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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING
Int. J. Adapt. Control Signal Process. 2015; 00:1–16
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs
Overview and New Results in Disturbance Observer based Adaptive
Vibration Rejection With Application to Advanced Manufacturing
Xu Chen1∗and Masayoshi Tomizuka2
1Department of Mechanical Engineering, University of Connecticut, Storrs, CT, U.S.A.
2Department of Mechanical Engineering, University of California, Berkeley, CA, U.S.A.
SUMMARY
Vibrations with unknown and/or timevarying frequencies signiﬁcantly aﬀect the achievable
performance of control systems, particularly in precision engineering and manufacturing applications.
This paper provides an overview of disturbanceobserver based adaptive vibration rejection schemes;
studies several new results in algorithm design; and discusses new applications in semiconductor
manufacturing. We show the construction of inversemodelbased controller parameterization, and
discuss its beneﬁts in decoupled design, algorithm tuning, and parameter adaptation. Also studied are
the formulation of recursive least squares and outputerror based adaptation algorithms, as well as
their corresponding scopes of applications. Experiments on a wafer scanner testbed in semiconductor
manufacturing prove the eﬀectiveness of the algorithm in highprecision motion control.
Copyright c
2015 John Wiley & Sons, Ltd.
Received . . .
KEY WORDS: adaptive vibration rejection, narrowband disturbances, advanced manufacturing,
inversebased control, semiconductor manufacturing, disturbance observer
1. INTRODUCTION
Identiﬁcation and rejection of vibrations are of fundamental importance in science and
engineering. Many problems—such as attenuation of repeatable and nonrepeatable runout
disturbances in hard disk drives (HDDs) [1], rejecting fan noise from cooling systems
[2,3], and suspension control [4]—fall into the category of vibration rejection. Although
many conventional problems have been relatively well solved, new and emerging challenges
continuously occur. For instance, to meet the everincreasing demand of storage volume in the
era of big data, the disk drive industry is moving towards the new bit patterned media (BPM)
recording. Each bit on BPM is stored in a fundamentally diﬀerent principle, to reach a 2030
times reduction in the unit storage size [5]. In the new architecture, vibrationrelated issues
have become more important than ever before, with various new challenges such as signiﬁcant
increase of harmonics, much wider span of disturbance frequencies, etc [6].
Fundamentally, the importance of highperformance vibration rejection roots in the fact
that practical systems, as long as they involve rotational or periodic motions, are inevitably
subjected to periodic commands and/or disturbances. The problem of adaptive narrow
band disturbance†rejection has thus attracted great research attention, and recently been
∗Correspondence to: Xu Chen, 191 Auditorium Road, Unit 3139, University of Connecticut, Storrs, CT, 06269
3139, U.S.A. Email: <xchen@engr.uconn.edu>
†Disturbances whose energies are concentrated within narrow frequency bands.
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2015 John Wiley & Sons, Ltd.
Prepared using acsauth.cls [Version: 2010/03/27 v2.00]
2
extensively studied in the multiband situation [7]. Many control design algorithms have been
studied in the related literature. Adaptive noise cancellation [8] uses sensors and stochastic
gradientbased adaptation to cancel the disturbance eﬀect. Adaptive feedforward cancellation
[9] composes an estimate of the sinusoidal disturbance using trigonometric functions. In
feedback control, it has been understood that for asymptotic disturbance rejection, controllers
should be customized to incorporate the model of the disturbance. This internalmodel
principle approach has been essential for the achieved results in [4,10,11,12,13,14,2]. In the
scope of direct adaptive controls, [12,11] used statespace designs; [4,10,13,14,2] applied
Youla Parameterization with adaptive ﬁniteimpulseresponse (FIR) ﬁlters. Alternatively,
indirect adaptive control can be formulated after online identiﬁcation of the disturbance
frequencies. Among the related literature on frequency identiﬁcation, MUSIC and ESPRIT
[15,16] are two of the most commonly used subspace spectrum estimation methods. Both
algorithms are batch processes that require an entire sequence (of suﬃcient length) of data to
ﬁnd the spectral peaks of the signals. Compared to Fast Fourier Transform (FFT), MUSIC
and ESPRIT require less data points to generate accurate identiﬁcation in noisy environments.
For adaptive control, more common than the batch process are the adaptive notch ﬁlters and
phase lock loops, where improved frequency identiﬁcation is provided at each time step; and in
addition, it is possible to track the change of the disturbance characteristics online. Algorithms
of this kind include [17,18,19,20,21,22,23,24], among which [18,19,20,21,23] are state
space constructions suitable for diﬀerent computation complexities; [22] is a phase locked loop
method that is commonly combined with adaptive feedforward cancellation; [17] is a popular
adaptive notch ﬁlter based on recursive prediction error methods.
Recently, focusing on the concept of selective model inversion, the authors designed a
class of adaptive narrowband disturbance observers for spectral peak suppression. Successful
applications have been achieved on HDDs in information storage systems [25,26], electrical
power steering [27], and the recent benchmark on active suspension [28,29]. Comparison to
algorithms of peer researchers has also been made in the benchmark summary report [7].
