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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−Nis 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

∂zz=−h=0, where his the

height of the powder bed.

Considering surface conduction, convection and radiation,

apply another boundary condition [26]:

−k∂T

∂zz=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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