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© 2023 The Japan Society of Mechanical Engineers. This is an open access
article under the terms of the Creative Commons Attribution-NonCommercial-
NoDerivslicense (https://creativecommons.org/licenses/by-nc-nd/4.0/).
Advance Publication by J-STAGE
Mechanical Engineering Journal
DOI: 10.1299/mej.22-00362
Received date : 6 September, 2022
Accepted date : 1 March, 2023
J-STAGE Advance Publication date : 11 March, 2023
© The Japan Society of Mechanical Engineers
Infinite-impulse-response periodic-disturbance observer for
harmonics elimination with wide band-stop bandwidths
Hiroki TANAKA∗and Hisayoshi MURAMATSU∗
∗Mechanical Engineering Program, Hiroshima University
1-4-1, Kagamiyama, Higashihiroshima, Hiroshima, 739-0046, Japan
E-mail: muramatsu@hiroshima-u.ac.jp
Abstract
Repetitive motions of automatic machines induce a periodic disturbance, and the periodic disturbance deteriorates
precision of motion control of the machines. A periodic-disturbance observer (PDOB) was proposed to suppress
the periodic disturbance, but variation of the periodic disturbance frequency deteriorates the periodic disturbance
suppression because its band-stop bandwidths in the sensitivity function for harmonics elimination are narrow. This
paper proposes an infinite impulse response (IIR)-PDOB. The IIR-filter realizes wider frequency bandwidths in
the sensitivity function than the PDOB, which achieves robustness against the variation of the periodic disturbance
frequency. We confirmed that the IIR-PDOB was robust against the variation of the periodic disturbance frequency
using a parallel robot.
Keywords : Disturbance observer, Motion control, Periodic disturbance, Position control, Time-delay element
1. Introduction
Factory automation has been proceeded for enhancement of productivity, improvement in quality, and reducing labor
costs. In process of the factory automation, automatic machines, such as electric motors (Cao and Low, 2009), industrial
robots (Umeno et al. , 1993), and power systems (Bykhovsky and Chow, 2003) are used. Some periodic disturbance
is generated by repetitive motions, others derive from neighboring machines. To improve the motion control precision,
suppression of the periodic disturbances is necessary.
A disturbance observer (DOB) can estimate disturbances (Ohishi et al. , 1983; Komada et al. , 1991; Sariyildiz
et al. , 2020). The DOB is a two-degree-freedom-controller that disturbance suppression and tracking characteristics
can be designed independently (Matsumoto et al. , 1993). The sensitivity function corresponding to the disturbance
suppression characteristic is determined by a Q-filter. Typically, the DOB employs a low-pass filter as the Q-filter to make
the sensitivity and complementary sensitivity functions high-pass and low-pass characteristics, respectively. The DOB
can only suppress disturbances whose frequencies are less than a cut-offfrequency of the low-pass filter (Sariyildiz and
Ohnishi, 2015). Then, the high-pass characteristic is insufficient to eliminate higher harmonics of the periodic disturbance
at specific frequencies.
Robust adaptive resonant controller was proposed to attenuate a sinusoidal disturbance (Minghe et al. , 2022),
while the controller has only one band-stop characteristic in the sensitivity function. The single band-stop characteristic
is insufficient for suppressing harmonics. To suppress the periodic disturbance including harmonics, repetitive control
(Inoue et al. , 1981; Hara et al. , 1988; Wu et al. , 2013; Mooren et al. , 2022; Jie et al. , 2021) and iterative learning
control (Uchiyama, 1978; Arimoto et al. , 1984) were proposed using a time-delay element. The time-delay element
enables to use errors in the previous repetition for improving precision of repetitive operation. Their sensitivity functions
have an infinite number of band-stop frequencies to suppress the harmonics, whereas the repetitive controls amplify a
nonperiodic disturbance.