The purposes of this paper are twofold. First, we provide an overview and new discussions on
diﬀerent design options in the adaptive narrowband disturbance observer. New results about
the selection (and the corresponding designs) between FIR and inﬁnite impulse response (IIR)
ﬁlters are discussed. We study the use of an extended internal model principle for IIRbased
disturbance rejection, and show the reduction of error ampliﬁcation from the waterbed eﬀect,
which has often been theoretically overlooked previously. Also discussed are the parameter
adaptation in nonideal practical implementations, especially for situations where the number
of disturbance components are unknown. Another contribution of the paper is the new
implementation results in advanced manufacturing, where vibration and disturbance rejection
have been key for nextgeneration technology development, due to the continuously updating
precision requirement that is driven by Moore’s law.
The remainder of the paper is organized as follows. Section 2provides an overview of
the deterministic control structure and discusses the design options. Section 3discusses the
parameter adaptation under diﬀerent application environments. Experimental results on a
wafer scanner in semiconductor manufacturing are provided in Section 4. Section 5concludes
the paper.
2. DETERMINISTIC SELECTIVE MODEL INVERSION
2.1. Disturbance cancellation in disturbance observers
The narrowband disturbance observer is an inversebased disturbance cancellation scheme as
shown in Fig. 1. Independent from the outerloop feedback controller C(z−1), the thick red
line builds an internal feedback of the disturbance d(k).
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
3
+
+

+
C
(z
¡
1
)
u(k
)
y(k
)
P
(z
¡
1
)
z
¡
m
Q
(z
¡
1
)
P
¡
1
n
(z
¡
1
)
d
(k
)
^
d
(k
)
+
+
0
d
(k
)
Figure 1. Disturbance estimation and feedback in disturbance observers: P(z−1)–the controlled plant,
C(z−1)–feedback controller, P−1
n(z−1) = z−mˆ
P−1(z−1)–nominal inverse of the plant, z−m–relative
degree of the plant, Q(z−1)–Q ﬁlter.
More speciﬁcally, focus ﬁrst on the signal ﬂow from d(k)to ˆ
d(k). If P−1
n(z−1)correctly inverts
the plant dynamics P(z−1), then ideally letting Q(z−1)=1 will create a internal negative
feedback of d(k), hence the cancellation of the disturbance at the input of the plant.
To analyze the ﬂows of the control command u(k), notice ﬁrst, that practical systems
inevitably contain uncertainties that are excluded in the transfer function model P(z−1).
In addition, P−1(z−1)is mostly acausal and not directly implementable. Hence P−1
n(z−1) =
P−1(z−1)is practically unachievable. In narrowband disturbance observer, P−1
n(z−1)is set
as z−mˆ
P−1(z−1), where mis the relative degree of P(z−1)for realizability of P−1
n(z−1), and
ˆ
P(z−1)is a nominal stable inverse that captures the main dynamics of P(z−1). This way,
P−1
n(z−1)P(z−1)approximately equals z−m. Observing now the two internal paths from u(k)
to Q(z−1)—through P−1
n(z−1)P(z−1)and z−m, respectively—one can see that u(k)actually
has null inﬂuence on ˆ
d(k).
From the discussions in the last paragraph, one may observe that the disturbance observer is
build on the idea of rejecting d(k)without inﬂuencing the output of C(z−1).Q(z−1)can thus
be designed independently from the feedback controller C(z−1). This is the decoupled design
principle in disturbanceobserverbased feedback design.
Remark. Ultimately, the control goal is to eliminate the eﬀect of disturbance from the plant
output. Certainly, actual disturbances can impact the system internally and externally, from
both the input and the output of the plant. Yet the control algorithm can only inﬂuence the
command at the input side of the plant. In Fig. 1, we have lumped the disturbance at the
plant input as a mathematical extraction. The algorithm also works if the disturbance enters
via the more general BoxJenkins model y(k) = P(z−1)u(k) + Md(z−1)d(k)(see [28,29]),
where Md(z−1)is a rational transfer function modeling the disturbance dynamics. In this
case, the disturbance observer maintains its eﬀect by estimating and canceling an equivalent
input disturbance.
Although the inverse architecture provides rich information of d(k)in the raw estimate
dr(k), the signaltonoise ratio in dr(k)is low due to uncertaintyinduced model mismatch
(common at high frequencies) and sensor noises. Conventionally, Q(z−1)was chosen as a low
pass ﬁlter to avoid highfrequency noise ampliﬁcations. This is suﬃcient for lowfrequency
servo enhancement, yet infeasible for highperformance vibration rejection. To see this, notice
that P−1
nP≈z−mand hence dr(k)≈z−md(k)(z−1as a delay operator here). Therefore dr(k)
is actually a delayed estimate of d(k). For vibration rejection, especially at high frequencies,
the delay will introduce large mismatch between dr(k)and d(k)(see one example in [26]). To
cope with the diﬃculty, design of Q(z−1)in narrowband disturbance observer is based on the
cancellation criterion:
d(k)−ˆ
d(k) = d(k)−Q(z−1)P−1
n(z−1)P(z−1)d(k)≈1−z−mQ(z−1)d(k).(1)
Based on (1), for vibration signals satisfying
A(z−1)d(k)=0(asymptotically),(2)
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2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
4
asymptotic perfect disturbance rejection is achieved if the feedback loop is constructed to
contain the internal model:
1−z−mQ(z−1) = Kc(z−1)A(z−1)(3)
where Kc(z−1)is a realizability ﬁlter for Q(z−1)to be causal.
