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Despite the advantages and emerging applications, broader adoption of powder bed fusion (PBF) additive manufacturing is challenged by insufficient reliability and in-process variations. Finite element modeling and control-oriented modeling have been identified fundamental for predicting and engineering part qualities in PBF. This paper first builds a finite element model (FEM) of the thermal fields to look into the convoluted thermal interactions during the PBF process. Using the FEM data, we identify a novel surrogate system model from the laser power to the melt pool width. Linking a linearized model with a memory-less nonlinear submodel, we develop a physics-based Hammer-stein model that captures the complex spatiotemporal thermomechanical dynamics. We verify the accuracy of the Hammerstein model using the FEM and prove that the linearized model is only a representation of the Hammerstein model around the equilibrium point. Along the way, we conduct the stability and robustness analyses and formalize the Hammerstein model to facilitate the subsequent control designs.
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Proceedings of the ASME 2020
Dynamic Systems and Control Conference
DSCC2020
October 4-7, 2020, Pittsburgh, Pennsylvania, USA
DSCC2020-XXXXX
NEW HAMMERSTEIN MODELING AND ANALYSIS FOR CONTROLLING MELT
POOL WIDTH IN POWDER BED FUSION ADDITIVE MANUFACTURING
Dan Wang
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: daw1230@uw.edu
Xinyu Zhao
Dept. of Mechanical Engineering
University of Connecticut
Storrs, Connecticut, 06269
Email: xinyu.zhao@uconn.edu
Xu Chen
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: chx@uw.edu
ABSTRACT
Despite the advantages and emerging applications, broader
adoption of powder bed fusion (PBF) additive manufacturing is
challenged by insufficient reliability and in-process variations.
Finite element modeling and control-oriented modeling have
been identified fundamental for predicting and engineering part
qualities in PBF. This paper first builds a finite element model
(FEM) of the thermal fields to look into the convoluted thermal
interactions during the PBF process. Using the FEM data, we
identify a novel surrogate system model from the laser power to
the melt pool width. Linking a linearized model with a memory-
less nonlinear submodel, we develop a physics-based Hammer-
stein model that captures the complex spatiotemporal thermome-
chanical dynamics. We verify the accuracy of the Hammerstein
model using the FEM and prove that the linearized model is only
a representation of the Hammerstein model around the equilib-
rium point. Along the way, we conduct the stability and robust-
ness analyses and formalize the Hammerstein model to facilitate
the subsequent control designs.
1 INTRODUCTION
Different from conventional subtractive machining, additive
manufacturing (AM) builds a part directly from its digital model
by joining materials layer by layer. In particular, by applying
high-precision lasers or electron beams as the energy source,
powder bed fusion (PBF) AM has enabled unprecedented fab-
rication of complex parts from polymeric and metallic powder
materials. However, broader adoption of the technology remains
challenged by insufficient reliability and in-process variations in-
duced by, for example, uncertain laser-material interactions, en-
Corresponding author
vironmental vibrations, powder recycling, imperfect interactions
of mechanical components, and recursive thermal histories of
materials [1–4].
A typical part in PBF is built from many thousands of thin
layers. Within each layer, the energy beam is regulated to follow
trajectories predefined by the part geometry in a slicing process.
After one layer is finished printing, a new thin layer of powder
will be spread on top, and then another cycle begins. Appropri-
ate modeling of this sophisticated dynamic system plays a fun-
damental role in understanding and regulating the PBF and PBF-
related techniques such as laser metal deposition (LMD). Current
researches employ the finite element model (FEM) to explore the
energy-deposition mechanisms and the control-oriented model-
ing to build mathematical models for regulating in-process varia-
tions. Particularly, [6, 9, 10] adopt FEM to investigate the effects
of various scan configurations on the thermal fields of the powder
bed, the geometries of the melt pool, and the mechanical proper-
ties of the printed parts. In control-oriented modeling, [2, 11–13]
apply low-order system models, and [12–14] further build non-
linear submodels of LMD from laser power and scan speed to
layer height and melt pool temperature to cover more process dy-
namics. Based on the attained models, subsequent control algo-
rithms such as PID control [2,15–17], sliding mode control [14],
predictive control [11], and iterative learning control [18] have
proved their efficiencies in improving the dimensional accuracy
of the printed parts.
