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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.
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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.
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: (Xu Chen?), (Tianyu Jiang), (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 Htheories. 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=ND1, (ii) N(∈ S)and D(∈ S )are
coprime transfer functions, and (iii) D1∈ 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
YNQ :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:
1 + P Call
1 + P C 1N
Literature often additionally normalizes the coprime
factorization, such that NX +DY = 1 and thus
S=D(YNQ). 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 1N 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 P1(z)and
C(z)are stable, then
Call (z) = C(z) + zmP1(z)Q(z)
1zmQ(z), Q(z)∈ S (3)
parameterizes all stabilizing controllers for P(z), and
the sensitivity function is
S(z) = 1zmQ(z)
1 + P(z)C(z)
Proof follows by letting X(z) = C(z),Y(z)=1,
N(z) = zm, and D(z) = zmP1(z)in Theorem 1.
2Y()denotes, respectively, Y(s)|s=and Y(z)|z=, in
the continuous- and the discrete-time cases.
Proposition 2 reduces the add-on module in the sensitiv-
ity function to 1zmQ(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 zmap-
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) + zmˆ
1zmQ(z), Q(z)∈ S (5)
where ˆ
P1(z)is chosen stable. If P(z) = ˆ
P(z), then
the new feedback loop consisting of P(z)and ˜
guaranteed stability. Otherwise, the new feedback loop is
stable if the roots of the following characteristic equation
are all inside the unit circle:
P(z)[AC(z)AP(z) + BC(z)BP(z)]
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),
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) + zmAC(z)Aˆ
P(z)AC(z) [AQ(z)zmBQ(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
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
As the baseline feedback loop, Q(z), and ˆ
P1(z)are all
stable, the new closed loop is thus stable. 2
Proposition 3 relaxes the requirements on stable Cand
P1by 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
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
P1and P1, 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) = 1zmQ(z)
1 + P(z)C(z) + zmQ(z) ( ˆ
P1(z)P(z)1) (9)
If ˆ
P1(e )P(e )=1, namely, at frequencies where the
inverse model is accurate, (9) gives
S(e ) = 1emjω Q(ejω)
1 + P(e )C(e )(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 emjωiQ(ej ωi)=1, which gives perfect distur-
bance rejection (S(ei)=0) and reference tracking
(T(ei) = 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(ek)0, to keep the influence
of the mismatch element zmQ(z)( ˆ
small in (9). More formally, as Q(z)and the base-
line sensitivity 1/(1 + P(z)C(z)) are both stable, if
|Q(e )(P(e )ˆ
P1(e )1)|<
1 + P(e )C(e )
(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 zmQ(z)|z=ei1,c(k)approximates d(k)at
ωi(hence canceling the disturbance). Such time-domain
intuition can be used for tuning during implementations.
? d(k)
Fig. 1. Block diagram of pseudo YK parameterization
3Indeed, inverse-based design has long been used in motion
control, e.g., in feedforward designs.
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
2ζcos ωiz1+ζ2z2)—or after multiplication—Aζ(z) =
1 + a1ζz1+· · · +anζnzn+· · · +a1ζ2n1z2n+1 +
ζ2nz2n, 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(ei) = ˆ
P(ei), and
Q(z) = BQ(z)
Aα(z):K(z)Aβ(z)+zmBQ(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
and Q(ei) = ej i.
We verify the solution along with a discussion on several
central concepts in LLS. (11) yields
1zmQ(z) = Aβ(z)
For each ωi,Aβ(z)/Aα(z)has a pair of damped pole
and zero αe±i, β e±i.αcontrols the width of the
attenuation range. The zero βei—on or close to the
unit circle—provides small gains to
1ejωm Qe
when ωis close to ωi. If β= 1, applying cos(ωi) =
(ei+ej ωi)/2gives Aβ=1(ei) = (1 2 cos ωiei+
j=1,j6=i(1 2 cos ωjej ωi+e2i)=0, and
hence Q(ei) = ej iin (12), which renders S(ei) =
0in (10) and therefore full disturbance rejection at ωi.