2.2. Extended internal model principle
The most intuitive internalmodel based approach is perhaps to design from the time domain.
For instance, consider the case where m= 1 and A(z−1) = 1 −2 cos ωoz−1+z−2, i.e. d(k)
is a singlefrequency vibration in the form of Cosin(ωok+ζo). Letting Kc(z−1)=1 and
1−z−1Q(z−1)=1−2 cos ωoz−1+z−2yields the ﬁrstorder FIR solution of (3):
Qz−1= 2 cos ωo−z−1.(4)
Although satisfying (3), the FIR solution will be seen to be not practical in general. The
narrowband disturbance observer assigns instead
1−z−1Q(z−1) = 1−2 cos ωoz−1+z−2
1−2αcos ωoz−1+α2z−2(5)
and for more general values of m,
1−z−mQ(z−1) = K(z−1)A(z−1)
A(αz−1)(6)
where A(αz−1)[α∈(0,1)] is a damped polynomial of z−1, obtained by replacing every z−1
in A(z−1)with αz−1. Based on (2) and (3), the disturbance is still asymptotically rejected
under the extended internal model A(z−1)/A(αz−1). The new solution to the example at the
beginning of this subsection is
Q(z−1) = −2 (α−1) cos ωo+α2−1z−1
1−2αcos ωoz−1+α2z−2.(7)
The choice of an IIR design (7) over the FIR solution (4) is based on the frequencydomain
properties of the ﬁlters. As shown in the example in Fig. 2, the damped IIR design provides
a bandpass ﬁlter characteristics. As the magnitude of A(z−1)is normalized by A(αz−1), the
frequency response of A(z−1)/A(αz−1)is approximately unity at low and high frequencies.
Hence from (5), the magnitude of Q(z−1)must be close to zero at the corresponding frequency
regions. On the contrary, the FIR design has no default constraints on the ﬁlter magnitude, and
gives a scaled highpass ﬁlter response—signiﬁcantly amplifying the addon noises in dr(k).
The involved magnitude constraint in the IIR design provides the fundamental control of the
waterbed eﬀect in narrowband disturbance observers.
Indeed, the FIR ﬁlter in (4) satisﬁes Qe−jω ω=0 = 2 cos ωo−1(DC gain),
Qe−jω ω=π= 2 cos ωo+ 1 (gain at Nyquist frequency). Regardless of the value of ωo,
Q(1) −Q(−1)equals 2, namely, it is not possible to keep the gains at DC and Nyquist
frequency small at the same time. For disturbances below one half of Nyquist frequency (typical
in practice), cos ωois positive and Q(−1) is thus larger than one, hence a highpass Q(z−1).
The bandpass Qﬁlter structure forms the idea of selective model inversion: only the plant
dynamics at the disturbance frequencies is actually inverted. At frequencies outside of the
passband of Q(z−1), the eﬀect of the inverse structure is neutralized by the small gains of
Q(z−1). Thus, the previously discussed plant uncertainties is acceptable in practice, as long as
the magnitude of Q(z−1)is suﬃciently small at the corresponding frequencies. This selective
model inversion principle is central for high performance of the algorithm in practice.
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
5
10
1
10
2
10
3
10
4
60
40
20
0
20
Gain (dB)
10
1
10
2
10
3
10
4
180
90
0
90
180
Phase (degree)
Frequency (Hz)
Q based on: A( z
1
)/ A(
α
z
1
)
Q based on: A( z
1
)
Figure 2. Example frequency responses of FIR and IIR Qﬁlter designs (design parameters: m= 1,
sampling time Ts= 1/26400 sec, vibration frequency ωo= 2πTs×800 rad, α= 0.998).
For general narrowband vibrations in the form of
d(k) =
n
X
i=1
Cisin(ωik+ψi) + no(k),(8)
‡the internal model is extended to
Az−1=
n
Y
i=1 1−2 cos (ωi)z−1+z−2(9)
=1 + a1z−1+a2z−2· · · +anz−n+· · · +a2z−2n+2 +a1z−2n+1 +z−2n.(10)
Equations (9) and (10) are respectively the (nonlinear) cascaded and the (linear) direct forms
of the polynomial Az−1.
Depending on the design of K(z−1)in (6), Q(z−1)takes two forms of structures.
Direct Q design approach: Letting Kz−1be an FIR ﬁlter Kz−1=ko+k1z−1+
· · · +knKz−nKand selecting Qz−1=BQz−1/A αz−1in (6) yield
Kz−1Az−1+z−mBQz−1=Aαz−1.(11)
Matching coeﬃcients of z−ion each side of (11), we can obtain Kz−1and BQz−1.