This paper establishes a new modeling and understanding of
PBF by taking advantage of the FEM and control-oriented mod-
eling. We first develop an FEM of the thermal fields to look
into the convoluted thermal interactions during the PBF process.
The developed FEM then serves as a simulation platform to pro-
vide data for verifying and identifying parameters of the pro-
1 Copyright © 2020 by ASME
500 1000 1500 2000 2500
Temperature (K)
0
10
20
30
40
50
Thermal conductivity (W/m .K)
500 1000 1500 2000 2500
Temperature (K)
2000
2500
3000
3500
4000
4500
Density (kg/m3)
500 1000 1500 2000 2500
Temperature (K)
0
1
2
3
4
5
6
7
Heat Capacity (J/g .K)
Figure 1. Temperature-dependent thermal properties of Ti6Al4V [5–8]. Solid line: solid and liquid materials. Dash-dotted line: powder material. The two
vertical dotted lines respectively indicate Tsol and Tm.
posed modeling schemes. In the control-oriented modeling of
PBF, stepping beyond commonly used low-order system mod-
els, this paper develops a physics-based Hammerstein model that
accommodates more of the complex spatiotemporal thermome-
chanical dynamics. The Hammerstein model is formulated by
concatenating a memoryless nonlinear submodel derived from
the Rosenthal equation to a linear model obtained from standard
system identification techniques with laser power as the input
and melt pool width as the output. We verify the accuracy of
the Hammerstein model using the FEM and prove that the iden-
tified model is only a linear representation of the Hammerstein
model around the equilibrium point. Along the way, we analyze
the stability and robustness properties of the models and present
a generic control scheme of the Hammerstein model.
The remainder of this paper is structured as follows. Sec-
tion 2 builds the FEM of the thermal fields in PBF. Section 3
identifies the linear plant model from the FEM. Section 4 derives
the closed-form expressions of the steady-state melt pool width
and furthermore develops and analyzes the main Hammerstein
model. Section 5 concludes the paper.
2 FEM OF THERMAL FIELDS IN PBF
In this section, we build and refine an FEM of the thermal
fields in PBF. The model considers surface convection, surface
radiation, conduction, and latent heat of fusion. For brevity and
without loss of generality, the effects of evaporation, fluid flow,
and Marangoni force are neglected when implementing the FEM
in COMSOL Multiphysics 5.3a software. The governing equa-
tion for the conduction heat flow is
ρcp
dT (x,y,z,t)
dt =·(kT(x,y,z,t)) + qs,(1)
where kis the thermal conductivity, cpthe specific heat capac-
ity, ρthe effective density, tthe time, Tthe temperature, and qs
the rate of local internal energy generated per unit volume [19].
When no confusion would arise in the context, T(x,y,z,t)is ab-
breviated to Tin the remaining of this paper.
2.1 Nonlinear Phase Change and Temperature-
dependent Thermal Properties
Fig. 1 shows the temperature-dependent thermal properties
used in this paper. We account for the latent heat of fusion Lfby
introducing the effective heat capacity [20]:
cp,e f f (T) =
cp1(T)T0<TTsol
Lf
TmTsol +cp1(Tsol )+cp2(Tm)
2Tsol <T<Tm
cp2(T)TTm
,(2)
where T0is the ambient temperature, Tsol the solidus temperature,
Tmthe melting point, cp1the heat capacity of the solid or powder,
and cp2the heat capacity of the liquid.
For the thermal properties, we adopt k,cp, and ρin [5,6] for
the solid and liquid materials. We generate the thermal proper-
ties of the powder material from those of the solid material by
considering the porosity φ[7, 8]:
kpowder =ksol id (1φ)4and ρpowder =ρsolid (1φ),
where φis expressed as
φ(T) =
φ0T0<TTsol
φ0
TsolTm(TTm)Tsol <T<Tm
0TTm
with φ0denoting the initial porosity. Here, the heat capacity is
assumed to be the same for the powder and solid materials [7].
