Frequency response data-based peak filter design
applied to MIMO large-scale high-precision scan stage
Masahiro Maea,∗, Wataru Ohnishia , Hiroshi Fujimotoa , Koichi Sakatab , Atsushi Haraba The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, Japanb Nikon Corporation, 471, Nagaodaicho, Sakae, Yokohama, Kanagawa, 244-8533, Japan

Abstract
A large-scale high-precision scan stage is important equipment in the industrial productions of micro-fabrication such as flat panel
display (FPD) lithography systems. Designing controllers for multi-input multi-output (MIMO) systems is time-consuming andneeds experience because of the interaction between each axis and many controller tuning parameters. The aim of this study
is to develop a peak filter design method based on frequency response data to reduce repetitive disturbance. This data-based
approach does not use the model and only uses the frequency response data of the controlled system and the disturbance spectrum
calculated from the scanning error data (Contribution 1). The peak filter is designed by convex optimization and satisfies robust
stability conditions for six-degree-of-freedom systems (Contribution 2). The control performance of the designed peak filter isexperimentally demonstrated with an industrial MIMO large-scale high-precision scan stage in reducing the scanning error of themain stroke of the translation along the x-axis (Contribution 3).
Keywords: Concave-convex procedure; Data-based design; Frequency responses; Disturbance rejection; Peak filter; MIMO
system

1. Introduction
Large-scale high-precision scan stages have an importantrole in industries such as manufacturing semiconductors andflat panel displays (FPD). To improve the throughput and the
product quality, fast and precise positioning control is required,
and these specifications become severe year by year because of
the growing need for TVs, PCs, and smartphones [1].
The large-scale scan stage has several challenges in position control such as low resonance modes because of the low
stiffness and many disturbances because of the wide scan range[2]. The large-scale scan stage is typically controlled with twodegree-of-freedom (DOF) control with a feedforward controller
for reference tracking such as a perfect tracking control basedon a multirate feedforward control [3, 4] and a feedback controller for disturbance rejection. In the scan region, the reference trajectory of the high-precision scan stage is with constantvelocity without acceleration, and the feedback controller plays
a major role in tracking control performance.
Classical scan stages move along (x, y, θz )-axes, and the interaction between each axis is ignored, and a single-input singleoutput (SISO) decentralized control is commonly employed.
However, in these applications, the high-precision scan stages
are supported by the magnetic force or air pressure and movingin 6-DOF with (x, y, θz , z, θ x , θy )-axes to reduce disturbances by
the friction and the vibration from the ground and to improve
tracking performance, and they become multi-input multi-output∗ Corresponding author

Email address: mmae@ieee.org (Masahiro Mae)
Preprint submitted to Mechatronics(MIMO) systems [5, 6, 7, 8]. The controller design of the
MIMO systems has several challenges such as stability analysis in a coupled system between each axis, modeling of MIMO
systems, and enormous tuning parameters of the controllers.
Especially, improving the tuning method of the controller parameters is important for the cost of time and effort of on-site
control engineers.
Based on these challenges in designing the feedback controller, several data-based controller design approaches with an
optimization method are proposed, such as genetic algorithm[9], Nelder-Mead method [10], particle swarm optimization [11],loop shaping method [12], bundle method [13, 14], sequential
linearization method [15, 16, 17, 18] using concave-convex procedure [19].
Among these methods, the sequential linearization method
using the concave-convex procedure has an advantage in monotonic convergence to a saddle point or a local optimum and suits
for controller design. Other methods also need the parametricmodel of the controlled system. The precise modeling is difficult when the system is complicated such as MIMO systems.
In this study, the sequential linearization method using theconcave-convex procedure is used with the frequency response
data of the controlled system and disturbance spectrum to design the optimal feedback controller.
The disturbance spectrum during the scanning motion withconstant velocity has a repetitive characteristic such as a motor
cogging and has a large amplitude in a certain frequency.
The repetitive control approaches are presented to reject theperiodic disturbances [20, 21, 22]. They reject the disturbanceon not only main disturbance frequency but also harmonic ones.