More generally, (11) is in the form of a polynomial Diophantine equation. Solutions exist as
long as the greatest common factor of A(z−1)and z−mdivides A(αz−1), which is always true.
Example 1. Consider the case of m= 2,n= 1, and Az−1= 1 −2 cos ωoz−1+z−2. Let
a=−2 cos ωo. We have Aαz−1= 1 + αaz−1+α2z−2. The Diophantine equation is then
Kz−11 + az−1+z−2+z−2BQz−1= 1 + αaz−1+α2z−2.Letting Kz−1=k0+
k1z−1gives z−2BQz−1= 1 −k0+ (αa −ak0−k1)z−1+α2−k0−ak1z−2−k1z−3.To
have a realizable BQz−1, we need 1−k0= 0 and αa −ak0−k1= 0. Solving the two
equations for k0and k1gives (after simpliﬁcations)
Qz−1=α2−1−a2(α−1)−(α−1) az−1
1 + αaz−1+α2z−2.(12)
‡Here no(k)is the additive noise; ωi= 2πΩiTsis the frequency in rad (Ωiis the frequency in Hz, Tsis
the sampling time); Ci(6= 0) and ψiare respectively the unknown magnitude and phase of each sinusoidal
component.
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
6
101102103104
−80
−60
−40
−20
0
Gain (dB)
101102103104
−180
−90
0
90
180
Phase (degree)
Frequency (Hz)
multi−Q approach
direct approach
Figure 3. Internalmodel based Qdesign example: frequency response of Qz−1.
MultiQ approach: For the case of IIR design on K(z−1), in [28,29] we have obtained the
solution:
Q(z−1) = "P2n
i=1(αi−1)aiz−i+1
A(αz−1)#m
;ai=a2n−i, a2n= 1 (13)
1−z−mQ(z−1) = A(z−1)
A(αz−1)
m
X
i=1 m
i−A(z−1)
A(αz−1)i−1
;m
i=m!
i!(m−i)!.(14)
where {ai}n
i=1 is as deﬁned in (10).
Equation (13) is a cascaded version of a baseline ﬁlter in the square bracket, hence the name
of the solution approach. Recall that mis the delay in the disturbance estimate. Intuitively,
instead of solving the Diophantine equation, the multiQ approach cascades mstandardized
sub ﬁlters in (13). Each sub ﬁlter is a special bandpass ﬁlter, and compensates one step of
delay,§contributing to a total of mstep phase advance for perfect disturbance cancellation.
2.3. Comparison of diﬀerent Q designs
To see the similarity and diﬀerences between the two design approaches, we provide now a case
study assuming that: the sampling time Ts= 1/26400 sec; the plant delay m= 2; the number
of vibration components n= 1; and the disturbance frequency Ω = 900 Hz.
From Example 1, the Q ﬁlter in the direct approach is given by (12). Letting α= 0.9882,
we obtain a Qz−1whose frequency response is shown in the dashed line in Fig. 3.
Correspondingly in Fig. 4, the dashed line provides the magnitude response of 1−z−mQz−1.
Recalling that 1−z−mQ(z−1)characterizes the dynamics of disturbance rejection in (1), we
observe the strong attenuation (zero gain) at 900 Hz, with a stopband width that is about 100
Hz. For the multiQ approach, we use (13) to get Qz−1=Qm=1 z−1mwhere Qm=1 z−1
is from (7). Letting α= 0.9765 yields the solid lines in Figs. 3and 4.
Both approaches give a desired notch shape of 1−z−mQz−1in Fig. 4, so that disturbance
attenuation is speciﬁcally focused at 900 Hz, and the waterbed eﬀect is well controlled. In the
phase response in Fig. 3, both ﬁlters have nonzero phases at 900 Hz—the center frequency of
the pass band. This is for compensating the delay eﬀect z−m. The multiQ approach provides
smaller magnitude response outside the pass band of Qz−1, which implies stronger ﬁltering
§Indeed, if m= 1 and n= 1, the multiQ solution simpliﬁes to (7).
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
7
102103104
−80
−70
−60
−50
−40
−30
−20
−10
0
Magnitude (dB)
Frequency (Hz)
multi−Q approach
direct approach
Figure 4. Internalmodel based Qdesign example: magnitude response of 1−z−mQz−1.
102103104
−40
−20
0
Magnitude (dB)
Frequency (Hz)
Q
102103104
−100
−50
0
Magnitude (dB)
Frequency (Hz)
1−z−mQ
Figure 5. Multiple narrowband Qdesign example.
of noises and model mismatches by Qz−1. Hence, under the same bandwidth of disturbance
attenuation, the multiQ approach is less sensitive to plant uncertainties than the direct
solution.
Remark (Choosing αin practice).In the example, we designed the two Q ﬁlters to
approximately have the same pass band in Fig. 3. The width of the pass band is controlled by
the parameter α[∈(0,1)]. A smaller αgives a wider pass band. For narrowband disturbance
rejection, αis usually selected to be close to one. Note that to achieve the same pass band in
the example above, a smaller αis used in the multiQ approach. This is because the multiQ
solution consists of a set of cascaded bandpass ﬁlters and αdirectly controls the pass band of
the individual sub ﬁlters.
