Control-Oriented Modeling and Repetitive Control in In-Layer and Cross-Layer Thermal Interactions in Selective Laser Sintering

Abstract

Although laser-based additive manufacturing (AM) has enabled unprecedented fabrication of complex parts directly from digital models, broader adoption of the technology remains challenged by insufficient reliability and in-process variations. In pursuit of assuring quality in the selective laser sintering (SLS) AM, this paper builds a modeling and control framework of the key thermodynamic interactions between the laser source and the materials to be processed. First, we develop a three-dimensional finite element simulation to understand the important features of the melt-pool evolution for designing sensing and feedback algorithms. We explore how the temperature field is affected by hatch spacing and thermal properties that are temperature-dependent. Based on high-performance computer simulation and experimentation, we then validate the existence and effect of periodic disturbances induced by the repetitive in- and cross-layer thermomechanical interactions. From there, we identify the system model from the laser power to the melt pool width and build a repetitive control algorithm to greatly attenuate variations of the melt pool geometry.

Full text

Control-oriented Modeling and Repetitive Control
in In-layer and Cross-layer Thermal Interactions
in Selective Laser Sintering
Dan Wang
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: daw1230@uw.edu
Tianyu Jiang
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: tjiang19@uw.edu
Xu Chen∗
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: chx@uw.edu
Although laser-based additive manufacturing (AM) has en-
abled unprecedented fabrication of complex parts directly
from digital models, broader adoption of the technology re-
mains challenged by insufficient reliability and in-process
variations. In pursuit of assuring quality in the selective
laser sintering (SLS) AM, this paper builds a modeling and
control framework of the key thermodynamic interactions be-
tween the laser source and the materials to be processed.
First, we develop a three-dimensional finite element simu-
lation to understand the important features of the melt-pool
evolution for designing sensing and feedback algorithms. We
explore how the temperature field is affected by hatch spac-
ing and thermal properties that are temperature-dependent.
Based on high-performance computer simulation and exper-
imentation, we then validate the existence and effect of pe-
riodic disturbances induced by the repetitive in- and cross-
layer thermomechanical interactions. From there, we iden-
tify the system model from the laser power to the melt pool
width and build a repetitive control algorithm to greatly at-
tenuate variations of the melt pool geometry.
1 Introduction
Different from conventional subtractive machining, ad-
ditive manufacturing (AM, also called 3D printing) builds
up a part from its digital model by adding together materi-
als layer by layer. This paper studies laser-based AM tech-
nologies, with a focus on the selective laser sintering (SLS)
subcategory. This AM technology applies laser beams as the
energy source to melt and join powder materials. A typi-
cal workpiece is built from many thousands of thin layers.
Within each layer, the laser beam is controlled to follow tra-
jectories predefined by the part geometry in a slicing process.
After the sintering of one layer is finished, a new thin layer
of powder is spread on top, and then another cycle begins.
SLS accommodates a broad range of materials (e.g., met-
als, polymers, and ceramics) and can build customized parts
with complex features and high accuracy requirements. De-
spite the advantages and continuously emerging applications,
∗Corresponding author
broader adoption of the technology remains challenged by
insufficient reliability and in-process variations. These vari-
ations are induced by, for example, environmental vibrations,
powder recycling, imperfect laser-material interactions, and
mechanical wears [1–3]. Predictive modeling and process
control have thus been key for mitigating the variations and
enhancing the energy deposition in SLS.
Several existing strategies employ numerical and
control-oriented modeling to understand SLS and other
laser-based AM processes such as laser metal deposition.
In numerical modeling, most researchers adopt finite ele-
ment analysis (FEA) to investigate thermal fields of the pow-
der bed and substrate, melt pool geometries, and mechan-
ical properties of the printed parts in response of various
scanning patterns, scan speeds, number of lasers, and over-
hanging structures [4–6]. In control-oriented modeling, cur-
rent researches often implement low-order system models
obtained from system identification techniques, taking laser
power or scan speed as the input and melt pool temperature
or geometry as the output [2, 7–9]. Furthermore, [8, 10] con-
nect a nonlinear memoryless submodel in series with the lin-
ear system model to account for nonlinearities. [9] builds a
spatial-domain Hammerstein model to identify the coupled
repetitive in- and cross-layer dynamics. The Rosenthal equa-
tions give the analytical solutions for a moving laser source
in thick and thin plates and have been used to predict the
temperature distribution of the powder bed [11–14].