2.2 Initial Condition, Boundary Conditions, and Laser
Beam Profile
The initial condition can take any physical configuration. In
this study, we assume a uniform distribution of the initial temper-
ature T(x,y,z,0) = T0. When the substrate (left plot of Fig. 2) is
designed to be large enough compared to the heat affected zone,
one boundary condition is established by assuming the bottom
2 Copyright © 2020 by ASME
X
Z
Y
Powder bed: F ree triangular
and Swept (2mm)
Substrate: Free
tetrahedral (3.5 mm)
10 mm
5mm
2mm
50 µm
Laser tracks
Powder bed: F ree quad
and Swept (60 µm)
(K)
Figure 2. Left: powder bed and substrate with selective meshing
scheme. Right: surface temperature distribution at t=0.14s. The lined
isotherm indicates T=Tm.
(z=h) of the substrate has no heat loss in the rapid laser scan-
ning: kT
z
z=h=0. The other boundary condition considers
surface conduction, convection, and radiation:
kT
z
z=0
=Q+hc(TT0) + εσB(T4T4
0),(3)
where Qis the input heat flux, hcthe convection heat transfer
coefficient, εthe emissivity, and σBthe Stefan-Boltzmann con-
stant. Here, we assume Qhas a Gaussian laser beam profile:
Q2q
πR2e2r2
R2, where qis the laser power, Rthe effective laser
beam radius, and rthe radial distance from a certain point to the
center of the laser spot. Then Appendix has listed the process
parameters used in this study.
2.3 Meshing and Scanning Schemes
The left plot of Fig. 2 shows the built FEM with a substrate
and a thin layer of powder bed. For melt pools in the scale of
around 248µm in diameter, we use a selective meshing scheme to
balance model accuracy with computation time: a fine quad-and-
swept mesh with a maximum element size of 60µm is applied
to the central powder bed region that directly interacts with the
energy beam, whereas less finer tetrahedral mesh (3.5mm) and
triangular-and-swept mesh (2mm) are applied to the substrate
and the peripheral powder bed, respectively. We consider both
single and multiple scans of the laser beam on the powder bed.
The left plot of Fig. 2 illustrates the bidirectional scan scheme
used in this study.
The developed FEM has been validated experimentally and
analytically in [21] and serves as a simulation platform in this pa-
per. Later on the data generated from the FEM such as melt pool
width will be used to identify and verify the accuracies of the
proposed models. In FEM, melt pool widths are generated from
the temperature distribution T(x,y,z,t)by searching around the
position of the laser beam to find the maximum width of the melt
pool geometry bounded by Tm.
10110210 3104
-160
-150
-140
-130
-120
Magnitude (dB)
10110210 3104
Frequency (Hz)
-200
-100
0
100
200
Phase (degree)
Measured system using sinusoidal and PRBS signals
identified system P = 0.001671/(s+1055)
Figure 3. Measured and identified system responses.
3 LINEAR MODEL
From the developed FEM, we identify the linear plant model
as P(s) = 0.001671/(s+1055)from laser power changes δq
to melt pool width changes δwaround the equilibrium point at
(q0=60W,w0=248.41 µm). Here, q=q0+δqand w=w0+δw
are the actual laser power fed to the FEM and the melt pool width
generated from the FEM, respectively. The input signals fed to
the FEM include a pseudorandom binary sequence (PRBS) sig-
nal and multiple sinusoidal signals (10~300 Hz), with a mag-
nitude of 20 W and an add-on DC component of q0=60W.
The frequency responses of the measured and identified systems
match well with each other (Fig. 3). Under the sampling time
tsof 0.5ms, the zero-order-hold equivalent of the plant model
is Pd(z) = 6.493 ×107/(z0.5901). As shown in the bottom
plot of Fig. 4, Pd(z)is further normalized to P(z) = Pd(z)/c=
0.4099/(z0.5901)that has unit DC gain, where cis the DC
gain of Pd(z).
4 HAMMERSTEIN MODEL IN PBF
In this section, we show the limit of the linear model sub-
ject to the complicated nonlinear thermomechanical dynamics
of PBF and build a new physics-based Hammerstein Model to
address the limitations. After that, we analyze the stability and
robustness of the models. The Hammerstein model is conven-
tionally employed in system identification for nonlinear systems,
consisting of a nonlinear static element followed by a linear dy-
namic element. Recent studies of the Hammerstein model tar-
get at parameter estimation and neural network based solutions
[22–24].