The experimental setup in this study does not have a characteristic of harmonic disturbance frequencies, and the repetitive
control approaches are not suitable for it.
Previous researches show that repetitive disturbance can beeffectively rejected by a peak filter, which is the same as inverse notch filter in other literature, with the same resonancefrequency, and it is applied in several industrial products suchas hard disk drives [23]. However, in the application of thehigh-precision scan stage, the repetitive disturbance rejection
by a peak filter has not been reported in the literature.
Since the peak filter has a large gain at a certain frequency, it
may easily deteriorate the closed-loop stability due to the interaction. Moreover, the combination of the controller parametercan blow up in such a multi-axis system. Hence, the heuristic
tuning approach [24] depends on experiences and efforts, and
it is not the optimal solution. To address this problem, the frequency response data-based peak filter design method considering both the SISO robust stability condition and the MIMO
stability condition is proposed in this study.
The optimal disturbance filter design method is also presented [25]. However, the peak filter is designed with nonlinear
optimization procedure not with convex optimization. The convergence of the nonlinear optimization procedure is not monotonic and it could take a long time for the optimization. Therefore, the data-based peak filter design method with convex optimization suitable to industrial applications is proposed in thisstudy.
The proposed peak filter design method has an advantage
in convex optimization without parametric modeling. The control performance of the designed peak filter is experimentallydemonstrated with an industrial MIMO large-scale high-precisionscan stage in reducing the scanning error of the main stroke ofthe translation along the x-axis. This study consists of mainly
these three following contributions:Contribution 1. The optimization problem of data-based peakfilter design for the MIMO system is formulated.

Figure 1: Experimental setup of FPD lithography system [26].

linear motors [6]. The positions and angles of the stage are measured by laser displacement sensors and linear encoders [24].
The frequency response of the experimental setup is shownin Figure 2.
The main stroke of the scan stage is the translation along
the x-axis. The scan trajectory of translation along the x-axis
is shown in Figure 3. The scan stage moves with the constantvelocity through four scan regions at the same scanning procedure. The same controllers are used through four scan regionsat the same scanning procedure.
2.2. Disturbance rejection with peak filter
Previous researches in high-precision positioning systems
such as a hard disk drive show that a repetitive disturbance thathas a large spectrum in a specific frequency is rejected by apeak filter that has the same resonance frequency [23] due tothe internal model principle [27]. The transfer function of thepeak filter is given byFn (jω, ρ)
s2 + 2ρωs + ω2=
Fd (jω, η)
s2 + 2ηωs + ω2
(0 ≤ η < ρ ≤ 1),

FPeak (jω, ρ, η) =(1)

where the resonance frequency is ω ∈ R, the damping coefficients are ρ ∈ R and η ∈ R, and ρ must be larger than η in thepeak filter. It is noted that it becomes a notch filter when ρ issmaller than η.
The scan stage moves with the constant velocity in the scanregion and has several repetitive disturbances such as cogging.
Therefore, the repetitive disturbance rejection approach withpeak filter is applied to the MIMO large-scale high-precisionscan stage. This approach has not been commonly used in the
MIMO high-precision scan stage because the MIMO system
high-gained by peak filter easily becomes unstable due to theinteraction between each axis. In this study, the peak filter isdesigned with SISO robust stability condition and MIMO stability condition not to make the controlled system unstable.

Contribution 2. The data-based peak filter design method withconvex optimization is presented.
Contribution 3. The designed peak filter is validated in the experiment with the industrial MIMO large-scale high-precisionscan stage.
2. Problem formulation
In this section, the control problem is formulated.
2.1. MIMO large-scale high-precision scan stage
The experimental setup is shown in Figure 1.
The experiment is conducted with the MIMO large-scale
high-precision scan stage in the FPD lithography system whichis 6-DOFs with six-inputs ( f x , fy , τz , fz , τ x , τy ) and six-outputs(x, y, θz , z, θ x , θy ).
The 6-DOFs stage is floating by the air bearing to cancel
the gravity and frictions and actuated by voice coil motors and2.3. Details of controlled system
The block diagram of the 6-DOF controlled system is shownin Figure 4. The 6-DOF controlled system P is given by frequency response data, as shown in Figure 2. The 6-DOF experimental setup is decoupled by the mechanical design and thethrust distribution.
2

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Figure 2: Bode magnitude plot of 6-DOF experimental setup.

