Frequency Response Data-based Multiple Peak
Filter Design Applied to High-Precision Stagein Translation and Pitching
Masahiro Mae, Wataru Ohnishi and Hiroshi Fujimoto

Koichi Sakata

The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, Japan
mmae@ieee.org, ohnishi@ieee.org, fujimoto@k.u-tokyo.ac.jp

Nikon Corporation
471, Nagaodaicho, Sakae, Yokohama, Kanagawa, Japankoichi.sakata@nikon.com

Abstract—Tuning of feedback controllers in high-precision
stages needs a lot of time and skills. In particular, multiple
filter design for multi-input multi-output systems is difficult
because of a large number of controller parameters in eachaxis and a coupling problem between each axis. In this study,
the frequency response data-based multiple peak filter design
method is proposed to reduce the error. The multiple peak filters
are designed by convex optimization with the objective function
to reduce the error. The effectiveness of the proposed method
is validated in the simulated data of the two-degree-of-freedom
high-precision stage in translation and pitching.
Index Terms—Concave-convex procedure, Data-based design,Frequency responses, Disturbance rejection, Peak filter, MIMO
system

I. I NTRODUCTION
Disturbance rejection is important in the high-precision
stages with the scan motion. It improves the tracking performance of the high-precision scan stage used in the industry
such as manufacturing semiconductors and flat panel displays(FPD) [1]. Many high-precision stages in the industry havemulti-degree-of-freedoms (multi-DOFs) such as moving withtranslation and pitching in each axis and are multi-input multioutput (MIMO) systems. They are typically controlled with
2-DOF control with a feedforward controller for reference
tracking and a feedback controller for disturbance rejection[2]. The model-based feedforward controller design approachfor the MIMO system with translation and pitching is proposed[3]. However, a feedback controller design for disturbance rejection in MIMO systems has several challenges that it is timeconsuming and needs experience because of the interaction
between each axis and many controller tuning parameters.
To design without the heuristic tuning approach depending
on experiences, several approaches of the frequency response
data-based feedback controller design with convex optimization are proposed such as using the concave-convex procedure[4]. In the data-based design approach, the structure of thecontroller is important for the physical interpretation and effort
of on-site control engineers.
The disturbance can be rejected by the peak filter with thesame resonance frequency [5] because of the internal model
principle [6] and, auto-tuning method for one peak filter is

proposed [7]. However, the multiple peak filter design using
convex optimization has not been reported yet. The convex
optimization has the challenge to formulate the controller inaffine to avoid multiplying the tuning parameters each other.
In this study, the frequency response data-based multiple peakfilter design method using convex optimization is presented.
The main contributions of this study are i) the structured
multiple peak filter is formulated, ii) the peak filter is directly
designed with frequency response data, and iii) the control
problem can be solved with convex optimization.
The outline of this paper is as follows. In Section II, the control problem is formulated. The frequency response data-based
multiple peak filter design is presented in Section III. The
advantages of the approach are demonstrated in Section IV.
Conclusions are presented in Section V.
II. P ROBLEM FORMULATION
A. Controlled system
The controlled system is given as a frequency response dataof nu -input ny -output open-loop system G(jωkf ) ∈ Cny ×nu
as follows:G(jωkf ) = G(ky ,ku ) (jωkf ),(1)

where the indices of inputs, outputs, and frequency response
data are ku = 1, . . . , nu , ky = 1, . . . , ny , and kf = 1, . . . , nf .
In this study, the square system (nu = ny ) is considered.
B. Designed controller
The ny -input ny -output diagonal peak filter F (jωkf , ρ) is
designed as follows:
F (jωkf , ρ) = diag(Fky (jωkf , ρky )),where the tuning parameter is ρ = [ρ1 , . . . , ρny ].

(2)

r +

e
−

d

u
F

G

where the disturbance frequency spectrum in each experimentdke (jωkf ) ∈ Cny ×1 is given byT
dke (jωkf ) = dke ,1 (jωkf ) · · · dke ,ny (jωkf ) . (8)+ y+Fig. 1. Block diagram of the controlled system. The given open-loop system
G is controlled by the designed diagonal peak filter F . r, e, u, y, andd denote reference, error, input, output, and disturbance, respectively. The
objective is to minimize the error e.