4.1 Core Physics of Melt Pool at Quasi Steady State
When a moving point laser source is acting on a large thick
plate, the analytical solution of (1) in the steady state is the
Rosenthal equation [19]:
T(ξ,y,z)T0=q
2πkr eux(r+ξ)
2κ,(4)
where (ξ,y,z)is a coordinate system attached to the moving
source, r=pξ2+y2+z2, and κ=k/(ρcp).
3 Copyright © 2020 by ASME
f(.)P(z)
δqδwss(i)δw
Hammerstein model H
cP(z)
δqδw
Pd(z)
Figure 4. Block diagrams of the Hammerstein model and the identified
linear model.
Some assumptions and simplifications in deriving the
Rosenthal equation are:
1. The material’s physical coefficients such as k, ρ, and cpare
independent of temperature. Using an average value pro-
vides a reasonable approximation and enables a closed-form
solution to be obtained.
2. The internal heat generation is neglected, i.e., qs=0.
3. The workpiece material is homogeneous and isotropic.
4. When the powder bed is processed long enough, a Quasis-
tationary state is reached, that is, the temperature undergoes
no change with time with respect to the coordinate system
attached to the heat source, i.e., (ξ,y,z).
5. A point heat source rather than Gaussian distribution is used.
6. The effect of latent heat of fusion is negligible since the ab-
sorbed latent heat evolves later on.
From the Rosenthal equation in (4), the closed-form equation re-
lating the steady-state melt pool width (wssi or wss) with the laser
power qis [25]:
q=πk(TmT0)wssi +eπρcp(TmT0)uxw2
ssi/8.(5)
Assumptions in deriving (5) are:
1. ln(rN)
rM0, that is, r1
eN, where ris the values of rat
the melt pool width, M=ux
2κ, and N=2πk(TmT0)
q.
2. rM1.
3. The approximation of qis found to be improved by account-
ing for the zero-speed power in (4), that is, the first term on
the right hand side of (5).
The first two assumptions are reasonably valid for all alloys ex-
cept AlSi10Mg under typical PBF configurations.
4.2 Infrastructure of the Hammerstein Model
We start to build the Hammerstein model by lumping the
memoryless nonlinear submodel in (5) with the identified linear
dynamics P(z)that has unit DC gain (see Section 3). As shown
in Fig. 4, the Hammerstein model upgrades Pd(z)by replacing
the constant cwith the nonlinear closed-form expression of the
steady-state melt pool width f(·). In (5), the values of parameters
k,ρ, and cpare to be determined. Substituting the equilibrium
point (q0,w0)to (5) gives q0=πk(TmT0)w0+eπρcp(Tm
T0)uxw2
0/8, that is,
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
-1
-0.5
0
0.5
1
w (m)
10-4
FEM result
Linear model Pd(z)
Hammerstein model w/o compensation
Hammerstein model w/ compensation
Figure 5. Parameter identification of the Hammerstein model with input
signal of 10 Hz.
eρcpux=8(Bq k)/w0>0,(6)
where B=1/[π(TmT0)w0]is a constant. In (5) and (6), ρand
cpare multiplied together and related to k. Based on the first
assumption in Section 4.1, we choose k=40 W/(m ·k) from Fig.
1. Substituting (6) to (5) yields
(Bq0k)w2
ssi +kw0wssi Bw2
0q=0.(7)
Omitting the negative root, we get
wssi =pk2+4(Bq0k)Bq k
2(Bq0k)w0.(8)
With all parameters determined, the Hammerstein model in
Fig. 4 is thus formalized around the equilibrium point by con-
necting (8) with P(z)and letting δwssi =wssi w0and δq=
qq0. Certainly, due to simplifications in deriving (5), its di-
rect solution (8) only works at specific input conditions. Under
the input signal of δq10 =20Wsin(2πf tst), where f=10Hz and
tsis the sampling time, we can tell from Fig. 5 that the output of
the Hammerstein model (dashed line) deviates greatly from the
FEM result (solid line).
To add more flexibility to the nonlinear block, we multiply
wssi in (8) with a compensation factor α(q):
wss =pk2+4(Bq0k)Bq k
2(Bq0k)w0α(q),(9)
where α(q)is a quadratic function that passes through three
points (60W,1)(i.e., no compensation at the equilibrium point),
(80W,α1)(i.e., the maximum laser power), and (40 W,α2)(i.e.,
the minimum laser power). We identify the parameters α1and
α2, respectively, as 0.8507 and 1.1973 using the Parameter Esti-
mation tool in MATLAB. The nonlinear least square regression
is used to minimize the sum of squared errors between the FEM
data and the output of the updated Hammerstein model with com-
pensation (solid and dash-dotted lines in Fig. 5).
