Hesam Zomorodi Moghadam
Department of Mechanical and Aerospace Manufacturing
Missouri University of Science and Technology, Rolla, MO 65409
Switching Model Predictive Control of Aqueous–Based Freeze Extrusion
Fabrication Processes
ABSTRACT
Controlling freeze extrusion fabrication of aqueous-based ceramics is one of the challenging
problems in additive manufacturing because of the high amount of uncertainties that exist in
these processes. In this work an innovative model predictive control strategy featuring a gain
switching law is introduced in order to overcome these challenges. A simulation study is
performed in order to investigate the effectiveness of this method at different conditions.
Because of the switching nature of the proposed method the controller is showing a good
performance over a wide range of system operating conditions.
INTRODUCTION
Using ceramics in manufacturing specialty components has received much attention recently.
However, building complex components out of ceramic materials is a time consuming process
and is still a challenge [1]. Solid Freeform Fabrication (SFF), which is a process of building parts
layer by layer, has been used in the fabrication of ceramic parts to reduce the time and effort
associated with these processes [2, 3]. However, most SFF techniques for ceramic part
fabrication involve the generation of environmentally harmful wastes due to the binder removal
stage [4]. Freeze-form Extrusion Fabrication (FEF) is one of the recent SFF techniques for
1
creating ceramic parts with 3D geometries using layer by layer extrusion of aqueous-based
ceramic pastes [5].
Most of the literature in modeling and control of liquid-solid material extrusion is
devoted to screw type extrusion [6-8]. However, for extruding ceramics screw type extrusion is
not recommended because of the corrosive nature of ceramics [9]. For ceramics, ram based
extrusion is used instead [5, 9-11]. Ram extrusion of ceramic pastes typically has uncertainties
and unpredictable disturbances such as air bubble release, nozzle clogging, and liquid phase
migration [9-11]. Few studies have addressed the modeling and control problem of the ram
extrusion processes. Mason et al. conducted empirical modeling of ram extrusion of aqueousbased alumina ceramic pastes [11]. They concluded that such processes can be modeled by a first
order dynamics. However, in their study they determined that the time constant was subject to
tremendous variations during extrusion. In another study by Li et al., an analytical model was
derived using numerical modeling methods for ram extrusion processes [10]. In this work some
of the uncertainties, such as air bubble release, were explained and modeled. In order to
compensate for such uncertainties, an on/off force controller was designed and tested
experimentally by Mason et al. [12]. Although their controller showed good performance, it
was based on trial and error and did not address other important issues in the ram extrusion
processes like velocity control. Zhao et al. [9] proposed an adaptive feedback controller to tackle
this problem more systematically. They used least squares to identify a force model based on
first order dynamics. Although most of the studies in ram extrusion have concentrated on
compensation of extrusion force, in some cases it is desired to control the ram velocity instead.
For instance, in a recent work by Leu et al. [13] a multiple ram extrusion device was developed
that was able to extrude pastes with different combinations of materials. When extruding
2
multiple pastes, the flow rate of each paste needs to be controlled, which is best accomplished
via ram velocity regulation. However, since most of the uncertainties in ram based extrusion,
such as air bubble release, are best detected by the extrusion force and not by ram velocity,
velocity control alone can lead to deficiencies in the final part [14]. When air bubbles that are
trapped in the paste during preparation and insertion in the paste reservoir, gather in the nozzle,
the system dynamics drastically change [9-11, 14]. As seen in Figure 1, when a large air bubble
releases, the extrusion force reduces dramatically. The best way to predict if extrusion is
occurring at this time is by monitoring the extrusion force. At these instances, when only a ram
velocity controller is used, as seen from Figure 1, it takes a long time (in this case 50 s) for
extrusion to begin again. This period is the time the ram needs to fill the air gap in the nozzle
caused by the air bubble release, and start to extrude again. Since the x-y table under the nozzle
will not stop when an air bubble releases, the section of the part under production at that moment
will not receive any paste and the part will be defective. Therefore, although it is desired to
control the ram velocity, in these instances there needs to be another control strategy that
regulates the extrusion force to return the extrusion force back to its desired value in a short time.
An adaptive predictive controller is proposed in this work to handle these situations.
3
Figure 1: Extrusion force during air bubble release [14].
APPROACH
The extrusion ram in this study is composed of a linear axis which pushes the paste inside a
reservoir into a nozzle. Therefore, in order to model the system dynamics, two levels should be
considered; servo and process dynamics, which are coupled and affect each other. The servo
level is composed of the servomotor, leadscrew, and slide, and the process level is composed of
the extrusion dynamics relating the extrusion force to the ram velocity. In this section the servo
and process dynamics are formulated and then the error dynamics of the total system is presented
in a state space format to be used in control design.
During extrusion, the servomechanism (i.e., the linear axis and the ram) velocity
dynamics can be modeled by
 s vr t   vr t   Ksc ec t   Ksf Fr t   f sf
(1)
where τs (s) is the time constant of the servo system, vr (mm/s), is the ram speed, Ksc ((mm/s)/V),
is the servo system control voltage gain, ec (V), is the control voltage, Ksf ((mm/s)/N), is the
4
servomechanism ram force gain, Fr (N), is the ram force, and fsf (mm/s) is the effect of Coulomb
friction on the servo dynamics.
It can be shown that during extrusion the force applied to the ram is dynamically related
to the ram velocity by [10]
Fr  t 
 F t   F