To eliminate the periodic disturbance including the harmonics, a periodic-disturbance observer (PDOB) was pro-
posed (Muramatsu and Katsura, 2018; Muramatsu and Katsura 2019). The sensitivity function of the PDOB has a
© The Japan Society of Mechanical Engineers
IIR-PDOB
Q-filter
Fig. 1 Block diagram of the proposed IIR-PDOB.
high-pass characteristic and an infinite number of band-stop frequencies. The high-pass characteristic and band-stop
frequencies suppress the periodic disturbance. In contrast to the repetitive control, the PDOB does not amplify nonperi-
odic disturbance in the case of plants with first relative order, where the Bode sensitivity integral does not affect. Moreover,
its implementation is simple. Similar to the PDOB, combinations of the DOB and repetitive control were proposed (Chen
and Tomizuka , 2014; Linlin et al. , 2022), which amplify the nonperiodic disturbance also but less than the repetitive
control. Nevertheless, the band-stop bandwidths of the DOB based repetitive control and PDOB are narrow; thus, the
suppression performance is not robust against variation of periodic disturbance. For the variation of periodic disturbance
frequency, there are adaptive controls that estimate the periodic disturbance frequency (Jie et al. , 2021; Muramatsu and
Katsura, 2018), but the estimation is slow and inaccurate under harmonics.
This paper proposes an infinite impulse response (IIR)-PDOB with wide band-stop bandwidths in the sensitivity
function for harmonics elimination. A Q-filter of the IIR-PDOB comprises a low-pass filter and feedback loop using a
time-delay element. The Q-filter of the IIR-PDOB is an IIR filter by the feedback loop using a time-delay element when
the low-pass filter is excepted. By using the IIR-filter including a time-delay element, the suppression performance of
the IIR-PDOB is robust against the variation of the periodic disturbance frequency than the PDOB. The nominal stability
of the IIR-PDOB with the proportional-derivative control system is guaranteed. The IIR-PDOB has three parameters α,
β, and γ, which are designed by desired gains of the sensitivity and complementary sensitivity functions. By turning
the time-delay element of the IIR-PDOB, mth harmonic suppression performance is optimized. In addition, desired first
harmonic suppression performance determines a lower limit of the cut-offfrequency, and a robust stability condition
determines a upper limit of the cut-offfrequency. We conducted experiments to compare the suppression performance
of the IIR-PDOB with the conventional PDOB and the DOB using a parallel robot. The suppression performance of the
IIR-PDOB was robust against the variation of periodic disturbance frequency than the conventional PDOB owing to the
wider band-stop bandwidths. In addition, we confirmed the IIR-PDOB can optimize any target harmonic suppression
performance.
2. IIR-PDOB
2.1. Q-filter
To estimate a periodic disturbance including harmonics, we set the Q-filter for the proposed IIR-PDOB as
Q(s)=gp
s+gp
γ1−β
α+e−Ls ,(1)
where s,gp, and e−Ls are the Laplace operator, cut-offfrequency, and time-delay element, respectively. α, β, and γare
real numbers to be designed. Fig. 1 shows the block diagram of proposed IIR-PDOB based on the Q-filter. Iref,d,ˆ
d,
θres,Kt,Ktn ,Jn, and Jdenote the current reference, disturbance, estimated disturbance, angle response, torque constant,
nominal torque constant, inertia, and nominal inertia, respectively. The torque constant and inertia include modeling error
∆(s)
Kt(s)1
J(s)=Ktn
1
Jn
(∆(s)+1).(2)
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(a) Sensitivity functions.
(b) Complementary sensitivity functions.
Fig. 2 Comparison of the sensitivity functions and complementary sensitivity functions of the IIR-PDOB with
those of the the conventional PDOB and DOB. The parameters for the PDOBs are gp=500 rad/s,
gpd =500 rad/s, L=2π/10 rad/s, α=2, β=1, γ=1.5, and ∆(s)=0. The cut-offfrequency of
the DOB is 10 rad/s.
PDOB
Fig. 3 Block diagram of the PDOB.
DOB
Fig. 4 Block diagram of the DOB.
where ∆(s) is the modeling error. The open-loop transfer function of the IIR-PDOB is
F(s)=Q(s)
1−Q(s)
gpd
s+gpd
∆(s).(3)
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(a) Sensitivity function.
(b) Complementary sensitivity function.
