1

Multi-Axis Resonant Filter Design using Frequency
Response Data applied to Industrial Scan Stage
Masahiro Mae, Member, IEEE, Wataru Ohnishi, Member, IEEE, Hiroshi Fujimoto, Fellow, IEEE,
and Koichi Sakata

Abstract—Disturbance rejection of the high-precision scan
stages is important in industrial lithography equipment. The aimof this paper is to develop an optimization method for designing
multi-axis resonant filters, that enhance the disturbance rejection
performance in scanning motion. The developed optimization
method explicitly defines resonant filters in structured representation and formulates the data-driven convex optimization problem.
The method enables the multi-axis resonant filter design with
iterative convex optimization using the frequency response dataof the six-degree-of-freedom experimental setup. Experimental
results on the industrial large-scale high-precision scan stage
demonstrate the performance improvement of the disturbancerejection in the scanning motion using the optimized resonant
filters.
Index Terms—Disturbance rejection, Loop shaping, Resonant
filter, MIMO system, Frequency response, Data-driven design,
Convex optimization

I. I NTRODUCTION
ISTURBANCE rejection in scanning motion has animportant role in the product quality of semiconductors
and Flat Panel Displays (FPD) in the lithography equipment[1], [2]. The disturbance comes from several kinds of sources
such as electrical or mechanical systems around the scannerin several frequencies, and difficult to model the disturbancesources in actual systems explicitly. The feedback controller
is usually used for disturbance rejection in the two-degreeof-freedom control scheme. The challenge is how to design
the feedback controller considering the disturbance frequency
characteristics. The structured definition of the feedback controller is also important in the physical meaning of interpretability and intuitive tuning of on-site control engineers.
The data-driven design method [3]–[5] such as using thefrequency response data is one of the solutions for uncertain
disturbances without modeling. It also has an advantage in
multi-input multi-output (MIMO) systems because of the difficulty of modeling the interaction between each axis comparedto single-input single-output (SISO) systems. The challenge indata-driven design is the convex formulation of the optimization problem because the non-convex optimization problem isdifficult to guarantee monotonic convergence and it could takea long time for the optimization calculation in practice.
The data-driven feedback controller design methods with theconvex optimization have been developed using the sequential
linearization with the concave-convex procedure [6] for designing PID controller [7], [8], FIR filter [9], decoupling ofD

M. Mae, W. Ohnishi, and H. Fujimoto are with the University of Tokyo,
Japan. K. Sakata is with Nikon Corporation, Japan. (e-mail: mmae@ieee.org)

MIMO systems [10], mechanical resonance cancellation [11],disturbance observer [12], [13], and peak filter [14].
Although several data-driven feedback controller design
methods have been developed, the multi-axis disturbance rejection in several frequencies has not been fully addressed
yet, and there is a gap between the theory of numerical
optimization and the actual implementation for performance
improvement in complex MIMO industrial systems. The aimof this paper is to develop a new approach for the multiaxis disturbance rejection in several frequencies during scanning motion using structured feedback controllers designedby convex optimization. As a consequence, the developed
approach is applied to the industrial MIMO large-scale highprecision scan stage, and performance improvement is experimentally validated. The only preliminary result is presented
in the previous study [15] and the approach is theoretically
improved, generalized, and successfully implemented in theactual industrial scan stage in this paper.
The main contributions of this paper are as follows.
Contribution 1: The optimization problem of multi-axis resonant filter design using frequency response data is formulatedwith structured representation and phase stabilization.
Contribution 2: The multi-axis resonant filter design problem is solved by iterative convex optimization with the objective function of the MIMO performance evaluation.
Contribution 3: The performance of the designed multi-axis
resonant filters is experimentally validated in the industrial
MIMO large-scale high-precision scan stage.
The outline is as follows. In Section II, the control problem with the experimental setup and the designed resonant
filter is formulated. In Section III, the multi-axis resonant
filter design for the MIMO system is formulated, constituting
Contribution 1. In Section IV, the resonant filter design
problem is solved in iterative convex optimization, constituting
Contribution 2. In Section V, the optimized resonant filter
is experimentally validated in the industrial MIMO largescale high-precision scan stage, constituting Contribution 3.
In Section VI, conclusions are presented.
II. P ROBLEM FORMULATIONThe setup of the industrial MIMO large-scale high-precisionscan stage is introduced, and the concept of the multi-axis
resonant filter design for disturbance rejection is formulated.
A. Experimental setup
Fig. 1 shows the experimental setup of the industrial
FPD lithography system which is used for the production

2model principle [18]. In the decentralized resonant filter for
the MIMO controlled system, resonant filters in each axis areshown in Fig. 5 and are defined as
Fky (jωkf , ρky )
nr,ky

=1+

X
kr =1

Fig. 1. Experimental setup of FPD lithography system [17].