2

PID controllers, disturbance observers, phase lead filters, andnotch filters is given beforehand. The diagonal peak filter F isdesigned with a proposed data-based design method.
The 6-DOF controlled system P, the fixed diagonal feedback controller C, and the diagonal peak filter F are defined asfollows:

1
0-1

P(jωk ) = Plm (jωk ),Cl (jωk ) (l = m)C(jωk ) = ,0(l , m) Fn (jωk ,ρl ) Fdll (jωk ,ηl ) (l = m)F(jωk , ρ, η) = ,0(l , m)

-2
0

20

40

60

80

Figure 3: Scan trajectory of translation along the x-axis. The scan stage moves
through four scan regions (■) at the same scanning procedure. The scanningvelocity of translation along the x-axis is set to 0.1 m/s.

(2)
(3)

(4)

where (l, m) ∈ {x, y, θz , z, θ x , θy } denotes the index of the 6-DOFs
and the subscript k represents the frequency point of the frequency response data. The damping coefficients ρ and η of thepeak filter are defined as follows:hiT(5)ρ = ρ x · · · ρθy ,hiTη = η x · · · ηθy .
(6)

The diagonal term in the frequency range less than about
10 Hz is modeled as a second-order rigid body system in eachaxis. The feedback controllers are conventionally designed withthe model of the second-order rigid body systems in the only diagonal terms for the ease of the on-site final tuning process. In
the proposed method, the peak filter can be designed with convex optimization using the frequency response data of the controlled system without the mathematical modeling of the largescale MIMO high-precision scan stage.
The fixed diagonal feedback controller C that consists of

2.4. Problem description and outline
In this study, the data-based peak filter design method ispresented for the industrial MIMO large-scale high-precision
3

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Pθ z θ y Pzθy 
Pθ x θ y Pθy θyd+y+Figure 4: Block diagram of 6-DOF controlled system.

The sensitive function S in Figure 4 is given byscan stage with respect to the following requirements:

S(jωk ) = (I + P(jωk )C(jωk )F(jωk ))−1

Requirement 1. The diagonal peak filter is designed with thefixed diagonal feedback controller and FRF data of the 6-DOF
controlled system.

= S lm (jωk ),(7)

and the output disturbance d and the tracking error e are also
given by

Requirement 2. The optimization problem is convex.
Requirement 3. The scanning error of the translation along
the x-axis that is the main stroke of the scan stage should be
reduced in the experiment.

hiTd(jωk ) = d x (jωk ) · · · dθy (jωk ) ,hiTe(jωk ) = e x (jωk ) · · · eθy (jωk ) .

(8)

(9)
As is outlined in Section 1, pre-existing approaches failThe output disturbance d does not deteriorate the tracking erto meet all requirements: the approaches with loop shapingror
e within the frequency in which the gain of the sensitivity
method [12] and bundle method [13, 14] do not satisfy Refunction
S is low. Therefore, the weighting function of the senquirement 1; the approaches with genetic algorithm [9], Neldersitivity
functionis designed from the disturbance spectrum of
Mead method [10], and particle swarm optimization [11] do notthescanning
motion.
satisfy Requirement 2; the approach with sequential linearizaThe output disturbance spectrum estimated from the tracktion method [15, 16, 17] using concave-convex procedure [19]ing
error of the scanning motion is given by
does not satisfy Requirement 3. In summary, only the proposed approach satisfies the structured, diagonal, and convex
d(jωk ) = S−1 (jωk )e(jωk ).
(10)
optimized characteristics compared with other preexisting approaches.
The weighting function of the sensitivity function using the outIn Section 3, the proposed data-based peak filter design method
put disturbance spectrum [28] is given byis presented that attains Requirement 1 through the weightinghiTfunction of the sensitivity function with estimated disturbanceWS (jωk ) = WS x (jωk ) · · · WS θy (jωk ) ,(11)
spectrum, the SISO robust stability condition, and the MIMO
stability condition, constituting Contribution 1. The convex
where
optimization problem is formulated that attains Requirement 2
through sequential linearization method [15, 16, 17] using concave|WS l (jωk )| = αl |dl (jωk )|,(12)
convex procedure [19], constituting Contribution 2. In Section 4, the benefit of the approach that attains Requirement 3 isand αl is a scaling parameter.
demonstrated through the experiment with the industrial MIMO
The feedback controller is designed to satisfy the condition
large-scale high-precision scan stage, forming Contribution 3.
of the sensitivity function as follows:
In Section 5, conclusions are presented.
|S ll (jωk )WS l (jωk )| ≤ 1.
(13)
3. Data-based design method of peak filter with convex optimization