Fig. 5 Bode diagrams of the sensitivity and complementary sensitivity functions in Eq. (4) and the nominal
sensitivity and complementary sensitivity functions in Eq. (5) of the IIR-PDOB. The parameters are
gp=500 rad/s, gpd =500 rad/s, L=2π/10 rad/s, α=2, β=1, γ=1.5, and ∆(s)=0.
Furthermore, its sensitivity and complementary sensitivity functions are
S(s)=1
1+F(s),T(s)=F(s)
1+F(s),(4)
Fig. 2 shows the Bode diagrams of the nominal sensitivity and the nominal complementary sensitivity functions of the
proposed IIR-PDOB, conventional PDOB (Muramatsu and Katsura, 2018), and DOB. Fig. 3 and Fig. 4 show the block
diagram of the PDOB and DOB, respectively. The nominal sensitivity functions of both the PDOBs have harmonics
suppression and high-pass characteristics. The DOB has only high-pass characteristic. In particular, the IIR-PDOB
has the wider band-stop bandwidths, which means the IIR-PDOB is less affected by the periodic disturbance frequency
changes.
Nominal sensitivity and complementary sensitivity functions are
Sn(s)=1−Q(s),Tn(s)=Q(s).(5)
If ∆(s)=0, and gpd → ∞; then, S(s)=Sn(s),T(s)=Tn(s). Fig. 5 shows the Bode diagrams of the sensitivity
and complementary sensitivity functions in Eq. (4) and those of the nominal sensitivity and complementary sensitivity
functions in Eq. (5). From the gains of the sensitivity functions in Fig. 5(a), the IIR-PDOB has band-stop frequency to
suppress harmonics.
This paper designs the parameters α,β,γ, delay L, and cut-offfrequency gpusing the nominal sensitivity and
complementary sensitivity functions in Eq. (5) focusing on the frequency range lower than the cut-offfrequency gpd .
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(a) m=1.
(b) m=3.
Fig. 6 Gains of the nominal sensitivity functions with and without the low-pass filter (LPF) in the cases of m=1
and m=3. The parameters for the IIR-PDOB are g=1000 rad/s, and ω0=100 rad/s.
2.2. Parameter design of α,β,γ
We determine the three parameters α,β, and γaccording to the gains of the nominal sensitivity and complementary
sensitivity functions in Eq. (5). By substituting nω0for the Fourier transformed time-delay element, e−jLωis rewritten
using L=2π
ω0as
e−jLnω0=e−j2π
ω0nω0=e−j2nπ=1,n∈Z+,(6)
where ω0is the first harmonic frequency of a periodic disturbance. In a similar manner, substituting (2n−1)ω0/2, e−jLω
is rewritten as
e−jL 2n−1
2ω0=e−j2π
ω0
2n−1
2ω0=e−j(2n−1)π=−1.(7)
We impose four nominal gain requirements
|Sn(jnω0)|=
(1 −γ)(α+1) +βγ
α+1
=0,(8)
|Tn(jnω0)|=
γ(α+1−β)
α+1
=1,(9)
Sn j2n−1
2ω0!
=
(1 −γ)(α−1) +βγ
α−1
=1,(10)
Tn j2n−1
2ω0!
=
γ(α−1−β)
α−1
=0,(11)
on the Q-filter. By calculating Eq. (8), Eq. (9), and Eq. (10), the three design parameters α,β, and γare determined
α=2, β =1, γ =1.5.(12)
The derived three parameters in Eq. (12) satisfy the unused condition Eq. (11).
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Fig. 7 Gains of the nominal sensitivity function with m=1, m=3, and m=5 at harmonic frequencies. The
parameter is gp=1000 rad/s.
2.3. Design of delay L
The delay Ldetermines the band-stop frequencies of the sensitivity functions for eliminating harmonic. To adjust
any mth band-stop frequency to its mth target harmonic frequency, the delay is calculated as
L=2π
ω0
1+1
2mπtan−1
−6mω04gp+5qg2
p+4m2ω2
0
−80m2ω2
0+9gpqg2
p+4m2ω2
0
,(13)
where mis the number of the target harmonic. Fig. 6(a) and (b) show the Bode diagram of the nominal sensitivity functions
with and without the low-pass filter gp/(s+gp) in the Q-filter. As shown in Fig. 6(a), by setting m=1 in Eq. (13), the first
band-stop frequency of the IIR-PDOB with the low-pass filter was adjusted to the first harmonic frequency (100 rad/s).
