Peak Filter Tuning based onDisturbance Spectrum for
MIMO High-Precision Scan Stage
Masahiro Mae ∗ Wataru Ohnishi ∗ Hiroshi Fujimoto ∗Koichi Sakata ∗∗ Atsushi Hara ∗∗∗The University of Tokyo, 5-1-5, Kashiwanoha, Kashiwa, Chiba,277-8561, Japan (e-mail: mmae@ieee.org, ohnishi@ieee.org,
fujimoto@k.u-tokyo.ac.jp)∗∗
Nikon Corporation, 47-1, Nagaodaicho, Sakae, Yokohama,
Kanagawa, 244-8533, Japan (e-mail: koichi.sakata@nikon.com,
atsushi.hara@nikon.com)

Abstract: In high-precision positioning systems such as scanning machines, the feedback
controller tuning needs a lot of time and skills. In particular, a feedback controller tuningwith six-degree-of-freedoms (x, y, θz , z, θx , θy ) is difficult because of the numbers of controller
parameters in each axis. The feedback controllers designed as single-input single-output
controllers in each axis may not achieve sufficient performance in multi-input multi-output
systems, and stability may not be satisfied because of a coupling problem between each axis.
The repetitive disturbance makes the performance worse in a constant velocity scanning motion,and they are conventionally rejected by a peak filter in other applications of high-precisionsystems such as a hard disk drive. In this paper, we propose a tuning method of a peak filter
to suppress a repetitive disturbance which satisfies robust stability conditions for six-degreeof-freedom systems by using convex optimization. The effectiveness of the proposed method isverified by the error in the constant velocity scanning motion.
Keywords: Concave-convex procedure, Data based tuning, Frequency responses, Disturbancerejection, Peak filter, Feedback control, MIMO system
1. INTRODUCTION
In high-precision motion control systems in industries,
the cost of tuning controllers becomes very large due tomass productions and demands of a factory automation.
In particular, the high-precision scanning systems such as
manufacturing semiconductors and liquid crystal panels
need much cost of tuning controllers because of the necessity of achieving very small tracking errors and growing
needs of PCs and smartphones.
In these applications, the high-precision stages have 6
degrees of freedoms (6-DOFs) (x, y, θz , z, θx , θy ) to achieve
high-precision tracking performance [Butler (2011); Oomen(2018); Mae et al. (2019)]. Therefore, these stages become multi-input multi-output (MIMO) systems, and thenumber of controller parameters in each axis becomesenormous. MIMO systems are difficult to model due to acoupling problem between each axis and modeling errors
occur frequently. Therefore, it is desirable to tune thecontroller from not the model but the frequency response
data obtained by the experiment directly. Several optimization methods for tuning controller parameters fromfrequency response data are proposed, such as genetic
algorithm [Tang et al. (2001)], Nelder-Mead method [Leeet al. (1985)], particle swarm optimization [Gaing (2004)].
Especially, the methods based on convex optimization areproposed such as loop shaping method [Karimi and Gal-dos (2010)], bundle method [Do and Artieres (2012); Kitayoshi and Fujimoto (2019)], and sequential linearization
method [Hast et al. (2013); Shinoda et al. (2017); Ohnishi(2019)] using Concave-Convex Procedure (CCCP) [Yuilleand Rangarajan (2003)]. Concave-Convex Procedure hasa characteristic that the solution will converge to a saddle
point or a local minimum. Even though, when we design
controllers of MIMO systems, Concave-Convex Procedure
is preferable because MIMO systems are difficult to modeland this method can be used to the frequency response
data, directly.
In these high-precision scan stages, disturbances due to themotor cogging and the vibration from the ground deteriorate the tracking performance. These disturbances should
be suppressed by feedback control. Several disturbancessuch as the cogging of the motor have repetitive characteristics. Previous studies show that repetitive disturbances
are effectively suppressed by using a peak filter with thesame frequency as the repetitive disturbance frequency,
mainly in the hard disk drive field [Atsumi et al. (2007)].
On the other hand, multi-axis high-precision scan stages
have a risk to become unstable because a coupling problem
between each axis occur due to other axis controllers thatare high gained by peak filter. Therefore, peak filters havenot been used for multi-axis high-precision stages so far.

