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Overview and new results in disturbance observer based adaptive vibration rejection with application to advanced manufacturing

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Vibrations with unknown and/or time-varying frequencies significantly affect the achievable performance of control systems, particularly in precision engineering and manufacturing applications. This paper provides an overview of disturbance-observer-based adaptive vibration rejection schemes; studies several new results in algorithm design; and discusses new applications in semiconductor manufacturing. We show the construction of inverse-model-based controller parameterization and discuss its benefits in decoupled design, algorithm tuning, and parameter adaptation. Also studied are the formulation of recursive least squares and output-error-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 high-precision motion control. Copyright © 2015 John Wiley & Sons, Ltd.
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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING
Int. J. Adapt. Control Signal Process. 2015; 00:116
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 Chen1and 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 time-varying frequencies significantly affect the achievable
performance of control systems, particularly in precision engineering and manufacturing applications.
This paper provides an overview of disturbance-observer based adaptive vibration rejection schemes;
studies several new results in algorithm design; and discusses new applications in semiconductor
manufacturing. We show the construction of inverse-model-based controller parameterization, and
discuss its benefits in decoupled design, algorithm tuning, and parameter adaptation. Also studied are
the formulation of recursive least squares and output-error 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 high-precision motion control.
Copyright c
2015 John Wiley & Sons, Ltd.
Received . . .
KEY WORDS: adaptive vibration rejection, narrow-band disturbances, advanced manufacturing,
inverse-based 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 ever-increasing 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 20-30
times reduction in the unit storage size [5]. In the new architecture, vibration-related 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 high-performance 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 disturbancerejection 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. E-mail: <xchen@engr.uconn.edu>
Disturbances whose energies are concentrated within narrow frequency bands.
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2015 John Wiley & Sons, Ltd.
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2
extensively studied in the multi-band situation [7]. Many control design algorithms have been
studied in the related literature. Adaptive noise cancellation [8] uses sensors and stochastic-
gradient-based 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 internal-model-
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 state-space designs; [4,10,13,14,2] applied
Youla Parameterization with adaptive finite-impulse-response (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 narrow-band 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 narrow-band 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 IIR-based
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 non-ideal 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 next-generation 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 narrow-band disturbance observer is an inverse-based disturbance cancellation scheme as
shown in Fig. 1. Independent from the outer-loop feedback controller C(z1), 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
¡
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(z1)–the controlled plant,
C(z1)–feedback controller, P1
n(z1) = zmˆ
P1(z1)–nominal inverse of the plant, zm–relative
degree of the plant, Q(z1)–Q filter.
More specifically, focus first on the signal flow from d(k)to ˆ
d(k). If P1
n(z1)correctly inverts
the plant dynamics P(z1), then ideally letting Q(z1)=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(z1).
In addition, P1(z1)is mostly acausal and not directly implementable. Hence P1
n(z1) =
P1(z1)is practically unachievable. In narrow-band disturbance observer, P1
n(z1)is set
as zmˆ
P1(z1), where mis the relative degree of P(z1)for realizability of P1
n(z1), and
ˆ
P(z1)is a nominal stable inverse that captures the main dynamics of P(z1). This way,
P1
n(z1)P(z1)approximately equals zm. Observing now the two internal paths from u(k)
to Q(z1)—through P1
n(z1)P(z1)and zm, 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(z1).Q(z1)can thus
be designed independently from the feedback controller C(z1). This is the decoupled design
principle in disturbance-observer-based 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 Box-Jenkins model y(k) = P(z1)u(k) + Md(z1)d(k)(see [28,29]),
where Md(z1)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 signal-to-noise ratio in dr(k)is low due to uncertainty-induced model mismatch
(common at high frequencies) and sensor noises. Conventionally, Q(z1)was chosen as a low-
pass filter to avoid high-frequency noise amplifications. This is sufficient for low-frequency
servo enhancement, yet infeasible for high-performance vibration rejection. To see this, notice
that P1
nPzmand hence dr(k)zmd(k)(z1as 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(z1)in narrow-band disturbance observer is based on the
cancellation criterion:
d(k)ˆ
d(k) = d(k)Q(z1)P1
n(z1)P(z1)d(k)1zmQ(z1)d(k).(1)
Based on (1), for vibration signals satisfying
A(z1)d(k)=0(asymptotically),(2)
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2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
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4
asymptotic perfect disturbance rejection is achieved if the feedback loop is constructed to
contain the internal model:
1zmQ(z1) = Kc(z1)A(z1)(3)
where Kc(z1)is a realizability filter for Q(z1)to be causal.