The error frequency spectrum matrix in the open-loopsystem G(jωkf )F (jωkf , ρ) is given by the calculation withthe disturbance frequency spectrum matrix as follows:E(jωkf , ρ) = S(jωkf , ρ)D(jωkf ),The designed peak filter in each output is given by
Fky (jωkf , ρky )
np,ky

=1+

X
kp =1

ρky ,(kp ,2) (jωkf )2 + ρky ,(kp ,1) (jωkf )2(jωkf )2 + 2ζp,ky ,kp ωp,ky ,kp (jωkf ) + ωp,k
y ,kp


1
(jωkf )




2

T 

(jωkf )2 + 2ζp,ky ,1 ωp,ky ,1 (jωkf ) + ωp,k


y ,1
1
2


(jωkf ) ρky ,(1,1)  
 


2
2
(jωkf ) + 2ζp,ky ,1 ωp,ky ,1 (jωkf ) + ωp,ky ,1 ρky ,(1,2)  
 


=
..
..
 





.
.
 



ρky ,(np,k ,1)  
(jω
)
k
f
y


2

2
ρky ,(np,ky ,2)  (jωkf ) + 2ζp,ky ,np,ky ωp,ky ,np,ky (jωkf ) + ωp,ky ,np,k y (jωkf )2
2
2
(jωkf ) + 2ζp,ky ,np,ky ωp,ky ,np,ky (jωkf ) + ωp,ky ,np,k


y

= ρT
ky φky (jωkf ),(9)where S(jωkf , ρ) = (I + G(jωkf )F (jωkf , ρ))−1 ∈ Cny ×ny
is the sensitivity function matrix with the open-loop system
G(jωkf )F (jωkf , ρ).
The control objective considered in this study is to minimize
the error frequency spectrum E(jωkf , ρ) with the designed
diagonal peak filter F (jωkf , ρ).
III. A PPROACH
A. Objective function
The objective function is designed to minimize the error
frequency spectrum matrix E(jωkf , ρ) with the designed
diagonal peak filter F (jωkf , ρ).
From (9), the weighted error frequency spectrum matrix is
given by(3)

where the index of the peak filters in each output is kp =1, . . . , np,ky , the tuning parameter in each output is ρky ∈R2np,ky +1 , and the resonance frequency and damping coefficient of each peak filter are ωp,(ky ,kp ) and ζp,(ky ,kp ) .
C. Control objective

W −1 E(jωkf , ρ) = W −1 (I + G(jωkf )F (jωkf , ρ))−1 D(jωkf ),where W ∈ Rny ×ny is the scaling matrix of the error.
The Moore–Penrose inverse of the weighted error frequency
spectrum matrix is given by(W −1 E(jωkf , ρ))+ = (W −1 (I + G(jωkf )F (jωkf , ρ))−1 D(jωkf )))+ ,The control objective considered in this study is to minimize
the error frequency spectrum of the given open-loop system
G with the designed diagonal peak filter F as shown in Fig.
1.
The error frequency spectrum matrix in the open-loopsystem G(jωkf ) is given by the experiment as follows:
E0 (jωkf ) = e0,1 (jωkf ) · · · e0,ne (jωkf ) ,(4)
where the experiment is conducted in ne times, the index of
the experiment is ke = 1, . . . , ne , and the error frequency
spectrum in each experiment e0,ke (jωkf ) ∈ Cny ×1 is givenbyTe0,ke (jωkf ) = e0,ke ,1 (jωkf ) · · · e0,ke ,ny (jωkf ) . (5)The disturbance frequency spectrum matrix is given fromthe error frequency spectrum matrix as follows:D(jωkf ) = S0−1 (jωkf )E0 (jωkf )−1(6)ny ×nywhere S0 (jωkf ) = (I + G(jωkf ))∈ C
is thesensitivity function matrix with the open-loop system G(jωkf )and I ∈ Rny ×ny is an identity matrix. The elements of the
disturbance frequency spectrum matrix is given byD(jωkf ) = d1 (jωkf ) · · · dne (jωkf ) ,
(7)