The design naturally extends to the case for multiple narrowband disturbance rejection.
Fig. 5shows the use of the directQ approach to attenuate ﬁve narrowband disturbances
based on (11). Such complex bandpass Qz−1is challenging to obtain using conventional
approaches, especially when the frequencies of diﬀerent bands are very close to each other.
Finally, examining Fig. 6—an enlarged version of Fig. 4—we see that 1−z−mQz−1has
slightly diﬀerent behaviors outside the narrow band at 900 Hz: in the multiQ approach, the
magnitude is almost strictly 1 (0 dB) for the majority of the frequency region, and slightly
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
8
102103104
−12
−10
−8
−6
−4
−2
0
Magnitude (dB)
Frequency (Hz)
multi−Q approach
direct approach
Figure 6. Qdesign example: magnitude response of 1−z−mQ(z)(enlarged view).
+
+

+
C
(
z
¡
1
)
u(k
)
y(k
)
P
(
z
¡
1
)
z
¡
m
P
¡
1
n
(z
¡
1
)
d
(k
)
^
d
(k
)
+
+
0
d
r
(k
)
Parameter
adaptation
algorithm (PA A)
Copy of parameters
Noise
reduction
w(k
)
A(z
¡1
)
A(®z
¡
1
)
Q
(z
¡
1
)
Figure 7. Structure of adaptive narrowband disturbance observer.
more ampliﬁed near 800 and 1000 Hz [about 1.185 (1.5dB) maximum]; in the direct approach,
1−z−mQz−1has ﬂatter magnitude response but slightly less robustness against noise and
uncertainties (recall Fig. 3). For narrowband loop shaping, the impact of the diﬀerence is quite
small. When the desired attenuation region is wider, designers will need to make engineering
judgments between the two Q designs for best control of the waterbed eﬀect.
3. PARAMETER ADAPTATION
Knowledge of the disturbance frequencies is needed in the disturbance model A(z−1). This
section studies the adaptive formulation when such information is not a priori available.
The overall adaptive narrowband disturbance observer scheme is shown in Fig. 7. Recall
that the input to the Q ﬁlter is a delayed and noisy estimate of the actual disturbance. We
can use this dr(k)signal for parameter adaptation on Qz−1. To see the reason of the noise
reduction block, one can derive the output dynamics from the block diagram:
y(k) = P(z−1)
1 + P(z−1)C(z−1)hd(k)−ˆ
d(k)i
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
9
which yields, after substituting in (1),
y(k)≈1−z−mQ(z−1)P(z−1)
1 + P(z−1)C(z−1)d(k),1−z−mQ(z−1)yo(k).
Here yo(k) = {P(z−1)/[1 + P(z−1)C(z−1)]}d(k)is the baseline output without the disturbance
observer. Using Pz−1/1 + Pz−1Cz−1 (or a nominal version of it) as the noise
reduction block makes w(k)≈yo(k)in Fig. 7. The parameter adaptation algorithm is thus
directly based on yo(k), which reﬂects the actual eﬀect of the disturbance on the output. By
tuning the coeﬃcients of Q(z−1), the adaptation aims to directly minimize the system output
under vibration.
Remark. Of course, if prior knowledge about a coarse region of the disturbances is available,
additional ﬁltering—using e.g., standard bandpass ﬁlters from the ﬁlter design toolbox in
MATLAB—should be applied in the noisereduction block. This will further improve the
signaltonoise ratio during adaptation.
In the case of m= 1,Kz−1equals unity for both designs of the Q ﬁlter [recall (5)]. Hence
we have 1−z−1Q(z−1) = A(z−1)/A(αz−1). Adaptation can be directly constructed based on
v(k) = Az−1
A(αz−1)w(k).(15)
Note that when w(k)is a pure sinusoidal signal, the output v(k)is zero for tuned value of
the ﬁlter. Thus, the goal of parameter adaptation is to drive the adaptive prediction of v(k)
toward zero.¶
Equation (15) turns out to be also applicable for the cases when m > 1, as 1−z−mQ(z−1)
in (6) always contains the cascaded factor A(z−1)/A(αz−1).
Under the directﬁlter form (10), (15) has the realization
v(k) = ψ(k−1)Tθ+w(k) + w(k−2n)−α2nv(k−2n)(16)
where the parameter vector θand the regressor vector ψ(k−1) are
θ= [a1, a2, . . . , an]T(17)
ψ(k−1) = [ψ1(k−1) , ψ2(k−1) , . . . , ψn(k−1)]T(18)
ψi(k−1) = w(k−i) + w(k−2n+i)(19)
−αiv(k−i)−α2n−iv(k−2n+i) ; i= 1, ..., n −1
ψn(k−1) = w(k−n)−αnv(k−n).(20)
The linear ﬁlter form is suitable for various parameter adaptation algorithms for diﬀerent
application environments. Recall that v(k)will be null if w(k)is composed of pure sinusoidal
signals and Az−1/A αz−1has the correct frequency parameters. Based on (16), the
simplest recursive least squares (RLS) PAA can then be constructed to ﬁnd
ˆ
θ(k) = arg min
θ∗
k
X
i=1 ( k−1
Y
j=i
λ(j)!hψ(i−1)Tθ∗+w(i) + w(i−2n)i2)(21)
where λ(k)is a forgetting factor—close or equal to 1—that puts diﬀerent weightings on the
quadratic terms at diﬀerent time steps. In designing the cost function, we used the result
that v(k)equals zero under the pure sinusoidal disturbance assumption. This observation also
simpliﬁes the regressor vectors in (19) and (20), by annihilating all the terms related to v(k).