2.2. Extended internal model principle
The most intuitive internal-model based approach is perhaps to design from the time domain.
For instance, consider the case where m= 1 and A(z1) = 1 2 cos ωoz1+z2, i.e. d(k)
is a single-frequency vibration in the form of Cosin(ωok+ζo). Letting Kc(z1)=1 and
1z1Q(z1)=12 cos ωoz1+z2yields the first-order FIR solution of (3):
Qz1= 2 cos ωoz1.(4)
Although satisfying (3), the FIR solution will be seen to be not practical in general. The
narrow-band disturbance observer assigns instead
1z1Q(z1) = 12 cos ωoz1+z2
12αcos ωoz1+α2z2(5)
and for more general values of m,
1zmQ(z1) = K(z1)A(z1)
A(αz1)(6)
where A(αz1)[α(0,1)] is a damped polynomial of z1, obtained by replacing every z1
in A(z1)with αz1. Based on (2) and (3), the disturbance is still asymptotically rejected
under the extended internal model A(z1)/A(αz1). The new solution to the example at the
beginning of this subsection is
Q(z1) = 2 (α1) cos ωo+α21z1
12αcos ωoz1+α2z2.(7)
The choice of an IIR design (7) over the FIR solution (4) is based on the frequency-domain
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(z1)is normalized by A(αz1), the
frequency response of A(z1)/A(αz1)is approximately unity at low and high frequencies.
Hence from (5), the magnitude of Q(z1)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 high-pass filter response—significantly amplifying the add-on noises in dr(k).
The involved magnitude constraint in the IIR design provides the fundamental control of the
waterbed effect in narrow-band disturbance observers.
Indeed, the FIR filter in (4) satisfies Qe ω=0 = 2 cos ωo1(DC gain),
Qe 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 high-pass Q(z1).
The bandpass Q-filter 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(z1), the effect of the inverse structure is neutralized by the small gains of
Q(z1). Thus, the previously discussed plant uncertainties is acceptable in practice, as long as
the magnitude of Q(z1)is sufficiently small at the corresponding frequencies. This selective
model inversion principle is central for high performance of the algorithm in practice.
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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 Q-filter designs (design parameters: m= 1,
sampling time Ts= 1/26400 sec, vibration frequency ωo= 2πTs×800 rad, α= 0.998).
For general narrow-band vibrations in the form of
d(k) =
n
X
i=1
Cisin(ωik+ψi) + no(k),(8)
the internal model is extended to
Az1=
n
Y
i=1 12 cos (ωi)z1+z2(9)
=1 + a1z1+a2z2· · · +anzn+· · · +a2z2n+2 +a1z2n+1 +z2n.(10)
Equations (9) and (10) are respectively the (nonlinear) cascaded and the (linear) direct forms
of the polynomial Az1.
Depending on the design of K(z1)in (6), Q(z1)takes two forms of structures.
Direct Q design approach: Letting Kz1be an FIR filter Kz1=ko+k1z1+
· · · +knKznKand selecting Qz1=BQz1/A αz1in (6) yield
Kz1Az1+zmBQz1=Aαz1.(11)
Matching coefficients of zion each side of (11), we can obtain Kz1and BQz1.
More generally, (11) is in the form of a polynomial Diophantine equation. Solutions exist as
long as the greatest common factor of A(z1)and zmdivides A(αz1), which is always true.