(10)

E (jωkf , ρ)W = D (jωkf )(I + G(jωkf )F (jωkf , ρ))W .
+

+

(11)
(12)

In this approach, 1-norm of each row of E + (jωkf , ρ)W is
evaluated in each frequency. The objective function is givenbymaximizeρ+min kD(k(jωkf )(I + G(jωkf )F (jωkf , ρ))W k1 .
e ,:)∀ k ,∀ kef(13)

B. Constraint of sensitivity function
The robust stability condition in each output is consideredwith the upper bound of the gain of the sensitivity function as
follows:|ws,ky (jωkf )| − G(ky ,ky ) (jωkf )Fky (jωkf , ρ) + 1 ≤ 0,

(14)

where ws,ky (jωkf ) is the weighting function of the upperbound of the gain of the sensitivity function.
The input saturation condition in each output is consideredwith the lower bound of the gain of the sensitivity function as
follows:|wu,ky (jωkf )| − G(ky ,ky ) (jωkf )Fky (jωkf , ρ) + 1 ≥ 0,

(15)

where wu,ky (jωkf ) is the weighting function of the lower
bound of the gain of the sensitivity function.

C. Optimization problem formulation

TABLE IPARAMETERS OF INITIAL PEAK FILTER Fx .

The optimization problem is formulated with the objectivefunction and the constraints of the sensitivity function as
follows.
maximizeρsubject to+min kD(k(jωkf )(I + G(jωkf )F (jωkf , ρ))W k1
e ,:)

(16)

|ws,ky (jωkf )| − G(ky ,ky ) (jωkf )Fky (jωkf , ρ) + 1 ≤ 0(17)

G(ky ,ky ) (jωkf )Fky (jωkf , ρ) + 1 − |wu,ky (jωkf )| ≤ 0(18)∀ k ,∀ kefkp

ωp,1,kpζp,1,kp

ρ1,kp ,1

ρ1,kp ,21234

2π × 4.78
2π × 5.86
2π × 6.65
2π × 8.81

0.10
0.10
0.10
0.10

0.67
0.61
2.80
0.92

−4.94 × 10−03
7.41 × 10−04
2.25 × 10−02
3.99 × 10−02∀ k ,∀ ,kyf

TABLE IIPARAMETERS OF INITIAL PEAK FILTER Fθy .

Because of the nonlinearity objective function in (16), theoptimization problem is reformulated with the linear objectivefunction γ as follows.
maximizeγ(19)subject to+γ − kD(k(jωkf )(I + G(jωkf )F (jωkf , ρ))W k1 ≤ 0
e ,:)

(20)

|ws,ky (jωkf )| − G(ky ,ky ) (jωkf )Fky (jωkf , ρ) + 1 ≤ 0(21)

G(ky ,ky ) (jωkf )Fky (jωkf , ρ) + 1 − |wu,ky (jωkf )| ≤ 0

(22)

ρ

∀k

∀
∀
e , kf , ,ky

D. Concave-convex procedure
The constraints (20) and (21) in the optimization problem
are the differences in convex functions and they are nonconvex. In this approach, the concave-convex procedure [4]is applied. A sequential linearization is applied around thecurrent solution and the new feasible sets (23) and (24) become
convex subsets of the original feasible sets. The parameter is
optimized by repeating the update of the optimization problem
and the convex optimization calculation.
In this approach, it is not secured to converge to the global
optimum solution. It is likely to converge local optimum
solution or saddle point. However, the optimization problem can be solved by convex optimization with monotonous
convergence secured. Therefore, the quasi-optimized solution
can be derived with easy and stable calculation compared tononlinear optimization.
IV. S IMULATION
A. Motion system
The 2-DOF motion system of a high-precision stage in thetranslation x and the pitching θy is illustrated in Fig. 2. It isnoted that the number of the output is ny = 2. The index
ky = 1 means the translation x and the index ky = 2 means
the pitching θy .
The given open-loop system G is consists of the 2-DOF
motion system and the given feedback controller. The given
open-loop system G is controlled by the designed diagonal
peak filter F as shown in Fig. 3.
B. Conditions
The number of frequency response data points is set to nf =1000, and they are arranged at linearly even intervals in therange from 3 Hz to 30 Hz. The frequency response data of thegiven open-loop system G is shown in Fig. 4.