Based on the reduced-order models, existing researches
[2, 15, 16] apply PID control to regulate the process parame-
ters and reduce the in-process errors. From there, [17] adds
a feedforward path for tracking improvement. Other con-
trollers have also been shown capable in improving the di-
mensional accuracy of the printed parts, including but not
limited to the sliding mode controller [10], predictive con-
troller [7], and iterative learning controller [18]. Note that
except for [2], which was developed for SLS, all the other
reviewed controllers were tailored for laser metal deposition.
Stepping beyond current architectures, this study builds
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C(z)P(z)
Q(z)
z−mˆ
P−1(z)z−m
d(k)
+
r(k)+e(k)
+u(k)+
y(k)
−
+
w(k)
+
+
Plug-in compensator
Figure 1: Block diagram of a plug-in RC design.
a modeling and control framework that accounts for FEA,
control-oriented modeling, and advanced control strategies
in a unified scheme. Focusing on the root cause and charac-
teristics of imperfect laser-material interactions, we first de-
velop a three-dimensional finite element simulation to char-
acterize the melt-pool evolution for designing sensing and
feedback algorithms. We explore how the temperature field
is affected by hatch spacing and thermal properties that are
temperature-dependent. Along the course of framing the
FEA and experimenting on an in-house built SLS testbed,
we identify highly periodic melt-pool-width variations rising
from the repetitive in- and cross-layer thermomechanical in-
teractions. Then, we introduce repetitive control (RC) to at-
tenuate these periodic variations that challenge conventional
PID control. After identifying the system model from the
laser power to the melt pool width, we build a plug-in RC
algorithm and validate that its disturbance-attenuation per-
formance is superior to that of PID control.
The remainder of this paper is structured as follows.
Section 2 reviews a plug-in RC design. Section 3 builds and
refines a numerical model of the SLS process. Section 4 veri-
fies the existence of the periodic disturbances induced by the
repetitive in- and cross-layer thermomechanical interactions.
An RC algorithm is built in Section 5 to attenuate these peri-
odic disturbances. Section 6 concludes the paper.
2 Preliminaries of Repetitive Control
RC is designed to track/reject periodic exogenous refer-
ences/disturbances in applications with repetitive tasks [19].
By learning from previous iterations, RC can extensively
enhance current control performance in the structured task
space. In digital RC, an internal model 1/(1−z−N)is in-
corporated in the controller, where zis the complex indeter-
minate in the z-transform. Nis the period of the signal and
equals the sampling frequency (denoted in this paper as 1/Ts
or fs) divided by the fundamental disturbance frequency ( f0).
Consider a baseline feedback system composed of the
plant P(z)and the baseline controller C(z)(Fig. 1 with the
dotted box removed). C(z)can be designed by common
servo algorithms, such as PID, H∞, and lead-lag compensa-
tion. The signals r(k),e(k),d(k), and y(k)represent, re-
spectively, the reference, the tracking error, the input dis-
turbance, and the system output. The sensitivity function
S(z)=1
1+P(z)C(z)is the transfer function from d(k)to y(k).
This section introduces a plug-in RC design [20] that uti-
lizes the internal signals e(k)and u(k)to generate a compen-
sation signal w(k)(Fig. 1). Let mdenote the relative degree
of ˆ
P(z), the nominal model of P(z). The transfer function of
the overall controller from e(k)to u(k)is
Call (z)=C(z)+z−mˆ
P−1(z)Q(z)
1−z−mQ(z).(1)
-40
-20
0
Magnitude (dB)
1-z-mQ(z)
100101102103
Frequency (Hz)
-100
-50
0
Q(z)
=0.999
=0.99
=0.9
Figure 2: Magnitude responses of 1−z−mQ(z)and Q(z)with dif-
ferent values of αand n0=1in the example of Section 5.
-60
-40
-20
0
Magnitude (dB)
1-z-mQ(z)
100101102103
-100
-50
0
Q(z) n0=0 n0=1 n0=2
Figure 3: Magnitude responses of 1−z−mQ(z)and Q(z)with dif-
ferent n0and α=0.99 in the example of Section 5.