K(z)Aβ(z) + zmBQ(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 zmare coprime.
The minimum-order solution satisfies BQ(z) = bQ,0+
bQ,1z1+· · · +bQ,2n1z2n+1 and
K(z) = k0+k1z1+· · · +km1zm+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+k1z1. Matching coefficients of zi’s in
the Diophantine equation (11) gives
Q(z)=(α1)(α+ 1 a2)az1
1 + aαz1+α2z2(14)
with a=2 cos(ω1)and K(z) = 1 + (α1)az1.
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|1zjmω Q(e )|(= ||1zmQ(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
ln |1emjω Q(e)|dω=π l
ln |γi| − ln |σ+ 1|!
where γi(l0) are the unstable (outside the unit circle)
zeros of 1zmQ(z), and σ= limz→∞{zmQ(z)/(1
The proof follows by invoking Bode’s Integral Theorem
[23] and treating LQ=zmQ(z)/(1 zmQ(z)) as the
open-loop transfer function in a negative feedback loop.
As a particular case, if m > 0and 1zmQ(z)has no
unstable zeros, then Rπ
0ln |1emjω Qe |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 1Q(z) = kAβ(z)/Aα(z).
Letting k(0,1) gives Q() = 1 kand then
ln |σ+ 1|= ln |1/k|>0in Corollary 4. Additionally,
1Qhas 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 m1. 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 m1.
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
Qe |,|1ejmω ˜
Qe |is closer to unity outside
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(ei). Then
|S(ei)|= (1 g)|So(ej ωi)|and ||1zm˜
1 + g(||1zm˜
Q(z)||1) <||1zmQ(z)||.
Proof The prototype LLS solution satisfies Q(ei) =
ejmωi. Hence 1ej i˜
Q(ei)=1g, and
|S(ei)|= (1 g)|So(ei)|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 ||1zm˜
Q||= supu6=0{||[1
Q]u||2/||u||2}= supu6=0{||[1zmgQ]u||2/||u||2} ≤
supu6=0{[||(1 g)u||2+||g(1 zmQ)u||2]/||u||2}=
1g+g||1zmQ||<||1zmQ||, where the last
inequality is due to ||1zmQ||>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
zmQ(z), if exist, can be pulled into the unit disk by
cascading the gain g. From root locus analysis, regard
zmgQ(z)as the open-loop transfer function, then as g
moves from 1 to 0, poles of 1/(1gzmQ(z))—i.e. zeros
of 1gzmQ(z)—move from poles of 1/(1 zmQ(z))
to poles of zmQ(z), which are all stable.
Besides the overall detuning to reduce ||1zmQ(z)||,
Q(z)can be locally manipulated to reduce |1
ejmω Q(e )|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
Designing B0(z)=1z1for 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
ejmω Qe |in the highlighted region.
Magnitude (dB)
Magnitude (dB)
Frequency (Hz)
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)=12ρcos ωpz1+ρ2z2, which places fixed
zeros ρe±pin (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 + 0.7z1)(1
0.86z1)(1 1.6 cos(2π×6000Ts)z1+ 0.82z2)(1
1.4 cos(2π×10000Ts)z1+0.72z2). There, reduced gain
is achieved at almost all frequencies outside the pass
bands of Q(z).
Magnitude (dB)
Magnitude (dB)
Frequency (Hz)
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) + zmB0(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,1z1+· · · +bQ,2n1z2n+1 ;K(z) = k0+
k1z1+· · · +k(m+nB01) z(m+nB01).
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 |1ejmωQ(ej ω)|, as
shown next.
Corollary 6 If Q(z)(∈ S)has an unstable zero zuthen
where pi’s are Np(0) unstable zeros of 1zmQ(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 ||1zmQ(z)||1.