r
pf
 Ar patm 
Ar p0l0
2
vr  t  103  vss  Fr  t  
(2)
where vr (mm/s), is the ram velocity, Fr (N), is the extrusion force, Fpf (N) is the effect of
reservoir-wall friction on the process dynamics, Ar (m3) is the ram cross sectional area, p0 (Pa), is
the initial pressure in the reservoir, l0 (m), is the effective thickness of air trapped in the paste,
and vss (m/s), is the steady state velocity for a constant extrusion force, which is a function of
paste rheology and can be explicitly determined from analytical models or experimental studies
[10]. For aqueous–based alumina paste, this steady state relationship can be approximated by
 F  21.2 
vss  Fr   10  r

 173 
2.33
6
(3)
Therefore, the extrusion force dynamics for aqueous–based alumina pastes is
Fr  t 
 F t   F

r
2.33

 Ar patm  
3
6  Fr  t   21.2 
vr  t  10  10 
 
Ar p0l0
173


 
2
pf
(4)
The error dynamics of the total extrusion dynamics is now derived. Initially equation (4) is
linearized around the operating force and ram-velocity,  Fr , vr  , at which extrusion occurs for
aqueous–based alumina pastes. The linearized extrusion force dynamic is
eF  t   
1
F
eF  t  
5
K Fv
F
ev  t 
(5)
where eF  t   Fr  t   Fr t  and ev  t   vr  t   vr  t  are the force and velocity deviation from
their operating points, respectively, and Fr and vr are the operating extrusion force and ram
velocity, respectively. For the material and system considered in this work, Fr = 650 N and vr =
0.0046 mm/s. At this operating point τF = 65.9 s and KFv = 51,885 N/(mm/s). The velocity error
dynamics can also be described by
ev  t   
where   t   ec  t  
1
s
ev  t  
K sf
s
K sf
f sf

1
vr  t  
Fr  t   s a  t  
K sc
K sc
K sc r
K sc
eF  t  
K sc
s
 t 
(6)
, and vr  t  , ar  t  and Fr are the reference
velocity (mm/s), acceleration (mm/s2) and extrusion force (N), respectively.
Combining equations (5) and (6), the total state space model is
 1

ev  t     s


eF  t    K Fv

 F
K sf 
 K 
 s  ev  t     sc 
 s  t 



1  eF  t   
 
 0 
F 
1 0  ev  t  
y t   


0 1  eF  t  
(7)
Ram velocity, numerically derived from position encoder signals using a first order backwards
finite difference, and extrusion force are the system outputs and are measured from an encoder,
with a resolution of 0.62 μm, embedded in a Kollmorgan AKM23D DC servo motor and an
Omega LC-305 load cell, respectively.
MODEL PREDICTIVE CONTROL OF EXTRUSION PROCESS
A model predictive controller is now implemented on the FEF process. First, the state space
model is transferred into the discrete domain using a Zero Order Hold
6
X  k  1  Ad  k  X  k   Bd  k  u  k 
(8)
The system is now augmented in order to relate the incremental state dynamics to the
incremental control signal. This will add integral action to the system dynamics resulting in
added robustness to the system and the rejection of constant disturbances. The incremental
discrete dynamics are
 x  k  1   Ad


