR Filters

From the frequency-response perspective, cascading two

bandpass ﬁlters with the same center frequency gener-

ates an enhanced one. Consider Q(z) = Q0(z)B0(z),

where Q0(z)is the prototype/basic solution from (11)

and B0(z)is a standard bandpass ﬁlter with the same

center frequencies as Q0(z). Any standard bandpass de-

sign with B0(ejωi)=1is applicable here. One candidate

choice is

B0(z) = 1 −ηAβ=1(z)

Aα(z), η ∈(0,1] (17)

Fig. 4 presents the Q(z)and 1−z−mQ(z)solved from the

discussed algorithm. The solid line is the basic solutions

from (11) (i.e., B0(z)=1). Both methods create the re-

quired attenuation at around 3000 Hz, while the Q ﬁl-

ter with cascaded IIR design has an enhanced bandpass

property. Despite a shifted waterbed concentration, the

maximum ampliﬁcation is still around 1.6dB (1.2023)

while the attenuation is as large as 50dB (not shown due

to limit of ﬁgure size) in a wide frequency region.

4 Simulation and Experimental Results

Recall the two examples at the beginning of the Intro-

duction Section. This section provides the experimental

veriﬁcation of the algorithm on one stage of a wafer scan-

ner [10], and the data-in-the-loop simulation on an HDD

benchmark [24] using actual audio-vibration test data.

103104

−8

−6

−4

−2

0

2

Magnitude (dB)

1−z−mQ(z)

103104

−40

−20

0

Magnitude (dB)

Frequency (Hz)

Q(z)

B0 = 1

IIR B0

Fig. 4. Eﬀect of cascaded IIR enhancement in Q(z)

4.1 Experimental Results on a Wafer-Scanner Testbed

Based on physics, the nominal model of the plant has

pure inertia dynamics and satisﬁes Pn(z)=3.129 ×

10−7(z+ 1) /[z(z−1)2](sampling frequency: 2500 Hz).

Pn(z)has a relative degree of two (i.e. m= 2), and a

102103

−160

−140

−120

−100

−80

Frequency (Hz)

Magnitude (dB)

nominal

measurement

Fig. 5. Plant model of the reticle stage in a wafer scanner

zero on the unit circle. Shifting this zero to be strictly in-

side the unit circle, and normalizing the gain, we obtain

the stable nominal inverse z−mˆ

P−1(z) = 107/3.47655 ×

(z−1)2/(z2+ 0.8z). One can verify that ˆ

Pand Pn

have almost identical frequency response. The dashed

line in Fig. 5 shows the frequency response of ˆ

P(z),

which matches with the solid line from system identiﬁ-

cation experiments up to 150 Hz. Fig. 6 shows the fre-

quency responses of diﬀerent LLS designs at 100 Hz.

The IIR enhancement uses η= 0.9in (17); the en-

hancement at 200 Hz and 700 Hz is achieved by ﬁxing

B0(z) = (1 −1.7 cos(2πTs×200)z−1+ 0.852z−2)(1 −

1.6 cos(2πTs×700)z−1+ 0.82z−2)(1 + 0.7z−1); the re-

maining designs use the prototype LLS solution.

The measured sensitivity functions in Fig. 7 demonstrate

the challenge of wide-band LLS compared to narrow-

band loop shaping. Compared to the solid line, which

shows no visible waterbed ampliﬁcation, a large gain

increase occurred between 200 Hz and 300 Hz in the

dashed line, due to |Q(ejω )|being not suﬃciently small

to accommodate the model mismatch beyond 200 Hz.

With the cascaded IIR enhancement, |Q(ejω )|in the dot-

ted line of Fig. 6 decreases rapidly after 200 Hz. Cor-

6

101102103

−60

−40

−20

0

Gain (dB)

101102103

−180

−90

0

90

180

Phase (degree)

Frequency (Hz)

basic Q

w/ enhancement @ {200, 700} Hz

w/ cascaded IIR enhancement

basic Q (narrow−band case)

Fig. 6. Frequency responses of diﬀerent Q-ﬁlter designs.