From these backgrounds, this paper proposes an autotuning method of a peak filter in order to suppress repetitive disturbance of multi-axis high-precision scan stages
which satisfies robust stability conditions including a coupling problem between each axis of MIMO systems byusing convex optimization.
2. STRUCTURE OF CONTROLLED SYSTEM
2.1 Details of plant
We think about a 6-input 6-output high-precision scan
stage with 6-DOFs (x, y, θz , z, θx , θy ). We consider suppressing repetitive disturbance by using peak filters in aconstant velocity region such as a scan motion. In thispaper, we mainly handle the translation along the x axis
which is the main stroke of the stage.
2.2 Details of controller
The block diagram of the controlled system is shown in
Fig. 1. The block P , F and C denote a plant, a peakfilter, and given fixed controllers respectively. In order tosimplify the controller structure, F and C are designed
only for diagonal terms. Details of the controller structure
are shown in Fig. 2. The given fixed controllers C consist of
PID controllers, disturbance observers, phase lead filters,
and notch filters. In this paper, the parameters of C are
fixed, and the peak filter F is designed to be added to theoriginal controlled system.
2.3 Repetitive disturbance rejection using peak filter
In high-precision positioning systems such as a hard disk
drive, a repetitive disturbance d shown in Fig. 3 is suppressed by the peak filter FPeak which has the same frequency to the repetitive disturbance frequency according
to the internal model principle [Atsumi et al. (2007)]. In
this paper, we design a peak filter to suppress repetitive
disturbances for a multi-axis high-precision scan stages inwhich a peak filter has not been used because the MIMO
systems whose gain becomes high by the peak filter may
be unstable due to a coupling problem between each axis.
The peak filter designed in this paper is shown in (1).
s2 + 2ρωs + ω 2
Fn (jω, ρ)FPeak (jω, ρ, η) = 2=
s + 2ηωs + ω 2
Fd (jω, η)(0 ≤ η < ρ ≤ 1)(1)
ω is the resonant frequency of the peak filter. ρ and η are
the damping coefficients, and ρ must be larger than η. It
is noted that when ρ is smaller than η it becomes a notch
filter.
3. AUTO TUNING METHOD OF PEAK FILTERUSING CONVEX OPTIMIZATION
In order to evaluate robust stability conditions for thefrequency response data of the system, Nyquist diagram isused. In this paper, the gain and phase margins are treatedas a circle condition and coupling effects are evaluatedby using generalized Gershgorin bands on the Nyquist
diagram for each axis, as shown in Fig. 4.

3.1 Robust controller design with circle condition of gain
mergin and phase margin
The center (−σ, 0j) and radius rm of a circle conditionon the Nyquist diagram can be calculated from the gain
margin gm and the phase margin Φm , as shown in (2) and(3) [Maeda and Iwasaki (2014)].
2
gm
−1
(2)
σ=
2gm (gm cos Φm − 1)
(gm − 1)2 + 2gm (1 − cos Φm )rm =(3)2gm (gm cos Φm − 1)
When the controller is stable and the open-loop frequency
response does not cross the circle condition on the Nyquist
diagram, the controller satisfies robust stability conditions
for the gain margin gm and the phase margin Φm .
3.2 Robust controller design with generalized Gershgorin
bands for MIMO system
The feedback controller may become unstable due toa coupling problem between each axis even when thecontroller satisfies the stability condition for each axis. In
this paper, we evaluate the coupling effect in the open-loop
frequency response of MIMO systems by using generalized
Gershgorin bands [Rosenbrock (1969); Araki and Nwokah(1975); Sebe (1996)].
P , C, and F which denote a plant, a peak filter, andgiven fixed controllers are defined as (4), (5), and (6),respectively. The parameters of peak filter ρ and η arealso defined as (7) and (8), respectively.
P (jωk ) = Pij (jωk ) (i, j = 1, 2, . . . , n)(4)Ci (jωk ) (i = j)C(jωk ) =(i, j = 1, 2, . . . , n)0(i 6= j)(5)
( F (jω ,ρ )nik i(i = j)F (jωk , ρ, η) = Fdi (jωk ,ηi )(i, j = 1, 2, . . . , n)0(i 6= j)(6)Tρ = [ρ1 ρ2 · · · ρn ](7)T