¶Absolute zero convergence is achievable only under ideal sinusoidal assumptions.
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Prepared using acsauth.cls DOI: 10.1002/acs
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In other words, ψisimpliﬁes to
ψi(k−1) = w(k−i) + w(k−2n+i); i= 1, . . . , n −1
ψn(k−1) = w(k−n).
The cost function (21) is quadratic, and actually convex, in θ∗. An analytic solution can be
found for the global minimum. The iterative solution is
ˆ
θ(k) = ˆ
θ(k−1) + F(k−1) ψ(k−1) eo(k)
1 + ψ(k−1)TF(k−1) ψ(k−1) (22)
eo(k) = −ψ(k−1)Tˆ
θ(k−1) −(w(k) + w(k−2n)) (23)
F(k) = 1
λ(k)"F(k−1) −F(k−1) ψ(k−1) ψ(k−1)TF(k−1)
λ(k) + ψ(k−1)TF(k−1) ψ(k−1) #.(24)
The RLS algorithm has the advantages of simplicity and guaranteed stability, yet also relies
heavily on the noisefree assumption on w(k). In practice the RLS PAA provides accurate
parameter convergence only when w(k)has a high signaltonoise ratio. A more accurate
algorithm is to construct adaptation based on the output error
e(k) =
ˆ
Aˆ
θ, z−1
ˆ
Aˆ
θ, αz−1w(k).(25)
Here, e(k)is a residual error signal that we aim to minimize. In the associated diﬀerence
equation,
e(k) = φ(k−1)Tˆ
θ(k) + w(k) + w(k−2n)−α2ne(k−2n)(26)
φi(k−1) = w(k−i) + w(k−2n+i)(27)
−αie(k−i)−α2n−ie(k−2n+i) ; i= 1, ..., n −1
φn(k−1) = w(k−n)−αne(k−n),(28)
all terms about e(k−i)are no longer treated as 0. Constructing PAAs based on such output
errors provides more accurate parameter convergence, but requires more signal processing
to guarantee stability, as the adaptation changes poles of the IIR transfer function in (25).
Based on extensive simulation and experiments in [28,25], the outputerror PAA (aka parallel
PAA) with a ﬁxed compensator is one algorithm that provides accurate convergence and
computational eﬃciency. The main equations are:
•a priori estimation error
eo(k) = −(−φ(k−1))Tˆ
θ(k−1) + w(k) + w(k−2n)−α2ne(k−2n)(29)
where −φ(k−1) indicates the negative of the gradient direction.
•a posteriori adaptation error
(k) = Cz−1e(k) = e(k) + α2ne(k−2n) + ϕ(k−1)Tθc(30)
θc= [c1, c2, . . . , cn]T(31)
ϕi(k−1) = αie(k−i) + α2n−ie(k−2n+i) ; i= 1, ..., n −1(32)
ϕn(k−1) = αne(k−n)(33)
where C(z−1)and θcare, respectively, estimates of A(αz−1)and θ.
•a priori adaptation error
o(k) = eo(k) + α2ne(k−2n) + ϕ(k−1)Tθc(34)
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•PAA
ˆ
θ(k) = ˆ
θ(k−1) + F(k−1) (−φ(k−1)) o(k)
1 + φ(k−1)TF(k−1) φ(k−1) (35)
F(k) = 1
λ(k)F(k−1) −F(k−1)φ(k−1)φT(k−1)F(k−1)
λ(k) + φT(k−1)F(k−1)φ(k−1) (36)
We focus next on several important aspects about practical implementations of the PAA
algorithm. Above all, the two types of PAAs can be combined, to take advantage of the
simplicity and global convergence of RLS, as well as the accuracy of the outputerror method
under noisy environments. This is central for successful applications when a good idea of the
disturbance characteristics is unavailable in practice. The basic principle is to run the RLS
PAA as an initialization of the outputerror PAA. After convergence of RLS, the identiﬁed
ˆaican be assigned to the coeﬃcients ciin the ﬁxed compensator. Some engineering tuning
has to be performed to conﬁrm convergence of RLS. A predictionerrorbased autoswitching
algorithm is provided in [25].
Additional implementation aspects during adaptation are discussed next.
Online Adaptation and Oﬄine Monitoring: The frequencies and the identiﬁed
parameters are mapped by
n
Y
i=1 1−2 cos(ωi)z−1+z−2= 1 + a1z−1+· · · +anz−n+· · · +a1z−2n+1 +z−2n.(37)
Although aiis the actual parameter updated online, knowing the frequencies of the
vibrations is often useful for understanding the problem and for algorithm tuning. For the
simplest case where n= 1, we have a1=−2 cos ω1, from which we can compute ω1= 2πΩ1Ts.