As shown in Figure 5, when the scaling parameter αl becomeslarge, 1/WS l goes down. Therefore, the effective disturbancerejection is achieved when the condition (13) is satisfied in alarger scaling parameter αl .

In this section, the data-based peak filter design method isformulated as a convex optimization problem. The proposed
method is formulated to design the peak filters for the MIMO
controlled system in each axis, independently.

3.2. SISO robust stability condition with circle condition forgain margin and phase margin
In this study, the stability condition of the controlled system
is analyzed by the Nyquist stability theorem. The SISO robust
stability condition is considered with circle condition calculated
from the gain margin gm and the phase margin Φm [29].

3.1. Weighting function design for sensitivity function using disturbance spectrum
As shown in Figure 4, a sensitivity function S is a transfer
function from the output disturbance d to the tracking error e.
4Magnitude [dB]

The open-loop frequency response of each axis may move within
the radius at each frequency point on the Nyquist diagram when
MIMO systems have the interaction between each axis. As
shown in Figure 6, when the controller is stable and the generalized Gershgorin bands do not include the point of (−1, 0j)
on the Nyquist diagram, the controlled system satisfies a MIMO
stability condition with the interaction between each axis.

1/WS
α → large

0S

Frequency [Hz]3.4. Optimization problem formulation
The optimization problem to design a peak filter with adisturbance spectrum, a SISO robust stability condition, and a
MIMO stability condition is formulated from (18) to (22).

Figure 5: Sensitivity function with disturbance spectrum weighting function.
Gain MarginIm1

maximize αlCircle Condition(18)

ρl ,ηl

−1 −σ

1
0

rm

subject toRe∀kPhase Margin|WS l (jωk )| −−1

Generalized Gershgorin Band
rg

rm −

Open Loop Frequency Response
Figure 6: Circle condition of gain margin, phase margin and generalized Gershgorin bands.

Pll (jωk )Cl (jωk )Fnl (jωk , ρl )+1 ≤0Fdl (jωk , ηl )Pll (jωk )Cl (jωk )Fnl (jωk , ρl )+σ ≤0Fdl (jωk , ηl )rgl (jωk ) −Pll (jωk )Cl (jωk )Fnl (jωk , ρl )+1 ≤0Fdl (jωk , ηl )0 ≤ βηl ≤ ρl ≤ 1
The center (−σ, 0j) and radius rm of a circle condition on
the Nyquist diagram is given by(β > 1)

(19)
(20)
(21)
(22)