η = [η1 η2 · · · ηn ](8)
It is noted that the numbers of inputs and outputs of theplant are both n in this paper.
The interaction index λ of the plant P is calculated in(9) as the maximum eigenvalue of the matrix M which isdefined as (10).
λ(jωk ) = max(|eig(jωk |M )|)(9)(0(i = j)(i, j = 1, 2, . . . , n) (10)M (jωk ) =Pij (jωk )(i 6= j)Pjj (jωk )The open-loop frequency response data at each frequency
on the Nyquist diagram of each axis can move within theradius rgi which is defined as (11) when MIMO plants havea coupling problem between each axis.
Pii (jωk )Fni (jωk , ρ)Ci (jωk )rgi (jωk ) = λ(jωk )Fdi (jωk , η)(i, j = 1, 2, . . . , n)
(11)

C
r +

e
−


Cx
0

0

0

0
0

0
Cy
0
0
0
0

0
0
Cθz
0
0
0

F

0
0
0
Cz
0
0

0
0
0
0
Cθx
0


Fx
0

0

0

0
0



0
0 

0 

0 

0 
C θy

0
Fy
0
0
0
0

0
0
Fθz
0
0
0

0
0
0
Fz
0
0

P
0
0
0
0
Fθx
0



0
0 

0 

0 

0 
Fθ y



u

Pxx
 Pxy

Pxθz

 Pxz

Pxθx
Pxθy

Pyx
Pyy
Pyθz
Pyz
Pyθx
Pyθy

Pθ z xPθ z yPθ z θ z
Pθ z zPθ z θ x
Pθ z θ y

Pzx
Pzy
Pzθz
Pzz
Pzθx
Pzθy

Pθ x xPθ x yPθ x θ z
Pθ x zPθx θxPθ x θ yPθ y xPθ y y Pθ y θ z Pθ y z Pθ y θ x Pθy θy

d
+

y

+

y

+

Fig. 1. Block diagram of 6-DOF controlled system.
C
r +

e

+

CPID

−

F
FPLF

FNotch

FPeak

d
u
P

+

Pn−1

Q

+

+

Q

+

Fig. 2. Block diagram of detail in each controller.
d
r

+

e
−

FPeak

C

+ y

u
P

+

S(jωk ) = (I + P (jωk )C(jωk ))−1
= Sij (jωk ) (i, j = 1, 2, . . . , n)

(12)
T

(13)

T

(14)

d(jωk ) = [d1 (jωk ) d2 (jωk ) · · · dn (jωk )]e(jωk ) = [e1 (jωk ) e2 (jωk ) · · · en (jωk )]Fig. 3. Block diagram of controlled system with peak filter.
Gain MarginIm1d(jωk ) = S −1 (jωk )e(jωk )Circle Condition
−1 −σ

1
0

rm

In this paper, the frequency spectrum of the disturbancein the constant velocity region is calculated from thefrequency spectrum of the error by the inverse of thesensitivity function, as shown in (15).

Re

Phase Margin

Then, the weighting functions of the sensitivity function
WS , defined as (16), is designed by adjusting the frequency
spectrum of the disturbance by the scaling parameter αi
as shown in (17) [Atsumi and Messner (2012)].

−1

Generalized Gershgorin Bandrg

Open Loop Frequency Response

Fig. 4. Circle condition of gain margin, phase margin andgeneralized Gershgorin bands.

(15)

WS (jωk ) = [WS1 (jωk ) WS2 (jωk ) · · · WSn (jωk )]|WSi (jωk )| = αi |di (jωk )| (i = 1, 2, . . . , n)

T

(16)
(17)

We design the controller so that the sensitivity function
S satisfies the robust stability condition (18) for theweighting function of the sensitivity function WS for eachaxis.
|Sii (jωk )WSi (jωk )| ≤ 1

The multiple circles in which the open-loop frequency response data can move are called as generalized Gershgorin
bands. When the controller is stable and generalized Gershgorin bands do not include the point of (−1, 0j) on the
Nyquist diagram, the system satisfies a stability condition
for MIMO systems with a coupling problem between eachaxis.