As shown in Fig. 6(b), by setting m=3, the third band-stop frequency matches the third harmonic frequency (300 rad/s).
Fig. 7 shows the gain of the nominal sensitivity function at harmonic frequencies with variations of m. From Fig. 7, m=1
takes smallest gain at the first harmonic frequency. Also, m=3 and m=5 take smallest gain at third and fifth harmonic
frequencies, respectively.
The delay calculation Eq. (13) is obtained by deriving a correction function τ(m) of
L=2π
ω0
(1+τ(m)), τ(m)∈[−0.5,0.5),m∈Z+(14)
that minimizes the gain of the sensitivity function at mth harmonic frequency. To this end, we calculate derivative and
second derivative of the gain with respect to τ(m). By substituting nω0and Eq. (14) for the Fourier transformed time-delay
element e−jLω, it is written as
e−jLω=e−j2π
ω0(1+τ(m))nω0=e−j2nπe−j2nπτ(m)=e−j2nπτ(m).(15)
Using Euler’s formula
e−j2nπτ(m)=cos 2nπτ(m)−jsin 2nπτ(m) (16)
and the gain |Sn(s)|of the nominal sensitivity function based on parameters Eq. (12)
|Sn(s)|=
1−3
2
gp
s+gp 1+e−Ls
2+e−Ls !
,(17)
the gain at nth harmonic frequency becomes
|Sn(jnω0)|=rN1
D1
,(18)
where
N1=g2
p+10n2ω2
0−(g2
p−8n2ω2
0) cos 2nπτ(m)+6gpnω0sin 2nπτ(m),(19)
D1=2(g2
p+n2ω2
0)(5 +4 cos 2nπτ(m)).(20)
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0
0
0.1 0.2
0.05
0.10
0.15
0.20
0.25
0.3 0.375
Gain
Fig. 8 Gain of the nominal sensitivity function with respect to the ratio ω0/gpin Eq. (36).
Fig. 9 Equivalent block diagram of Fig. 1.
Then, the first-order partial derivative of |Sn(jnω0)|2with respect to τ(m) is
∂
∂τ(m)|Sn(jnω0)|2=N2
D2
,(21)
where
N2=3gpnπ(8nω0+10nω0cos 2nπτ(m)+3gpsin 2nπτ(m) (22)
D2=(g2
p+n2ω2
0)(5 +4 cos 2nπτ(m))2,(23)
and the solutions of ∂
∂τ(m)|Sn(jnω0)|2=0 are
τ1(m)=1
2mπtan−1
−6mω04gp+5qg2
p+4m2ω2
0
−80m2ω2
0+9gpqg2
p+4m2ω2
0
,(24)
τ2(m)=1
2mπtan−1
6mω0−4gp+5qg2
p+4m2ω2
0
−80m2ω2
0−9gpqg2
p+4m2ω2
0
.(25)
The second derivative of |Sn(jnω0)|2is
∂2
∂τ(m)2|Sn(jnω0)|2=N3
D3
,(26)
where
N3=6gpn2π2(18gp+15gpcos 2nπτ(m)−6gpcos 4nπτ(m)+14nω0sin 2nπτ(m)+20nω0sin 4nπτ(m)),(27)
D3=(g2
p+n2ω2
0)(5 +4 cos 2nπτ(m))3.(28)
By substituting Eq. (24) and Eq. (25) for Eq. (26), the signs of ∂2
∂τ(m)2|Sn(jnω0)|2are positive and negative, respectively.
Hence, Eq. (18) is convex downward if τ(m)=τ1(m), the gain of the sensitivity function |Sn(jnω0)|takes the minimum
value at τ1(m). Therefore, we obtain Eq. (13) with Eq. (14) and Eq. (24).
