Synthesis and Analysis of Multirate Repetitive Control for Fractional-order Periodic Disturbance Rejection in Powder Bed Fusion

Abstract

This paper studies control approaches to advance the quality of repetitive energy deposition in powder bed fusion (PBF) additive manufacturing. A key pattern in the nascent manufacturing process, the repetitive scanning of the laser or electron beam can be fundamentally improved by repetitive control (RC) algorithms. An intrinsic limitation, however, appears in discrete-time RC when the exogenous signal frequency cannot divide the sampling frequency. In other words, the disturbance period in the internal model is not an integer. Such a challenge hampers high-performance applications of RC to PBF because periodicity of the exogenous signal has no guarantees to comply with the sampling rate of molten-pool sensors. This paper develops a new multirate RC and a closed-loop analysis method to address such fractional-order RC cases by generating high-gain control signals exactly at the fundamental and harmonic frequencies. The proposed analysis method exhibits the detailed disturbance- attenuation properties of the multirate RC in a new design space. Numerical verification on a galvo scanner in laser PBF reveals fundamental benefits of the proposed multirate RC.

Full text

Proceedings of 2018 International Symposium on Flexible Automation
ISFA 2018
July 15-19, 2018, Kanazawa, Japan
[ISFA2018-L094]
SYNTHESIS AND ANALYSIS OF MULTIRATE REPETITIVE CONTROL FOR
FRACTIONAL-ORDER PERIODIC DISTURBANCE REJECTION IN POWDER BED
FUSION
Dan Wang and Xu Chen∗
Dept. of Mechanical Engineering
University of Connecticut
Storrs, CT, 06269, U.S.A.
Email: dan.wang@uconn.edu, xchen@uconn.edu
ABSTRACT
This paper studies control approaches to advance the qual-
ity of repetitive energy deposition in powder bed fusion (PBF)
additive manufacturing. A key pattern in the nascent manu-
facturing process, the repetitive scanning of the laser or elec-
tron beam can be fundamentally improved by repetitive control
(RC) algorithms. An intrinsic limitation, however, appears in
discrete-time RC when the exogenous signal frequency cannot
divide the sampling frequency. In other words, Nin the internal
model 1/1−z−Nis not an integer. Such a challenge hampers
high-performance applications of RC to PBF because periodic-
ity of the exogenous signal has no guarantees to comply with
the sampling rate of molten-pool sensors. This paper develops a
new multirate RC and a closed-loop analysis method to address
such fractional-order RC cases by generating high-gain control
signals exactly at the fundamental and harmonic frequencies.
The proposed analysis method exhibits the detailed disturbance-
attenuation properties of the multirate RC in a new design space.
Numerical verification on a galvo scanner in laser PBF reveals
fundamental benefits of the proposed multirate RC.
1 Introduction
Repetitive control (RC) [1] is a key feedback control method
for tracking/rejecting periodic exogenous signals. By learning
from previous iterations, RC can greatly enhance control perfor-
mance of the current iteration in a repetitive task space. This
∗Corresponding author
principal property has benefited various application domains, in-
cluding, for instance, tracking controls in robotic manipulators
[2], wafer scanners [3], and optical disk drives [4], as well as
regulation controls in unmanned aerial vehicles [5], power con-
verters [6], and wind turbines [7].
This paper studies RC in powder bed fusion (PBF) additive
manufacturing (AM) processes that apply laser or electron beams
to melt and join powder materials. In this AM family, thousands
of thin layers build up a typical workpiece. Within each layer,
the molten pool is controlled to follow trajectories predefined
by a “slicing” step. This process contains highly repetitive ther-
momechanical interactions [8]. As a result, periodic errors are
introduced by the beam-material interaction and path planning.
Indeed, other AM technologies [9] have validated and leveraged
such periodicity to enhance servo performances.
To fully release the capability of RC to fundamentally im-
prove the repetitive beam scanning in PBF, the internal model
principle [10] must be carefully configured in the control de-
sign. More specially, digital RC implements an internal model
1/(1−z−N), where zis the complex indeterminate in the z-
transform and N, the period of the disturbance/reference, equals
the sampling frequency (1/Tsor fs) divided by the fundamental
signal frequency ( f0). For Nbeing a non-integer, that is, f0can-
not divide fs, RC with the approximated Ncan no longer aim at
the fundamental and harmonic frequencies, resulting in degraded
servo performances.