Fig. 2. Scan trajectory of translation along the x-axis. The scan stage moves
through 8 scan regions (■) at the same scanning motion. The scanning velocityof translation along the x-axis is set to 0.5 m/s.

of flat panel displays. In the setup, the MIMO large-scale
high-precision scan stage is implemented and it has 6-DOFs(x, y, θz , z, θx , θy ). The 6-DOF stage is supported by the air
bearing against gravity and friction, actuated by voice coilmotors and linear motors, and measured by laser displacement
sensors and linear encoders. The main scan stroke is thetranslation along the x-axis. The scanning motion is conducted
in 8 scan regions shown in Fig. 2. The stage is moving witha constant velocity of 0.5 m/s in scan regions.
The open-loop frequency response data is given for the setup
with pre-designed decentralized feedback controllers which
consist of PID controllers, disturbance observers, phase leadfilters, and notch filters. The open-loop frequency response
data is acquired by the system identification using the chirp
signal excitation. The Bode magnitude plot of the open-loop
frequency response data shown in Fig. 3 is defined as
G(jωkf ) = G(ky ,ku ) (jωkf ),(1)

where ku , ky ∈ {x, y, θz , z, θx , θy } are the indices of inputs
and outputs, and the numbers of the input and the output
are nu = ny = 6. The index of frequency response data
is kf = 1, . . . , nf . Note that the variation of the controlled
system can be used as the set of the frequency response datafor robustness. The details can be seen in [16].
The 6-DOF controlled system is shown in Fig. 4 In thispaper, the decentralized multi-axis resonant filters F are
designed to reject the disturbance d, and are defined as
F (jωkf , ρ) = diag(Fky (jωkf , ρky )),(2)

where the tuning parameters are ρ = [ρx , . . . , ρθy ].
B. Disturbance rejection with multi-axis resonant filters
The resonant filter has high-gained characteristics at thedesigned resonance frequency and effectively rejects the disturbance at the same frequency as explained in an internal

ρky ,(kr ,2) (jωkf )2 + ρky ,(kr ,1) (jωkf ),2(jωkf )2 + 2ζr,ky ,kr ωr,ky ,kr (jωkf ) + ωr,k
y ,kr

(3)

where the number of resonant filter in each axis is nr,ky andthe index of the resonant filters is kr = 1, . . . , nr,ky . The
tuning parameters, the resonance frequency, and the damping
coefficient in each axis are ρky ,(kr ,:) , ωr,ky ,kr , and ζr,ky ,kr .
The problem addressed in this paper is the design of thedecentralized multi-axis resonant filters with respect to thefollowing requirements.
Requirement 1: The resonant filter is designed in structured
representation in which the parameters have physical meaning.
Requirement 2: The resonant filter is designed by the datadriven method in which the frequency response data is directly
used and the parametric model is not needed.
Requirement 3: Convex optimization is used for designing
the resonant filter in the MIMO system.
In this paper, the optimization method is developed to satisfy
these requirements.
III. F ORMULATION OF MULTI - AXIS RESONANT FILTERDESIGN USING FREQUENCY RESPONSE DATAIn this section, the design method of multi-axis resonant
filters is formulated. First, the objective function is designed
as the error frequency spectrum evaluation, and the constraints
are defined in gain and phase stabilization conditions. The
initial condition of the optimization is designed by the preexisting approach of a resonant filter with stable resonantmodes. It results in Contribution 1.
A. Objective function to minimize error frequency spectrum
The objective function to optimize multi-axis resonant filters
is designed to minimize the error. The challenge of the objective function design is the integrated performance evaluationof multiple axes and multiple scanning regions.
The error frequency spectrum matrix without the resonant
filter is measured in the pre-experiment and is defined as
E0 (jωkf ) = e0,1 (jωkf ) · · · e0,ne (jωkf ) ,(4)
where ke = 1, . . . , ne is the index of the scanning regions. The error frequency spectrum in each scanning region
e0,ke (jωkf ) ∈ Cny ×1 is defined asTe0,ke (jωkf ) = e0,ke ,1 (jωkf ) · · · e0,ke ,ny (jωkf ) . (5)From the reproducible error frequency spectrum matrix, thedisturbance frequency spectrum matrix is assumed to be invariant for the controller design and is given byD(jωkf ) = S0−1 (jωkf )E0 (jωkf ),(6)

where the sensitivity function matrix is S0 (jωkf ) = (I +G(jωkf ))−1 ∈ Cny ×ny with the open-loop frequency response

3Fig. 3. Bode magnitude plot of open-loop frequency response in the 6-DOF experimental setup.