3.5. Concave-convex procedure
There are two challenges in solving this optimization probg2m − 1
lem with convex optimization. First, the peak filter has tuning
,
(14)
σ=
parameters not only in the numerator but also in the denomi2gm (gm cos Φm − 1)
nator. Second, the constraints of the formulated optimization2(gm − 1) + 2gm (1 − cos Φm )
.
(15)
rm =
problem are non-convex functions because of the difference of
2gm (gm cos Φm − 1)the convex functions. To solve these challenges, a sequential
linearization method using concave-convex procedure [19] withAs shown in Figure 6, when the controller is stable and thetuning parameters in both the numerator and the denominatoropen-loop frequency response does not cross the circle condi[33] is applied to this optimization problem.
tion on the Nyquist diagram, the controller satisfies the SISOFirst, the denominator |Fd (jωk , η)| is multiplied on both sides
robust stability condition for the gain margin gm and the phaseof the constraints, as shown in (23), (24), and (25). Second,
margin Φm .
a first-order approximation of the difference term of the con3.3. MIMO stability condition with generalized Gershgorin bands straints is calculated, as shown in (26), (27), and (28). Fromthese two procedures, the constraints of the optimization probThe MIMO controlled system may become unstable due tolem become convex in the current operating point. It is noted
the interaction between each axis even if the SISO controlledthat ρli−1 and ηli−1 are the values of the optimization result in thesystem in each axis is stable. The MIMO stability conditionprevious
iteration, and this optimization problem can be solved
is considered in the Direct Nyquist Array (DNA) method withasaconvexoptimization problem with iterative calculations. In
generalized Gershgorin bands [30, 31, 32].
thisstudy,adichotomy method is used for an iterative calculaIn the DNA method, the interaction index λ(jωk |P) of thetionalgorithm.
controlled system P is defined as a maximum eigenvalue ofM(jωk |P), where the matrix M(jωk |P) is given by
4. Experimental validation(l = m)0.
(16)
M(jωk |P) = 
In this section, experimental validation is conducted. Thek) PPlm (jω(l , m)mm (jωk )aim of this study is to develop a peak filter design method basedon frequency response data. The control performance of the
The radius rgl of the generalized Gershgorin bands is given by
designed peak filter is experimentally demonstrated with an industrial MIMO large-scale high-precision scan stage in reducPll (jωk )Cl (jωk )Fnl (jωk , ρ)rgl (jωk ) = λ(jωk ).
(17)
ing the scanning error of the main stroke of the translation along
Fdl (jωk , η)
the x-axis.
5

|WS l (jωk )Fdl (jωk , ηl )| − |Pll (jωk )Cl (jωk )Fnl (jωk , ρl ) + Fdl (jωk , ηl )| ≤ 0(23)

rm |Fdl (jωk , ηl )| − |Pll (jωk )Cl (jωk )Fnl (jωk , ρl ) + σFdl (jωk , ηl )| ≤ 0(24)

rgl (jωk )|Fdl (jωk , ηl )| − |Pll (jωk )Cl (jωk )Fnl (jωk , ρl ) + Fdl (jωk , ηl )| ≤ 0(25)!
(Pll (jωk )Cl (jωk )Fnl (jωk , ρli−1 ) + Fdl (jωk , ηli−1 ))∗(Pll (jωk )Cl (jωk )Fnl (jωk , ρli ) + Fdl (jωk , ηli )) ≤ 0 (26)
|WS l (jωk )Fdl (jωk , ηl )| − Re
|Pll (jωk )Cl (jωk )Fnl (jωk , ρli−1 ) + Fdl (jωk , ηli−1 )|!
(Pll (jωk )Cl (jωk )Fnl (jωk , ρli−1 ) + σFdl (jωk , ηli−1 ))∗rm |Fdl (jωk , ηli )| − Re
(Pll (jωk )Cl (jωk )Fnl (jωk , ρli ) + σFdl (jωk , ηli )) ≤ 0 (27)|Pll (jωk )Cl (jωk )Fnl (jωk , ρli−1 ) + σFdl (jωk , ηli−1 )|!
(Pll (jωk )Cl (jωk )Fnl (jωk , ρli−1 ) + Fdl (jωk , ηli−1 ))∗(Pll (jωk )Cl (jωk )Fnl (jωk , ρli ) + Fdl (jωk , ηli )) ≤ 0 (28)rgl (jωk )|Fdl (jωk , ηl )| − Re
|Pll (jωk )Cl (jωk )Fnl (jωk , ρli−1 ) + Fdl (jωk , ηli−1 )|In the MIMO stability condition, the generalized Gershgorin bands are considered only in the frequency range fc /2 ≤f < 2 fc , when fc is the gain cross over frequency of the openloop transfer function without using the peak filter. The MIMO
stability condition with the generalized Gershgorin bands is asufficient condition, and the controlled system should not be designed too conservative. Therefore, only this frequency range
in which the stability condition is most affected is considered in
the MIMO stability condition. In other frequencies, the interference index is set to λ(jωk ) = 0.
The dichotomy method is used to solve the convex optimization problem with a sequential linearization method using
concave-convex procedure. The output disturbance spectrum
along the x-axis used for the weighting function of the sensitivity function with the scaling parameter α x is shown in Figure 9.
The main reason of the disturbance in the frequency range offrom 5 Hz to 10 Hz is the vibrations from a coarse stage and aground transmitted through the gravity canceller [24].
The objective function α x is searched in the range of αmin=x
= 1×107 . The sensitivity function without peak
1×105 and αmaxxfilter and the weighting functions with the initial conditions areshown in Figure 10. The Nyquist diagram without a peak filter
is shown in Figure 11. The Nyquist diagram with the initial
peak filter is shown in Figure 12.
In the proposed peak filter design method, parametric modelof the controlled system and the given controller are not needed,
and the frequency response data of them are enough for designing the peak filter.
The iterative optimization by the dichotomy method is rengng
peated until ααokx ≤ 1.01, where αokx and α x are defined as thexvalues of α x in the feasible and infeasible solutions, respectively. The optimization calculation searches the feasible setof the parameter with larger ρ in the current operating point todesign a peak filter with a wider resonance peak. The optimization problem is calculated by YALMIP [34] and Mosek [35].