Several strategies exist to potentially address such
fractional-order RC cases. [11] employs spatial RC to ob-

tain time-invariant disturbance periods in a spatial domain. [12]
and [13] propose adaptive RC to adjust the sampling rate to
get an integer N. [14] and [15] introduce high-order RC with
delay elements to widen the high-gain regions near the harmonic
frequencies. [16] presents a delay-varying RC that uses the repet-
itive variable to continuously adjust the time-varying delay. [6]
and [17] design different filters to approximate the fractional-
order delays. [18] uses a correction factor to correct the deviated
poles of the fractional-order repetitive controller. [19] introduces
two wide-band and quasi RCs together with a multirate RC in a
plug-in configuration; the full closed-loop properties, however,
are not investigated.
Despite the existing literature, it remains not well under-
stood how to create RC exactly at the harmonic frequencies in the
presence of fractional-order periods and how to systematically
analyze the closed-loop performances. To bridge this knowledge
gap, the proposed multirate RC algorithm introduces a second di-
visible sampling frequency f0
ssuch that N=f0
s/f0is an integer,
and embeds a new zero-phase low-pass filter design to address
multirate closed-loop robustness. Along the course of formulat-
ing the algorithm, an unexpected selective loop-shape modula-
tion is discovered in the multirate digital control design. This
fundamental behavior, prone to be neglected in the design phase,
inspires in the first instance a closed-loop analysis method that
exhibits the complete disturbance-attenuation properties of the
multirate RC. This analysis method also enables a new design
space for applying RC to general systems with the mismatched
sampling and task periodicity. This paper will discuss the per-
formance benefit and implementation guidance of the proposed
algorithm. A case study on a galvo scanner in laser PBF verifies
the theoretical analyses.
The remainder of this paper is structured as follows. Sec-
tion 2 reviews a conventional RC design. Two examples in Sec-
tion 3 elucidate the existence of fractional-order disturbances in
PBF. Section 4 builds the proposed multirate fractional-order RC
algorithm. Section 5 provides the numerical verification of the
algorithm. Section 6 concludes the paper.
2 Preliminaries of repetitive control
The proposed multirate RC algorithm is based on a plug-
in RC design in Fig. 1 [3]. The baseline feedback system here
consists of the plant P(z)and the baseline controller C(z)(Fig.
1 without the plug-in compensator). Common servo algorithms,
such as PID, H∞, and lead-lag compensation, can apply to the
baseline controller design. Throughout this paper, we assume
1) coefficients of all transfer functions are real; 2) the baseline
feedback loop including P(z)and C(z)is stable; and 3) P(z)and
C(z)are proper, linear, rational, and time-invariant.
Let mdenote the relative degree of P(z), whose nominal
model is ˆ
P(z). With the plug-in compensator, the overall con-
C(z)P(z)
Q(z)
z−mˆ
P−1(z)z−m
d(k)
+
r(k)+e(k)
+ud(k)+
yd(k)
−
+
w(k)
+
+
Plug-in compensator
FIGURE 1. Block diagram of a plug-in RC design.
troller from e(k)to ud(k)is
Call (z) = C(z) + z−mˆ
P−1(z)Q(z)
1−z−mQ(z).(1)
If Q= (1−αN)zm−N/(1−αNz−N), 1 −z−mQ(z) = (1−
z−N)/(1−αNz−N), where α∈[0,1)determines the attenua-
tion bandwidth of 1 −z−mQ(z). At the harmonic frequencies
(ωk=k2πf0Ts,k∈Z+, the set of positive integers), the mag-
nitude responses of 1 −z−mQ(z)are zero because 1 −e−jωkN=
1−e−jk2πf0Ts/(f0Ts)=1−e−jk2π=0. Hence, |Call (z)| → ∞
and Gd→yd(z) = P(z)[1−z−mQ(z)]/[1+P(z)C(z)] = 0 when
z=ejωk. At the intermediate frequencies, Q(ejω)≈0, and
|1−z−mQ(z)|z=ejω≈1 when αis close to 1; thus Call(z)≈C(z),
and the original loop shape is maintained. Choosing a smaller α
can yield a wider attenuation bandwidth, at the cost of deviating
from the baseline loop shape.
For robustness against high-frequency plant uncertain-
ties, Q(z)is additionally designed to contain zero-phase pairs
qj(z−1)qj(z)(j∈Z):
Q(z) = (1−αN)z−(N−m)
1−αNz−N
M
∏
j=0
qj(z−1)qj(z),(2)
where M∈Zis determined according to the design requirements.