F
r +

e
−


Fx
0

0

0

0
0

0
Fy
0
0
0
0

0
0
Fθz
0
0
0

0
0
0
Fz
0
0

G
0
0
0
0
Fθx
0


0
0 

0 

0 

0 
Fθ y

u



Gxx
 Gyx

Gθz x Gzx
Gθx x
Gθy x

Gxy
Gyy
Gθz y
Gzy
Gθx y
Gθy y

Gxθz
Gyθz
Gθz θzGzθzGθx θzGθy θz

Gxz
Gyz
Gθz z
Gzz
Gθx z
Gθy z

Gxθx
Gyθx
Gθz θxGzθxGθx θxGθy θx
Gxθy
Gyθy 
Gθz θy Gzθy 
Gθx θy Gθy θy

d

+ y

+

(8)

The error frequency spectrum matrix with the designed
resonant filter is given by

Fig. 4. Block diagram of 6-DOF controlled system.

E(jωkf , ρ) = S(jωkf , ρ)D(jωkf ),(9)where the sensitivity function matrix is S(jωkf , ρ) = (I +G(jωkf )F (jωkf , ρ))−1 ∈ Cny ×ny with the designed resonant
filter F (jωkf , ρ). In the MIMO systems, the unit of each
output is different in many actual applications such as translation and pitching. Therefore, from (9), the normalized error
frequency spectrum matrix is given by

F (s)ρ(1,2) s2 + ρ(1,1) s
2
s2 + 2ζr,1 ωr,1 s + ωr,1..
.
ρ(nr ,2) s2 + ρ(nr ,1) s2
2
s + 2ζr,nr ωr,nr s + ωr,nr

region dke (jωkf ) ∈ Cny ×1 is defined asT
dke (jωkf ) = dke ,1 (jωkf ) · · · dke ,ny (jωkf ) .

+W −1 E(jωkf , ρ) = W −1 (I + G(jωkf )F (jωkf , ρ))−1 D(jωkf ), (10)++where the scaling matrix is W ∈ Rny ×ny . In the optimization calculation, the norm of the normalized error frequency
spectrum matrix can be used to minimize error.

+
Fig. 5. Block diagram of resonant filters in each axis.

B. Constraints of stability condition
data G(jωkf ) and an identity matrix I ∈ Rny ×ny . The elementof the disturbance frequency spectrum matrix is defined asD(jωkf ) = d1 (jωkf ) · · ·dne (jωkf ) ,(7)

where the disturbance frequency spectrum in each scanning

The gain stability condition shown in Fig. 6 is defined as|ws,ky (jωkf )| − 1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ) ≤ 0, (11)where the weighting function of the upper bound gain in thesensitivity function is ws,ky (jωkf ). Although the constraint of
the sensitivity function is commonly used for the frequency
response data-based design, the stability cannot be guaranteed4|S| [dB]×vsvs |S|max

Im
ψ
κ

ws

×K ×1

Fky (jωkf , ρky ) = ρTky ϕky (jωkf )ws


1(jωkf )




2

T 

(jωkf )2 + 2ζr,ky ,1 ωr,ky ,1 (jωkf ) + ωr,k


y ,1
1
2


(jω
)
kf

 ρky ,(1,1)  
 


2
2
(jωkf ) + 2ζr,ky ,1 ωr,ky ,1 (jωkf ) + ωr,ky ,1 ρky ,(1,2)  
 


= 
..
..
 
,
 

.
.
 



ρky ,(nr,k ,1)  
(jω
)
kf
y


2

2
ρky ,(nr,ky ,2)  (jωkf ) + 2ζr,ky ,nr,ky ωr,ky ,nr,ky (jωkf ) + ωr,ky ,nr,k y (jωkf )2
2
2
(jωkf ) + 2ζr,ky ,nr,ky ωr,ky ,nr,ky (jωkf ) + ωr,ky ,nr,k−1

G(jωr ) G(jωr )F (jωr )0Re

f [Hz]
Fig. 6. Constraints of sensitivity
function in Bode magnitude plot.

Fig. 7. Vector locus with resonant
filter and robust stability condition.

only with the gain stability condition in the use of a controllerthat changes the gain and phase with resonant modes, andthe phase stability condition is introduced. The phase stability
condition [16] shown in Fig. 7 is defined as−Using these initial conditions for optimization, the designed
resonant filters in each axis are linearly parameterized intuning parameters of the numerator and are defined as ππ≤ ∠ 1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ) − ∠ 1 + G(ky ,ky ) (jωkf ) ≤ .
2
2

(12)

The phase stability condition represents that the vector locus
with the resonant filter is between the angle ±90◦ from theangle with the origin of (−1, j0) without resonant filters. By
integrating the gain and phase stabilization conditions, thevector locus with resonant filters must be on the same sideagainst (−1, j0) and at the outside of the modulus margin.

(16)

y

where the tuning parameters are ρky ∈ R

2nr,ky +1.