The weighting function of the sensitivity function WS are
designed from the frequency spectrum of the disturbance.
The controller is tuned so that the sensitivity function
satisfies a robust stability condition for larger scaling
parameters αi .

3.3 Design of weighting function of sensitivity function

3.4 Optimization problem formulation

The sensitivity function S, defined as (12), should be small
in the main frequency range of the disturbance because itis the transfer function from disturbance d, defined as (13),to error e, defined as (14), in Fig. 1.

From the considerations above, the optimization problem
of designing a peak filter which satisfies robust stability
conditions for MIMO systems can be described from (19)to (23).

(i = 1, 2, . . . , n)(18)

maximize αi
ρi ,ηi(i = 1, 2, . . . , n)(19)subject to∀kPii (jωk )Ci (jωk )Fni (jωk , ρi )+σ ≤0Fdi (jωk , ηi )Pii (jωk )Ci (jωk )Fni (jωk , ρi )rgi (jωk ) −+1 ≤0
Fdi (jωk , ηi )Pii (jωk )Ci (jωk )Fni (jωk , ρi )|WSi (jωk )| −+1 ≤0Fdi (jωk , ηi )0 ≤ ηi ≤ ηimax < ρmin≤ ρi ≤ 1
i
rm −

(20)
(21)
(22)
(23)

3.5 Concave-Convex Procedure (CCCP)
The constraints of (20), (21), and (22) are concave functions because they are the difference of the convex functions, respectively. Therefore, they are not solved as convex
optimization. In addition, it is also the problem that thepeak filter has tuning parameters not only in the numerator but also in the denominator. In this paper, we use asequential linearization method by using Concave-Convex
Procedure with tuning parameters in the numerator andthe denominator is used to transform from the concave optimization problem into the convex optimization problem[Nakamura et al. (2017)].
First, we consider a constraint of the circle condition (20).
We multiply the denominator |Fd (jωk , η)| on both sides of(20), and we get (24). Second, we consider a first-orderapproximation of the difference term in (24), and we get
the convex optimization problem (25).
The same procedure is applied to a constraint of therobust stability condition for MIMO systems (21) and aconstraint of sensitivity functions (22). First, we multiply
the denominator |Fd (jωk , η)| on both sides of (21) and (22),and we get (26) and (28), respectively. Second, we consider
a first-order approximation of the difference term in (26)and (28), and we get the convex optimization problem (27)and (29), respectively.
By using the sequential linearization method, it is possible
to transform from a concave optimization problem into a
convex optimization problem. It is noted that ρil−1 andηil−1 are the parameters of the optimization results inthe previous iteration, and this optimization problem canbe solved as a convex optimization problem by iterative
calculations. In this paper, we use a dichotomy method as
an iterative algorithm.
4. SIMULATION EVALUATIONThe controllers can be designed independently at each axis
for MIMO systems by using the proposed method. In thispaper, we consider tuning a peak filter to reject a repetitive
disturbance in the translation along the x axis which is themain stroke of the stage.