Example 1. Consider the case of m= 2,n= 1, and Az1= 1 2 cos ωoz1+z2. Let
a=2 cos ωo. We have Aαz1= 1 + αaz1+α2z2. The Diophantine equation is then
Kz11 + az1+z2+z2BQz1= 1 + αaz1+α2z2.Letting Kz1=k0+
k1z1gives z2BQz1= 1 k0+ (αa ak0k1)z1+α2k0ak1z2k1z3.To
have a realizable BQz1, we need 1k0= 0 and αa ak0k1= 0. Solving the two
equations for k0and k1gives (after simplifications)
Qz1=α21a2(α1)(α1) az1
1 + αaz1+α2z2.(12)
Here no(k)is the additive noise; ωi= 2πiTsis the frequency in rad (iis the frequency in Hz, Tsis
the sampling time); Ci(6= 0) and ψiare respectively the unknown magnitude and phase of each sinusoidal
component.
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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. Internal-model based Q-design example: frequency response of Qz1.
Multi-Q approach: For the case of IIR design on K(z1), in [28,29] we have obtained the
solution:
Q(z1) = "P2n
i=1(αi1)aizi+1
A(αz1)#m
;ai=a2ni, a2n= 1 (13)
1zmQ(z1) = A(z1)
A(αz1)
m
X
i=1 m
iA(z1)
A(αz1)i1
;m
i=m!
i!(mi)!.(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 multi-Q 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 m-step 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 Qz1whose frequency response is shown in the dashed line in Fig. 3.
Correspondingly in Fig. 4, the dashed line provides the magnitude response of 1zmQz1.
Recalling that 1zmQ(z1)characterizes the dynamics of disturbance rejection in (1), we
observe the strong attenuation (zero gain) at 900 Hz, with a stop-band width that is about 100
Hz. For the multi-Q approach, we use (13) to get Qz1=Qm=1 z1mwhere Qm=1 z1
is from (7). Letting α= 0.9765 yields the solid lines in Figs. 3and 4.
Both approaches give a desired notch shape of 1zmQz1in 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 non-zero phases at 900 Hz—the center frequency of
the pass band. This is for compensating the delay effect zm. The multi-Q approach provides
smaller magnitude response outside the pass band of Qz1, which implies stronger filtering
§Indeed, if m= 1 and n= 1, the multi-Q solution simplifies to (7).
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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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102103104
−80
−70
−60
−50
−40
−30
−20
−10
0
Magnitude (dB)
Frequency (Hz)
multi−Q approach
direct approach
Figure 4. Internal-model based Q-design example: magnitude response of 1zmQz1.
102103104
−40
−20
0
Magnitude (dB)
Frequency (Hz)
Q
102103104
−100
−50
0
Magnitude (dB)
Frequency (Hz)
1−z−mQ
Figure 5. Multiple narrow-band Q-design example.
of noises and model mismatches by Qz1. Hence, under the same bandwidth of disturbance
attenuation, the multi-Q 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 narrow-band 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 multi-Q approach. This is because the multi-Q
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 narrow-band disturbance rejection.
Fig. 5shows the use of the direct-Q approach to attenuate five narrow-band disturbances
based on (11). Such complex bandpass Qz1is 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 1zmQz1has
slightly different behaviors outside the narrow band at 900 Hz: in the multi-Q approach, the
magnitude is almost strictly 1 (0 dB) for the majority of the frequency region, and slightly
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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. Q-design example: magnitude response of 1zmQ(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 narrow-band disturbance observer.
more amplified near 800 and 1000 Hz [about 1.185 (1.5dB) maximum]; in the direct approach,
1zmQz1has flatter magnitude response but slightly less robustness against noise and
uncertainties (recall Fig. 3). For narrow-band 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(z1). This
section studies the adaptive formulation when such information is not a priori available.
The overall adaptive narrow-band 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 Qz1. To see the reason of the noise-
reduction block, one can derive the output dynamics from the block diagram:
y(k) = P(z1)
1 + P(z1)C(z1)hd(k)ˆ
d(k)i
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which yields, after substituting in (1),
y(k)1zmQ(z1)P(z1)
1 + P(z1)C(z1)d(k),1zmQ(z1)yo(k).
Here yo(k) = {P(z1)/[1 + P(z1)C(z1)]}d(k)is the baseline output without the disturbance
observer. Using Pz1/1 + Pz1Cz1 (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(z1), 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 noise-reduction block. This will further improve the
signal-to-noise ratio during adaptation.