IV. O PTIMIZATION OF MULTI - AXIS RESONANT FILTERIn this section, the optimization approach to design themulti-axis resonant filter is formulated. The MIMO performance is evaluated by the Frobenius norm of the normalized
error frequency spectrum matrix. The original non-convex
optimization problem is reformulated to the iterative convex
optimization problem by the Moore-Penrose inverse and sequential linearization. It results in Contribution 2.
A. Non-convex optimization problem formulation

C. Resonant filter design with stable resonant mode
For the initial condition with stable resonant modes [19],the resonant mode of the designed resonant filter is defined as
2

Fr (s) =κs + κψs,s2 + 2ζr ωr s + ωr2

(13)

−1

Re{T (jωr )},Im{T −1 (jωr )}(14)

where the complementary sensitivity function at the resonanceG(jωr )
frequency ωr is given by T (jωr ) = 1+G(jω. The parameterr)κ can be derived geometrically [19] and is given by±2ζr ωr
κ= p(K − 1)|T −1 (jωr )|ψ 2 + ωr2


±ωr
−1
when ∠T (jωr ) = atan2,±ψminimize max∥W −1 (I + G(jωkf )F (jωkf , ρ))−1 D(jωkf )∥2F (17a)∀ρsubject tokf|ws,ky (jωkf )| − 1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ) ≤ 0

(17b)


 π
π
≤ ∠ 1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ) − ∠ 1 + G(ky ,ky ) (jωkf ) ≤ ,
2
2

(17c)

∀ k ,∀ ,kyf−where the tuning parameters are the gain κ and the phase ψ.
The resonance frequency ωr and the damping coefficient ζr
are usually pre-defined by the shape of the error frequency
spectrum. The parameters in this representation have physicalmeaning and satisfy Requirement 1.
The initial resonant filters are designed as K times larger
gain at the resonance frequency ωr and the vector locus
recedes from (−1, j0) with a resonance circle as shown in
Fig. 7. The parameter K is tuned by the users based on theerror frequency spectrum. In this condition, the parameter ψ
is given byψ = ωrFrom the objective function (10), and the constrains (11)and (12), the optimization problem is formulated aswhere the MIMO performance is evaluated as the maximum
square Frobenius norm of the normalized error frequency
spectrum matrix at each frequency. The optimization problem
is formulated by the data-driven design method in which
the frequency response data is directly used and satisfies
Requirement 2. The challenge is how to formulate a convex
optimization problem that satisfies Requirement 3.
B. Objective function formulation by Moore–Penrose inverse
To deal with the non-convex objective function that includes
the parameter ρ in the inverse, the Moore–Penrose inverse isapplied. The Moore–Penrose inverse of the normalized error
frequency spectrum matrix (10) is given by
E + (jωkf , ρ)W = D + (jωkf )(I + G(jωkf )F (jωkf , ρ))W , (18)and the reformulated objective function is given by
maximize min∥D + (jωkf )(I + G(jωkf )F (jωkf , ρ))W ∥2F , (19)
∀
ρ

(15)

where the order of plus and minus corresponds to each other.

kf

where the performance value is evaluated as the minimum
square Frobenius norm of the normalized Moore–Penrose inverse error frequency spectrum matrix at each frequency. Fromthe property of the Frobenius norm and the singular value, the5objective function becomes the sum of the inverse square singular value of the normalized error frequency spectrum matrix.
The difference between (17a) and (19) is that the evaluation of
the singular values of the error frequency spectrum matrix iswhether the arithmetic mean or the harmonic mean. The linear
parameterization for convex optimization is maintained in thisformulation. Note that the number of the scanning regions neshould be greater than or equal to that of the output ny andsatisfying ne ≥ ny to compute the Moore-Penrose inverse inthis matrix order.
The optimization problem is reformulated as
maximize γ(20a)subject toγ − ∥D + (jωkf )(I + G(jωkf )F (jωkf , ρ))W ∥2F ≤ 0(20b)|ws,ky (jωkf )| − 1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ) ≤ 0

(20c)

ρ

∀ k ,∀ ,k
y
f


 π
π
− ≤ ∠ 1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ) − ∠ 1 + G(ky ,ky ) (jωkf ) ≤ , (20d)22where the objective function (19) is defined in the constraint(20b) with the parameter γ ∈ R+
0 of the objective function.
C. Iterative convex optimization using sequential linearization
The constraints of the non-convex optimization problem(20) can be dealt with by a sequential linearization using
the concave-convex procedure [6]. The derivatives around theoperating point of the current iteration in the variable ρ canbe solved by an iterative convex optimization calculation asshown in (21). Finally, the design procedure is summarizedin Algorithm 1. It can be solved in an iterative convex
optimization calculation and has an advantage in monotonic
convergence to a saddle point or a local optimum. It resultsin satisfying Requirement 3.
Algorithm 1: Multi-axis resonant filter design
Input: G(jω), E0 (jω), W , ωr , ζr , K, ws , γend /γend−1
Output: ρopt
1 obtain ρini from (13), (14), (15), and (16)
2 compute D(jω) from (6)3 while (γki /γki −1 ) ≥ (γend /γend−1 ) do
4
maximize γ subject to convex constraints (21)ρ5∀ k ,∀ ,kyf