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2.4. Design of cut-off frequency gp
The lower limit of a cut-offfrequency of the low-pass filter gp/(s+gp) in the Q-filter is determined by required
suppression performance. The gain of the nominal sensitivity function at the first harmonic frequency ω0is
|Sn(jω0)|=
1−3
2
gp
gp+jω0 1+e−jLω0
2+e−jLω0!
,(29)
=
1−3
2
gp
gp+jω0 1+e−jϕ
2+e−jϕ!
,(30)
where
ϕ=tan−1
−6ω04gp+5qg2
p+4ω2
0
−80ω2
0+9gpqg2
p+4ω2
0
,(31)
using the delay Lin Eq. (13). The first-order Taylor approximation provides the approximate gain
|Sn(jω0)| ≈
1−3
2
gp
gp+jω0 1+1−jϕ
2+1−jϕ!
.(32)
The gain of the sensitivity function in Eq. (32) is transformed into
|Sn(jω0)|=rN4
D4
,(33)
where
N4=0.25g2
p(tan−1ϕ)2+3gpω0(tan−1ϕ)+(9 +(tan−1ϕ)2)ω2
0(34)
D4=(g2
p+ω2
0)(9 +(tan−1ϕ)2).(35)
We define the ratio of the first harmonic frequency and cut-offfrequency as
µ=ω0
gp
, µ ∈(0,0.375),(36)
where 0.375 is owing to the range of tan−1ϕin Eq. (31). Using µ,ϕis expressed as
ϕ=tan−1
−6µ4+5p1+4µ2
−80µ2+9p1+4µ2
.(37)
By using Eq. (37), Eq. (32) is expressed by
|Sn(jω0)|=s0.25(tan−1ϕ)2+3(tan−1ϕ)µ+(9 +(tan−1ϕ)2)µ2
(1 +µ2)(9 +(tan−1ϕ)2).(38)
Fig. 8 shows the gain of the nominal sensitivity function at the first harmonic frequency ω0with respect to the ratio ω0/gp.
Required first harmonic suppression performance determines the lower limit of the cut-offfrequency with Fig. 8.
Next, the upper limit of the cut-offfrequency is determined by the modeling error ∆(s) which is
∆(s)=W(s)δ(s),(39)
using the weighting function W(s) and variation δ(s). The variation satisfies ∥δ(s)∥∞≤1. Fig. 9 is an equivalent block
diagram of Fig. 1 assuming gpd → ∞. Suppose the modeling error ∆(s) is stable; then, the robust-stability condition based
on Fig. 9 and the small-gain theorem is
∥W(s)Q(s)∥∞<1.(40)
According to the relationship between the gain of the nominal complementary sensitivity function and that of the low-pass
filter
gp
s+gp
1.5 1+e−sL
2+e−sL !
≤
gp
s+gp
(41)
and Eq. (40), the upper limit of the cut-offfrequency gpis determined to satisfy
gp
s+gp
<
1
W(s)
.(42)
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Fig. 10 Block diagram of the position control system using the IIR-PDOB.
Fig. 11 Poles of the transfer function An/Ad,Bn/Bd,Cn/Cd, and Dn/Ddin Eq. (45) to Eq. (52) with using
parameters in Table 1 the cut-offfrequencies gp=5 rad/s, 50 rad/s, 500 rad/s, and 5000 rad/s.
2.5. Nominal stability analysis
We consider nominal stability of the system shown in Fig. 10 with ∆(s)=0. Then, the transfer functions from θcmd
and dto θres and ˆ
dare
θres(s)=An(s)
Ad(s)θcmd(s)+Bn(s)
Bd(s)d(s),(43)
ˆ
d(s)=Cn(s)
Cd(s)θcmd(s)+Dn(s)
Dd(s)d(s),(44)
where An(s), Ad(s), Bn(s), Bd(s), Cn(s), Cd(s), Dn(s), and Dd(s) are shown in Eq. (45) to Eq. (52). Figs. 11 and 12
show poles of the transfer functions An/Ad,Bn/Bd,Cn/Cd, and Dn/Ddwith variations of the cut-offfrequency gpand the
proportional and derivative gains Kpand Kv. According to Figs. 11 and 12, the system is nominally stable as the real parts
of the poles are negative.