For instance, qi(z)(i∈Z+) in the first line of (3) places four ze-
ros of Q(z)at e±jΩiT0
sto make its frequency response equal zero
at Ωi, and q0(z−1)q0(z)adds n0(∈Z)zero pairs at the Nyquist
frequency:
qi(z) = (1−2cos(ΩiTs)z+z2
2−2cos(ΩiTs),i∈Z+
(1+z)n0
2n0,i=0.(3)
Note the Q-filter in (2) and (3) is designed assuming an integer
Nunder the sampling time of Ts.
3 Fractional-order disturbances in PBF
This section introduces two examples regarding the fun-
damental applicability of fractional-order RC to PBF. The first
example, a numerical simulation, verifies the existence of the
fractional-order periodic disturbances in the PBF process. The
second example shows the intrinsic fractional-order disturbances
in the beam scanning mechanism used in laser PBF.

3.1 Example one: periodic thermal cycles in PBF
The PBF is built upon repeated scanning of high-energy
beam on a bed of powder feedstock. The scan trajectories deter-
mine the periodicity of the beam-material interactions (see, e.g.,
Fig. 2). Here, the laser or electron beam melts the powder mate-
rial following predefined tracks, and monitoring sensors, such as
cameras and imaging systems, are applied to obtain the molten
pool information. To get a uniform part quality, the molten pool
width is desired to be kept at a user-defined reference value [20].
To quantitatively demonstrate the periodic thermal cycles,
the COMSOL Multiphysics 5.3 software is used to stimulate a
proof-of-concept benchmark problem. The process parameters,
governing equation, initial condition, and boundary conditions
used in the simulation are listed in the Appendix. The physics-
controlled meshing method is used in the finite element model.
The time step Tsis 2ms, that is, the sampling frequency of the
camera is simulated to be fs=1/Ts=500Hz. Eight tracks are
sintered bidirectionally with transitions. The path planning of the
first five tracks is shown in the left plot of Fig. 2. The right plot of
Fig. 2 illustrates the simulation result of the surface temperature
distribution of the powder bed at t=0.834s.
...
4
5
Track No.
Scan
direction
3
2
Unfused
powder
1
x
y
FIGURE 2. Schematic of path planning and thermal simulation result
at t=0.834s.
After a short transient, the average molten pool width
reaches a steady state as a result of balanced heat influx and dif-
fusion. The molten pool width varies over time and fluctuates
around the average value (0.25 mm in the top plot of Fig. 3). In
the bidirectional scanning (Fig. 2), when the energy source ap-
proaches the end of one track, the large latent heat does not have
enough time to dissipate out before the next track starts. This ac-
cumulated heat effect results in a higher initial temperature at the
beginning of the track to be sintered. Therefore, the molten pool
width, directly associated with the initial temperature, generates
a periodic increasing spike at the beginning of each track (the top
plot of Fig. 3) when the input heat flux keeps constant. Those
undesired increasing spikes in the time domain form a periodic
disturbance with a repetitive spectrum in the frequency domain
(the bottom plot of Fig. 3). The fundamental frequency f0of the
disturbance is defined by the time taken to scan one single track
t0:f0=1/t0=v/L, where vis the scan speed and Lis the track
length. In this example, f0=100/10 =10 Hz, and frequency
spikes at {n f0}(n∈Z+) appear in the fast Fourier transform
(FFT) of the disturbance.
In this sample simulation, period N(= fs/f0=50)is an inte-
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
Time (s)
2
3
4
Molten pool width (m)
10-4
0 50 100 150 200 250
Frequency (Hz)
0
1
2
3
4
Spectrum amplitude (dB)
10-5
Track 5
Track 4 Track 6 Track 7 Track 8
X: 29.53
X: 19.69
X: 80.71
X: 70.87
X: 9.843
FIGURE 3. Simulated example molten pool width in the time domain
and the disturbance in the frequency domain.
ger because v/Ldivides 1/Ts. However, the scan speed vand the
track length Lare tailored to the required energy density but not
the speed of the monitoring sensors (which is restricted for cam-
eras and general integrated imaging systems). For instance, if
Ts=3ms, N=100/3 will become a non-integer. Therefore, the
disturbance periodicity—defined by the scan speed, part geome-
try, and path planning—has no guarantees to be an integer mul-
tiple of the sampling time of the molten pool sensors. It is also
important to recognize that besides the proof-of-concept bidirec-
tional trajectory, other scanning patterns yield repetitive distur-
bance components in a similar fashion (see, e.g., experimental
results in [21]). These fractional-order disturbances challenge
conventional RC and demand new theoretical designs for RC to
maximize performance in PBF.