Proof Applying all-pass factorization such that
1zmQ(z) = Mm(z)QNp
i=1(zpi)/(z¯pi1), where
Mm(z)is minimum-phase. Then ||1zmQ(z)||=
||Mm(z)||and Mm(zu)QNp
i=1(zupi)/(zu¯pi1) = 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(zpi)/(z¯pi1) = 1.
Otherwise straightforward complex analysis gives that
|zu¯pi1|/|zupi|>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(ei)=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 1zmQ(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.
Magnitude (dB)
Magnitude (dB)
Frequency (Hz)
B0 = 1
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 ×
107(z+ 1) /[z(z1)2](sampling frequency: 2500 Hz).
Pn(z)has a relative degree of two (i.e. m= 2), and a
Frequency (Hz)
Magnitude (dB)
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 zmˆ
P1(z) = 107/3.47655 ×
(z1)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
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)z1+ 0.852z2)(1
1.6 cos(2πTs×700)z1+ 0.82z2)(1 + 0.7z1); 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(e )|being not sufficiently small
to accommodate the model mismatch beyond 200 Hz.
With the cascaded IIR enhancement, |Q(e )|in the dot-
ted line of Fig. 6 decreases rapidly after 200 Hz. Cor-
Gain (dB)
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.
Magnitude (dB)
Frequency (Hz)
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(e )|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(e )|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
Magnitude (dB)
Frequency (Hz)
basic Q
w/ cascaded IIR enhancement
(a) based on IIR enhancement
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
Normalized Magnitude
baseline: 3σ = 33.54 %TP
500 1000 1500 2000 2500 3000
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
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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... A narrow notch in the error-rejection function can no longer provide sufficient attenuation; yet a wide notch tends to cause undesired amplification at other frequencies due to the fundamental waterbed limitation of feedback control [16]- [18]. In view of such challenges, the authors proposed an infinite-impulseresponse (IIR) filter design in DOB to control the waterbed effect manually [15], [19] and optimally [20]; such a design also benefits narrow-band disturbance rejection, and underpins first-tier results [21], [22] in an international benchmark on adaptive regulation [12]. [23] provides additional comparison of the DOB framework with peak filter algorithms. ...
... Careful stability guarantees must be enforced for algorithms that directly update C [3], [6], [8], [9], [13]. By using a control architecture that intrinsically guarantees stability, algorithms built on Youla parameterization [10], [19], [24]- [26] have the benefit of directly shaping the closed-loop sensitivity function S with, e.g., S = (1 − N Q/Y )/(1 + P C), where transfer functions N and Y come from coprime factorizations of P and C, and transfer function Q is the affine design parameter. For plants with a stable inverse, we have shown that N and Y are reducible to yield the minimumorder factorization S = (1 − z −m Q)/(1 + P C). ...
... For plants with a stable inverse, we have shown that N and Y are reducible to yield the minimumorder factorization S = (1 − z −m Q)/(1 + P C). This is the approach proposed in the aforementioned DOB [1], [19], [21], but has been infeasible for plants with nonminimum-phase zeros. Multiple strategies exist to create stable and realizable model inversions. ...
Full-text available
Closed-loop disturbance rejection without sacrificing overall system performance is a fundamental issue in a wide range of applications from precision motion control, active noise cancellation, to advanced manufacturing. The core of rejecting band-limited disturbances is the shaping of feedback loops to actively and flexibly respond to different disturbance spectra. However, such strong and flexible local loop shaping (LLS) has remained underdeveloped for systems with nonminimum-phase zeros due to challenges to invert the system dynamics. This article proposes an LLS with prescribed performance requirements in systems with nonminimum-phase zeros. Pioneering an integration of the interpolation theory with a model-based parameterization of the closed loop, the proposed solution provides a filter design to match the inverse plant dynamics locally and, as a result, creates a highly effective framework for controlling both narrowband and wideband vibrations. From there, we discuss methods to control the fundamental waterbed limitation, verify the algorithm on a laser beam steering platform in selective laser sintering additive manufacturing, and compare the benefits and tradeoffs over the conventional direct inverse-based loop-shaping method. The results are supported by both simulation and experimentation.