The internal model is integrated in Call if the Qfil-
ter is designed as Q(z)=(1−αN)zm−N/(1−αNz−N), that
is, 1−z−mQ(z)=(1−z−N)/(1−αNz−N), where α∈ [0,1)
is a tuning factor that determines the attenuation width of
1−z−mQ(z). Then at the harmonic frequencies ωk=k2πf0Ts
(k∈Z+, the set of positive integers), the magnitude re-
sponses of 1−z−mQ(z)are zero because 1−e−jωkN=1−
e−jk 2πf0Ts/( f0Ts)=1−e−jk 2π=0. Hence, we have |Call (z)| →
∞and the new sensitivity function S0(z)=1
1+P(z)Cal l (z)≈
1−z−mQ(z)
1+P(z)C(z)=0with z=ejωk. At the intermediate frequen-
cies ω,k2πf0Ts,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. A smaller αcan yield a wider
attenuation width at the cost of deviating from the baseline
loop shape, as shown in Fig. 2.
During implementation, zero-phase pairs q0(z−1)q0(z)
are multiplied into Q(z)for robustness against plant uncer-
tainties at high-frequency regions:
Q(z)=(1−αN)z(m−N)
1−αNz−Nq0(z−1)q0(z),(2)
where q0(z)=(1+z)n0/2n0and n0∈Z+is the number of the
zero-phase pairs (Fig. 3).
3 Numerical Modeling in SLS
We use the COMSOL Multiphysics 5.3a software to
build and refine a finite element model for simulating the in-
and cross-layer thermal cycles in SLS. The simulation con-
siders surface convection, surface radiation, and conduction,
while the effects of fluid flow and Marangoni force are ne-
glected. We have also evaluated the effect of phase change,
which is out of the scope of this paper and is omitted here.
3.1 FEA
The governing equation for the conduction heat flow in
SLS is ρcpdT (x,y,z,t)
dt =∇ · (k∇T(x,y,z,t)) +qs, where kis
the thermal conductivity, cpthe specific heat, ρthe effective
density, tthe time, Tthe temperature, and qsthe rate of local
internal energy generated per unit volume [11]. T(x,y,z,t)
is abbreviated to Tin the following derivation. The process
parameters used in the simulation are listed in Table 1 unless
otherwise specified. Specially, the thermal properties k,cp,
and ρare temperature-dependent (Fig. 4).
The initial condition is T(x,y,z,0)=T0, where T0is the
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500 1000 1500 2000 2500
Temperature (K)
0
50
k (W/m.K)
500 1000 1500 2000 2500
Temperature (K)
3500
4000
4500
(kg/m3)
500 1000 1500 2000 2500
Temperature (K)
0.5
1
1.5
cp (J/g.K)
Powder
Solid
Liquid
Figure 4: Temperature-dependent thermal properties of Ti6Al4V [21]. The vertical dotted line indicates the melting point.
Table 1: Parameters for numerical simulation.
Parameters Value Parameters Value
Powder bed size 15 mm ×15 mm ×50 µm Material Ti6Al4V
Substrate size 15 mm ×15 mm ×5mm Track length L5mm
R220 µm/2 Time step Ts0.5ms
Absorptance 0.25 Emissivity 0.35
Tso l 1873 K Scan speed ux100 mm/s
Lf295 kJ/kg Laser power P60 W
T0/Tm293.15K/1923 .15 Kφ00.4
hc12.7W/(m2·K) k,cp, and ρsee Fig. 4
X
Z
Y
Powder be d: Free tr iangular
(72.6 µm) and Swept (50µm)
Powder be d: Free tr iangular
(2mm) and Swept (50 µm)
Substrate: Free
tetrahedral (3.5 mm)
…
Laser tracks
Figure 5: Selective meshing and in-layer laser sintering.
initial temperature. Since it is assumed that the bottom has
no heat loss, one boundary condition is −k∂T
∂z

z=h=0, where
hindicates the location of the bottom surface of the sub-
strate. Considering surface conduction, convection and ra-
diation, the other boundary condition is −k∂T
∂z

z=0=−Q+
hc(T−Te)+εσB(T4−T4
e), where hcis the convection heat
transfer coefficient, Tethe ambient temperature, εthe emis-
sivity, σBthe Stefan-Boltzmann constant, and Qthe input
heat flux. Here, we assume Qhas a Gaussian laser beam
profile: Q≈2P
πR2e−2r2(x,y,z,t)
R2, where Pis the laser power, R
the effective laser beam radius, and rthe radial distance from
the center of the laser spot.