102103

−30

−20

−10

0

10

Magnitude (dB)

Frequency (Hz)

baseline

w/ basic Q (narrow−band case)

w/ basic Q (wide−band case)

Fig. 7. Magnitude response of the sensitivity functions in

narrow- and wide-band LLS (only ﬁnite gridding of the fre-

quency can be obtained during experiments; both the solid

and the dashed lines actually have zero gain at 100 Hz)

respondingly, the large ampliﬁcation after 200 Hz is re-

moved in the sensitivity function in Fig. 8a. To match

the shape of the pass band in Fig. 8, the IIR enhance-

ment yields a larger |Q(ejω )|between 100 Hz and 200

Hz. This explains the shift of the waterbed eﬀect in the

sensitivity function. Applying the zero modulation to re-

duce |Q(ejω )|between 100 Hz and 200 Hz (dashed line

in Fig. 6), one can remove the ampliﬁcation below 200

Hz, as shown in Fig. 8.

4.2 Audio-vibration rejection in HDDs

The plant and baseline servo design of the HDD servo

benchmark have been described in [24,10]. The top plot

of Fig. 9 shows the spectrum of the position error signal

(PES) without LLS. After an LLS design that is similar

to the dashed line in Fig. 3, the bottom plot of Fig. 9

shows the resulted PES, where the spectrum has been

greatly ﬂattened compared to the baseline result. The

three-sigma value (sigma is the standard deviation) has

reduced from 33.54% TP (Track Pitch) to 22.79% TP

(here 1 TP = 254 nm), yielding a 29.07 percent improve-

ment. Notice that 1600 Hz is above the bandwidth of

102103

−40

−30

−20

−10

0

10

Magnitude (dB)

Frequency (Hz)

basic Q

baseline

w/ cascaded IIR enhancement

(a) based on IIR enhancement

102103

−40

−30

−20

−10

0

10

Magnitude (dB)

Frequency (Hz)

(a) basic Q

(b) baseline

(c) w/ enhancement @ {200, 700} Hz

(b) based on zero modulations

Fig. 8. Control of the waterbed eﬀect in wide-band LLS

the servo system, where disturbance rejection was not

feasible in the original baseline design.

500 1000 1500 2000 2500 3000

0

0.01

0.02

0.03

Normalized Magnitude

baseline: 3σ = 33.54 %TP

500 1000 1500 2000 2500 3000

0

0.01

0.02

0.03

Frequency (Hz)

w/ LLS: 3σ = 23.79 %TP

Normalized Magnitude

Fig. 9. Spectra (FFT) of the PES with and without LLS

5 Conclusions and Discussions

This paper has discussed a pseudo inverse-based Youla-

Kucera parameterization scheme for selectively enhanc-

ing the closed-loop servo performance at wide frequency

ranges. Simulation and experiments have been con-

ducted to validate the proposed designs. In the presence

of the fundamental limitation of feedback control, anal-

ysis and design methodologies have been presented to

control the waterbed eﬀect and minimize the negative

impact based on the servo task and the disturbance

7

spectra. Such design ﬂexibility is particularly needed

in precision systems, or applications where the distur-

bances are composed of rich frequency components.

References

[1] X. Chen, W. Xi, Y. Kim, and K. Tu, “Methods for closed-loop

compensation of ultra-high frequency disturbances in hard

disk drives and hard disk drives utilizing same,” U.S. Patent

8 630 059, January, 2014.

[2] M. Tomizuka, “Dealing with periodic disturbances in controls

of mechanical systems,” Annu. Rev. in Control, vol. 32, no. 2,

pp. 193 – 199, 2008.

[3] I. D. Landau, A. C. Silva, T.-B. Airimitoaie, G. Buche,

and M. Noe, “Benchmark on adaptive regulation–rejection of

unknown/time-varying multiple narrow band disturbances,”

European J. Control, vol. 19, no. 4, pp. 237 – 252, 2013.

[4] R. de Callafon, J. Zeng, and C. E. Kinney, “Active noise

control in a forced-air cooling system,” Control Engineering

Practice, vol. 18, no. 9, pp. 1045–1052, Sep. 2010.

[5] T. Oomen, R. van Herpen, S. Quist, M. van de

Wal, O. Bosgra, and M. Steinbuch, “Connecting system

identiﬁcation and robust control for next-generation motion

control of a wafer stage,” IEEE Trans. Control Syst.

Technol., vol. 22, no. 1, pp. 102–118, Jan 2014.

[6] X. Chen and M. Tomizuka, “A minimum parameter adaptive

approach for rejecting multiple narrow-band disturbances

with application to hard disk drives,” IEEE Trans. Control

Syst. Technol., vol. 20, no. 2, pp. 408 –415, march 2012.