... Given a linear time-invariant system model G, the inversion of G has numerous practical implementations including but not limited to iterative learning control (ILC) [1], repetitive control [2], two-degree-of-freedom servo in feedforward control [3,4], as well as Youla parameterization and disturbance observer in feedback control [5,6,7,8,9,10]. Here, G can be an open-loop plant model or a closed-loop control system. For a minimumphase system, G −1 is stable and ready to be implemented. ...
... The optimal inverse model given by (8) preserves accurate model information in the frequency region specified by W 1 (z) and, on the other hand, penalizes excessive high gains of F (z) at frequencies determined by W 2 (z). Typically, W 1 (z) is a low-pass filter, and W 2 (z) is a high-pass one, as shown in an example in Fig. 10. ...
... Via standard system identification methods, the system model G(z) is identified with a sampling rate of 800 Hz and has an order of 22. Four NMP zeros show up in G(z) (Fig. 9). Implementing the optimization principle in (8) gives the optimal inverse F (z). Figure 10 shows the frequency responses of the two weighting functions. We reduce the order of F (z) to 23 by applying the model-reduction function reduce in MATLAB. ...
Stably inverting a dynamic system model is fundamental to subsequent servo designs. Current inversion techniques have provided effective model matching for feedforward controls. However, when the inverse models are to be implemented in feedback systems, additional considerations are demanded for assuring causality, robustness, and stability under closed-loop constraints. To bridge the gap between accurate model approximations and robust feedback performances, this paper provides a new treatment of unstable zeros in inverse design. We provide first an intuitive pole-zero-map-based inverse tuning to verify the basic principle of the unstable-zero treatment. From there, for general nonminimum-phase and unstable systems, we propose an optimal inversion algorithm that can attain model accuracy at the frequency regions of interest while constraining noise amplification elsewhere to guarantee system robustness. Along the way, we also provide a modern review of model inversion techniques. The proposed algorithm is validated on motion control systems and complex high-order systems.
... To relax these conditions, the disturbance observer control [23][24][25][26][27] is reviewed and discussed for Tip-Tilt mirror control system in this paper. Furthermore, the error-based DOBC controller [28][29][30][31] of Tip-Tilt mirror only based on CCD is proposed here. It can be plugged into an existing feedback loop that leads to a generalized version of 1-Q (Q is the designed filter) brought in the numerator of the original sensitivity function, resulting in the overall sensitivity function equal to zero in theory at the expected frequencies. ...
... In the image sensor-based control system, the position sensor Y(s) cannot measure the disturbance D(s) directly. Thus, a new control block diagram 29,30 in the presence of the disturbance is proposed in Fig. 5. The new closed-loop sensitivity transfer functions illustrated in Fig. 5 can be expressed as follows: ...
Full-text available
Structural vibrations in Tip-Tilt modes usually affect the closed-loop performance of astronomically optical telescopes. In this paper, the state of art control methods-proportional integral (PI) control, linear quadratic Gaussian (LQG) control, disturbance feed forward (DFF) control, and disturbance observer control (DOBC) of Tip-Tilt mirror to reject vibrations are first reviewed, and then compared systematically and comprehensively. Some mathematical transformations allow PI, LQG, DFF, and DOBC to be described in a uniform framework of sensitivity function that expresses their advantages and disadvantages. In essence, feed forward control based-inverse model is the main idea of current techniques, which is dependent on accuracies of models in terms of Tip-Tilt mirror and vibrations. DOBC can relax dependences on accuracy model, and therefore this survey concentrates on concise tutorials of this method with clear descriptions of their features in the control area of disturbance rejections. Its applications in various conditions are reviewed with emphasis on the effectiveness. Finally, the open problems, challenges and research prospects of DOBC of Tip-Tilt mirror are discussed.