3.2 Example two: collaborative control in galvo scan-
ner
As a key component in laser PBF, the dual-axis galvo scan-
ner (Fig. 4) consists of two sets of motors, mirrors, and control
systems, here referred to as the X channel and the Y channel, re-
spectively. The two rotating mirrors reflect the input laser beam
to follow a scanning trajectory at high speed with high precision.
Encoders, mounted coaxially with the motor shaft, measure the
mirror rotation angles.
Power supply
Monitor
Servo driver x
Servo driver y
ADC
ADC
DAC
DAC
Laser source
Galvo scanne r
White scr een
Server with
dSPACE and
Matlab
FIGURE 4. Schematic of the hardware platform.

In practice, periodic disturbances appear in the dual-axis
galvo sets. First, we examine one single channel (e.g., Y chan-
nel) with a simple harmonic signal Asin(2πf0t+φ). Frequency
spikes at odd multiples of f0, instead of a single spike at f0, show
up in the FFT of the channel output (Fig. 5). This is because
signal conditioning boards in the servo driver limit the rate of
change in the output signal when the slope of the input signal
is faster than the predefined slew rate [22]. The slewed output
waveform is thus not a pure sine waveform and results in har-
monics at odd multiples of the fundamental frequency.
0 2 4 6 8 10 12 14 16 18 20
Frequency (kHz)
0
1
2
3
4
Spectrum amplitude
3 4 5 6 7 8 9 10 11
0
0.1
0.2
0.3
X: 3
Y: 0.2995 X: 5
Y: 0.06968 X: 7
Y: 0.02306 X: 9
Y: 0.007625
FIGURE 5. (Experimental result) FFT of the Y output with a simple
harmonic input.
Second, the collaborative control of the two channels also
introduces periodic disturbances. The mechanical motion of one
rotating mirror can transmit to the other mirror as disturbances.
High currents in the ground lines of the two servo drivers can
also cause the channels to crosstalk [23]. When one channel
is actuated with a simple harmonic signal at f0, the FFT of the
non-actuated channel output was observed to contain a frequency
spike at f0caused by mechanical vibrations and frequency spikes
at 2n f0(n∈Z+) due to crosstalk. The crosstalk is more obvious
with increased amplitudes and frequencies of the input signals.
For both single- and cross-channel disturbances, the dis-
turbance frequencies vary with the input signal frequencies
and are not guaranteed to divide the sampling frequency of
the galvo scanner. For instance, when Ts=1/16ms, conven-
tional RC fails in eliminating the crosstalk-induced harmonics at
{1200iHz}(i∈Z+)since N=16000/1200 in the internal model
is not an integer. Without loss of generality, in this paper, the pro-
posed multirate RC algorithm is evaluated on the dual-axis galvo
scanner as a case study to reduce the crosstalk.
4 Proposed multirate fractional-order RC algorithm
The new multirate RC is proposed to tackle a non-integer
Nin the internal model. For concreteness, we will use the col-
laborative control example in Sections 3.2 and 5 throughout the
discussions and generalize the algorithm along the course of de-
sign and analysis.
The proposed multirate RC addresses the fractional-order
period by introducing a second divisible sampling frequency f0
s
that equals the least common multiple (LCM) of the sampling
and fundamental frequencies, namely, f0
s=LCM(fs,f0). With-
out changing the sampling frequency of the plant, we design the
repetitive controller under the newly introduced sampling fre-
quency. Since N=f0
s/f0is now an integer, the multirate repeti-
tive controller can thus generate high-gain control signals exactly
at the fundamental frequency and its harmonics.
Cdh(z)↓FP(z)↑F
Q(z)
z−mˆ
P−1(z)z−m
d(k)
+
r(k)+e
+
udh(k)ud
+
yd
ydh(k)
−
+
w(k)
+
+
Call(z)
FIGURE 6. Block diagram for multirate RC.
More specially, the proposed multirate RC (Fig. 6) adds the
upsampler and downsampler into Fig. 1 before and after the over-
all controller. In Fig. 6, the solid and dashed lines stand for the
slow and fast signals sampled by Tsand T0
s(,1/f0
s), respectively.