In the case of m= 1,Kz1equals unity for both designs of the Q filter [recall (5)]. Hence
we have 1z1Q(z1) = A(z1)/A(αz1). Adaptation can be directly constructed based on
v(k) = Az1
A(αz1)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 1zmQ(z1)
in (6) always contains the cascaded factor A(z1)/A(αz1).
Under the direct-filter form (10), (15) has the realization
v(k) = ψ(k1)Tθ+w(k) + w(k2n)α2nv(k2n)(16)
where the parameter vector θand the regressor vector ψ(k1) are
θ= [a1, a2, . . . , an]T(17)
ψ(k1) = [ψ1(k1) , ψ2(k1) , . . . , ψn(k1)]T(18)
ψi(k1) = w(ki) + w(k2n+i)(19)
αiv(ki)α2niv(k2n+i) ; i= 1, ..., n 1
ψn(k1) = w(kn)αnv(kn).(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 Az1/A αz1has 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 ( k1
Y
j=i
λ(j)!hψ(i1)Tθ+w(i) + w(i2n)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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2015 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. (2015)
Prepared using acsauth.cls DOI: 10.1002/acs
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In other words, ψisimplifies to
ψi(k1) = w(ki) + w(k2n+i); i= 1, . . . , n 1
ψn(k1) = w(kn).
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(k1) + F(k1) ψ(k1) eo(k)
1 + ψ(k1)TF(k1) ψ(k1) (22)
eo(k) = ψ(k1)Tˆ
θ(k1) (w(k) + w(k2n)) (23)
F(k) = 1
λ(k)"F(k1) F(k1) ψ(k1) ψ(k1)TF(k1)
λ(k) + ψ(k1)TF(k1) ψ(k1) #.(24)
The RLS algorithm has the advantages of simplicity and guaranteed stability, yet also relies
heavily on the noise-free assumption on w(k). In practice the RLS PAA provides accurate
parameter convergence only when w(k)has a high signal-to-noise ratio. A more accurate
algorithm is to construct adaptation based on the output error
e(k) =
ˆ
Aˆ
θ, z1
ˆ
Aˆ
θ, αz1w(k).(25)
Here, e(k)is a residual error signal that we aim to minimize. In the associated difference
equation,
e(k) = φ(k1)Tˆ
θ(k) + w(k) + w(k2n)α2ne(k2n)(26)
φi(k1) = w(ki) + w(k2n+i)(27)
αie(ki)α2nie(k2n+i) ; i= 1, ..., n 1
φn(k1) = w(kn)αne(kn),(28)
all terms about e(ki)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 output-error 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(k1))Tˆ
θ(k1) + w(k) + w(k2n)α2ne(k2n)(29)
where φ(k1) indicates the negative of the gradient direction.
a posteriori adaptation error
(k) = Cz1e(k) = e(k) + α2ne(k2n) + ϕ(k1)Tθc(30)
θc= [c1, c2, . . . , cn]T(31)
ϕi(k1) = αie(ki) + α2nie(k2n+i) ; i= 1, ..., n 1(32)
ϕn(k1) = αne(kn)(33)
where C(z1)and θcare, respectively, estimates of A(αz1)and θ.
a priori adaptation error
o(k) = eo(k) + α2ne(k2n) + ϕ(k1)Tθc(34)
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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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PAA
ˆ
θ(k) = ˆ
θ(k1) + F(k1) (φ(k1)) o(k)
1 + φ(k1)TF(k1) φ(k1) (35)
F(k) = 1
λ(k)F(k1) F(k1)φ(k1)φT(k1)F(k1)
λ(k) + φT(k1)F(k1)φ(k1) (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 output-error 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 output-error 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 prediction-error-based auto-switching
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 12 cos(ωi)z1+z2= 1 + a1z1+· · · +anzn+· · · +a1z2n+1 +z2n.(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= 2π1Ts.
The parameter a1is online updated and 1can be computed offline for algorithm tuning.
For n > 1, as 12 cos(ωi)z1+z2=1eiz11ej ωiz1in (37), the values of
ωican be computed offline via calculating the angle of the complex roots of 1 + a1z1+· · · +
anzn+· · · +a1z2n+1 +z2n= 0.