... The use of YK parametrization Q in adaptive noise and vibration rejection context covered different applications requiring high control precision and low noise sensitivity as: wafer scanning in semiconductors [46], data storage systems (reading/writing) [39,47,48], mechatronics [49], active suspension systems [40,50] and biochemistry [51] where the regulation problem is to maximize the biomass productivity in the fed-batch fermentation of a specie of yeasts and the cell growth is an undesirable consequence and considered as an unstable disturbance. The specificity of this application is that the YK parameter Q represents an extra degree of freedom in the controller considering the disturbance model, and allowing the rejection of this unstable. ...
Automated vehicles are getting more and more attention because of their potential to improve drivers' lives, ensuring road safety, increasing highway capacity, or reducing carbon emissions. Proper autonomous driving requires vehicle stability, precise motion, and natural behavior guaranteeing comfort for passengers inside the vehicle. However, driving situations change depending on the road layout and potential interactions with other traffic agents. Furthermore, vehicle capabilities can be degraded because of the on-board sensors' limitations, or the complexity of the algorithm processing the perception data. This thesis proposes a multi-objective control architectures that can adapt the vehicle behavior to overcome the changes in the operating conditions and assure vehicle performance and stability. The automated control system should be able to address any circumstances ranging from a sudden change in the driving situation (i.e. lane change, obstacle avoidance) to an inaccurate measurement. This thesis uses Youla-Kucera (YK) parametrization to design control structures able to recognize the driving situation changes, adapting the controller response to satisfy the required performance level, and keeping the motion stability with a natural vehicle behavior. In this thesis we propose novel control structures based on controller reconfiguration, improving both lateral and longitudinal control state-of-the-art by solving the following problems: 1) The trade-off between precision in trajectory tracking and comfort when the driving situation changes in lateral motion; 2) The trade-off between robustness and performance when noise measurement appears in Adaptive Cruise Control (ACC) systems. The stability of the proposed controller is guaranteed thanks to YK parametrization. The validation of the proposed control structures is provided in both simulation and real-time experimentation using a Renault ZOE vehicle. The adaptability of the controllers to autonomous driving tasks is proved in different operating conditions.
... The use of YK parametrization Q in adaptive noise and vibration rejection context covered different applications requiring high control precision and low noise sensitivity as: wafer scanning in semiconductors Chen, Jiang, and Tomizuka (2015) , data storage systems (reading/writing) Chen and Tomizuka (2013) ; Martinez and Alma (2012) ; Wu, Zhang, Chen, and Wang (2018) , mechatronics Tomizuka (2008) , active suspension systems Doumiati et al. (2017) ; Landau, Constantinescu, and Rey (2005) , and biochemistry Valentinotti et al. (2003) where the regulation problem is to maximize the biomass productivity in the fed-batch fermentation of a specie of yeasts and the cell growth is considered as an unstable disturbance rejected by the YK parameter Q. In Luca, Rodriguez-Ayerbe, and Dumur (2011) the feedback YK noise rejection controller is extended to control a LPV plant. ...
Youla-Kucera (YK) parametrization was formulated decades ago for obtaining the set of controllers stabilizing a linear plant. This fundamental result of control theory has been used to develop theoretical tools solving many control problems ranging from stable controller switching, closed-loop identification, robust control, disturbance rejection, adaptive control to fault tolerant control. This paper collects the recent work and classifies them according to the use of YK parametrization, Dual YK parametrization or both, providing the latest advances with main applications in different control fields. A final discussion gives some insights on the future trends in the field.
... Then an improved error-based disturbance observer based on Youla parameterization is proposed to reduce this problem. Youla parameterization is a method that Youla et al proved in the late 1970s to parameterize a stable controller while ensuring the closed-loop stability of the control system [25]. The basic principle of the method is shown as follows. ...