T0
s=Ts/F(F>1 and F∈Z+). Note that the transfer func-
tions inside the Call(z)block are all implemented at T0
s. Base on
multirate signal processing [19], the frequency response of the
open-loop transfer function from ydto the summing junction be-
fore P(z)is ˜
C(ejΩTs) = 1
F∑F−1
k=0Call (ej(ΩT0
s−2πk
F)). Thus, when the
reference r(k)is zero (i.e., in regulation problems), the Fourier
transform of the plant output yd(k)is
Yd(ejΩTs) = P(ejΩTs)D(ejΩTs)
1+1
FP(ejΩTs)∑F−1
k=0Call (ej(ΩT0
s−2πk
F))
=P(ejΩTs)D(ejΩTs)
1+P(ejΩTs)˜
C(ejΩTs).(4)
Before discussing the detailed full multirate closed-loop
properties, we provide a conceptual example and an overall
disturbance-attenuation principle. Consider Ts=1/16ms and
f0=1200Hz. Multirate RC gives T0
s=1/LCM(16000,1200) =
1/48ms. The plug-in compensator is designed under T0
ssuch
that small gains of 1 −z−mQ(z)are generated at Ω0=2πn×
1200rad/s (n∈Z+) (Fig. 7). Since the Q-filter design in Section
2 yields Call (ejΩ0T0
s)→∞,˜
C(ejΩTs)in the summation form of
Call also goes to infinity. Thus, in (4), Yd(ejΩ0Ts)→0, yielding
yd(kTs) = 0 at Ω0.
4.1 Multirate closed-loop analysis
In Fig. 6, the transfer function from the disturbance d(k)
to the output yd(k)equals S(z) = S0(z)P(z), where S0(z)is the
closed-loop sensitivity function with S0(ejΩTs) = 1/G(ejΩTs)and
G(ejΩTs) = 1+1
FP(ejΩTs)
F−1
∑
k=0
Call (ej(ΩT0
s−2πk
F)).(5)
To reject disturbances at Ω0, when the plant dynamics is
fixed, |S0(ejΩ0Ts)|in the multirate RC is desired to be small at
Ω0, that is, |G(ejΩ0Ts)| → ∞. With the direct Q-filter design un-
der the sampling time of T0
s(Fig. 7), |S0(ejΩTs)|has the desired

100101102103104
Frequency (Hz)
-80
-60
-40
-20
0
Magnitude (dB)
Q(z)
-30
-20
-10
0
Magnitude (dB)
1-z-mQ(z)
3600
4800
2400
1200
FIGURE 7. Magnitude responses of 1 −z−mQ(z)and Q(z)in multi-
rate RC.
small gains at the target frequencies, as discussed in the para-
graph after (4). However, small spikes also appear in |S0(ejΩTs)|,
that is, decreasing notches show up in |G(ejΩTs)|(Fig. 8). The
undesired selective small gains imply potential amplification of
other error sources. The complete disturbance-attenuation prop-
erties of the proposed multirate RC will be deciphered next to
assist in eliminating those error amplifications.
100101102103
-60
-40
-20
0
20
Magnitude (dB)
S0(z)
100101102103
Frequency (Hz)
-20
0
20
40
60
Magnitude (dB)
G(z)
Multirate RC
Baseline control
undesired selective small gains
desired high gains
FIGURE 8. Frequency responses of S0(z)and G(z)in Multirate RC
with Ts=1/16ms, T0
s=1/48ms, F=3, and f0=1200 Hz.
Note that G(ejΩTs)in (5) contains hybrid frequency re-
sponses of P(z)under the sampling time of Tsand Call (z)under
T0
s, and the frequency index satisfies the periodicity property:
ej(ΩT0
s−2πk
F)=ej(Ω−2πk
Ts)T0
s=ej2π(f−k
Ts)T0
s,(6)
where f=Ω/(2π)is in Hz. Take the previous example (Ts=
1/16ms, T0
s=1/48ms, and F=Ts/T0
s=3). Then
G(ej2πf Ts) = 1+1
3P(ej2πf Ts)
2
∑
k=0
Call (ej2π(f−k
Ts)T0
s).(7)
Let Gk(ej2πf T 0
s) = 1+P(ej2πf Ts)Cal l (ej2π(f−k
Ts)T0
s). Then
(7) is decomposed to
G(ej2πf Ts) = 1
3hG0(ej2πf T 0
s) + G1(ej2πf T 0
s) + G2(ej2πf T 0
s)i.(8)
With ej2πf Ts=ej2π(f−k
Ts)Ts, the relationship between G0
and G1is G1(ej2πf T 0
s) = G0(ej2π(f−1
Ts)T0
s), and similarly
G2(ej2πf T 0
s) = G0(ej2π(f+1
Ts)T0
s).
