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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 =∇·(k∇T(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<T≤Tsol

Lf

Tm−Tsol +cp1(Tsol )+cp2(Tm)

2Tsol <T<Tm

cp2(T)T≥Tm

,(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 (1−φ)4and ρpowder =ρsolid (1−φ),

where φis expressed as

φ(T) =

φ0T0<T≤Tsol

φ0

Tsol−Tm(T−Tm)Tsol <T<Tm

0T≥Tm

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: −k∂T

∂z

z=h=0. The other boundary condition considers

surface conduction, convection, and radiation:

−k∂T

∂z

z=0

=−Q+hc(T−T0) + εσB(T4−T4

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:

Q≈2q

πR2e−2r2

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 ×10−7/(z−0.5901). As shown in the bottom

plot of Fig. 4, Pd(z)is further normalized to P(z) = Pd(z)/c=

0.4099/(z−0.5901)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 e−ux(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(Tm−T0)wssi +eπρcp(Tm−T0)uxw2

ssi/8.(5)

Assumptions in deriving (5) are:

1. −ln(r∗N)

r∗M≈0, that is, r∗≈1

eN, where r∗is the values of rat

the melt pool width, M=ux

2κ, and N=2πk(Tm−T0)

q.

2. r∗M1.

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(Tm−T0)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(Tm−T0)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

(Bq0−k)w2

ssi +kw0wssi −Bw2

0q=0.(7)

Omitting the negative root, we get

wssi =pk2+4(Bq0−k)Bq −k

2(Bq0−k)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=

q−q0. 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(Bq0−k)Bq −k

2(Bq0−k)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

0 20 40 60 80 100 120 140

Frequency of input signal (Hz)

4

6

8

10

12

RMS error (m)

10-6

Hammerstein model w/ compensation

Linear model Pd(z)

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

cδq−1

(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 Ωin Hz,

∆·T(ejΩTs)

<1, that is, |∆|<1/

T(ejΩTs)

, 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 Ω, we need to make

sure the maximum |∆|is less than 1/

T(ejΩTs)

. 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 f−1(·)

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)f−1(.)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.

7 Copyright © 2020 by ASME