end

V. E XPERIMENTAL VALIDATIONIn this section, the experimental validation is conductedin the industrial MIMO large-scale high-precision scan stage.
The MIMO stability analysis is conducted before the implementation. The experimental results demonstrate the scanning
performance improvement with the optimized resonant filters.
It results in Contribution 3.
A. Conditions
The 6-DOF experimental setup is shown in Fig. 1 and thestage is moving along the main stroke x-axis with constantvelocity 0.5 m/s and the number of the scanning regions is

ne = 8 as shown in Fig. 2. The number of frequency response
data points is nf = 1000 and the data points are arranged atlinearly even intervals in the range from 1 Hz to 20 Hz. The
Bode magnitude plot of the open-loop frequency response dataG(jωkf ) is shown in Fig. 3.
The resonance frequency ωr of the designed resonant filter
is pre-defined by the shape of the inverse disturbance spectrum
as shown in Fig. 8 and the damping coefficient is set toζr = 0.01. The initial resonant filter is designed with stable
resonant modes using the approach presented in [19]. The
initial resonant filters are designed as K = 2 times larger gainat the resonance frequency ωr and the vector locus recedesfrom (−1, j0) using (14) and (15). Note that the resonant
filters are designed in (x, z, θx , θy )-axes because (y, θz )-axes
are originally too high sensitivity peak before designing theresonant filters and not afford to design additional resonant
filters with robustness in the actual implementation. The
scaling matrix W is designed in the diagonal matrix withthe root mean square error of 8 scan regions for each axis in
the pre-experiment without the resonant filter.
The gain stabilization condition is designed by a datadriven approach with the scaled sensitivity function without
the resonant filter as shown in Fig. 6. The weighting functionof the sensitivity function is given by((1 + G(ky ,ky ) (jωkf ))(ωkf ≤ ωky ,smax )ws,ky (jωkf ) = vs ×
, (22)(1 + G(ky ,ky ) (jωky ,smax )) (ωkf > ωky ,smax )where the scaling parameter is set to vs = 1/1.1 andmin |1 + G(ky ,ky ) (jωkf )| = |1 + G(ky ,ky ) (jωky ,smax )|. (23)∀kfNote that the initial resonant filters do not satisfy this gainstabilization condition although they are designed with stable
resonant modes.
The iterative calculation in the optimization is continued
until the improvement of the objective function is less than
0.1 % set as γend /γend−1 = 1.001. The optimization problem
is computed in several minutes by the laptop using MATLAB,YALMIP [20], and MOSEK [21].
B. Optimization result
The Bode magnitude plot of the sensitivity function in
the diagonal SISO open-loop systems without the resonant
filters (w/o), with the initial resonant filters (ini), and with the
optimized resonant filters (opt) are shown in Fig. 9. It showsthat the optimized resonant filters satisfy the constraints of
the robust stability condition that is not satisfied by the initialresonant filters.
The Nyquist plot of the diagonal SISO open-loop systems
without the resonant filters (w/o), with the initial resonant
filters (ini), and with the optimized resonant filters (opt) are
shown in Fig. 10. It shows that both the gain and phase of theresonant filters are tuned to improve the disturbance rejection
performance by optimization.
C. MIMO stability analysis using eigenvalue loci
The stability analysis of the closed-loop system with the designed controller is always important in the actual implementation. The challenge is the stability in MIMO systems because

6

"
γ−

X

(

2|D + (jωkf )(I + G(jωkf )F (jωkf , ρ(ki −1) ))W |(ke ,ky ) Re

ke ,ky|ws,ky (jωkf )| − Re
∓ atan2Im(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Re(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Im(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Re(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))+(D + (jωkf )(I + G(jωkf )F (jωkf , ρ(ki −1) ))W )∗(ke ,ky )

)

#

(21a)

(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))∗(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρki )) ≤ 0 (21b)|1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) )||D + (jωkf )(I + G(jωkf )F (jωkf , ρ(ki −1) ))W |(ke ,ky )(D + (jωkf )(I + G(jωkf )F (jωkf , ρki ))W )(ke ,ky )− |D + (jωkf )(I + G(jωkf )F (jωkf , ρ(ki −1) ))W |2(ke ,ky ) ≤ 0

Re(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Im(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρki )) − Im(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Re(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρki ))1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) )!

± atan22

Im(1 + G(ky ,ky ) (jωkf ))Re(1 + G(ky ,ky ) (jωkf ))


−

π
≤0
2

when ± Re(1 + G(ky ,ky ) (jωkf )) ≥ 0 (21c)± atan2+Re(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Im(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρki )) − Im(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ))Re(1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρki ))1 + G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) )!