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Vibration rejection is a key technology of practical engineering, especially in optical telescopes with a stable accuracy of urad level. The closed-loop performance of optical telescopes is largely determined by the control bandwidth, while it is severely limited by the low sampling rate and large time delay of the image sensor, so it is difficult to mitigate structural vibrations in optical telescopes, especially wideband vibrations, because they exist universally and greatly influence the stability of the system. This paper develops an improved error-based disturbance observer (EDOB) based on the Youla parameterization approach to mitigate wideband vibrations in optical telescopes. This novel method can greatly improve the vibration rejection ability of the system by designing a proper Q-filter to accommodate wideband vibrations when their frequencies can be acquired. Because wideband vibrations in optical telescopes can be considered as multiple narrow-band vibrations with similar central frequencies, a novel Q-filter instead of a single wideband notch filter is proposed to mitigate wideband vibrations when considering the stability and closed-loop performance of the system. Moreover, this method only relies on a low frequency model, leading to a reduction in model dependence. Both the simulations and experimental results show that the error-based disturbance observer based on Youla parameterization can greatly improve the closed-loop performance of the system compared with the traditional feedback control loop.
Linear Parameter-Varying concept has been formulated decades ago for model linearization and control. This fundamental result of control theory has been used to develop theoretical tools solving many control problems as closed-loop stability, robustness, and optimized performance. This review is dedicated to the LPV switching/interpolating control techniques that have been investigated in the literature. It collects the recent works and classifies them according to the use of gain-scheduling, switching, or interpolating, providing the latest advances with main applications in different control fields. A final discussion gives the timeline from the past to present trends in the field.
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Constant in gain Lead in phase (CgLp) compensators, which are a type of reset elements, have shown high potential to overcome limitations of linear control systems. There are few works which investigate the tuning of these compensators. However, there are some significant drawbacks which make those methods unreliable. First, their analyses are performed in the open-loop configuration which do not guarantee the existence of steady-state response of the closed-loop. If it is guaranteed, unlike linear control systems, open-loop analyses cannot precisely predict the closed-loop steady-state performance. In addition, the stability condition could not be assessed during the tuning process. These significant challenges have been separately solved in our recent works by proposing frequency-domain frameworks for analyzing the closed-loop performance and stability of reset control systems. However, they are not formulated and implemented for tuning CgLp compensators. In this paper, based on the loop-shaping approach, the recent frequency-domain framework and the frequency-domain stability method are utilized to provide a reliable frequency-domain tuning method for CgLp compensators. Finally, different performance metrics of a CgLp compensator, tuned by the proposed method, are compared with those of a PID controller on a precision positioning stage. The results show that this method is effective, and the tuned CgLp can achieve more favorable dynamic performance than the PID controller.
Tip-Tilt mirrors play an important role in astronomical telescopes requiring the tracking performance at the level of microradian or sub-microradian. However, the closed-loop performance suffers a lot from the low-sample rate and time delay of image sensors. Especially, this issue is under the condition of vibrations, because dynamic behaviors are complex and the models are difficult to be obtained accurately. Another challenging issue comes from the measurement of vibrations and its extraction for the closed-loop control. This paper proposes a new method based on an add-on controller of the Tip-Tilt mirror to mitigate telescope vibrations. The proposed method only uses Tip-Tilt errors from an image sensor to implement a disturbance observer, which is not being restricted by an accurate model. As a result, the closed-loop performance can be optimized by designing of a proper Q-filter. To suppress the low-frequency and high-frequency vibrations, a novel Q-filter combining a lowpass filter and a bandpass filter is proposed here. The improved control method is validated by both simulation and experiment in the tip-tilt mirror control system under the condition of vibrations.
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In repetitive control (RC), the enhanced servo performance at the fundamental frequency and its higher order harmonics is usually followed by undesired error amplifications at other frequencies. In this paper, we discuss a new structural configuration of the internal model in RC, wherein designers have more flexibility in the repetitive loop-shaping design, and the amplification of nonrepetitive errors can be largely reduced. Compared to conventional RC, the proposed scheme is especially advantageous when the repetitive task is subject to large amounts of nonperiodic disturbances. An additional benefit is that the transient response of this plug-in RC can be easily controlled, leading to an accelerated transient with reduced overshoots. Verification of the algorithm is provided by simulation of a benchmark regulation problem in hard disk drives, and by tracking-control experiments on a laboratory testbed of an industrial wafer scanner.