G1(ej2πf T 0
s)and G2(ej2πf T 0
s)are thus shifted versions of
G0(ej2πf T 0
s). Note S0(ej2πf Ts)and G(ej2πf Ts)in (5) are evalu-
ated from 0 to the slower Nyquist frequency corresponding to
Ts, namely, f∈[0,8 kHz]in this example. G1(ej2πf T 0
s)at f∈
[0,8]kHz thus maps to G0(ej2πf T 0
s)at f∈[−16,−8]kHz, which
is symmetric to G0(ej2πf T 0
s)at f∈[8,16]kHz with respect to
the line f=0. Similarly, G2(ej2πf T 0
s)with f∈[0,8]kHz maps
to G0(ej2πf T 0
s)with f∈[16,24]kHz. Therefore, G0(ej2πf T 0
s)
evaluated at f∈[0,24]kHz (0 to 0.5/T0
s, the faster Nyquist fre-
quency corresponding to T0
s) includes all the desired informa-
tion of G0(ej2πf T 0
s),G1(ej2πf T 0
s), and G2(ej2πf T 0
s)under f∈
[0,8kHz], as shown in Fig. 9.
It can now be understood that because G(ej2πf Ts)in (8) is
the average of G0(ej2πfT 0
s),G1(ej2πf T 0
s), and G2(ej2πf T 0
s), the
undesired small gains of G(ej2πf Ts)in the bottom plot of Fig. 8
are inherited from G1(ej2πf T 0
s)and G2(ej2πf T 0
s)or, equivalently,
from G0(ej2πf T 0
s)at frequencies larger than 8kHz (Fig. 9).
2000 4000 6000 8000
-20
0
20
40
60
Magnitudes (dB)
G0 with f [0, 8] kHz
2000 4000 6000 8000
-20
0
20
40
60 G1 with f [0, 8] kHz
2000 4000 6000 8000
-20
0
20
40
60 G2 with f [0, 8] kHz
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
Frequency (Hz) 104
-20
0
20
40
60
Magnitude of G0 (dB)
Symmetric
FIGURE 9. The relationships between G0(ej2πf T 0
s),G1(ej2πf T 0
s), and
G2(ej2πf T 0
s).
100101102103104
Frequency (Hz)
-100
-50
0
50
Magnitude (dB)
FIGURE 10. Magnitude responses of P(ejΩTs)and ˆ
P(ejΩT0
s).
4.2 Implicit model mismatch
We show in this subsection that the undesired magnitude
characteristics of G0(ej2πf T 0
s)at high frequencies arise from an

implicit model mismatch. Recall that
G0(ej2πf T 0
s) = 1+P(ej2πf Ts)Cal l (ej2πf T 0
s).(9)
Substituting the frequency response of (1) into (9) gives
G0(ejΩT0
s) = [P(ejΩTs)ˆ
P−1(ejΩT0
s)−1]e−jmΩT0
sQ(ejΩT0
s)
1−e−jmΩT0
sQ(ejΩT0
s)+1+P(ejΩTs)Cdh(ejΩT0
s)
1−e−jmΩT0
sQ(ejΩT0
s).
(10)
Fig. 10 presents the frequency responses of P(ejΩTs)and
ˆ
P(ejΩT0
s).At low frequencies, P(ejΩTs)≈P(ejΩT0
s)≈ˆ
P(ejΩT0
s),
and (10) reduces to
G0(ejΩT0
s) = 1+P(ejΩTs)Cdh (ejΩT0
s)
1−e−jmΩT0
sQ(ejΩT0
s).(11)
G0(ejΩT0
s)thus generates high gains where the magnitude
responses of the denominator 1 −z−mQ(z)are designed to be
small (Fig. 7). At high frequencies, intrinsic model mis-
matches exist between P(ejΩTs)and ˆ
P(ejΩT0
s)due to different
sampling frequencies. Even though the magnitude response of
1−e−jmωQ(ejω)ω=ΩT0
sis small, the first term of (10) must
be carefully considered. To eliminate the undesired magni-
tude shapes, Q(ejΩT0
s)should be designed small enough at high
frequencies to reduce the effect of the model mismatches in
[P(ejΩTs)ˆ
P−1(ejΩT0
s)−1]e−jmΩT0
sQ(ejΩT0
s).
102103104
-400
-300
-200
-100
0
Magnitude (dB)
Q(z)
The second Q-filter design
The original Q-filter design
100101102103
Frequency (Hz)
-40
-20
0
Magnitude (dB)
S0(z)
Multirate RC
Baseline control
FIGURE 11. Magnitude responses of the second Q-filter design.
Compared with the original Q-filter design used in Fig. 8,
the multirate RC thus demands an enhanced design with re-
duced Q(ejΩT0
s)at high frequencies (the top plot in Fig. 11).