kp

ωp,2,kpζp,2,kp

ρ2,kp ,1

ρ2,kp ,21234

2π × 4.78
2π × 5.86
2π × 6.65
2π × 8.81

0.10
0.10
0.10
0.10

0.65
0.27
0.94
1.13

−6.02 × 10−03
1.78 × 10−05
7.93 × 10−03
8.26 × 10−02

The error data is measured by pre-experiment in the constantscan velocity of the translation x with 0.5 m/s for 12 times(ne = 12). The scaling matrix of the error is given by the meanerror of all experiments at all frequencies in each output asfollows:


nf
ne X
X
1
W = diag |e0,ke ,ky (jωkf )| .
(25)
ne nf
ke =1 kf =1

The weighting function of the upper bound of the gain of
the sensitivity function is given by
ws,ky (jωkf ) = vs ×(
(1 + G(ky ,ky ) (jωkf ))(ωkf ≤ ωky ,smax ), (26)(1 + G(ky ,ky ) (jωky ,smax )) (ωkf > ωky ,smax )where vs = 1/1.01 andmin |1 + G(ky ,ky ) (jωkf )| = |1 + G(ky ,ky ) (jωky ,smax )|. (27)∀kf

The weighting function of the lower bound of the gain of
the sensitivity function is given by
wu,ky (jωkf ) = vu × (1 + G(ky ,ky ) (jωkf ))(ωkf ≤ ωky ,smax ),(28)

where vu = 3.
The parameters of the initial peak filter is given by theheuristic approach [5] with the constraints of the sensitivity
function as shown in TABLE I and TABLE II.
The iterative calculation in the optimization is continued
until the improvement of the objective function is less than
1 %. The optimization problem is calculated by YALMIP [8]and Mosek [9].
C. Optimization result
The parameters of the optimized peak filter is shown in
TABLE III and TABLE IV.
The Bode diagram of the initial and optimized peak filter
is shown in Fig. 5.
The sensitivity functions of the single-input single-output(SISO) open-loop system without the peak filter, with theinitial peak filter, and with the optimized peak filter are shownin Fig. 6. It is shown that the optimized peak filter satisfied
the constraints of the SISO sensitivity function.

+(D(k(jωkf )(I + G(jωkf )F (jωkf , ρ(ki −1) ))W )∗e ,:)γ − Re!
+(D(k(jωkf )(I + G(jωkf )F (jωkf , ρki ))W )e ,:)≤ 0 (23)+|D(k(jωkf )(I + G(jωkf )F (jωkf , ρ(ki −1) ))W |
e ,:)
1


(G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) ) + 1)∗|ws,ky (jωkf )| − Re(G(ky ,ky ) (jωkf )Fky (jωkf , ρki ) + 1) ≤ 0 (24)|G(ky ,ky ) (jωkf )Fky (jωkf , ρ(ki −1) )|z

Output
Input

x

Input

ux

6
4
2
0
-2

Output

y

uθy

x θy
Stage

Fig. 2. 2-DOF motion system in translation x and pitching θy .

xref + − ex 
Fx0
θyref + − eθy

F
0
Fθy

 ux 
uθy

Gxx
Gθy x

G
Gxθy
Gθy θy

dx
 + +

x

+ + θy
dθy

180
90
0
-90
-180
101

Fig. 3. Block diagram of 2-DOF given open-loop system G in translation x
and pitching θy with designed diagonal peak filter F .