stability condition for MIMO system (21), we consider
generalized Gershgorin bands only in the frequency range
fc /2 ≤ f < 2fc , when fc is the gain cross over frequencyof the open-loop transfer function without using the peakfilter and the stability condition is most affected in thisfrequency range, so that the controllers should not bedesigned too conservative. In other frequencies, the interference index set to λ(jωk ) = 0. The gain margin and thephase margin are set to 4 dB and 20 deg, respectively. Thiscondition is satisfied in the controller without a peak filter.
The number of designed peak filters is one, and theresonant frequency ω of the peak filter is set to a constantfrequency in which the power spectra of the error at eachaxis is the maximum. In this paper, we design the peakfilter only in the translation along the x axis which is themain stroke of the stage, and the peak filters in other
DOFs are set to 1. The resonant frequency ωx of thepeak filter in the translation along the x axis is set toωx = 44.9 rad/s (fx = 6.67 Hz). The parameters of thegiven fixed controllers C are fixed. From the differenceof the error spectrum at the peak and the low frequency,
the gain of peak filter should be larger than 60 dB = 103 .
Therefore, the parameter constraints of the peak filter (23)
= 1×10−6 and ηxmax = 1×10−9 ,is set to the range in ρminx
and the initial value of each parameter is set to ρini
x = 1×10−6 and ηxini = 0, respectively. A Nyquist diagram using
the initial peak filter is shown in Fig. 5.
The dichotomy method is used to solve the convex
optimization problem sequentially linearized by using
Concave-Convex Procedure in the range of αxmin ≤ αx ≤αxmax . The range of αx is set to αxmin = 1 × 105 andαxmax = 1×107 . The sensitivity function and the weighting
fuction of the sensitivity function are shown in Fig. 6.
The iterative optimization by the dichotomy method is
αng
repeated until αxok × 100 = 1 %, when αxok and αxng arex
defined as the value of αx in the feasible and infeasible solutions, respectively. The optimization problem is calculated
by using YALMIP [Lofberg (2004)] and Mosek [MosekApS(2019)].
4.2 Peak filter design
The parameters of the optimal peak filter and the evaluaopt= 0, andtion function are designed as ρopt
x = 0.0065, ηx
αxopt = 941113, respectively.
The Nyquist diagram with the optimal peak filter is shownin Fig. 7. It is confirmed that the designed controller
satisfies the robust stability conditions.
The sensitivity function and the weighting function of
the sensitivity function with the optimal peak filter are
shown in Fig. 8. It is confirmed that the controller gainat the frequency with the maximum disturbance spectrum
becomes high due to using the peak filter, and the gain of
the sensitivity function becomes low.
4.3 Evaluation of repetitive disturbance rejection

4.1 Condition of optimization
The number of the frequency response data of the plantis 1000. The data are arranged at logarithmically evenintervals in the range of 1 Hz to 500 Hz. In the robust

The performance of the high-precision scan stage in aconstant velocity scanning motion is evaluated by thedifference between the maximum value and the minimum
value of the error. The error with the peak filter eopt isrm |Fdi (jωk , ηi )| − |Pii (jωk )Ci (jωk )Fni (jωk , ρi ) + σFdi (jωk , ηi )| ≤ 0 (i = 1, 2, . . . , n)(24)
rm |Fdi (jωk , ηil )|(Pii (jωk )Ci (jωk )Fni (jωk , ρil−1 ) + σFdi (jωk , ηil−1 ))∗− Re(Pii (jωk )Ci (jωk )Fni (jωk , ρil ) + σFdi (jωk , ηil )) ≤ 0|Pii (jωk )Ci (jωk )Fni (jωk , ρil−1 ) + σFdi (jωk , ηil−1 )|(i = 1, 2, . . . , n)(25)
rgi (jωk )|Fdi (jωk , ηi )| − |Pii (jωk )Ci (jωk )Fni (jωk , ρi ) + Fdi (jωk , ηi )| ≤ 0 (i = 1, 2, . . . , n)(26)rgi (jωk )|Fdi (jωk , ηi )|(Pii (jωk )Ci (jωk )Fni (jωk , ρil−1 ) + Fdi (jωk , ηil−1 ))∗− Re(Pii (jωk )Ci (jωk )Fni (jωk , ρil ) + Fdi (jωk , ηil )) ≤ 0|Pii (jωk )Ci (jωk )Fni (jωk , ρil−1 ) + Fdi (jωk , ηil−1 )|(i = 1, 2, . . . , n)(27)
|WSi (jωk )Fdi (jωk , ηi )| − |Pii (jωk )Fni (jωk , ρi )Ci (jωk ) + Fdi (jωk , ηi )| ≤ 0 (i = 1, 2, . . . , n)(28)
|WSi (jωk )Fdi (jωk , ηi )|(Pii (jωk )Fni (jωk , ρil−1 )Ci (jωk ) + Fdi (jωk , ηil−1 ))∗− Re(Pii (jωk )Fni (jωk , ρil )Ci (jωk ) + Fdi (jωk , ηil )) ≤ 0|Pii (jωk )Fni (jωk , ρil−1 )Ci (jωk ) + Fdi (jωk , ηil−1 )|(i = 1, 2, . . . , n)
(29)
2