4 Copyright © 2020 by ASME
Figure 6. Root mean squared (RMS) errors with respect to different input
frequencies.
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Time (s)
-4
-2
0
2
w (m)
10-5
FEM result Hammerstein model w/ compensation
Figure 7. Melt pool width changes with input signal of 60 Hz.
The compensated Hammerstein model is achieved by using
(9) instead of (8) and letting δwss =wss w0. From Fig. 5, we
can also tell that this Hammerstein model (dash-dotted line) gives
a better approximation (41% increasing) of the system dynamics
(solid line) than the identified linear model Pd(z)(dotted line).
More generally, as shown in Fig. 6, under different input frequen-
cies, the compensated Hammerstein model yields smaller root
mean squared errors (e.g., 4.67 µm at 10 Hz) with respect to the
FEM result than the linear model (7.89 µm at 10 Hz) and achieves
increasing model accuracies (41% increasing at 10 Hz). Fig. 7 il-
lustrates the results of the compensated Hammerstein model and
the FEM under the input frequency of 60 Hz. Following the pro-
cedures from (6) to (9), we can adaptively build the Hammerstein
model for each specific equilibrium point and structurally draw
the complete model map for the entire task space of PBF.
We compare in Fig. 8 how δwss changes with respect to δq
under different modeling schemes. In the identified linear model
Pd(z), the gradient of the dash-dotted line that links δwss with δq
is the constant c(Fig. 4). From Fig. 8, we can tell that Pd(z)is
only a linear representation of the nonlinear Hammerstein model
(solid line) near the equilibrium point. It is remarkable how
Pd(z)identified from the FEM data coincides tangentially with
the Hammerstein model derived from the governing equation.
Next we will conduct the stability and robustness analyses to in-
vestigate when Pd(z)would fail in representing the Hammerstein
model.
-30 -25 -20 -15 -10 -5 0 5 10 15 20
q (W)
-10
-5
0
5
wss (m)
10-5
Hammerstein model w/ compensation
Hammerstein model w/o compensation
Linear model Pd(z)
Figure 8. Steady-state melt pool width changes of different models.
4.3 Stability and Robustness
Based on the Hammerstein model, we evaluate the robust-
ness and stability properties of the linear models Pd(z)that are
commonly used in practice. Let H=Pd(z)(1+), where His
the Hammerstein model and is the bounded model uncertainty.
From Fig. 4, we have δwss ·P(z) = δq·cP(z)(1+), which gives
||=
δwss
cδq1
(10)
that is specified by the distance between the dash-dotted and solid
lines in Fig. 8.
Standard robust-stability analysis gives that a closed-loop
system consisting Pd(z)is stable if and only if both of the fol-
lowing conditions hold:
1. Nominal stability condition is satisfied, that is, the closed
loop is stable when =0.
2. Robust stability requirement is met by applying the
small gain theorem [26]: for any frequency in Hz,
·T(ejTs)
<1, that is, ||<1/
T(ejTs)
, where T(z)
is the complementary sensitivity function [1].
Note that ||in (10) is positively correlated to the control signal
δq, that is, more laser power deviation from the equilibrium point
yields a larger ||. Under a certain frequency , we need to make
sure the maximum ||is less than 1/
T(ejTs)
. When the con-
dition is violated, Pd(z)will no longer be a valid representation
of the Hammerstein model.
A sufficient condition for the stability of the Hammerstein
model is the BIBO stability of the linear model P(z)(Fig. 4)
[23]. In practice, the linear model is typically a rational transfer
function, whose stability can be easily examined.