(a) x

6
4
2
0
-2

20
0
-20
-40
-60
180
90
0
-90
-180

180
90
0
-90
-180
101

20
0
-20
-40
-60

(b) θyFig. 5. Bode diagram of initial peak filter (
).
(

180
90
0
-90
-180

) and optimized peak filter

D. Evaluation of disturbance rejection
101

101

The effectiveness of the optimized peak filter is evaluatedin reducing the error.
The frequency response data of the error for the evaluation is
given by the pre-experiment data in the worst case as follows:

Fig. 4. Bode diagram of 2-DOF given open-loop system G.

emax (jωkf ) = emax,ky (jωkf ),The Nyquist diagrams of the SISO open-loop system without the peak filter, with the initial peak filter, and with the
optimized peak filter are shown in Fig. 7. It is shown that the
SISO open-loop system with the optimized peak filter satisfied
the Nyquist stability condition.

(29)

where |emax,ky (jωkf )| = max∀ ke |e0,ke ,ky (jωkf )|.
The frequency response data of the error with designed peakfilter is calculated as follows:e(jωkf , ρ) = S(jωkf , ρ)S0−1 (jωkf )emax (jωkf ).

(30)

20

2

10
1
0
0
-10
-1

-20
-30

-2
-3

101

-2

(a) x

-1

0

1

0

1

(a) x

2

20
10

1
0
0
-10
-1

-20
-30

-2
-3

101

-2

-1

(b) θy(b) θyFig. 6. Bode magnitude diagram of SISO sensitivity function without peakfilter (), with initial peak filter (), and with optimized peak filter(). Constraints of sensitivity function are shown in ().

Fig. 7. Nyquist diagram without peak filter (), with initial peak filter), and with optimized peak filter (). Sensitivity peaks are shown
(
in (
).

TABLE III
PARAMETERS OF OPTIMIZED PEAK FILTER Fx .

with other cases.
The cumulative amplitude spectrums of the error without thepeak filter, with the initial peak filter, and with the optimized
peak filter are shown in Fig. 9, where fkf = 2πωkf andv
u X
u1|eky (jωkf )|2 .
(31)
Eky (fkf ) = t2kpωp,1,kpζp,1,kp

ρ1,kp ,1

ρ1,kp ,21234

2π × 4.78
2π × 5.86
2π × 6.65
2π × 8.81

0.10
0.10
0.10
0.10

5.51
3.36
1.74
0.66

−6.02 × 10−02
1.39 × 10−01
9.72 × 10−02
1.25 × 10−01

kf

TABLE IVPARAMETERS OF OPTIMIZED PEAK FILTER Fθy .
kp

ωp,2,kpζp,2,kp

ρ2,kp ,1

ρ2,kp ,21234

2π × 4.78
2π × 5.86
2π × 6.65
2π × 8.81

0.10
0.10
0.10
0.10

1.63
11.64
−3.59
−3.02

−1.14 × 10−01
4.07 × 10−02
2.01 × 10−01
2.37 × 10−02

It is noted that the final value of Eky (fkf ) is equal to the root
mean square of the inverse Fourier transform of eky (jωkf ). It
is shown that the root mean square error is reduced by the
optimized peak filter compared with other cases.
The results demonstrate the advantage of the proposed
approach in the objective of minimizing the error.
V. C ONCLUSION

The amplitude spectrums of the error without the peak filter,with the initial peak filter, and with the optimized peak filter
are shown in Fig. 8. It is shown that the maximum amplitudeof the error is reduced by the optimized peak filter compared

The frequency response data-based multiple peak filter design is developed. The proposed approach improves the design
method of the peak filter to reduce the error using convex
optimization compared with other heuristic approaches. The

5

10-7
2.5

10-6

4
2
3
1.5
2
1
1
0.5
0
101

0
101

(a) x

1.2

10

(a) x

-6

6

10-6

1
5
0.8
4
0.6
3
0.4
2
0.2
1
0
101

0
101

(b) θy(b) θyFig. 8. Amplitude spectrum of error without peak filter (), with initial
peak filter (), and with optimized peak filter (). Maximum amplitude
is shown in horizontal dashed line. Peak filter is designed at peak error
frequencies (×).

advantages of the approach are validated in the simulation
with the 2-DOF motion system of a high-precision stage
in the translation x and the pitching θy . Ongoing research
focuses on application to the 6-DOF high-precision stage andexperimental validation.
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