4.1. Experimental setup
The experimental setup is shown in Figure 1. In this setup,
the MIMO large-scale high-precision scan stage moves with theconstant velocity through four scan regions at the same scanning procedure as shown in Figure 3. In this study, the scanning velocity of translation along the x-axis is set to 0.1 m/s.
The frequency response of the 6-DOF experimental setup, asshown in Figure 2, and the scanning error data are collected by
pre-experiment and used for the peak filter design.
4.2. Data-based peak filter design conditions
The peak filter is designed by the convex optimization withthe frequency response of the 6-DOF experimental setup andthe scanning error data. The number of frequency response
data points is set to 1000, and they are arranged at logarithmically even intervals in the range from 1 Hz to 500 Hz. The error
spectrum and disturbance spectrum are averaged in four scanregions to consider four scan regions with the same designed
peak filter.
The flowchart of the proposed peak filter design method isshown in Figure 7. In this study, only one peak filter is designed in translation along the x-axis that is the main stroke ofthe scan stage, and the peak filters in other axes are set to 1.
The frequency response data of C x that is fixed feedback controller along the x-axis consisting of PID controller, disturbanceobserver, phase lead filter, and notch filter is shown in Figure 8.
The resonance frequency of the peak filter in translation
along the x-axis is set to a constant frequency ω x = 41.1 rad/s( f x = ω x /2π = 6.54 Hz) in which the power spectra of the scanning error is the maximum. From the difference of the disturbance spectrum at the peak and peripheral frequency, the gainof the peak filter should be larger than 6 dB (≈ 2). Therefore,
the minimum gain of the peak filter is set to β = 6 dB (≈ 2) inthe parameter constraints (22). The initial value of each param−2−3
eter is set to ρiniand ηini
x = 1 × 10
x = 1 × 10 , respectively. It
is noted that the initial condition is set to be a feasible solution.
The gain margin and the phase margin in the SISO robust
stability condition are set to gm = 4 dB and Φm = 20 deg, respectively. This condition is satisfied in the controlled system
without a peak filter.

4.3. Optimization results
By the optimization calculation, the values of the objecopt
tive function and the optimal peak filter are given by α x =opt
opt
1482520, ρ x = 0.02325, and η x = 0.011653, respectively.
6

DisturbanceSpectrumFig. 9, Eq. (26)
SISO Robust
Stability
Eq. (27)WSlωlFrequency
Responce Data
Fig. 2, Fig. 8

Pll

-140

Cl

rm
Convex Optimization

-120

αl =

αlok + αlng
2-160

Maximize ρl

MIMO
Stability
Eq. (28)

-180

rgl

(ρl , ηl )Feasible

No

-200
100

101

102

Yes
αlok = αlUpdate (ρl , ηl )Figure 9: Output disturbance spectrum along x-axis used for weighting functionof sensitivity function with scaling parameter α x .