The parameter a1is online updated and Ω1can be computed oﬄine for algorithm tuning.
For n > 1, as 1−2 cos(ωi)z−1+z−2=1−ejωiz−11−e−j ωiz−1in (37), the values of
ωican be computed oﬄine via calculating the angle of the complex roots of 1 + a1z−1+· · · +
anz−n+· · · +a1z−2n+1 +z−2n= 0.
Parameter Initialization: If the outputerror PAA is run without the RLS initialization,
proper assignments should be placed on ˆ
θ(0) and ci’s. One can ﬁrst estimate the disturbance
frequencies {ˆωi(0)}n
i=1 based on physics or existing data, and then expand the product on the
left side of (37), to obtain {ˆai(0)}n
i=1 for implementation in the PAA.
Underdetermined and Overdetermined Adaptation: We have been assuming that the
number of the adaptation parameters in {θi}n
i=1 is the same as the number of frequency
components in the actual disturbance signal. If there are actually more than n, say r(> n),
narrowband signals in w(k), the parameters in the outputerror PAA will converge to a local
optimal point,kas demonstrated in the example in Fig. 8. Here there are four independent
vibration components between 500 Hz and 1000 Hz. The adaptation was constructed using
only n= 2 in the PAA. From Fig. 8b, the PAA correctly found the two main components at
500 Hz and 900 Hz.
If, on the contrary, r < n (which is rare in practice), we will still be able to identify r
frequency components in w(k). Notice however, that in this case the adaptation model is
kThe location of the local optima depends on the initialization of the parameter estimate.
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Prepared using acsauth.cls DOI: 10.1002/acs
12
500 1000 1500 2000 2500
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency (Hz)
Magnitude
(a) Spectrum of actual disturbance
0 0.02 0.04 0.06 0.08 0.1
−1000
−500
0
500
1000
1500
2000
2500
3000
3500
4000
Estimated frequencies [Hz]
Time [sec]
(b) Identiﬁed frequencies (n= 2)
Figure 8. Narrowband disturbance identiﬁcation example: adaptation input has more frequency
components than the adaptation model.
overdetermined and the optimum parameter estimate will not be unique. We will have
1 + a1αz−1+· · · +anαnz−n+· · · +a1α2n−1z−2n+1 +α2nz−2n
=
r
Y
i=1 1−2αcos(ωi)z−1+α2z−2×
n
Y
i=r+1 1−2αcos(ωi)z−1+α2z−2.(38)
The PAA will conduct the adaptation to form a factor of Qr
i=1 1−2αcos(ωi)z−1+α2z−2
after convergence, and ωi’s in the last n−rterms on the right side of (38) will converge to
some region around the minimum.
4. EXPERIMENTAL RESULTS ON A WAFER SCANNER
Wafer scanners are key components for motion control in the photolithography process in
semiconductor manufacturing. Billions of transistors need to be printed on the integrated
circuit in silicon wafers. Wafer scanners are hence very sensitive to vibration disturbances.
This section provides implementation and evaluation of the narrowband disturbance observer
for vibration rejection on a laboratory wafer scanner testbed. The testbed is controlled via
LabVIEW programming language; equipped with air bearings; and uses DC linear permanent
magnet motors for precision motion. Additional hardware description and system modeling
have been described in [30]. All results in this section are from experiments on the actual
system.
Fig. 9shows the rejection of external vibrations. Such disturbances come from the operation
environments such as cooling systems, adjacent machines, etc. On the testbed, an 18Hz
vibration presents as shown in Fig. 9b. There are also other lowfrequency disturbances after
a spectrum analysis in Fig. 10. RLS alone thus has limited achievable performance here. The
adaptive directQ ﬁlter based disturbance observer was implemented for improved precision.
From Fig. 9, the vibration component was successfully identiﬁed online. After a transient
period of about 1000 time steps (1 time step = 0.4ms), the converged PAA provides a 72.8%
reduction (from 1.764 ×10−7m to 4.800 ×10−8m) of the 3σ(σis the standard deviation)
value of the position errors. The strong disturbance rejection can also be observed from the
frequency spectra in Fig. 10 (there is a small residual peak at 18 Hz, from the transient
response).
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2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
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0 500 1000 1500 2000 2500 3000 3500 4000
0
10
20
30
40
50
60
70
80
90
100
Time step
Identified frequency (Hz)
(a) Calculated frequency adaptation
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
−4
−2
0
2
4
6x 10−7
Position (m)
w/o compensation
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
−5
0
5x 10−7
Time step
Position (m)
w/ compensation
(b) Online time trace of position error
Figure 9. Rejection of environmental disturbance.
10
0
10
1
10
2
10
3
240
220
200
180
160
140
Magnitude (dB)
w/o compensation 3σ = 1.7642e07
10
0
10
1
10
2
10
3
240
220
200
180
160
140
Frequency (Hz)
Magnitude (dB)
w/ compensation 3σ = 4.7997e08
Figure 10. Frequency spectra of the position errors in rejection of environmental disturbance.