This second Q-filter is designed with α=0.999, n0=2, Ω1=
2π×(12kHz), and Ω2=2π×(22 kHz)(see Section 2). As a
result, in the multirate RC using the second Q-filter design, the
undesired selective small gains of |G0(ej2πfT 0
s)|disappear (the
bottom plot of Fig. 12), which yields a clear magnitude response
of the closed-loop sensitivity function with no visible error am-
plifications, as shown in the bottom plot in Fig. 11.
For more general cases of the multirate RC, the analysis
steps are:
2000 4000 6000 8000
-20
0
20
40
60
Magnitudes (dB)
G0 with f [0, 8] kHz
2000 4000 6000 8000
-20
0
20
40
60 G1 with f [0, 8] kHz
2000 4000 6000 8000
-20
0
20
40
60 G2 with f [0, 8] kHz
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
Frequency (Hz) 104
-20
0
20
40
60
Magnitude of G0 (dB)
Symmetric
FIGURE 12.G0(ej2πf T 0
s),G1(ej2πf T 0
s), and G2(ej2πf T 0
s)of the sec-
ond Q-filter design.
1. Given fsand f0, identify the second sampling frequency
f0
s=LCM(fs,f0)for multirate RC design. Let F=Ts/T0
s=
f0
s/fs.
2. Design the repetitive controller in (1) under the deviated
sampling frequency to get desired disturbance-attenuation
properties.
3. Calculate and plot the closed-loop sensitivity function
S0(ejΩTs)and G(ejΩTs)in (9) to check if undesired selective
small gains show up.
4. Look into Gk(ej2πf T 0
s)(k=0,1,2,·· · ,F−1) with f∈
[0,fs/2]to disentangle G(ejΩTs)in the summation form.
Since all Gk(ej2πf T 0
s)’s map into G0(ej2πfT 0
s), it suffices to
analyze G0(ej2πf T 0
s)under f∈[0,f0
s/2]to identify the fre-
quencies of the undesired notches.
5. Redesign the Q-filter in the repetitive controller, and repeat
steps 2–4 to reduce the undesired selective small gains until
the design requirements are satisfied.
5 Numerical verification in a dual-axis galvo scanner
This section provides the implementation guidance and nu-
merical verification of the theoretical analyses. As a case study,
we employ the multirate RC algorithm to reduce the crosstalk in
the collaborative control of the galvo scanner (see Section 3.2).
In the following Q-filter designs, n0in (3) is chosen to be 3, and
Min (2) equals zero.
In the X channel, the conventional RC in Section 2 and the
multirate RC in Section 4 are implemented on top of the base-
line controller to attenuate crosstalk-induced disturbances with
fractional-order periods. The same periodic disturbances with
five frequency components are introduced into the X-channel
loop (Figs. 1 and 6): d(k) = A∑5
n=1sin(2πn f0Tsk)with A=
4mV (corresponding to 3◦of the Y-channel mirror rotation) and
f0=1200Hz.

-100
-50
0
50
Magnitude (dB)
Measured system
Identified system
102103104
Frequency (Hz)
-200
-100
0
100
200
Phase (deg)
FIGURE 13. Bode plot of P0(z)sampled at T0
s.
The identified plant model (Fig. 13) with T0
s=1/48ms is
P0(z) = 0.061z2+0.103z+0.061
z5−1.485z4+1.032z3−0.433z2−0.057z−0.061 .(12)
The stable plant model under Ts=1/16ms is:
P(z) = 0.061z4+0.737z3+0.351z2+0.034z+0.0001
z5+0.144z4−0.773z3−0.359z2−0.034z−0.0001 .(13)
A factory-set PID-type controller is already embedded in the
plant models. Thus, we design the baseline feedback loop (Fig.
1 without the plug-in compensator) under Ts=1/16 ms by ap-
plying P(z)in (13) and C(z) = 1. Such a design provides a band-
width of 4400Hz in the complementary sensitivity function T(z).
However, because the PID controller is generic and not tailored
to the repetitive disturbance, the frequency-domain result in Fig.
14 shows that the baseline controller barely attenuates the fre-
quency spike at 1200Hz and provides limited attenuation to the
other four spikes.
0 1000 2000 3000 4000 5000 6000 7000 8000
0
2
4
Spectrum amplitude
10-3
Baseline control 3 = 0.014742
0 1000 2000 3000 4000 5000 6000 7000 8000
0
2
410-3
Conventional RC 3 = 0.0095682
0 1000 2000 3000 4000 5000 6000 7000 8000
Frequency(Hz)
0
2
410-3
Multirate RC 3 = 0.0053453
35%
64%
FIGURE 14. FFT of plant output sampled at Ts.