The built model of a thin layer of powder bed and the
substrate is depicted in Fig. 5. In this model, we use a se-
lective mesh scheme to balance model accuracy with compu-
tation time: a fine triangular-and-swept mesh with the max-
imum element size of 72.6µm (33% of the laser diameter,
i.e., 0.33d) is applied to the central powder bed region that
directly interacts with the energy beam, whereas less finer
triangular-and-swept mesh (3.5mm) and tetrahedral mesh
(2mm) are applied to the substrate and peripheral powder
bed, respectively. We apply a bidirectional scan scheme, as
illustrated in Fig. 5.
3.2 Mesh Refinement
In FEA, a correct model will move towards the exact so-
lution as the mesh size decreases towards zero. To ensure the
property of the built model, we conduct a mesh refinement
study, as shown in Fig. 6. The horizontal axis denotes the el-
ement size that equals the laser diameter ddivided by n. With
nincreasing, the mesh gets finer. We design 16 simulations
012345678
n (relating to element size)
-200
0
200
Tpeak (K)
Figure 6: Mesh refinement. Element size=laser diameter/n.
with different element sizes ranging from n=0.5(coarser) to
n=8(finer). Each simulation computes one sintering track
of length 5mm located in the middle of the powder bed. At
the end of each simulation, the peak temperature of the melt
pool already reaches the steady-state value. We then com-
pare each steady-state value with that of the finest simulation
(n=8), and the difference is plotted in the vertical axis of
Fig. 6. As we can see, beyond n=3, the peak tempera-
ture starts converging to the exact solution, and the benefit
of keeping reducing the element size becomes insignificant.
Throughout this paper, we select 0.33das the element size
of the central power bed where lie the laser tracks.
3.3 Hatch Spacing
Implementing the developed finite element model, we
will examine next how hatch spacing affects the melt pool
variation especially during the transition from the end point
of one track (named P1) to the start point of the adjacent
track to be sintered (P2). Here, hatch spacing is defined as
the distance between two adjacent laser tracks.
When the hatch spacing (e.g., 47 µm in (a1)-(a4) of Fig.
7) is much less than half of the melt pool width (around the
laser spot radius 110 µm), the laser spot at P2 will be centered
inside the melt pool region of P1 and thus can take advan-
tage of the accumulated heat, yielding a well-developed melt
pool at t=0.0505 s. Otherwise, when the hatch spacing (e.g.,
100 µm in (b1)-(b4) of Fig. 7) is close to or larger than half of
the melt pool width, the laser spot at P2 will be centered out
of the melt pool region of P1, yielding a lower initial tem-
perature at P2. Thus, the melt pool evolves slower, and two
immature states show up at t=0.5505 s and t=0.551 s.
In SLS, to get a consistent part quality, a stable melt
pool is desired during the transition. The immature melt
pool states can be eliminated by decreasing the hatch spac-
ing. However, there is a trade-off between melt pool stabil-
ity and sintering efficiency since simply shortening the hatch
spacing increases printing time.
4 Periodic Thermal Interaction in SLS
This section employs the developed finite element simu-
lation and the experimentation in an in-house built SLS ma-
chine to validate the existence of periodic disturbances in
SLS. These periodic disturbances come from the repetitive
in- and cross-layer thermomechanical interactions, as will be
investigated in Sections 4.1 and 4.2, individually. The com-
bined thermal effect will be elaborated in Section 4.3.
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0.0505 s 0.051 s 0.0515 s 0.052 s
(a1) (a2) (a3) (a4)
0.5505 s 0.551 s 0.5515 s 0.552 s
(b1) (b2) (b3) (b4)
(K)
Figure 7: Melt pool variation at the start of the 2nd track with hatch
spacing of 47 µm in (a1)-(a4) and the start of the 12th track with
hatch spacing of 100 µm in (b1)-(b4).