 y  k  1  Cd Ad
y  k   0
0   x  k    Bd 
u  k 


I   y  k   Cd Bd 
(9)
 x  k  
I

 y k  
where x  k   x  k   x  k  1 and u  k   u  k   u  k  1 . A receding horizon controller is
applied to the system with Nc as the control horizon and Np as the length of the optimization
window, where Np ≥ Nc. Defining y  ki  i | ki  to be the predicted output at time step ki  i for a
measured output at time step k i , the vector of Np predicted system outputs in equation (9) is
 x  k  
Y k   F 
  U  k 
 y k  
(10)
where
Y  k    y  ki  1| ki  y  ki  2 | ki 
U  k    u  k  u  k  1
y  ki  N p | ki 
u  k  N c  1 
T
T
(11)
and
 CA 
 CA2 

F 


 Np 
CA 
0
 CB
 CAB
CB


 N p 1
N 2
CA B CA p B
7


0


N p  Nc 
CA
B 
0
(12)
Now, considering the system in equation (10) an optimization problem is constructed to find the
optimal ΔU vector to minimize the output deviations from their desired trajectories. The
following cost function is now defined to minimize the deviation of extrusion force and ram
velocity from their desired trajectories
J   EFr  EF   F I  EFr  EF    Evr  Ev   v I  Evr  Ev  
T
T
(13)
U T u I U
where
EF  eF  ki  1| ki 
Ev  ev  ki  1| ki 
eF  ki  N p | ki  
,
EFr  01 p
,
ev  ki  N p | ki  , and Evr  01 p . The parameters, αF, αv, and βu are the
weighting constants for extrusion force tracking, ram velocity tracking, and control signal usage,
respectively. The necessary condition for this cost function to be minimal is
dJ
0
d U
(14)
However, from equation (13) it can be shown that
J  Yr  Y  Q Yr  Y   U T u I U
T
 v

F

where Q  



v
(15)