2∓ atan2Im(1 + G(ky ,ky ) (jωkf ))Re(1 + G(ky ,ky ) (jωkf ))


−

π
≤0
2

when ± Re(1 + G(ky ,ky ) (jωkf )) ≥ 0 (21d)

(a) x

(b) z

(c) θxFig. 8. Inverse disturbance spectrum of 8 scan regions. Vertical black dotted lines (

(a) x

(b) z

(d) θy) correspond to resonance frequencies of designed resonant filters.

(c) θx(d) θyFig. 9. Bode magnitude plot of SISO sensitivity function without resonant filters ( ), with initial resonant filters ( ), and with optimized resonant filters( ). Constraints of sensitivity function are shown in ( ). Vertical black dotted lines ( ) correspond to resonance frequencies of designed resonant filters.

(a) xFig. 10. Nyquist plot without resonant filters (in ( ).

(b) z(c) θx), with initial resonant filters (the stability condition in SISO systems is not necessary andsufficient for that in MIMO systems. The sufficient conditionof MIMO stability such as using a generalized Gershgorin
band in [14] results in conservative controller design. In thispaper, the eigenvalue loci [22] are used for MIMO stability
analysis in necessary and sufficient conditions. Note that thenecessary and sufficient condition of MIMO stability includes), and with optimized resonant filters ((d) θy). Sensitivity peaks are shown

the computation of eigenvalue and it is difficult to implement
in convex optimization. The eigenvalue loci without the resonant filters (w/o), with the initial resonant filters (ini), andwith the optimized resonant filters (opt) are shown in Fig. 11.
It shows that the MIMO stability condition is satisfied in allcases. Nyquist stability analysis using frequency response data
is reliable in linear systems because of no modeling error ex-

7(a) w/o(b) ini(c) opt

Fig. 11. Eigenvalue loci without resonant filters (w/o), with initial resonant filters (ini), and with optimized resonant filters (opt).

Fig. 12. Experimental time series errors of 8 scan regions in 6-DOFs without resonant filters (resonant filters ( ).

(a) x

(b) z

), with initial resonant filters ((c) θx), and with optimized(d) θyFig. 13. Experimental cumulative amplitude spectrum errors of 8 scan regions from high to low frequency without resonant filters ( ), with initial resonant
filters ( ), and with optimized resonant filters ( ). Vertical black dotted lines ( ) correspond to resonance frequencies of designed resonant filters.

cept for the assumption of dominant linear dynamics. Although
the eigenvalue loci are close to (−1, j0) as shown in Fig. 11, it
is because the controllers are designed with less robust margin

to satisfy the severe performance requirement. All controllers
are successfully implemented in the experimental setup andthe scanning performance is validated.

8

TABLE IE XPERIMENTAL ROOT MEAN SQUARE ERRORS OF 8 SCAN REGIONS IN
6-DOF S WITHOUT RESONANT FILTERS ( W / O ), WITH INITIAL RESONANTFILTERS ( INI ), AND WITH OPTIMIZED RESONANT FILTERS ( OPT ).
[count]
w/o
ini
opt

ex
49
88
59

ey
120
88
65

eθz
28
41
34

ez
116
111
107

eθx
61
60
48

eθy
75
108
116

R

t ∥W

−1 E∥
F

100 %
104 %
87 %

D. Experimental result
The scanning performance is validated in the last 0.5 s of 8
scan regions as a steady-state scanning error. The experimental
time series errors of 8 scan regions in 6-DOFs without resonant
filters, with initial resonant filters, and with optimized resonant
filters are shown in Fig. 12. From Fig. 12, the experimentalroot mean square errors of 8 scan regions in 6-DOFs areshown in TABLE I. From the identity of Parseval’s theorem
between the time domain signal and the frequency domain
signal in square integral, the integral normalized Frobenius
norm of the Rtime series error signal matrix of 8 scan regionsin 6-DOFs t ∥W −1 E∥F is used to evaluate the MIMO
performance of each approach. The results show that the
MIMO performance with optimized resonant filters outperforms that without resonant filters in 13 % and that withinitial resonant filters in 17 %. The initial resonant filters aredesigned without considering the MIMO performance and it
may worsen the MIMO performance because of the interaction
between each axes. The optimized resonant filters are designedwith optimization for the gain and phase of the resonant filters
to improve the MIMO performance. Note that the RMS errors
with the resonant filters are not improved in all axes but
become worse in several axes such as in x and θy axes becauseof the SISO performance trade-off and the nonlinear interaction dynamics between translation and rotation. Although thecontroller design using the linear frequency spectrum has alimitation in considering nonlinearity, it is a reasonable linear
approximation around the operating points.
The experimental cumulative amplitude spectrum errors of
8 scan regions are shown in Fig. 13. The disturbance rejection
performance is improved in z and θx axes but high-gainedinteraction deteriorates low-frequency performance in x andθy axes. Note that the final value of the cumulative amplitude
spectrum in Fig. 13 is the same as the root mean square error
in TABLE I from the identity of Parseval’s theorem.
VI. C ONCLUSION
The disturbance rejection in the scanning motion has animportant role in the lithography equipment. In this paper,the decentralized multi-axis resonant filters are formulated instructured representation to compensate for disturbances. The
resonant filter design problem is solved by iterative convex
optimization. Experiments on the industrial MIMO large-scale
high-precision scan stage demonstrate effective disturbancerejection performance with the optimized resonant filters.
Ongoing research focuses on global optimization with theinitial parameter dependence, nonlinear optimization with thecoefficients of the denominator, and total optimization withother pre-designed feedback controllers.