200
150

1

10-6

3
2

100

1

50

0

-1

0

-1

-2

-50
100

0

-3

-2

-1

0

1

-2
101

102

-3
0

Fig. 5. Nyquist diagram Fig. 6. Sensitivity function andweighting function of iniof initial condition intial condition in translatranslation along xtion along x axis.
axis.
2

200
150

1

100

0
50

-1

0

-2
-3

-2

-1

0

1

-50
100

101

102

Fig. 7. Nyquist diagram Fig. 8. Sensitivity function andweighting function of opof optimal condition
timal condition in translain translation along
tion along x axis.
x axis.
calculated from the error without the peak filter e, as
shown in (30).
eopt (jωk ) =S opt (jωk )S −1 (jωk )e(jωk )=(I + P (jωk )F (jωk , ρ, η)C(jωk ))−1
(I + P (jωk )C(jωk ))e(jωk )(30)
A comparison of the error without and with the peak filterin the translation along the x axis in a constant velocity
region is shown in Fig. 9. It is confirmed that the difference
between the maximum and minimum errors is reduced.

2

4

6

8

Fig. 9. Position error ex in translation along x axis without
and with peak filter in constant velocity region.
A comparison of the power spectra of the error without
and with the peak filter in the translation along the xaxis in a constant velocity region is shown in Fig. 10. It is
confirmed that the error around 6.67 Hz which is the peakfrequency of the error power spectra is reduced.
The difference between the maximum and minimum errors
in 6-DOFs (x, y, θz , z, θx , θy ) are shown in Table 1. The
difference between the maximum and minimum error in
the translation along the x axis is reduced about 37 %,while those of other DOFs are changed about 1 %. It is
confirmed that the error in the translation along the x axis
can be reduced without affecting other DOFs. From theevaluation above, the effectiveness of the proposed method
is verified.
5. CONCLUSION
In this paper, we proposed a method of tuning a peak filter
automatically to reject repetitive disturbances by using
convex optimization for a multi-axis high-precision scan
stage in a constant velocity scanning motions. By using thecircle condition of the gain margin and the phase margin
and generalized Gershgorin bands of the MIMO system,a designed controller satisfied robust stability conditions
for a multi-input multi-output system with a coupling
problem between each axis. In addition, the weighting
function of the sensitivity function is designed from theTable 1. Peak to peak error without and with peak filter in constant velocity region.
Axis

ex [m]

ey [m]eθz [rad]ez [m]eθx [rad]eθy [rad]w/o FPeak
w/ FPeak

4.65 × 10−6
2.92 × 10−6

8.74 × 10−7
8.85 × 10−7

8.48 × 10−7
8.48 × 10−7

6.85 × 10−6
6.86 × 10−6

1.16 × 10−6
1.15 × 10−6

1.06 × 10−5
1.06 × 10−5

−100
−130
−135
−140

−150

−145

6

6.5

7

7.5

8

−200

−250
10

0

10

2

Fig. 10. Power spectra of position error ex in translation
along x axis without and with peak filter in constantvelocity region.
disturbance spectrum calculated from the error spectrum
so that the actual characteristic of the disturbance is considered. The concave optimization problem is transformed
into the concave optimization problem with the iterative
calculation by the sequential linearization method using
Concave-Convex Procedure. The peak filter which satisfies
the robust stability conditions for the MIMO system isdesigned in the translation along the x axis, automatically,
and the reduction of the error in the translation along thex axis is confirmed by the simulation.
The problems of the initial value dependency and the
avoidance from the local optimum in Concave-Convex Procedure, selection of the number of the peak filter, tuning
the resonant frequency ω of the peak filter, overall optimization including other controllers are future research.
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