 . Substituting equation (10) into equation (15) and solving


 F 
equation (14) for ΔU, it can be shown that
U   T Q  u I  T Q Yr  FX  ki  
1
(16)
Since in receding horizon control only the first m elements of ΔU are actually fed into the system
the incremental control signal is
u  ki    I mm
0mm
0mm m N U Nc 1  k y yr  ki   k mpc x  ki 
c
8
(17)
where m is the number of inputs. The physical control signal is
u  ki   u  ki   u  ki  1
(18)
When the extrusion force stays in an acceptable range around the reference value, only the ram
velocity deviations will affect the cost function and, when the extrusion force is outside an
acceptable range, the cost function is modified such that more emphasis is placed on the force
deviations. The proposed update law is
 v  5000
v  0
 F =1
if
 F =100 if
eF  k   30
30  eF  k 
(19)
In the next section the proposed adaptive predictive controller strategy is implemented in
simulation studies.
IMPLEMENTATION
The triple-extruder mechanism considered in this study (see Figure 2) consists of three extruders
mounted on a gantry system, with three orthogonal linear drives (MN10-0020-E01-21
Velmex BiSlide), each with a 508 mm travel range. Four Pacific Scientific PMA22B DC servo
motors are used with a maximum speed of 127 mm/s, each with an embedded encoder providing
a position measurement with a resolution of 2.54 µm. All axes are controlled with a Delta-Tau
Turbo programmable multi-axis controller (PMAC) card providing G&M code interface for axis
motion control. As can be seen in Figure 2, the extruder mechanism is composed of three ram,
each having a plunger driven with a Kollmorgan AKM23D DC servo motor with a Servostar300
amplifier and a built in encoder which is sampled through a National Instrument’s PXI-6602
counter/timer card with 32 bits and provides the plunger position measurements with a resolution
of 0.62 μm. Three Omega LC-305-1K load cells are used to provide extrusion force
9
measurement with a resolution of 2 mV/V with an excitation of 10Vdc and maximum force
capacity of 4.448 kN. The load cell signals are sampled with a National Instrument’s PXI-6025E
card with 12 bits resolution providing a force measurement with 4.43 N resolution. The Gantry
and extruder system can communicate with each other through analog signals from the PMAC
card to the National Instrument’s PXI-6025E card to provide the reference velocity and extrusion
force from the PMAC system to the extrusion system using G&M codes.
Figure 2: FEF system with triple extruder.
RESULTS AND DISCUSSION
In this section a simulation case study is conducted for 35 seconds. Extrusion starts at the first
second and during the extrusion a large amount of air bubble is released from the nozzle outlet at
15 s. At 20 s the extrusion is stopped and restarts after 5 s. Due to system limitations the
10
extrusion force cannot exceed 1000 N, and the motor amplifiers will saturate at ±10 V. A
reference set point Fr = 600 N and vr = 0.018 mm/s are used for the simulations, which are
normally used for the extrusion of aqueous-based alumina ceramic pastes. It is assumed that
when the air bubble shows up the system dynamics changes from  F  21.3 (1/s), K Fv  18300
(N∙s/mm) to  F  100 , K Fv  1 (N∙s/mm). A sampling time of 0.001 s is used for all simulations.
For this simulation the prediction horizon is Np=500 steps (i.e., 0.5 seconds) and the control
horizon is Nc=4.
The resulting ram velocity, extrusion force, control signal and a history of switching
parameters are shown in Figure 3. In the switching parameters section, a high value for gain
selection refers to high emphasis on force compensation and a low value refers to a high
emphasis on velocity compensation. Also, a high value of in Free/Engaged history line indicates
that the ram is separated from the plunger and is moving freely while a low value indicates the
ram is pushing on the plunger. A zoomed in view of the velocity and force results is presented in
Figure 4. It can be seen that to start the extrusion at the first second, the reference velocity is
changed to 0.018 mm/s and reference extrusion force is changed to 600 N. At this stage, since
the force deviation is more than 30 N, the gain selection is switched to high emphasis on
extrusion force. At this configuration the controller ignores any error in velocity and uses the
motors maximum ability to reach the desired force. As the motors saturate at 10 V it will take
approximately 8 s for the force to reach the desired range. During this time the command voltage
and velocity are at the maximum values possible for the system. However at 8 s when the
extrusion force enters the allowable deviation range, the gain selection is switched to high
emphasis on the velocity and extrusion with the desired rate starts. It is assumed that a large air
bubble is developed in the nozzle and released at 15 s. It can be seen that the extrusion force is
11
reduced drastically to approximately 200 N. At this point it can be seen that the ram dynamics
are also changed to free movement. While this change in dynamics is an uncertainty for the
designed controller it can be seen that when the gains are switched to high emphasis on force,
since the velocity errors are ignored, the controller drives the extrusion force back to the normal
range in 3 s. It should be noted that when the system is recovering from an air bubble, it can be
seen from the Free/Engaged history line that after approximately 0.5 s the nozzle is filled up with
paste and the system dynamics are switched back to engaged configuration and at this point the
ram shows some oscillations for about one s because of the sudden transition between two
different dynamics with very large differences in the time constants. After this stage it takes the