2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5
2.2
2.4 10-4
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time (s)
2.2
2.4
2.6
Melt pool width (m)
10-4
8
7
6
5
4
3
2
1
(a)
(b)
0
5
Magnitude (dB)
10-6
Disturbance 3 = 1.9952e-05
0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)
0
2
10-6
(c)
(d)
Figure 8: Evolution of melt pool width. In-layer thermal interaction
in (a) time domain and in (c) Fast Fourier transform (FFT). Com-
bined in- and cross-layer thermal interaction in (b) time domain and
in (d) FFT.
4.1 In-layer Numerical Modeling
Using the finite element model developed in Section 3,
80 tracks are bidirectionally sintered within one layer (Fig.
5). From the simulation results in Fig. 8a, we observe
that the melt pool width fluctuates around the average value
216 µm (close to the laser spot diameter 220 µm) after reach-
ing the steady state. Most importantly, the start of each track
has larger melt pool widths than the rest of the track. This is
because in bidirectional scanning, when the energy beam ap-
proaches the end of one track, the large latent heat does not
have enough time to dissipate out before the next track starts.
The resulting increased melt pool widths at the beginning of
each track form a periodic disturbance with a repetitive spec-
trum in the frequency domain (Fig. 8c).
Note that besides the bidirectional scan used here, other
scan patterns yield similar repetitive disturbances (see, e.g.,
experimental results in [22]). Here, automatic control algo-
rithms [1, 20] can be brought in to handle those undesired
repetitive spectra, as will be discussed in Section 5.
4.2 Cross-layer Experimental Verification
Similar to the in-layer case where previously sintered
tracks generate thermal disturbances to the new track, the
heating-cooling cycles of previously sintered layers also cre-
0 5 10 15
Time (s)
0
10
20
Melt pool width
(pixel)
Scan through heated part
21 3 4 5 6 7 8 9 10
Figure 9: Illustration of cross-layer disturbance. Top left: thermal
image during sintering. Top middle: processed image in grayscale.
Top right: in-house built SLS testbed. Bottom: temporal evolution
of melt pool width showing cross-layer thermal interaction.
ate disturbances to the temperature profile of the top layer.
This cross-layer thermal disturbance is particularly harmful
to parts with abrupt geometric changes, such as overhang
structures. We verify in this section the existence of the
cross-layer thermal patterns based on the experimental re-
sults collected from an in-house built SLS testbed (Fig. 9).
In the experimental setup, an aluminum part is buried in
the Nylon 12 powder bed and directly attached to the build
plate that is heated by heat resistors from underneath. A
few thin layers of the powder are spread on top of the flat
rectangular surface of the part. At the stage of pre-heating,
since aluminum has a higher thermal conductivity than Ny-
lon 12, the temperature of the powders on top of the part
surface is significantly higher than that of the powders else-
where (top plots in Fig. 9). The surface thermal profile of the
powder bed is recorded by a FLIR A325sc infrared camera.
A2.8Watt 445 nm laser diode that gives the desired energy
density is used to fuse the powder materials bidirectionally.
Also, the laser scan pattern has a larger length than the alu-
minum part sitting below. This configuration imitates the
sintering process of parts with overhang structures, where
the aluminum part corresponds to the previously fused lay-
ers. Note that we intentionally select a large hatch spacing of
10 mm such that the in-layer thermal interaction is negligible.
From the bottom plot in Fig. 9, we recognize signifi-
cant repetitive variations of the melt pool width (evaluated
by pixel numbers in the video frames). When the laser scans
through the powder right on top of the aluminum part, larger
melt pool widths are detected due to higher initial tempera-
tures of the powders therein (Tracks 4-7).
4.3 In- and Cross-layer Numerical Modeling
This section employs the developed finite element
model in Section 3 to demonstrate the combined effect of
periodic in- and cross-layer thermal interactions. We put un-
der the powder bed a Ti6Al4V part (4.45×1×1mm3) that is
preheated to 1200 K. Due to the high initial temperature of
the added part, the powder on top of the part has a higher ini-
tial temperature than the powder elsewhere. We also use the
selective mesh scheme here: triangular-and-swept (72.6µm)
for the central powder bed, triangular-and-swept (1.5mm)
for the peripheral powder bed, and free tetrahedra (2mm) for
the substrate and the added part. The scan strategy is the
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0.0135 s 0.025 s 0.05 s
Preheated part
(K)
Figure 10: Surface temperature distributions in K (top view) during
sintering the first track.
same as that in Fig. 5. Eight tracks are sintered bidirection-
ally with the hatch spacing of 50 µm. The length of the laser
track (5mm) is greater than that of the added part (1mm).