Table I. Three sigma values of the position errors before and after turning on the proposed narrowband
disturbance observer
3σ: before 3σ: after
freq. overall/steadystate overall reduction steadystate reduction
(Hz) (m) (m) (m)
40 2.0E4 1.8E5 90.8% 4.0E7 99.8%
53 2.5E4 2.7E5 89.4% 5.2E7 99.8%
80 8.6E5 1.1E5 87.1% 3.2E7 99.6%
90 9.3E5 1.2E5 86.8% 3.6E7 99.6%
To evaluate the algorithm performance at diﬀerent frequencies, a series of external
disturbances were added to the system. Table Isummarizes the reductions of 3σvalues. The
overall 3σvalues measure the timedomain error reduction over the entire experiment length.
The steadystate results were computed after the initial transient period. All numbers indicate
strong vibration rejection at diﬀerent frequencies. Based on the steadystate results of more
than 99.6% error reduction, one can say that practically all disturbances were fully removed.
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
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0 500 1000 1500
20
40
60
80
100
120
140
160
180
200
Time steps
Identified frequency (Hz)
(a) Eﬀects of diﬀerent initial conditions
0 500 1000 1500 2000 2500 3000 3500 4000
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5 x 10−5
samples
tracking error (m)
(b) Online Position error (compensation starts at the
1000th sample)
Figure 11. Performance test under diﬀerent PAA initializations.
2000 4000 6000 8000 10000 12000
−4
−2
0
2
4x 10−5
Position (m)
w/o compensation
2000 4000 6000 8000 10000 12000
−4
−2
0
2
4x 10−5
Time step
Position (m)
w/ compensation
(a) Time traces
100101102103
−180
−160
−140
−120
Magnitude (dB)
100101102103
−180
−160
−140
−120
Frequency (Hz)
Magnitude (dB)
(b) Frequency spectra
Figure 12. Example rejection of two vibration components.
Fig. 11 studies performance of the adaptive algorithm under diﬀerent parameter
initializations. From Fig. 11a, the algorithm maintained accurate convergence even when the
PAA was initialized far away from the actual optimal point. From Fig. 11b, the disturbance
compensation successfully rejected the vibrations in full. The signaltonoise ratio in this case
is suﬃciently large to allow a more aggressive adaptation gain. We were able to obtain a
faster convergence (less than 200 time steps) than the results in Fig. 9. In practice, it is
recommended to always increase the signaltonoise ratio as much as possible, by preﬁltering
during adaptation.
Fig. 12 extends the evaluation to the rejection of two vibration components. In Fig. 12a,
external vibrations were injected in the ﬁrst 6000 time steps. Similar to the singlefrequency
case, the vibrations were signiﬁcantly attenuated. In the frequency domain, the two spectral
peaks at 40 Hz and 53 Hz were removed without virtually distinguishable error ampliﬁcations
at other frequencies. There were some residual peaks due to the transient at 0th and 6000th
time steps. After removal of the transient, Table II again reveals that the disturbances were
fully rejected at steady state. Also shown in Table II are the performance evaluations at
diﬀerent frequencies. In all cases, the algorithm achieved more than 87% overall 3σreduction
and 99.4% steadystate improvement.
Copyright c
2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
15
Table II. Performance evaluation at diﬀerent frequencies: two vibration component.
3σ: before 3σ: after
freq. overall/steadystate overall reduction steadystate reduction
(Hz) (m) (m) (m)
18, 40 2.1E4 1.4E5 93.2% 3.0E7 99.9%
40, 53 4.0E5 3.7E6 90.7% 2.6E7 99.4%
53, 80 5.3E5 5.6E6 89.4% 2.3E7 99.6%
80, 90 6.3E5 8.1E6 87.0% 2.6E7 99.6%
5. CONCLUSIONS AND DISCUSSIONS
This paper has discussed the design of adaptive narrowband disturbance observers and
the implementation for vibration rejection in semiconductor manufacturing. Overall, the
algorithm is an (inverse) modelbased controller parameterization, with a special internal model
absorbed in the Qﬁlter design. Modelbased design has been well recognized to be essential for
highprecision highperformance control systems. The inversebased design further decouples
the baseline control system with the adaptive vibration rejection scheme. This provides
conveniences in controller tuning in practice, especially for vibration rejection applications
using bandpasstype Q ﬁlters.
The practical importance and historical investigations have formed adaptive vibration
rejection as a rich research ﬁeld. Many challenges and new applications remain to be explored.
For instance, current results have been mostly considering adaptations of no more than three
frequency components. Depending on the algorithm conﬁgurations, order of the adaptation
vector ranges from three to six for the case of rejecting three vibration components [7].
Various engineering considerations (e.g. stability, algorithm tuning, monitoring, and the eﬀects
of numerical quantization) are needed for online adaptation of highorder IIR ﬁlters. Also,
control of the transient response in closedloop adaptive control schemes remains an unsolved
issue.
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