R EFERENCES[1] M. Heertjes, H. Butler, N. Dirkx, S. van der Meulen, R. Ahlawat,
K. O’Brien, J. Simonelli, K.-T. Teng, and Y. Zhao, “Control of Wafer
Scanners: Methods and Developments,” in 2020 American Control
Conference (ACC), vol. 2020-July. IEEE, jul 2020, pp. 3686–3703.
[2] M. Steinbuch, T. Oomen, and H. Vermeulen, “Motion Control, Mechatronics Design, and Moore’s Law,” IEEJ Journal of Industry Applications, vol. 2, no. 4, p. 21006010, 2021.
[3] R. de Rozario, A. Fleming, and T. Oomen, “Finite-Time Learning
Control Using Frequency Response Data With Application to a Nanopositioning Stage,” IEEE/ASME Transactions on Mechatronics, vol. 24,
no. 5, pp. 2085–2096, oct 2019.
[4] L. Blanken, S. Koekebakker, and T. Oomen, “Multivariable Repetitive
Control: Decentralized Designs with Application to Continuous MediaFlow Printing,” IEEE/ASME Transactions on Mechatronics, pp. 1–1,
2019.
[5] E. Kuroda, Y. Maeda, and M. Iwasaki, “Autonomous Parameter Design
for Cascade-Structure Feedback Controller Based on Cooperative Optimization Method,” IEEJ Journal of Industry Applications, vol. 10, no. 4,
p. 20011750, jul 2021.
[6] A. L. Yuille and A. Rangarajan, “The Concave-Convex Procedure,”
Neural Computation, vol. 15, no. 4, pp. 915–936, apr 2003.
[7] A. Karimi and G. Galdos, “Fixed-order H infinity controller design for
nonparametric models by convex optimization,” Automatica, vol. 46,
no. 8, pp. 1388–1394, aug 2010.
[8] S. Boyd, M. Hast, and K. J. Åström, “MIMO PID tuning via iterated
LMI restriction,” International Journal of Robust and Nonlinear Control,
vol. 26, no. 8, pp. 1718–1731, may 2016.
[9] K. Nakamura, K. Yubai, D. Yashiro, and S. Komada, “Fully Parameterized Controller Design Method for High Control Bandwidth Using
Frequency Response Data Sets,” in The 3rd IEEJ international workshop
on Sensing, Actuation, Motion Control, and Optimization (SAMCON
2017), 2017.
[10] S. Shinoda, K. Yubai, D. Yashiro, and J. Hirai, “Multivariable Controller
Design Achieving Diagonal Dominance Using Frequency Response
Data,” Electronics and Communications in Japan, vol. 100, no. 10, pp.
12–23, oct 2017.
[11] W. Ohnishi, “Data-based feedback controller tuning utilizing collocatedand non-collocated sensors,” in Joint 8th IFAC Symposium on Mechatronic Systems and 11th IFAC Symposium on Nonlinear Control Systems,
2019.
[12] X. Wang, W. Ohnishi, and T. Koseki, “Linear Matrix Inequality
Based Data-driven Disturbance Observer Design: Application to a NonMinimum Phase Motion Stage,” IEEJ Journal of Industry Applications,p. 21005770, 2021.
[13] ——, “Frequency Response Data Based Disturbance Observer Design:
With Application to a Nonminimum Phase Motion Stage,” IEEE/ASME
Transactions on Mechatronics, pp. 1–9, 2022.
[14] M. Mae, W. Ohnishi, H. Fujimoto, K. Sakata, and A. Hara, “Frequency
response data-based peak filter design applied to MIMO large-scale
high-precision scan stage,” Mechatronics, vol. 83, no. December 2020,
p. 102733, may 2022.
[15] M. Mae, W. Ohnishi, H. Fujimoto, and K. Sakata, “Frequency Response
Data-based Multiple Peak Filter Design Applied to High-Precision
Stage in Translation and Pitching,” in 2022 IEEE 17th International
Conference on Advanced Motion Control (AMC), vol. 83, no. 2. IEEE,feb 2022, pp. 100–105.
[16] M. Mae, W. Ohnishi, and H. Fujimoto, “Frequency Response Databased Resonant Filter Design considering Phase Stabilization and Stroke
Limitation applied to Dual-Stage Actuator Hard Disk Drives,” in The
22nd IFAC World Congress, vol. 56, Jul. 2023, pp. 10 614–10 619.
[17] Nikon, “FPD Lithography Systems Lineup,” https://www.nikon.com/
business/fpd/lineup/, 2024.
[18] B. A. Francis and W. M. Wonham, “The internal model principle forlinear multivariable regulators,” Applied Mathematics & Optimization,
vol. 2, no. 2, pp. 170–194, jun 1975.
[19] T. Atsumi, A. Okuyama, and M. Kobayashi, “Track-Following Control
Using Resonant Filter in Hard Disk Drives,” IEEE/ASME Transactions
on Mechatronics, vol. 12, no. 4, pp. 472–479, aug 2007.
[20] J. Lofberg, “YALMIP : a toolbox for modeling and optimization in
MATLAB,” in 2004 IEEE International Conference on Robotics andAutomation (IEEE Cat. No.04CH37508). IEEE, 2004, pp. 284–289.
[21] Mosek, “MOSEK 9.3,” 2021. [Online]. Available: https://www.mosek.
com[22] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control,
2nd ed. John Wiley & Sons, Ltd, 2005.