[7] S. Hara, Y. Yamamoto, T. Omata, and M. Nakano,

“Repetitive control system: a new type servo system for

periodic exogenous signals,” IEEE Trans. Autom. Control,

vol. 33, no. 7, pp. 659 –668, Jul 1988.

[8] M. Tomizuka, T.-C. Tsao, and K.-K. Chew, “Analysis and

synthesis of discrete-time repetitive controllers,” ASME J.

Dyn. Syst., Meas., Control, vol. 111, no. 3, pp. 353–358, 1989.

[9] M. Steinbuch, “Repetitive control for systems with uncertain

period-time,” Automatica, vol. 38, no. 12, pp. 2103–2109,

2002.

[10] X. Chen and M. Tomizuka, “New repetitive control

with improved steady-state performance and accelerated

transient,” IEEE Trans. Control Syst. Technol., vol. 22, no. 2,

pp. 664–675, March 2014.

[11] L. Sievers and A. von Flotow, “Comparison and extensions

of control methods for narrow-band disturbance rejection,”

IEEE Trans. Signal Process., vol. 40, no. 10, pp. 2377–2391,

1992.

[12] M. Bodson, “Adaptive

algorithms for the rejection of sinusoidal disturbances with

unknown frequency,” Automatica, vol. 33, no. 12, pp. 2213–

2221, Dec. 1997.

[13] X. Chen and M. Tomizuka, “Selective model inversion and

adaptive disturbance observer for time-varying vibration

rejection on an active-suspension benchmark,” European J.

Control, vol. 19, no. 4, pp. 300 – 312, 2013.

[14] I. D.

Landau, A. Constantinescu, and D. Rey, “Adaptive narrow

band disturbance rejection applied to an active suspension–

an internal model principle approach,” Automatica, vol. 41,

no. 4, pp. 563–574, Apr. 2005.

[15] Q. Zhang and L. Brown, “Noise analysis of an algorithm

for uncertain frequency identiﬁcation,” IEEE Trans. Autom.

Control, vol. 51, no. 1, pp. 103–110, 2006.

[16] F. Ben-Amara, P. T. Kabamba, and a. G. Ulsoy, “Adaptive

sinusoidal disturbance rejection in linear discrete-time

systems–part I: Theory,” ASME J. Dyn. Syst., Meas.,

Control, vol. 121, no. 4, pp. 648–654, 1999.

[17] Z. Wu and M. Liu, “Adaptive regulation against unknown

narrow band disturbances applied to the ﬂying height control

in data storage systems,” Asian J. Control, vol. 16, no. 5, pp.

1532–1540, 2014.

[18] R. A. de Callafon and H. Fang, “Adaptive regulation via

weighted robust estimation and automatic controller tuning,”

European J. Control, vol. 19, no. 4, pp. 266 – 278, 2013.

[19] B. Widrow, J. Glover Jr, J. McCool, J. Kaunitz, C. Williams,

R. Hearn, J. Zeidler, E. Dong Jr, and R. Goodlin, “Adaptive

noise cancelling: principles and applications,” Proc. IEEE,

vol. 63, no. 12, pp. 1692–1716, 1975.

[20] D. Youla, J. J. Bongiorno, and H. Jabr, “Modern wiener–hopf

design of optimal controllers part i: The single-input-output

case,” IEEE Trans. Autom. Control, vol. 21, no. 1, pp. 3–13,

1976.

[21] V. Kucera, “Stability of discrete linear feedback systems,” in

Proc. 6th IFAC World Congress, paper 44.1, vol. 1, 1975.

[22] I. D. Landau, R. Lozano, and M. M’Saad, Adaptive Control,

J. W. Modestino, A. Fettweis, J. L. Massey, M. Thoma, E. D.

Sontag, and B. W. Dickinson, Eds. Springer-Verlag New

York, Inc., 1998.

[23] B.-F. Wu and E. Jonckheere, “A simpliﬁed approach to bode’s

theorem for continuous-time and discrete-time systems,”

IEEE Trans. Autom. Control, vol. 37, no. 11, pp. 1797–1802,

Nov 1992.

[24] IEEJ, Technical Commitee for Novel Nanoscale Servo

Control, “NSS benchmark problem of hard disk drive

systems,” http://mizugaki.iis.u-tokyo.ac.jp/nss/, 2007.

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