αlng = αlisfies both the SISO robust stability condition and the MIMO
stability condition.
αlng≤ 1.01
αlok

No4.4. Simulation evaluation
The simulation evaluation is conducted before the experiment.
The error with the peak filter ew (jωk ) is calculated from thesensitivity function with and without the peak filter, Sw (jωk ) andS(jωk ), and the error without the peak filter e(jωk ) as follows:Yes
Fl (jω, ρl , ηl )Figure 7: Flowchart of proposed peak filter design method.

ew (jωk ) =Sw (jωk )S−1 (jωk )e(jωk )=(I + P(jωk )C(jωk )F(jωk , ρ, η))−1(I + P(jωk )C(jωk ))e(jωk ).

150

The time series of the simulated scanning errors of translation along the x-axis in four scan regions are calculated byinverse Fourier transform as shown in Figure 15. The unit“[count]” means the resolution of the measurement sensor in
the setup. From the simulation results, the optimized peak filter
outperforms the without and with the initial peak filters.

100
100
0
-90
-180
-270
-360
100

101

(29)

102

4.5. Experimental results
101

The experimental validation is conducted without and withthe peak filter. In the conventional methods, the peak filter cannot be designed with convex optimization using the frequency
response data of the controlled system, not using the model ofthe controlled system. Therefore, the feedback controller without peak filter is used as the conventional method in the experimental validation.
The time series of the scanning errors of translation along
the x-axis in four scan regions are shown in Figure 16. The
amplitude spectra of the scanning errors of translation along
the x-axis in four scan regions are also shown in Figure 17.
The scanning error is effectively reduced in scan region 1
and 4, as shown in Figure 16(a) and Figure 16(d). The disturbance around 6.54 Hz is effectively rejected by the peak filter
and the scanning error is also reduced, as shown in Figure 17(a)102

Figure 8: Frequency response data of C x that is fixed feedback controller along
x-axis consisting of PID controller, disturbance observer, phase lead filter, andnotch filter.

The sensitivity function and the weighting function with theinitial and the optimal peak filter are shown in Figure 13. It is
confirmed that the controller gain at the frequency of the maximum error spectrum becomes high due to using the peak filter,
and the gain of the sensitivity function becomes low.
The Nyquist diagram with the optimal peak filter is shownin Figure 14. It is confirmed that the designed controller sat7

100

2
1.5

50
1
0.5

0

0

-50

-0.5

100

101

-1

102

-1.5
-2
-3

Figure 10: Sensitivity function and weighting function without peak filter oftranslation along the x-axis.

-2

-1

0

1

2
Figure 12: Nyquist diagram with initial condition of translation along the xaxis.

1.5
1

action of the control input from the x-axis is increased by the
feedback controller high-gained by the peak filter. From these
discussions, the peak filter has the advantage to reject the disturbance and to reduce the scanning error in the MIMO large-scale
high-precision scan stage.
In this study, the feedback controller is designed diagonallyas a setup restriction and human friendliness for the manual tuning in the industrial final introduction process. In this case, thecoupling effect cannot be completely suppressed by the feedback controller because of not enough degree of freedom. Therefore, the disturbance rejection performance of the main scan
x-axis is mainly concerned, and only the MIMO stability condition is comprised in the peak filter design. Several decoupling
control techniques are also proposed from both sides of the linear and the nonlinear control approach [36, 37]. The physicaland mathematical analysis of coupling effect by the designed
peak filter and decoupling controller design is ongoing study.
Further improvement is possible with additional peak filters in the x-axis or other axes. The proposed peak filter design
method can be applied to other axes with the MIMO stability
condition. However, MIMO robust performance improvement
is not guaranteed. Therefore, the decoupling controller also
should be designed at the same time with designed peak filtersin multi-axes. It also should be concerned that the additional
stability margin is needed to design the additional peak filters
because of the phase lag after the resonance frequency. To deal
with this problem, the phase compensator should be designed
simultaneously as the peak filter, or all controllers should be
optimized at once.