9

Masahiro Mae (Member, IEEE) received the B.E.,
M.S., and Ph.D. degrees in electrical engineering andinformation systems from The University of Tokyo,
Tokyo, Japan, in 2018, 2020, and 2023, respectively.
He is currently an Assistant Professor with the
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The
University of Tokyo. His research interests include
multivariable control and data-driven learning formechatronics and energy systems in industrial applications.
Dr. Mae is a Member of Institute of Electrical Engineers of Japan, Societyof Instrument and Control Engineers, Society of Automotive Engineers ofJapan, and Japan Society of Energy and Resources.

Wataru Ohnishi (Member, IEEE) received the B.E.,
M.S., and Ph.D. degrees in electrical engineering andinformation systems from The University of Tokyo,
Tokyo, Japan, in 2013, 2015, and 2018, respectively.
He held a visiting position with the Eindhoven
University of Technology, Eindhoven, Netherlands.
He is currently an Associate Professor with the
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The
University of Tokyo. His research interests include
high-precision motion control and optimization.
Dr. Ohnishi is a Senior Member of the Institute of Electrical Engineers ofJapan.

Hiroshi Fujimoto (Fellow, IEEE) received the Ph.D.
degree in electrical engineering from the Department
of Electrical Engineering, The University of Tokyo,
Japan, in 2001.
In 2001, he joined the Department of ElectricalEngineering, Nagaoka University of Technology, Niigata, Japan, as a Research Associate. From 2002
to 2003, he was a Visiting Scholar with the Schoolof Mechanical Engineering, Purdue University, West
Lafayette, IN, USA. In 2004, he joined the Department of Electrical and Computer Engineering,
Yokohama National University, Yokohama, Japan, as a Lecturer, and becamean Associate Professor in 2005. From 2010 to 2020, he was an Associate
Professor with The University of Tokyo, where has been a Professor since2021. His interests include control engineering, motion control, nanoscaleservo systems, electric vehicle control, motor drive, visual servoing, andwireless power transfer.
Dr. Fujimoto is a Senior Member of Institute of Electrical Engineersof Japan. He is also a Member of the Society of Instrument and Control
Engineers, Robotics Society of Japan, and Society of Automotive Engineers of Japan. He was the recipient of the Best Paper Award from IEEE
TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 2001 and 2013,
Isao Takahashi Power Electronics Award in 2010, Best Author Prize of SICE
in 2010, the Nagamori Grand Award in 2016, First Prize Paper Award forIEEE TRANSACTIONS ON POWER ELECTRONICS in 2016, and IEEJ
Industry Applications Society Distinguished Transaction Paper Award in 2018and 2023. He has been a Senior Editor for IEEE/ASME TRANSACTIONSON MECHATRONICS since 2022 and an Associate Editor for the IEEE
Industrial Electronics Magazine since 2006. He has been a Chairperson of
the Society of Automotive Engineers of Japan (JSAE) Technology Board
since 2022 and was a past Chairperson of the IEEE/IES Technical Committee
on Motion Control from 2012 to 2013, and the JSAE vehicle electrificationcommittee from 2014 to 2020.

Koichi Sakata received the B.S., M.S., and Ph.D.
degrees in physics, electrical, and computer engineering from Yokohama National University, Yokohama, Japan, in 2006, 2008, and 2011, respectively.
He was a Research Student with The Universityof Tokyo, Tokyo, Japan, from 2010 to 2011. He wasalso a Research Fellow (DC2) of Japan Society for
the Promotion of Science, from 2009 to 2011. Since
2011, he has been with Nikon Corporation, Yokohama, Japan. His research interests include motion
control and nano-scale servo control. He works on
the design of a motion control and high-precision servo control of large-scalestage.
Dr. Sakata is a Senior Member of the Institute of Electrical Engineers ofJapan.