3. Experiments
3.1. Setup
We used a parallel robot shown in Fig. 13. This study does not consider the higher order dynamics, including reso-
nance. The parallel robot had the six direct-drive servo motors: SGM7E-04BFA41 from Yaskawa Electric Corporation.
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Fig. 12 Poles of the transfer function An/Ad,Bn/Bd,Cn/Cd, and Dn/Ddin Eq. (45) to Eq. (52) with using
parameters in Table 1 the proportional gains Kp=2500, 25000, and 250000 with same the damping ratio.
An(s)=Jns4g2
pd(s(gp−e−L sgp+2(2 +e−Ls )s)
+2(2 +e−Ls )(gp+s)gpd)+2(2 +e−Ls )Jn(gp+s)(s+gpd)2((s+gpd)Kp+sgpd Kv) (45)
Ad(s)=Jn(s+gpd)2(s3(gp−e−L sgp+2(2 +e−Ls )s)+2(2 +e−Ls )s(gp+s)Kp+2(2 +e−Ls )(gp+s)gpd(Kp+s(s+Kv))),(46)
Bn(s)=(s(gp−e−Lsgp+2(2 +e−L s)s)+2(2 +e−L s)(gp+s)gpd ),(47)
Bd(s)=Jn(s3(gp−e−Lsgp+2(2 +e−L s)s)+2(2 +e−L s)s(gp+s)Kp+2(2 +e−L s)(gp+s)gpd (Kp+s(s+Kv))),(48)
Cn(s)=3(1 +e)gpJns3((gpd +s)2Kp+gpd s(gpd s+(gpd +s)Kv)) (49)
Cd(s)=(gpd +s)2(s2(3(1 +e)gps+2(2 +e)(gp+s)(gpd +s)) +2(2 +e−Ls )(gp+s)((gpd +s)Kp+gpd sKv)) (50)
Dn(s)=3(1 +e−Ls )gp((gpd +s)Kp+gpd s(s+Kv)) (51)
Dd(s)=2(2 +e−Ls )(gp+s)(gpd +s)Kp+s(s(2(2 +e−Ls)s(gpd +s)
+gp(2(2 +e−Ls )gpd +s−e−Ls s)) +2(2 +e−Ls )gpd(gp+s)Kv).(52)
We provided a sinusoidal position command and zero attitude commands to the end-effector:
pcmd =
0.14 cos(ω0t) m
0.14 cos(ω0t) m
0.08 cos(ω0t) m
0 rad
0 rad
0 rad
.(53)
The position command pcmd were converted to the joint angle command θcmd by inverse kinematics. Fig. 10 is the block
diagram of the position control system for each motor of the parallel robot. The joint angle command θcmd in Fig. 10
is an element of the vector θcmd. We implemented the controller using the Advanced Robot Control System (ARCS)
https://github.com/sidewarehouse/ARCS-PUBLIC with Linux OS. Errors are defined as the difference between
the position command and the end-effector position. Table 1 shows the experimental parameters. The delay Lin Fig. 10
was the integer multiple of the sampling time Tkin Table 1.
We conducted two experiments. First, we validated the suppression performance of the IIR-PDOB with the con-
ventional PDOB and the DOB while the periodic disturbance frequency was changed. Second, we validated that the
band-stop frequency of the IIR-PDOB can be adjusted to any target harmonic frequency.
3.2. Experiment 1
In Experiment 1, we compared the suppression performance of the IIR-PDOB with the conventional PDOB and
the DOB while the frequency ω0of the position command was changed from 4.5 rad/s to 5.5 rad/s. To compare the
suppression performance, we obtained the errors. By measuring the first harmonic frequency, we set the first frequency
˜ω05.03 rad/s for PDOB’s setting.
Fig. 14 shows the root-mean-square errors of the IIR-PDOB, conventional PDOB, and DOB. Fig. 15(a) and (b) show
the experimental errors of the x-axis with the frequency 5.1 rad/s. Fig. 15(c) shows the discrete Fourier transformed errors
from 15 s to 100 s eliminating the transient response. From Fig. 15(a), the position errors converged to steady state around
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750 mm
Fig. 13 Parallel robot.