The conventional RC algorithm is also implemented at Ts=
1/16ms, i.e. fs=16 kHz. N=fs/f0=16000/1200 ≈13. The
relative degree of P(z)in (13), namely m, is 1. The Q-filter tar-
gets frequency spikes at fs/N=1230.77 Hz and its integer mul-
tiples. A wider attenuation width is demanded in 1 −z−mQ(z)to
cover the adjacent harmonics at 1200Hz, 2400 Hz, 3600 Hz, etc.
To achieve this goal, αin (2) is set as 0.8.
1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 1.0045 1.005
Time(s)
-0.01
-0.005
0
0.005
0.01
yd(t)(V)
Baseline control
Conventional RC
Multirate RC
FIGURE 15. Plant outputs under baseline control, conventional RC,
and multirate RC.
In the proposed multirate RC, the plug-in compensator is
designed at T0
s=1/48 ms ( f0
s=48kHz) with F=Ts/T0
s=3. The
relative degree of P0(z)in (12) is 3 (m=3). αis chosen to be
0.999 to reduce the waterbed effect. N=f0
s/f0=48000/1200 =
40. In Fig. 7, small gains of 1 −z−mQ(z)are generated exactly
at 1200Hz and its integer multiples.
The numerical results of the three control systems are com-
pared in Fig. 14. As a performance metric, the 3σvalue of the
time-domain result in each control system is provided, where σ
represents the standard deviation. The performance gains of the
conventional and multirate RCs are 35% and 64%, respectively.
The output signals yd(t)in Fig. 15 show the clear performance
difference of multirate RC>conventional RC>baseline control.
Indeed, frequency-domain analysis reveals that the conventional
RC method can reduce the first two but not other high-frequency
spikes above 3600Hz. The multirate RC algorithm, on the other
hand, can effectively attenuate the periodic frequency spikes by
generating frequency notches in 1 −z−mQ(z)and enhanced con-
trol efforts exactly at those frequencies.
6 Conclusion
This paper proposes a new multirate repetitive control (RC)
algorithm to overcome the intrinsic limitation of the RC inter-
nal model 1/(1−z−N)under fractional-order situations. Illus-
trative examples in this paper demonstrate the applicability of
the fractional-order RC to powder bed fusion additive manufac-
turing. To apply repetitive error rejection when the fundamental
disturbance frequency does not divide the sampling frequency,
the multirate RC applies a new fast sampling frequency that al-
lows for exact attenuation at the desired periodic frequencies.
A new closed-loop analysis method reveals the full disturbance-
attenuation properties of the multirate RC. Numerical results ver-
ify that the multirate RC outperforms the conventional RC by
providing more systematic, precise servo enhancement, particu-
larly at high frequencies.

Acknowledgment
This research is supported in part by NSF Award 1750027
and by an award from UTC Institute of Advanced Systems Engi-
neering.
Appendix
Finite element model of PBF
The governing equation for heat flow in PBF is
ρcp
dT (x,y,z,t)
dt =∇·(k∇T(x,y,z,t)) + qs,(14)
where ρis the density of the material being sintered, cpis the
specific heat, Tis the temperature, tis the time, kis the thermal
conductivity, and qsis the rate of local internal energy generated
per unit volume [24]. ρ,cp, and kare assumed to be temperature-
dependent (see Table 5.3 in [25]).
The initial condition is specified as T(x,y,z,0) = T0, where
T0is the initial temperature. The timescale of sintering one layer
is orders of magnitude faster than the heat transfer in the building
(z) direction. The bottom is thus assumed to have no heat loss,
and one boundary condition is −k∂T
∂zz=−h=0, where his the
height of the powder bed.
Considering surface conduction, convection and radiation,
apply another boundary condition [26]:
−k∂T
∂zz=0
=−Q+hc(T−Ta) + εσ(T4−T4
a),(15)
where εis the powder bed emissivity, σis the Stefan-Boltzmann
constant, hcis the convection heat transfer coefficient, and Tais
the ambient temperature. Qis the input heat flux (assuming a
Gaussian laser beam profile): Q≈2Pe−2r2
R2/(πR2), where Pis
the laser power, Rdenotes the effective laser beam radius, and r
is the radial distance from the center of the laser spot.
Parameters for numerical simulation of PBF
Dimensions of powder bed 10 mm ×10 mm ×100 µm
Powder material Ti–6Al–4V
Laser power 30 W
Scan speed 100 mm/s
Laser spot diameter 35 µm
Emissivity 0.35
Ambient and initial temperature 23 °C
Convection heat transfer coefficient 12.7 W/(m2·K)
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