This configuration similar to that in Section 4.2 simulates
the sintering of the overhang structure.
Fig. 10 illustrate the top views of the surface tempera-
ture profiles during the first-track sintering from 0 to 0.05 s.
When the laser is passing through the preheated powder at
0.025 s, we get a larger melt pool width, compared to when
the laser is approaching (t=0.0135 s) or leaving (t=0.05 s)
the preheated region. The larger melt pool width is due to
the higher initial temperature of powder on top of the pre-
heated part. This result coincides with that in Section 4.2.
Moreover, as time goes by, the temperature difference be-
tween the preheated part and the ambient material is getting
smaller, as shown by the blurrier border of the preheated re-
gion at t=0.05 s in Fig. 10.
During the evolution of the melt pool width in Fig. 8b,
at the beginning of each track, there is a large increase of the
melt pool width caused by the in-layer thermal interaction, as
explained in Section 4.1. Furthermore, we can tell from the
arrowed peaks between two adjacent dotted lines that larger
melt pool widths will be created every time the laser scans
through the powder on top of the preheated part (correspond-
ing to previously fused layers). Also, these peaks caused by
the cross-layer thermal interaction get smaller as the heat ac-
cumulated by the preheated part dissipates out (Tracks 7 and
8 in Fig. 8b). Compared with the bottom plot in Fig. 9, the
amplitudes of these peaks fall faster because Ti6Al4V has a
larger thermal conductivity than Nylon 12.
In summary, we have demonstrated that the periodic
evolution of the melt pool width is a lumped output of the
repetitive in- and cross-layer heat transfer dynamics. When
comparing the frequency spectra in Figs. 8c and 8d, we can
tell that the cross-layer variations change the magnitudes of
the spectral peaks but not the main harmonic frequency val-
ues. These major repetitive variations can thus be attenuated
by the same feedback control algorithms. As a case study,
we implement the plug-in RC presented in Section 2 to re-
duce the periodic disturbances caused by the repetitive in-
layer thermomechanical interaction.
5 Repetitive Control in SLS
In this section, we first employ a pseudorandom binary
sequence (PRBS) (with a magnitude of 10 W and an add-
on DC component of 60 W) as the input signal to identify
the plant model between the laser power and the melt pool
width: P(s)=0.0009013
s+945.8.
-150
-100
Magnitude (dB)
102103
Frequency (Hz)
-200
0
200
Phase (deg)
Measured system
Identified system
Figure 11: Measured and identified system responses.
101102103
Frequency (Hz)
-80
-60
-40
-20
0
Magnitude (dB)
Baseline RC =0.999 RC =0.99 RC =0.9
0 0.5 1 1.5 2 2.5 3 3.5 4
n0
20
25
30
3 decrease%
=0.9 =0.99 =0.999
Figure 12: Top: Magnitude responses of sensitivity functions S(z)
in baseline control and S0(z)in RC with different values of αand
n0=1. Bottom: 3σdecreases with varying αand n0.
0
510-6
Baseline control 3 = 1.1498e-05
0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)
0
5
Magnitude (dB)
10-6
Repetitive control 3 = 7.5598e-06
decreasing 34.25% compared to baseline control
Figure 13: FFT of plant outputs ( y(k)in Fig. 1) with baseline con-
trol and RC (α=0.99 and n0=1).
As shown in Fig. 11, the frequency responses of the
measured and identified systems match well with each other.
We thereupon design a PI controller as C(s)=Kp+Ki/s
with Kp=1.56 ×106and Ki=2.48 ×109. Under the sam-
pling time Tsof 0.5ms (i.e., fs=2kHz), the zero-order-hold
equivalents of the plant and controller models respectively
are P(z)=3.591×10−7
z−0.6232 and C(z)=1.56×106z−3.2×105
z−1.
The solid line in the top plot of Fig. 12 shows the mag-
nitude response of the sensitivity function S(z)in the base-
line feedback loop composed of P(z)and C(z). Such a de-
sign provides a bandwidth at 184 Hz, which approximates
the limit of 20% of the Nyquist frequency (1000 Hz) and in-
dicates that the controller is well tuned.