4.4 Control Implementation
Although the focus of this paper is on the modeling of the
complex physics in PBF, we have additionally validated the pro-
posed model in closed-loop controls. Limited in space, we will
briefly discuss the key concept. Fig. 9 presents a typical feed-
back loop when applying the Hammerstein model. An f1(·)
block is added and connected with the block of the Hammerstein
model. Combining these two blocks together yields the linear
model P(z). It is thereafter standard practice to design the control
algorithms for P(z). To find the inverse of the nonlinear element,
5 Copyright © 2020 by ASME
C(z)f1(.)H
d(k)
+
r(k)+e(k)
+
y(k)
Figure 9. Block diagram of feedback control for a Hammerstein model.
we can use a high-order polynomial to approximate f(solid line
in Fig. 8). Besides, [22] proposes an approximate method for the
cases when fis not invertible.
5 CONCLUSION
In the paper, we first build a finite element model (FEM) to
simulate the thermal fields during the powder bed fusion (PBF)
process. Using the FEM data, we identify around the equilibrium
point the linear system model from the laser power to the melt
pool width. In addition, deriving from the Rosenthal equation,
we reach a nonlinear closed-form expression of the steady-state
melt pool width. Concatenating the nonlinear expression to the
identified linear model, we develop the main Hammerstein model
that captures more of the convoluted thermomechanical dynam-
ics of PBF. We prove that the Hammerstein mode gives a better
approximation (e.g., 41% increasing at 10 Hz) of the FEM result
than the linear model. From there, we analyze the stability and
robustness properties of the models and present a generic control
scheme for the Hammerstein Model.
ACKNOWLEDGEMENT
This material is based upon work supported in part by the
National Science Foundation under Award No. 1953155.
APPENDIX
Defined parameters of the FEM
Parameters Value Parameters Value
Powder bed size 15 mm×15 mm ×50 µm Material Ti6Al4V
Substrate size 15 mm ×15mm ×5 mm Track length L5mm
R220µm/2 Time step Ts0.5ms
Absorptance 0.25 Emissivity 0.35
Tsol 1873K Scan speed ux100mm/s
Lf295kJ/kg Laser power P60 W
T0/Tm293.15K/1923.15 K φ00.4
hc12.7W/(m2·K) k,cp, and ρsee Fig. 1
REFERENCES
[1] D. Wang and X. Chen, “A multirate fractional-order repet-
itive control for laser-based additive manufacturing,Con-
trol Engineering Practice, vol. 77, pp. 41–51, 2018.
[2] J.-P. Kruth, P. Mercelis, J. Van Vaerenbergh, and
T. Craeghs, “Feedback control of selective laser melting,
in Proceedings of the 3rd international conference on ad-
vanced research in virtual and rapid prototyping, 2007, pp.
521–527.
[3] V. Seyda, N. Kaufmann, and C. Emmelmann, “Investiga-
tion of aging processes of ti-6al-4 v powder material in laser
melting,” Physics Procedia, vol. 39, pp. 425–431, 2012.
[4] D. Wang, T. Jiang, and X. Chen, “Control-Oriented
Modeling and Repetitive Control in In-Layer and Cross-
Layer Thermal Interactions in Selective Laser Sinter-
ing,” ASME Letters in Dynamic Systems and Control,
vol. 1, no. 1, 03 2020, 011003. [Online]. Available:
https://doi.org/10.1115/1.4046367
[5] A. N. Arce, Thermal modeling and simulation of elec-
tron beam melting for rapid prototyping on Ti6Al4V alloys.
North Carolina State University, 2012.
[6] M. Masoomi, S. M. Thompson, and N. Shamsaei, “Laser
powder bed fusion of ti-6al-4v parts: Thermal modeling
and mechanical implications,” International Journal of Ma-
chine Tools and Manufacture, vol. 118, pp. 73–90, 2017.
[7] K. Karayagiz, A. Elwany, G. Tapia, B. Franco, L. Johnson,
J. Ma, I. Karaman, and R. Arróyave, “Numerical and ex-
perimental analysis of heat distribution in the laser powder
bed fusion of ti-6al-4v,IISE Transactions, vol. 51, no. 2,
pp. 136–152, 2019.
[8] J. Yin, H. Zhu, L. Ke, W. Lei, C. Dai, and D. Zuo, “Simu-
lation of temperature distribution in single metallic powder
layer for laser micro-sintering,” Computational Materials
Science, vol. 53, no. 1, pp. 333–339, 2012.
[9] A. Hussein, L. Hao, C. Yan, and R. Everson, “Finite ele-
ment simulation of the temperature and stress fields in sin-
gle layers built without-support in selective laser melting,
Materials & Design (1980-2015), vol. 52, pp. 638–647,
2013.