Full-text available
This paper presents an adaptive control scheme for identifying and rejecting unknown and/or time-varying narrow-band vibrations. We discuss an idea of safely and adaptively inverting a (possibly non-minimum phase) plant dynamics at selected frequency regions, so that structured disturbances (especially vibrations) can be estimated and canceled from the control perspective. By taking advantage of the disturbance model in the design of special infinite-impulse-response (IIR) filters, we can reduce the adaptation to identify the minimum amount of parameters, achieve accurate parameter estimation under noisy environments, and flexibly reject the narrow-band disturbances with clear tuning intuitions. Evaluation of the algorithm is performed via simulation and experiments on an active-suspension benchmark.
Two algorithms are presented for rejecting sinusoidal disturbances with unknown frequency. The first is an indirect algorithm in which an estimate of the frequency of the disturbance is used in another adaptive compensator that adjusts the in-phase and quadrature components of the input signal to attenuate the offending disturbance. The second is a direct algorithm in which frequency estimation and disturbance attenuation are performed simultaneously. Approximate analyses are presented for both schemes and can be used to select the design parameters. Simulations demonstrate the ability of the algorithms to reject sinusoidal disturbances with unknown frequency.
Periodic disturbances are common in control of mechanical systems. Such disturbances may be due to rotational elements such as motors and vibratory elements. When the period of a periodic disturbance is fixed and known in advance, repetitive control can be used for attenuating their effect. The most popular repetitive controller is based on the internal model principle. When the period is not fixed and unknown, adaptation capability must be introduced. This paper presents some fundamental issues and new challenges in the design of controllers to deal with periodic disturbances along with applications to mechanical systems.
This paper presents an adaptive regulation approach in linear systems against exogenous narrow band inputs such as disturbances or reference signals consisting of a linear combination of biased sinusoids with unknown amplitudes, frequencies, and phases. The design of the regulator is based on considering a Q-parameterized set of stabilizing controllers for the linear system, where an adaptive FIR filter with fixed IIR filtering is adopted as the Q parameter. The goal of the adaptation is to search within the set of stabilizing controllers for a controller, or equivalently a Q parameter, that yields regulation in the closed loop system. The proposed adaptive regulation algorithm is applied to an active suspension beam system, which is motivated by the flying height control problem in data storage systems. The experimental result of the closed loop system shows the effectiveness of the proposed adaptive regulator in achieving the desired tracking performance under unknown exogenous disturbances.
This paper proposes a methodology to adaptively reduce time-varying additive disturbances in a feedback system, comprised a plant with uncertainty and an adaptable linear feedback controller. Adaptive regulation is done via the direct estimation of a perturbation on the feedback controller in a Youla–Kucera parametrization. Uncertainty on the plant dynamics bounds the size of the allowable controller perturbation for adaptation to maintain stability robustness. By simultaneously minimizing the variance of the plant output signal and a control output signal, the direct estimation of the controller perturbation is formulated as a weighted robust estimation problem that is implemented recursively for a real-time implementation. The methodology is applied to a vibration control benchmark to demonstrate how the proposed adaptive feedback regulation can effectively reduce unknown harmonic disturbances with a time-varying frequency for a mechanical system with unmodelled dynamics.
Next-generation precision motion systems are lightweight to meet stringent requirements regarding throughput and accuracy. Such lightweight systems typically exhibit lightly damped flexible dynamics in the controller cross-over region. State-of-the-art modeling and motion control design procedures do not deliver the required model complexity and fidelity to control the flexible dynamical behavior. The aim of this paper is to develop a combined system identification and robust control design procedure for high performance motion control and apply it to a wafer stage. Hereto, new connections between system identification and robust control are employed. The experimental results confirm that the proposed procedure significantly extends existing results and enables next-generation motion control design.