In the in-layer sintering, a periodic disturbance exist-
ing in the evolution of the melt pool width is induced by
the repetitive thermal interaction between current track and
previous tracks (Section 4.1). The repetitive spectrum of
this disturbance in the frequency domain is exhibited in Fig.
8c. The fundamental frequency f0of the disturbance is de-
termined by the duration of scanning one track t0, that is,
f0=1/t0=ux/L, where uxis the scan speed and Lis the
track length. In this example, f0=100/5=20 Hz, and fre-
quency spikes at n f0(n∈Z+)appear in the fast Fourier trans-
form (FFT) of the disturbance. We first build the baseline
feedback loop consisting of P(z)and C(z)to attenuate the
disturbance. From Fig. 13, we have that the baseline PI con-
trol can attenuate to some extent the frequency spikes below
the closed-loop bandwidth (184 Hz) but leave the other high-
frequency spikes untouched. To enhance the disturbance-
attenuation performance, the plug-in RC compensator intro-
duced in Section 2 is added on top of the baseline controller.
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2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.5
0
20
40
60
80
Control signal (W)
0 0.5 1 1.5 2 2.5 3
Time (s)
-2
0
2
Melt pool width
variation (m)
10-5
Figure 14: Top plot: control signals (u(k)in Fig. 1) of baseline
control (dashed) and RC (solid). Bottom plot: melt pool width vari-
ations (y(k)in Fig. 1) of baseline control (dashed) and RC (solid).
In the design of the Qfilter in (2), the relative degree
mof ˆ
P(z)is 1, and the disturbance period N=fs/f0=
2000/20 =100. High-gain control efforts are generated ex-
actly at 20 Hz and its harmonics (see S0(z)in the top plot of
Fig. 12). We then search around to find the optimal pair of
αand n0that gives the best disturbance-attenuation result.
With αdecreasing, the decreasing spikes of 1−z−mQ(z)and
S0(z)at harmonic frequencies get wider and deeper, but the
intermediate frequency spikes are amplified (Fig. 2 and top
plot of Fig. 12). With n0increasing, frequency spikes at high
frequencies are further attenuated at the cost of shallower de-
creasing spikes at the harmonics (Fig. 3). To balance be-
tween attenuations at the harmonics and amplifications else-
where, we plot in Fig. 12 the decrease of the 3σvalue of
the output (y(k)in Fig. 1) with varying values of αand n0,
where σdenotes the standard deviation. With the plug-in
RC added to the baseline PI control, the more the 3σvalue
decreases, the further the disturbance is attenuated. Based on
the bottom plot of Fig. 12, we choose the pair of α=0.99
and n0=1that has the maximum 3σdecrease. Substituting
the values of m,N,α, and n0into (2), we thus get the transfer
function of the Qfilter.
The feedback loop with the plug-in RC compensator is
designed according to Fig. 1 with r(k)=0and d(k)com-
ing from the cross-scan thermodynamics interaction (Fig. 8a
with the mean removed). The control signals u(k)of the
baseline control and RC are shown in the top plot in Fig.
14. We can tell from Fig. 13 that compared with the base-
line control, the application of RC further lowers the peri-
odic frequency spikes especially at high frequencies beyond
the closed-loop bandwidth and decreases the 3σvalue by
34.25%. Similarly, in the time domain, the increased control
efforts of RC at the harmonic frequencies yield a further-
attenuated output y(k)(bottom plot in Fig. 14).
6 Conclusion
In this paper, we first developed a finite element model
to simulate the temperature response of the repetitive energy
deposition in selective laser sintering (SLS). We briefly dis-
cussed the effect of hatch spacing on the property of the
model. Employing high-performance computation and ex-
perimentation, we validated the existence of the periodic
disturbances in the evolution of melt pool width. The dis-
turbance periodicity is closely related to the recurring laser
scanning trajectories and the repetitive in- and cross-layer
thermomechanical interactions. From there, we identified the
system model from the laser power to the melt pool width
and built a repetitive control algorithm to advance the part
quality in SLS. We validated that the repetitive control algo-
rithm attenuates the periodic disturbances more substantially
compared to the PI control.
Acknowledgement
This material is based upon work supported in part by
the National Science Foundation under Grant No. 1953155.
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