system about 1.5 s to completely return the extrusion force to the reference value.
In order to investigate the ability of the controller for a fast extrusion stop, an extrusion
stop command is activated at 20 s (i.e., the reference force and velocity are reduced to 200 N and
0.2 µm/s, respectively). Since the prediction horizon is equal to 0.5 s, the controller reacts to the
force reference change at 19.5 s. However, since reducing the extrusion force at this point results
in a possible defective part. At the end the extrusion force is brought back to the acceptable
range. After this it can be seen that the stopping the extrusion requires 0.5 s and, after the
extrusion ceases, the controller maintains the extrusion force at 200 N until at 25 s the extrusion
starts again.
12
V [mm/s]
Simulation
1
r
0
-1
0
Fr [N]
Reference
5
10
15
20
25
30
35
5
10
15
20
25
30
35
5
10
15
20
25
30
35
600
400
200
0
V c [V]
10
0
-10
0
Switching
parameters
Gain Selection
Free/Engaged
4
2
0
0
5
10
15
20
time [s]
25
30
35
Reference
Simulation
0.05
r
V [mm/s]
Figure 3: Simulation results from proposed predictive controller.
Fr [N]
0
0
5
10
15
20
25
30
35
5
10
15
20
time [s]
25
30
35
500
0
Figure 4: Magnified results from Figure 3.
13
It can be seen that although the mentioned extruder system includes huge amounts of uncertainty
and time constant variations, using the gain switching law increases the robustness and improves
the performance of the controller during normal extrusion.
CONCLUSIONS
A predictive controller with adaptive gain switching was proposed and implemented on an
extrusion process of aqueous-based alumina pastes. The predictive controller showed good
performance until the dynamics of the paste was changed because of an air bubble. At this stage
an offset shows up in the velocity tracking but with the proposed adaptive method added to the
predictive controller when the force decreases to 200 N because of an air bubble release, the
error is rapidly regulated by increasing the emphasis of force error in the cost function. When the
error enters an acceptable range the gains are switched back to place more emphasis on velocity
error. This can help with a more robust extrusion of the paste and save a lot of paste. Because
when an air bubble shows up in the paste normally the part is unusable and the extrusion process
should start from the beginning. For future work, this method will be extended to consider the
dynamics of the system during an air bubble release in the control design for better performance
of air bubble recovery during the extrusion. Also different gain laws will be developed for
smoother response of the system at extrusion stop and start.
REFERENCES
[1]
J. Chae, S. S. Park, and T. Freiheit, "Investigation of micro-cutting operations,"
International Journal of Machine Tools and Manufactureing, vol. 46, pp. 313-332, 2006.
14
[2]
G. E. Hilmas, J. L. Lombardi, and R. A. Hoffman, "Advances in the Fabrication of
Functionally Graded Materials using Extrusion Freeform Fabrication," in Functionally
Graded Materials 1996, S. Ichiro and M. Yoshinari, Eds., ed Amsterdam: Elsevier
Science B.V., 1997, pp. 319-324.
[3]
J. W. Wang, Shaw, L.L., Xu, A. and Cameron, T.B., "Solid freeform fabrication of
artificial human teeth," presented at the Solid Freeform Fabrication Proceedings, 2004.
[4]
G. M. Lous, I. A. Cornejo, T. F. McNulty, A. Safari, and S. C. Danforth, "Fabrication of
piezoelectric ceramic/polymer composite transducers using fused deposition of
ceramics," Journal of the American Ceramic Society, vol. 83, pp. 124-28, 2000.
[5]
T. Huang, "Fabrication of ceramic components using freeze-form extrusion fabrication,"
PhD, Department of Materials Science and Engineering, University of Missouri Science
and Technology,Disseration, Department of Materials Science and Engineering, Rolla,
2007.
[6]
G. A. Hassan and J. Parnaby, "Model reference optimal steady-state adaptive computer
control of plastics extrusion processes," Polymer Engineering and Science, vol. 21, pp.
276-284, 1981.
[7]
M. McAfee and S. Thompson, "A novel approach to dynamic modelling of polymer
extrusion for improved process control," Proceedings of the Institution of Mechanical
Engineers. Part I: Journal of Systems and Control Engineering, vol. 221, pp. 617-628,
2007.
[8]
F. Previdi, S. M. Savaresi, and A. Panarotto, "Design of a feedback control system for
real-time control of flow in a single-screw extruder," Control Engineering Practice, vol.
14, pp. 1111-1121, 2006.
15
[9]
X. Zhao, R. G. Landers, and M. C. Leu, "Adaptive extrusion force control of freeze-form
extrusion fabrication processes," ASME Journal of Manufacturing Science and
Engineering, vol. 132, 2010.
[10]
M. Li, L. Tang, F. Xue, and R. Landers, "Numerical Simulation of Ram Extrusion
Process for Ceramic Materials," Reviewed manuscript, 2011.
[11]
M. S. Mason, T. Huang, R. G. Landers, M. C. Leu, and G. E. Hilmas, "Freeform
Extrusion of High Solids Loading Ceramic Slurries. Part 1: Extrusion Process Modeling,"
presented at the Solid Freeform Fabrication Symposium, Austin, TX, 2006.
[12]
M. S. Mason, T. Huang, R. G. Landers, M. C. Leu, and G. E. Hilmas, "Freeform
Extrusion of High Solids Loading Ceramic Slurries, Part II: Extrusion Process Control,"
presented at the Annual Solid Freeform Fabrication Symposium, Austin, TX, 2006.
[13]
M. C. Leu, L. Tang, B. Deuser, R. G. Landers, G. E. Hilmas, S. Zhang, et al., "Freezeform extrusion fabrication of composite structures," in International Solid Freeform
Fabrication Symposium, Austin, Texas, 2011.
[14]
M. S. Mason, T. Huang, R. G. Landers, M. C. Leu, and G. E. Hilmas, "Aqueous-based
extrusion of high solids loading ceramic pastes: Process modeling and control," Journal
of Materials Processing Technology, vol. 209, pp. 2946-2957, 2009.
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