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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.
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