Table 1 Experimental parameters.
Parameters Symbol Value
Sampling time Tk0.1 ms
Proportional gain Kp2500 /s−2
Derivative gain Kv200 s−1
Nominal torque constant Ktn 1.18 Nm/A
Nominal inertia Jn8.18 ×10−3kgm2
Cut-offof pseudo derivs. gpd 500 rad/s
Cut-offof Q-filter gp50 rad/s
Freq. of repetitive motions ω05 rad/s
1st harmonic frequency ˜ω05.03 rad/s
4th cycle. According to Figs. 14 and 15, the suppression performance of the IIR-PDOB was better than the conventional
PDOB while frequency of the position command was less than 4.8 rad/s or greater than 5.1 rad/s. For the conventional
PDOB, the frequency variations of the position commands impaired the periodic disturbance suppression performance.
In contrast, the suppression performance of the proposed IIR-PDOB was robust against the variation of the periodic
disturbance frequency owing to the wider band-stop bandwidths. As a limitation, the IIR-PDOB has mismatches in the
sensitivity function between the band-stop and harmonic frequencies, which caused the larger errors of the IIR-PDOB
than the conventional PDOB at the frequencies 4.9 and 5.0 rad/s, as shown in Fig. 14.
3.3. Experiment 2
In Experiment 2, we validated that the mth band-stop frequency of the mth IIR-PDOB can be adjusted to its target
harmonic frequency. The frequency of the position command was set to 5 rad/s, and we changed the target harmonic
number mof (13) as m=1,3, and 5. Fig. 16 shows the discrete Fourier transformed error results.
According to Fig. 16(a) with m=1, the amplitude of the errors of the first harmonic was smallest compared to
Fig. 16(b) with m=3 and (c) with m=5. Also, according to Fig. 16(b) with m=3, the amplitude of the errors at third
harmonic was smallest compared to Fig. 16(a) with m=1 and (c) with m=5. Lastly, the amplitude of the errors at fifth
harmonic in Fig. 16(c) with m=5 was smallest compared to Fig. 16(a) with m=1 and (b) with m=3. From the three
results, the IIR-PDOB performed the optimized mth harmonic suppression.
4. Conclusion
This paper proposed the IIR-PDOB realizing the wider band-stop bandwidths in the sensitivity function for sup-
pressing a periodic disturbance, which is robust against variation of a periodic disturbance frequency. The three design
parameters α,β, and γof the IIR-PDOB were determined by desired gains for the nominal sensitivity and complementary
sensitivity functions, and the IIR-PDOB’s nominal stability was guaranteed. Additionally, the delay Lwas designed to
adjust any band-stop frequency to its target mth harmonic frequency, and the cut-offfrequency gpwas determined by
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Fig. 14 Root-mean-square errors which is the sums of xaxis error, yaxis error, and zaxis error with variations
of the position command frequency from 15 s to 100 s. The IIR-PDOB and PDOB were designed to
suppress a periodic disturbance whose first harmonic frequency was 5.03 rad/s regardless of the change
of the command frequency.
(a) Errors. (b) Enlarged view from 90 s to 95 s. (c) Discrete Fourier transformed errors.
Fig. 15 Experimental errors of x-axis. The command frequency was 5.1 rad/s.
required suppression performance at the first harmonic and robust stability. We conducted the experiments to validate the
IIR-PDOB. According to the experimental results, the errors of the IIR-PDOB were smaller than those of the conventional
PDOB and DOB while the periodic disturbance frequencies were from 4.5 to 4.8 rad/s and from 5.1 to 5.5 rad/s. Moreover,
the experiments validated that the IIR-PDOB was possible to optimize mth harmonic suppression performance.
Acknowledgment
This work was supported by JSPS KAKENHI Grant Number 22K14205.
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IIR-PDOB
Frequency [rad/s]
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2.0
0
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5 10 20 250 15
(a) m=1.
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