[10] A. Foroozmehr, M. Badrossamay, E. Foroozmehr et al.,
“Finite element simulation of selective laser melting pro-
cess considering optical penetration depth of laser in pow-
der bed,” Materials & Design, vol. 89, pp. 255–263, 2016.
[11] L. Song and J. Mazumder, “Feedback control of melt pool
temperature during laser cladding process,” IEEE Transac-
tions on Control Systems Technology, vol. 19, no. 6, pp.
1349–1356, 2011.
[12] X. Cao and B. Ayalew, “Control-oriented mimo modeling
of laser-aided powder deposition processes,” in American
Control Conference (ACC), 2015. IEEE, 2015, pp. 3637–
3642.
[13] P. M. Sammons, D. A. Bristow, and R. G. Landers,
“Repetitive process control of laser metal deposition,” in
ASME 2014 Dynamic Systems and Control Conference.
American Society of Mechanical Engineers, 2014, pp.
V002T35A004–V002T35A004.
[14] A. Fathi, A. Khajepour, M. Durali, and E. Toyserkani,
“Geometry control of the deposited layer in a nonplanar
laser cladding process using a variable structure controller,
Journal of manufacturing science and engineering, vol.
130, no. 3, p. 031003, 2008.
[15] J. Hofman, B. Pathiraj, J. Van Dijk, D. de Lange, and
J. Meijer, “A camera based feedback control strategy for
the laser cladding process,” Journal of Materials Process-
ing Technology, vol. 212, no. 11, pp. 2455–2462, 2012.
6 Copyright © 2020 by ASME
[16] D. Salehi and M. Brandt, “Melt pool temperature control
using labview in nd: Yag laser blown powder cladding pro-
cess,” The international journal of advanced manufacturing
technology, vol. 29, no. 3, pp. 273–278, 2006.
[17] A. Fathi, A. Khajepour, E. Toyserkani, and M. Durali,
“Clad height control in laser solid freeform fabrication us-
ing a feedforward pid controller,The International Journal
of Advanced Manufacturing Technology, vol. 35, no. 3, pp.
280–292, 2007.
[18] L. Tang and R. G. Landers, “Layer-to-layer height control
for laser metal deposition process,” Journal of Manufactur-
ing Science and Engineering, vol. 133, no. 2, p. 021009,
2011.
[19] E. Kannatey-Asibu Jr, Principles of laser materials pro-
cessing. John Wiley & Sons, 2009, vol. 4.
[20] I. Yadroitsev, Selective laser melting: Direct manufacturing
of 3D-objects by selective laser melting of metal powders.
LAP LAMBERT Academic Publishing, 09 2009.
[21] D. Wang and X. Chen, “Closed-loop simulation integrating
finite element modeling with feedback controls in powder
bed fusion additive manufacturing,” in 2020 International
Symposium on Flexible Automation (ISFA). IEEE, 2020.
[22] Z. Rayouf, C. Ghorbel, and N. B. Braiek, “A new ham-
merstein model control strategy: feedback stabilization and
stability analysis,” International Journal of Dynamics and
Control, vol. 7, no. 4, pp. 1453–1461, 2019.
[23] F. J. Doyle, R. K. Pearson, and B. A. Ogunnaike, Identifi-
cation and control using Volterra models. Springer, 2002.
[24] X. Ren and X. Lv, “Identification of extended hammerstein
systems using dynamic self-optimizing neural networks,
IEEE Transactions on Neural Networks, vol. 22, no. 8, pp.
1169–1179, 2011.
[25] M. Tang, P. C. Pistorius, and J. L. Beuth, “Prediction of
lack-of-fusion porosity for powder bed fusion,Additive
Manufacturing, vol. 14, pp. 39–48, 2017.
[26] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feed-
back control theory. Courier Corporation, 2013.
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Book
1. Introduction.- 2. Qualitative Behavior.- 3. Restrietions & Extensions.- 4. Determination of Volterra Model Parameters.- 5. Practical Considerations in Volterra Model Identification.- 6. Model-Based Controller Synthesis.- 7. Advanced Direct Synthesis Controller Design.- 8. Model Predictive Control Using Volterra Series.- 9. Application Case Studies.- 10. Summary.