0.5
0
-0.5
-1
-1.5
-2
-3

-2

-1

0

1

Figure 11: Nyquist diagram without peak filter of translation along the x-axis.

and Figure 17(d). From this result, the effectiveness of the peakfilter in disturbance rejection is validated.
On the other hand, the scanning error is not dramaticallychanged in scan regions 2 and 3, as shown in Figure 16(b) and
Figure 16(c). This is because the frequency of the peak amplitude of the scanning error spectra in scan region 2 and 3 isaround 7.5 Hz, and the disturbance in this frequency range can
not be effectively rejected by the designed peak filter with theresonance frequency f x = 6.54 Hz.
The Root Mean Square (RMS) errors of a total of four scanregions in 6-DOFs (x, y, θz , z, θ x , θy ) are shown in Figure 18.
Figure 18(a) shows that the RMS error in translation along thex-axis that is the main stroke of the scan stage is reduced. Itmeans that the average scan quality is improved and the effectiveness of the designed peak filter is validated. It is also noteworthy that the RMS errors in (y, θz , z)-axes are also improved.
It is because that the interaction of the disturbance from the xaxis is reduced by the peak filter. On the other hand, the RMS
errors in (θ x , θy )-axes are increased. It is because that the inter-

5. Conclusion
The frequency response data-based peak filter design in thisstudy enables reducing tracking errors in the scanning motion.
The main underlying idea of this study is the combination of
8

100

2000

50

1000
0

0

-1000
-2000

-50

0
100

101

5

10

15

102
(a) Scan region 1.

2000

Figure 13: Sensitivity function and weighting function with optimal conditionof translation along the x-axis.

1000

2

0
1.5

-1000
1

-2000
0

0.5
0

5

10

15

(b) Scan region 2.

-0.5

2000
-1

1000

-1.5
-2
-3

0
-1000
-2

-1

0

1

-2000
0
Figure 14: Nyquist diagram with optimal condition of translation along the xaxis.

5

10

15

(c) Scan region 3.

the frequency response data-based design approach with convex optimization and the robust design method of the MIMO
controlled systems.
The effectiveness of the designed peak filter is demonstrated
in the experiment with the industrial MIMO large-scale highprecision scan stage. The disturbance spectrum of the maximum error frequency is effectively rejected by the designed
peak filter, and as a result, it is confirmed that the scanning error spectrum of that frequency is reduced.
Ongoing research focuses on designing additional peak filters with phase compensators and total optimization that includes not only peak filters but also other controllers.

2000
1000
0
-1000
-2000
0

5

10

15

(d) Scan region 4.
Figure 15: Simulated position error of translation along the x-axis in four scanregions. No peak filter is used in w/o (), one peak filter of translation
along the x-axis with initial condition is used in (), and one peak filterof translation along the x-axis with optimized condition is used in w/ ().
Errors in one peak filter of translation along the x-axis with initial and optimized
conditions are overlapped but a little improved in the optimized condition.

9

200
2000
150
1000
100
0
50

-1000

0
100

-2000
0

5

10

15

101

102

(a) Scan region 1.

(a) Scan region 1.

2000
100
1000
50

0
-1000

0
100

-2000
0

5

10

101

102

15
(b) Scan region 2.

(b) Scan region 2.

250
200

2000

150

1000

100
0
50
-1000

0
100

-2000
0

5

10

101

102

15
(c) Scan region 3.

(c) Scan region 3.

200

2000

150

1000

100

0

50

-1000

0
100

101

102

-2000
0

5

10

15
(d) Scan region 4.

(d) Scan region 4.

Figure 17: Amplitude spectrum of position error of translation along the x-axis
in four scan regions. No peak filter is used in w/o () and one peak filter oftranslation along the x-axis with optimized condition is used in w/ (). Thefrequency range around the resonance frequency of the designed peak filter(6.54 Hz) is highlighted (■).

Figure 16: Position error of translation along the x-axis in four scan regions.
No peak filter is used in w/o () and one peak filter of translation along thex-axis is used in w/ (
).

10

350
120
300
100
250
80
200
60

150
100

40

50

20

0

0
w/o

w/

w/o

(a) x

w/

(b) y

350

60

300

50

250
40
200
30
150
20

100

10

50

0

0
w/o

w/

w/o

(c) θz

w/

(d) z

800
150
600
100
400
50
200

0

0
w/o
(e) θ x

w/

w/o

w/

(f) θyFigure 18: Root Mean Square (RMS) errors of a total of four scan regions in
6-DOFs (x, y, θz , z, θ x , θy ). No peak filter is used in w/o (red bar ■) and one
peak filter of translation along the x-axis with optimized condition is used in w/(blue bar ■).

11

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