Parameter Initialization: If the output-error 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),
narrow-band signals in w(k), the parameters in the output-error 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.
Copyright c
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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. Narrow-band disturbance identification example: adaptation input has more frequency
components than the adaptation model.
over-determined and the optimum parameter estimate will not be unique. We will have
1 + a1αz1+· · · +anαnzn+· · · +a1α2n1z2n+1 +α2nz2n
=
r
Y
i=1 12αcos(ωi)z1+α2z2×
n
Y
i=r+1 12αcos(ωi)z1+α2z2.(38)
The PAA will conduct the adaptation to form a factor of Qr
i=1 12αcos(ωi)z1+α2z2
after convergence, and ωi’s in the last nrterms 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 narrow-band 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 low-frequency disturbances after
a spectrum analysis in Fig. 10. RLS alone thus has limited achievable performance here. The
adaptive direct-Q 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 ×107m to 4.800 ×108m) 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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Prepared using acsauth.cls DOI: 10.1002/acs
13
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 = 1.7642e-07
10
0
10
1
10
2
10
3
-240
-220
-200
-180
-160
-140
Frequency (Hz)
Magnitude (dB)
w/ compensation 3σ = 4.7997e-08
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 narrow-band
disturbance observer
3σ: before 3σ: after
freq. overall/steady-state overall reduction steady-state reduction
(Hz) (m) (m) (m)
40 2.0E-4 1.8E-5 90.8% 4.0E-7 99.8%
53 2.5E-4 2.7E-5 89.4% 5.2E-7 99.8%
80 8.6E-5 1.1E-5 87.1% 3.2E-7 99.6%
90 9.3E-5 1.2E-5 86.8% 3.6E-7 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 time-domain error reduction over the entire experiment length.
The steady-state results were computed after the initial transient period. All numbers indicate
strong vibration rejection at different frequencies. Based on the steady-state results of more
than 99.6% error reduction, one can say that practically all disturbances were fully removed.
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Prepared using acsauth.cls DOI: 10.1002/acs
14
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
1000-th 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 signal-to-noise 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 signal-to-noise 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 single-frequency
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% steady-state improvement.
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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/steady-state overall reduction steady-state reduction
(Hz) (m) (m) (m)
18, 40 2.1E-4 1.4E-5 93.2% 3.0E-7 99.9%
40, 53 4.0E-5 3.7E-6 90.7% 2.6E-7 99.4%
53, 80 5.3E-5 5.6E-6 89.4% 2.3E-7 99.6%
80, 90 6.3E-5 8.1E-6 87.0% 2.6E-7 99.6%
5. CONCLUSIONS AND DISCUSSIONS
This paper has discussed the design of adaptive narrow-band disturbance observers and
the implementation for vibration rejection in semiconductor manufacturing. Overall, the
algorithm is an (inverse) model-based controller parameterization, with a special internal model
absorbed in the Q-filter design. Model-based design has been well recognized to be essential for
high-precision high-performance control systems. The inverse-based 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 bandpass-type 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 high-order IIR filters. Also,
control of the transient response in closed-loop adaptive control schemes remains an unsolved
issue.
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... In recent years, the RC has attracted a great deal of attentions due to its design simplicity and capabilities in rejecting periodic disturbances. The versatility of the RCs has been reported in several practical applications [18,43,44]. The main idea of the conventional RC is to include a 1 À e ÀTs (T is the period of the repetitive disturbance) in continuous-time or 1 À z ÀN (N denotes the period of the repetitive disturbance) in discrete-time into the overall feedback loop. ...
... This implies that the numerator in (26) should contain the term 1 À e ÀTs . In this paper, the infinite impulse response (IIR) Q filter proposed by [43,44] is employed: ...
... This involves feeding the output y of a plant through an inverse model of the plant and subtracting the input signal u to estimate the disturbance signal. Disturbance observers have been shown to be effective in high precision motion control for mechatronic stages in semiconductor processes including lithography and chip packaging [93][94][95][96]. The disturbance observer concept has also been applied to run-to-run control to deal with some of the shortcomings of EWMA control. ...
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